A Mathematical Study of Hawking Radiation on
Collapsing, Spherically Symmetric Spacetimes
Frederick Alford
Supervisor: Prof. Mihalis Dafermos
Department of Applied Mathematics and Theoretical Physics
University of Cambridge
This thesis is submitted for the degree of
Doctor of Philosophy
Jesus College September 2021

Declaration
This thesis is the result of my own work and includes nothing which is the outcome of work done in collaboration
except as declared in the Preface and specified in the text. I further state that no substantial part of my thesis has
already been submitted, or, is being concurrently submitted for any such degree, diploma or other qualification
at the University of Cambridge or any other University or similar institution except as declared in the Preface
and specified in the text. It does not exceed the prescribed word limit for the relevant Degree Committee.
Frederick Alford
September 2021
iv
A Mathematical Study of Hawking Radiation on Collapsing, Spherically
Symmetric Spacetimes
Summary
In this thesis, we give a mathematical treatment of the late time Hawking radiation of massless bosons emitted by
a family of collapsing, spherically symmetric, charged models of black hole formation, including both extremal
and sub-extremal black holes. We further bound the rate at which the late time behaviour is approached. This
treatment relies heavily on analysing the behaviour of the linear scattering map for massless bosons (solutions
to the wave equation), which will be discussed further in this thesis. The thesis will be split into three chapters.
The first chapter will be an introduction and derivation of the underlying spacetime models, known as
Reissner–Nordström Oppenheimer–Snyder (RNOS) models. We will discuss the derivation of the Oppenheimer–
Snyder model [39], before moving on to the more general charged case. We will then summarise the interesting
and useful properties of these models.
The second chapter will cover the analysis of the scattering map for the wave equation on RNOS backgrounds.
The main results will be the forward boundedness and backwards non-boundedness of the scattering map on the
original Oppenheimer–Snyder space-time [39], and then the subsequent generalisation of this to RNOS models.
These results will be achieved primarily by using vector field methods: by considering different energy currents
and how they interact with the collapsing dust cloud, we will show that solutions of the linear wave equation
have bounded energy when going from past null infinity up until a spacelike hypersurface which intersects the
point of collapse of the dust cloud. Previous works allow us to extend this result to one on the whole spacetime.
The final chapter of this thesis will apply the above results to the calculation first considered by Stephen
Hawking [27, 28], in order to obtain the rate of radiation emitted by collapsing black holes. This result will
further make use of some high frequency approximations and also an r∗p weighted energy estimate. In particular,
we will prove that for late times, the radiation given off by any RNOS model approaches its predicted Hawking
radiation limit, that of a black body of fixed temperature. We will also prove a bound on the rate at which this
limit is approached.
Frederick Alford
Acknowledgements
There are many people who have had a large impact of my research, and the individuals listed below are only a
small subset of these people. I apologise to the many people whom I’ve not mentioned by name - I am very
grateful to you all, I can not come close to thanking you all enough.
Firstly, I would like to thank my supervisor, Mihalis Dafermos, who has been very supportive and endlessly
patient with me throughout my PhD, both mathematically and with my somewhat questionable grasp of the
English language. I would also like to thank Owain Salter Fitz-Gibbon, Dejan Gajic and Yakov Shlapentokh-
Rothman for many insightful mathematical discussions. I would like to thank all the individuals from Princeton
for their interesting comments and new perspectives on my work, and making my time there memorable.
I am very grateful all my friends and colleagues in DAMTP for making the past four years enjoyable and
productive, with a special mention to Miren Radia for all his tech-support. I would like to thank all my friends
at Jesus College for preventing my PhD from consuming my life entirely, most notably my housemates during
the lock-down periods.
I would like to thank Alice for somehow being supportive, helpful, motivating and distracting at exactly the
right times. Finally, I would like to thank my family. Words cannot express how grateful I am for their support
throughout my life.
Formal Acknowledgements
This work was funded by EPSRC DTP [1936235].

Contents
Introduction 1
0.1 Summary of the Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1
0.2 Physical Derivation of Hawking Radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2
0.2.1 Quantum Field Theory on a Curved Background . . . . . . . . . . . . . . . . . . . . 2
0.2.2 Bogoliubov Coefficients . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3
0.2.3 The Collapsing Setting . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 4
0.2.4 The Hawking Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
0.2.5 Goal of this Thesis . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5
1 RNOS Spacetimes 7
1.1 The Oppenheimer–Snyder Spacetime . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.1 Exterior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 7
1.1.2 Interior . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 8
1.1.3 Global Coordinates and the Definition of the Oppenheimer–Snyder Manifold . . . . . 10
1.1.4 Penrose Diagram of (M ,g) . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2 Generalising to the RNOS model . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 11
1.2.1 Derivation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 12
1.2.2 T ∗ < 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.3 T ∗ ≥ 1 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 14
1.2.4 Definition of the RNOS Manifold and Global Coordinates . . . . . . . . . . . . . . . 15
2 The Scattering Map 17
2.1 Introduction and Overview of Main Theorems . . . . . . . . . . . . . . . . . . . . . . . . . . 17
2.2 Previous Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 18
2.3 Notation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 20
2.4 Existence and Uniqueness of Solutions to the Wave Equation . . . . . . . . . . . . . . . . . . 23
2.4.1 The Reflective Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 23
2.4.2 The Permeating Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 27
2.5 Boundedness of the Wave Equation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.1 The Reflective Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 29
2.5.2 The Permeating Case . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 33
2.6 Higher Order Boundedness . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 38
2.7 The Scattering Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 45
2.7.1 Existence of Radiation Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7.2 Backwards Scattering from Σt∗c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 46
2.7.3 Forward Scattering from Σt∗c . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 54
2.7.4 The Scattering Map . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 55
3 Hawking Radiation 57
3.1 Overview . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 57
3.2 Previous Works . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 58
viii Contents
3.3 Classical Scattering and Transmission and Reflection Coefficients of Reissner–Nordström
Spacetimes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 59
3.4 The Hawking Radiation Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 61
3.4.1 The Set-up and Reduction to Fixed l . . . . . . . . . . . . . . . . . . . . . . . . . . . 62
3.4.2 Summary of the Proof . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 63
3.4.3 Evolution in Pure Reissner–Nordström . . . . . . . . . . . . . . . . . . . . . . . . . 64
3.4.4 The Reflection . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 75
3.4.5 High Frequency Transmission . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 80
3.4.6 Treatment of the I.E. Terms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 86
3.4.7 Final Calculation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 100
3.5 Future Work . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 104
Bibliography 107
Appendix A High Frequency Behaviour of the Reflection Coefficient 109
Introduction
In this thesis, we will be studying the behaviour of solutions to the linear wave equation
□gφ =
1√−g∂µ(
√−ggµν∂νφ) = 0, (1)
on spherically symmetric solutions to the Einstein equations
Rµν − 12Rgµν = 8πT µν . (2)
In general in this thesis, Tµν in (2) will be given by either the Maxwell energy momentum tensor (see already
(1.31)), or the energy momentum tensor of dust (1.2).
The overall goal of this thesis will be to give a mathematically rigorous treatment of Hawking radiation,
first considered in [27, 28]. This quantity is the rate at which radiation is emitted by black holes predicted by
quantum field theories on curved background spaces.
Classically, black holes, once formed, are permanent and (conjecturally) stable. The discovery of Hawking
Radiation was therefore a major breakthrough in understanding how black holes can vary radically over time, as
it describes a mechanism to decrease the mass of black holes and potentially cause them to evaporate entirely.
Since Hawking proposed this phenomenon in 1974 [27], there have been hundreds of papers on the topic within
the physics literature. For an overview of the physical aspects of Hawking radiation, we refer the reader to
[46] . Concerning a mathematically rigorous treatment of Hawking radiation, however, there are substantially
fewer works (see already [6] and Section 3.2 for a discussion of further references), and the mathematical status
of Hawking radiation still leaves much to be desired. As a result, it has not been possible yet to ask more
quantitative questions about Hawking radiation, which are necessary if one wants to eventually understand this
phenomenon in the non-perturbative setting. This thesis hopes to contribute towards solving this problem by
giving a new physical space approach to Hawking’s calculation allowing one to obtain also a rigorous bound on
the rate at which emission approaches black body radiation.
0.1 Summary of the Thesis
We begin in Chapter 1, by considering the possible spherically symmetric models of collapsing spacetimes.
We first consider the original model studied by Hawking, the Oppenheimer–Snyder model [39]. This model
considers chargeless, pressureless dust collapsing to form a black hole. We then proceed to add charge to the
dust and consider the effects on the behaviour of the surface of the dust cloud. This allows us to generalise to
the RNOS (Reissner–Nordström Oppenheimer–Snyder) models, which will be the topic of this thesis. The main
result of this chapter is the derivation itself of the metric of these models, and of the behaviour of the surface of
the dust cloud.
In Chapter 2, we construct the Scattering map. That is, we define the map taking initial data of solutions to
the wave equation from past null infinity, I −, to future null infinity and the event horizon, I +∪H +. The
main result of this Chapter, Theorem 2.7.2 is that this map is a linear, bounded map (with respect to an energy
norm defined in Section 2.3), but that its inverse (where it is defined) is not bounded. In the process of proving
this, we will also determine exactly where this inverse map ‘goes wrong’, which will allow us to proceed with
our treatment of Hawking radiation, despite the difficulty that non-invertibility imposes.
2 Contents
In Chapter 3, we will perform the Hawking radiation calculation originally done in [27, 28] in a mathematic-
ally rigorous manner. This calculation determines the change in frequency of a solution from I + to I −. In
particular, the calculation supposes the solution on I + contains only positive frequencies, and determines the
size of the negative frequency components of the solution on I −. Here ‘size’ is with respect to the particle
current, which we will explain in more detail in Section 0.2. This chapter will culminate in Theorem 4, showing
that the size of these negative frequency components approaches a fixed limit, and will also prove a bound on
the rate at which this limit is approached.
There is also one appendix, concerned with pure Reissner–Nordström spacetime. Appendix A derives a
result bounding reflection coefficients, which is used in the Hawking calculation.
Throughout this thesis, we will be using the signature convention {−,+,+,+}, with summation convention
(repeated indices are summed over unless otherwise stated). Summations over Greek characters and early Latin
characters (a,b,c) will be over all 4 dimensions, where as summations over i, j,k are summations over the 3
spatial dimensions. The Fourier transform of a function f will be denoted by fˆ , and will use the convention
fˆ (ω) =
1√
2π
∫ ∞
−∞
e−iωx f (x)dx. (3)
Fourier transform of a function on a cylinder will always be with respect to the non-angular variable.
Other conventions and notations are covered in Section 2.3.
0.2 Physical Derivation of Hawking Radiation
Before turning to the mathematical study of Hawking radiation, we briefly review the physical derivation of the
Hawking calculation. This section is not intended to be rigorous, it is only intended to give an overview of the
motivation for this thesis. For more on making the framework of quantum field theories on curved backgrounds
rigorous, we refer the reader to [30], [46]. In this section, we will be imposing h¯ = 1, as well as the usual
G = c = 1. This section will be closely following Chapter 10 of [40].
0.2.1 Quantum Field Theory on a Curved Background
Let us consider a metric of the form
g =−N2dt2+hi j(dxi+Nidt)(dx j +N jdt), (4)
known as a 3+1 decomposition.
On this background, let us consider a massless scalar field, with action
S =
∫
M
√−g
2
∇aψ∇aψdtd3x. (5)
This has equation of motion given by (1). We can consider the momentum conjugate of this scalar field
(using our t coordinate) to obtain
Π=
δS
δ (∂tψ)
=
√−ggtµ∂µψ. (6)
As in all quantum mechanics, we now promote ψ and Π to operators, and impose the following commutation
relations:
[ψ(t,x),Π(t,x′)] = iδ (x− x′) [ψ(t,x),ψ(t,x′)] = 0 = [Π(t,x),Π(t,x′)]. (7)
We now consider what these operators act on. Assuming our manifold is globally hyperbolic, we know
that any solution is uniquely determined by data on Σ0 = {t = 0}. Let α,β ∈ S, where S is a space of suitable
0.2 Physical Derivation of Hawking Radiation 3
complex solutions to (1). We define the particle current “inner product" as follows:
(α,β ) =−i
∫
Σ0
√−ggtµ(α¯∇µβ −β∇µ α¯)d3x. (8)
Note we have not defined the function space on which this is an inner product yet.
We note from (1) that this current is conserved, as
∇µ
(
α¯∇µβ −β∇µ α¯
)
= 0. (9)
Thus, the integral (8) over any surface of constant t, Σt , is independent of t. Also note the following properties:
(α,β ) =−(β¯ , α¯) (10)
(α,β ) = 0 ∀β ∈ S =⇒ α = 0. (11)
However, as (α,α) =−(α¯, α¯), we have that this inner product (8) is not positive definite on S. We would
like to consider a subset of Sp on which (,) is positive definite, denoted Sp. By (10), we know that (,) is negative
definite on S¯p = {α¯ : α ∈ Sp}. We would like to pick an Sp such that
S = Sp⊕ S¯p. (12)
In general, there will be many ways to do this. For now, we will just pick one, though we will return to this
choice later.
In quantum theory, we define creation and annihilation operators (associated to f ∈ Sp) of a real scalar field
φ by
a( f ) = ( f ,φ) a( f )† =−( f¯ ,φ). (13)
We then have the following commutation relations
[a( f ),a(g)†] = ( f ,g) [a( f ),a(g)] = 0 = [a( f )†,a(g)†] = 0. (14)
As usual in quantum theories, we define the vacuum state, |0⟩, by
a( f )|0⟩= 0 ∀ f ∈ Sp ⟨0|0⟩= 1. (15)
An N-particle state is defined by
a†( f1)a†( f2)...a†( fn)|0⟩, fi ∈ Sp. (16)
We now finally define our Hilbert space, Hp, to be the Fock space spanned by the vacuum state, the 1-particle
states, the 2-particle states, etc.
If we consider particles created by the creation operator a( f )†, the expected number of these particles
measured in state |ϕ⟩ is
⟨ϕ|N f |ϕ⟩= ⟨ϕ|a( f )†a( f )|ϕ⟩. (17)
0.2.2 Bogoliubov Coefficients
We have defined Hp using Sp, for which there are many options. Therefore, there are in fact many different
options for this Fock space (these spaces would be isomorphic, but we already have an obvious map between
them, and they are not the same space using this map). We will now consider how to transform between two
different sets of creation and annihilation operators.
4 Contents
Let Sp and S′p be two different choices for a space of positive definite solutions to (12). Let {φi} ⊂ Sp be an
orthonormal basis for Sp, i.e.
(φi,φ j) = δi j, (18)
and let {φ ′i } be an orthonormal basis for S′p (these exist, as Sp and S′p restricted to any Cauchy surface are
subsets of L2(R3)).
As {φi}∪{φ¯i} is a basis of S, we can write
φ ′i =∑
j
(
Ai jφ j +Bi jφ¯ j
)
, (19)
for some A,B, known as Bogoliubov coefficients. We can rearrange for the Bi j coefficient to get
Bi j =−(φ¯ j,φ ′i ). (20)
Annihilation and creation operators are then related by
a(φ ′i ) =∑
j
(
A¯i ja(φ j)− B¯i ja(φ j)†
)
. (21)
Now we wish to consider the number of particles given by a(φ ′i ) expected to be in the vacuum state given by
Sp.
⟨0|a(φ ′i )†a(φ ′i )|0⟩=∑
j,k
⟨0|(−Bi j)a(φ j)(−B¯ik)a(φk)†|0⟩=∑
j,k
Bi jB¯ik = (BB†)ii. (22)
0.2.3 The Collapsing Setting
As stated previously, we have many choices of Sp. However, if we have a preferential choice of timelike
coordinate, say t, we have a preferential choice of Sp: Let φp be the eigenfunction of the operator ∂t with
positive imaginary eigenvalue, that is
∂tφp = ip0φp p0 > 0. (23)
Thus, if we have a preferred timelike derivative, we define Sp to be the span of all such positive imaginary
eigenfunctions of ∂t .
We now consider an asymptotically flat gravitational collapse. Then near to I − and I +, we have a
preferential choice of timelike coordinate, given by the definition of asymptotically flat. This allows us to define
a canonical choice of S±p on I ±. These will now take the part of Sp and S′p in equations (18) to (22).
Now suppose we wish to calculate the expected number of particles of frequency ω emitted by this
gravitational collapse. Let ψ ′ω be a function on future null infinity with frequency approximately ω . Note that
here, approximately means that ψˆ ′ is supported on [ω−ε,ω+ε]×S2 for some small ε . Here S2 is the 2-sphere.
This is required, as if ψˆ ′ was only supported at ω , then ψ ′ω ∝ eiωu, and we cannot have (ψ ′ω ,ψ ′ω) = 1.
Let φ ′ω be the solution to (1) with future radiation field ψ ′ω , and which vanishes at H + (as we are not
interested in the expected number of particles crossing the event horizon). Denote the past radiation field of ψ ′ω
by ψI −ω . We can split ψI
−
ω up into positive and negative frequency components:
ψI
−
ω = ψ
I −+
ω +ψ
I −−
ω , (24)
Where ψˆI −+ω is supported in [0,∞]×S2, and ψˆI
−−
ω is supported in [−∞,0]×S2.
0.2 Physical Derivation of Hawking Radiation 5
Let φi be a basis of S−p . Then the number of expected particles emitted by the formation of the black hole is
∑
j,k
Bi jB¯ik = (∑
j
B¯i jφ j,∑
k
B¯ikφk) = (ψ¯I
−−
ω , ψ¯
I −−
ω ) (25)
=−i
∫
ψI
−−
ω ∂vψ¯
I −−
ω − ψ¯I
−−
ω ∂vψ
I −−
ω sinθdvdθdϕ
= 2
∫ ∞
σ=−∞
σ |ψˆI −−ω |2 sinθdσdθdϕ
= 2
∫ 0
σ=−∞
σ |ψˆI −ω |2 sinθdσdθdϕ.
Here σ is the Fourier space variable.
0.2.4 The Hawking Calculation
Let ψ+ be a Schwartz function on the cylinder, with ψˆ+ supported in [ω− ε,ω+ ε], with (ψ+,ψ+) = 1. Let φ
be the solution to (1) which vanishes on the future event horizon, and has future radiation field equal to ψ+. Let
ψI − be the past radiation field of φ . Then we must calculate
2
∫ 0
σ=−∞
σ |ψˆI −|2 sinθdσdθdϕ. (26)
In his original paper [27], Hawking argued that at late times (to be defined more rigorously later) in the
formation of the Schwarzschild black hole the rate of radiation of frequency ω emitted would tend towards
that of a black body of temperature κ/2π , where κ is the surface gravity of the black hole. Since then, there
have been many heuristic arguments for this result to extend to all Reissner–Nordström black holes, along with
several more rigorous papers exploring this phenomenon (see Section 3.2 for a further discussion of these).
There are two ways to obtain similar results by considering quantum states on a non-collapsing Reissner–
Nordström background. Firstly, one can construct the Unruh state [44]. If one considers quantum states on the
permanent Reissner–Nordström black hole, one can show that there is a unique state that coincides with the
vacuum state on I − and is well behaved atH + (i.e. is a Hadamard state). This state evaluated on I + is a
thermal state of temperature κ/2π (see [17], for example).
The second similar result can be obtained by constructing the Hartle–Hawking–Israel state. If one again
considers quantum states on the permanent Reissner–Nordström black hole, one can show that there is a unique
state that is well behaved (Hadamard) at I +, I −,H +,H − [31]. This state is that of a thermal black body,
again of temperature κ/2π . The interpretation of this is that the black hole is in equilibrium with this level of
thermal radiation, and is therefore emitting the radiation of a black body of this temperature. This result has
been considered in a mathematically rigorous manner on Schwarzschild [29, 17], and more recently in a more
general setting [43, 23]. This thesis, however, will be focused on the collapsing setting, as it is believed that this
method will generalise more readily, as the Hartle–Hawking–Israel state has been shown not to exist in Kerr
spacetimes [31].
One of the reasons this result has gained so much traction in both the mathematics and physics communities
is that it provides the only known mechanism for black holes to lose mass. Without the ability to lose mass, any
black hole that has formed would be a permanent fixture of the universe, and could only grow in size. However,
if the black holes are able to emit radiation at a fixed rate, even if only in very small doses, then given enough
time, isolated black holes will disappear entirely, known as black hole evaporation.
0.2.5 Goal of this Thesis
In this thesis, we take (26) to be our starting point. We will impose the radiation field ψ+(u−u0,θ ,ϕ) at I +,
and 0 atH +. We will rigorously define ψˆI − in terms of classical scattering theory, using results from Chapters
1 and 2. We will then rigorously calculate the limit that (26) approaches as u0 → ∞, on a family of collapsing
6 Contents
models forming Reissner–Nordström blackholes. We will further include a rigorous bound on the rate at which
this result is approached, obtaining the result of Theorem 4:
Theorem 4 (Late Time Emission of Hawking Radiation). Let ψ+(u,θ ,ϕ) be a Schwartz function on the 3-
cylinder, with ψˆ+ only supported on positive frequencies. Let φ be the solution of (2.1), as given by Theorem
2.4.1, such that
lim
v→∞r(u,v)φ(u,v,θ ,ϕ) = ψ+(u−u0,θ ,ϕ) (27)
lim
u→∞r(u,v)φ(u,v,θ ,ϕ) = 0 ∀v≥ vc, (28)
Define the function ψ−,u0 by
lim
u→−∞r(u,v)φ(u,v,θ ,ϕ) = ψ−,u0(v,θ ,ϕ). (29)
Then for all |q|< 1, n ∈ N, there exist constants An(M,q,T ∗,ψ+) such that∣∣∣∣∣
∫ 0
ω=−∞
∫ 2π
ϕ=0
∫ π
θ=0
|ω||ψˆ−,u0(ω,θ ,ϕ)|2 sinθdωdθdϕ−
∫
H −
|ω||ψˆH −(ω,θ ,ϕ)|2
e
2π|ω|
κ −1
sinθdωdθdϕ
∣∣∣∣∣≤ Anu−n0 ,
(30)
for sufficiently large u0.
Here, ψH − is the reflection of ψ+ in pure Reissner–Nordström spacetime (as will be discussed in Section
3.3), and κ is the surface gravity of the Reissner–Nordström black hole.
In the case |q|= 1, there exists a constant A(M,q,T ∗) such that∣∣∣∣∫ 0ω=−∞
∫ 2π
ϕ=0
∫ π
θ=0
|ω||ψˆ−,u0(ω,θ ,ϕ)|2 sinθdωdθdϕ
∣∣∣∣≤ A
u3/20
, (31)
for sufficiently large u0.
This result is restated in Theorem 3.4.1 and Corollary 3.4.6, including the precise relationship of An,A on
ψ+.
In terms of the Unruh state mentioned previously, one can view this result as the following statement: Let
us impose the vacuum state on past null infinity and evolve forward to future null infinity. Then at late times,
any number operator acting on this future null infinity state approaches the same number operator acting on the
Unruh state at future null infinity (and again a rate is included).
This result is especially physically relevant for the extremal case. As the surface gravity of the extremal
Reissner–Nordström black hole is 0, this is the only current model of black hole which may be stable to Hawking
radiation (though it may be unstable to classical perturbations [4, 5]). That is, any black hole with non-zero
surface gravity will emit radiation at a constant rate, and will therefore, if left alone, eventually evaporate.
However, this thesis will show that the extremal Reissner–Nordström black hole will emit only a finite amount
of radiation over time, and may therefore not evaporate, becoming a permanent fixture to the universe.
Chapter 1
RNOS Spacetimes
1.1 The Oppenheimer–Snyder Spacetime
Any discussion of collapsing spacetimes must start with the Oppenheimer–Snyder spacetime [39]. The
Oppenheimer–Snyder spacetime is a homogeneous, spherically symmetric collapsing dust star. That is to
say, a spherically symmetric solution of the Einstein equations:
Rµν − 12Rgµν = 8πTµν (1.1)
where for dust, we have
Tµν = ρuµuν (1.2)
∇µTµν = 0.
Here the vector uµ is the 4-velocity of the dust, and ρ is the density of the dust. On our initial timelike
hypersurface, this density is a positive constant inside the star, but 0 outside the star. The case of the non-
homogeneous dust cloud was studied by Christodoulou in [11].
As this density is not continuous across the boundary of the star, the Oppenheimer–Snyder model is only a
global solution of the Einstein equations in a weak sense. However, it is a classical solution on both the interior
and the exterior of the star.
We therefore have two specific regions of the space-time to consider: inside the star (section 1.1.2), and
outside the star (section 1.1.1). We will finally give the definition of our manifold and global coordinates in
section 1.1.3. If the reader is uninterested in the derivation of the metric, they may want to skip to that section.
Finally in section 1.1.4, we discuss the Penrose diagram for this space-time.
1.1.1 Exterior
We will first consider the exterior of the star. This region is a spherically symmetric vacuum space-time, thus by
Birkhoff’s theorem, [40], this is a region of Schwarzschild space-time. It is bounded by the timelike hypersurface
r = rb(t∗). This hypersurface will be referred to in this thesis as the boundary of the star. We will be using the
following coordinate system in the exterior of the star:
g =−
(
1− 2M
r
)
dt∗2+
4M
r
dt∗dr+
(
1+
2M
r
)
dr2+ r2gS2 (t
∗,r,θ ,ϕ) ∈ R× [R∗(t∗),∞)×S2 (1.3)
where gS2 is the usual metric on the unit sphere, and t
∗ is defined by
t∗ = t+2M log
( r
2M
−1
)
, (1.4)
in terms of the usual t and r coordinates on Schwarzschild.
8 RNOS Spacetimes
As the surface of the star is itself free-falling and massive, we may assume that the surface of the star follows
timelike geodesics. This assumption is true in the Oppenheimer–Snyder model, but also generalises to other
models, provided the matter remains well behaved. Thus if a particle on the surface has space-time coordinates
xα(τ), then these coordinates satisfy
−1 = gab
(
dx
dτ
a)(dx
dτ
b)
=−
(
1− 2M
r
)(
dt∗
dτ
)2
+
4M
r
dt∗
dτ
dr
dτ
+
(
1− 2M
r
)(
dr
dτ
)2
=
(
−
(
1− 2M
r
)
+
4M
r
r˙b+
(
1− 2M
r
)
r˙b2
)(
dt∗
dτ
)2
. (1.5)
Here we are using the t and r coordinates in equation (1.3), and using the fact that this space-time is spherically
symmetric to ignore dθdτ and
dϕ
dτ terms. Note that r˙b =
drb
dt∗ .
Now, as rb(t∗) is to be timelike and is the surface of a collapsing star, we assume r˙b < 0, and that the surface
emanates from past timelike infinity. Again, this is true in Oppenheimer–Snyder space-time, but also in many
other models of gravitational collapse. At some time, t∗c , we have rb(t∗c ) = 2M (note that rb(t∗) does not cross
r = 2M in finite t coordinate, as t → ∞ as r → 2M on any timelike curve). For t∗ > t∗c and r ≥ 2M, we have that
the space-time is standard exterior Schwarzschild space-time, with event horizon at r = 2M.
In the exterior region, we define our outgoing and ingoing null coordinates as follows:
v = t∗+ r (1.6)
u = t∗− r−4M log
( r
2M
−1
)
(1.7)
g =−
(
1− 2M
r
)
dudv+ r(u,v)2gS2. (1.8)
1.1.2 Interior
We now move on to considering the interior of the star. One thing that is important to note here is that as we go
from considering the exterior of the star to considering the interior, i.e. as our coordinates cross the boundary of
our star, our metric changes from solving the vacuum Einstein equations to solving the Einstein equations with
matter. Thus across the boundary, our metric will not be smooth, so we must be careful when wishing to take
derivatives of the metric. This will have implications on the regularity of solutions of (2.1) across the surface of
the star.
This derivation will closely follow the original Oppenheimer–Snyder paper, [39].
We first consider taking a spatial hypersurface in our space time, which is preserved under the spherical
symmetry SO3 action. We can therefore parametrise this by some R, θ , ϕ , where θ and ϕ are our spherical
angles. Then we locally extend this coordinate system to the space-time off this surface by constructing the
radial geodesics through each point with initial direction normal to the surface. In these coordinates, our metric
must be of the form
g =−dτ2+ eω¯dR2+ eωgS2. (1.9)
for ω = ω(τ,R) and ω¯ = ω¯(τ,R).
Now our matter is moving along lines of constant R, θ and φ , as we assume our matter follows radial
geodesics which are normal to our initial surface. In these coordinates the dust’s velocity uµ is therefore
proportional to ∂τ . Thus, we have from equation (1.2) that T ττ = −ρ , for density ρ . We also have that all
other components of the energy momentum tensor T vanish. Then the Einstein equations (1.1) imply that the
following is a solution:
eω¯ =
1
4
ω ′2eω (1.10)
eω = (Fτ+G)
4
3 , (1.11)
1.1 The Oppenheimer–Snyder Spacetime 9
where ′ denotes derivative with respect to R, and F , G are arbitrary functions of R. Then we can rescale R to
choose G = R
3
2 . We now assume that at τ = 0, ρ is a constant density ρ0 inside the star, and vacuum outside the
star, i.e.
ρ(0,R) =
ρ0 R≤ Rb0 R > Rb , (1.12)
for Rb > 0 constant. Then the equation for T ττ gives:
FF ′ =
9πρ0R2 R≤ Rb0 R > Rb (1.13)
where, in these coordinates, {R = Rb} is the boundary of the star. This has the particular solution
F =
−32
√
2M
(
R
Rb
) 3
2
R≤ Rb
−32
√
2M R > Rb
, (1.14)
for M = 4πρ0R3b/3. This gives us a range for which our coordinate system is valid, as the angular part of the
metric, eω has to be greater than or equal to 0. Thus we obtain τ ≤ 2R
3
√
2M
. Now, if we transform to a new radial
coordinate, r = e
ω
2 , then we obtain a metric of the form:
g =

−
(
1− 2Mr2
r3b
)
dτ2+2
√
2Mr2
r3b
drdτ+dr2+ r2gS2 r < rb
−(1− 2Mr )dτ2+2√2Mr drdτ+dr2+ r2gS2 r ≥ rb (1.15)
where
rb(τ)
3
2 = R
3
2
b −
3τ
2
√
2M. (1.16)
In the region r ≥ rb, (1.15) are known as Gullstrand–Painlevé coordinates.
Once rb(τ)≤ 2M, i.e. τ ≥ τc = 4M3
((
Rb
2M
) 3
2 −1
)
, we have r = 2M is the surface of an event horizon, and
the r ≥ 2M section of our space-time is exterior Schwarzschild space-time.
Thus any point which can be connected by a future directed null geodesic to a point outside r = 2M at τ ≥ τc
is outside our black hole, and any point which cannot reach r > 2M at τ ≥ τc is inside our black hole. The future
directed, outgoing radial, null geodesic which passes through r = 2M, τ = τc is given by:
r = rb(τ)
(
3−2
√
rb(τ)
2M
)
. (1.17)
Thus the set of coordinates obeying both (1.17) and τ ∈ [τc−,τc] is part of the boundary of our black hole, where
τc− = 2M
(
2
3
(
Rb
2M
) 3
2
− 9
4
)
. (1.18)
Before τc− , no part of the star is within a black hole, and for τ > τc, all of the collapsing star is inside the black
hole region.
10 RNOS Spacetimes
Thus, we define our ingoing and outgoing null geodesics by defining their derivative:
du =
dτ− (1−
√
2M/r)−1dr r ≥ rb
α(dτ− (1−
√
2Mr2/r3b)
−1dr) r < rb
(1.19)
dv =
dτ+(1+
√
2M/r)−1dr r ≥ rb
β (dτ+(1+
√
2Mr2/r3b)
−1dr) r < rb
. (1.20)
These coordinates exist, thanks to Frobenius’ theorem (see for example [40]) with α and β real functions on the
manifold, bounded both above and away from 0. However, we may not be able to write α and β explicitly.
Remark 1.1.1. Note that when using different coordinates across the boundary of the star, r = rb(τ), such as
in (1.19) and (1.20) compared to (1.15), one should be concerned that these coordinates may define different
smooth structures onM . For example, the function f (τ,r,θ ,ϕ) = r− rb(τ) is smooth on r = rb(τ) with respect
to (τ,r,θ ,ϕ), but is not smooth with respect to coordinates (τ,x := (r− rb(τ))3,θ ,ϕ).
However, when considering (in the exterior) the coordinates in (1.3) compared to (1.15), the change of
coordinates is smooth with bounded (above and away from 0) Jacobian. Thus a function is smooth with respect
to (1.3) if and only if it is smooth with respect to (1.15), so this is not a concern in this case.
1.1.3 Global Coordinates and the Definition of the Oppenheimer–Snyder Manifold
We summarise the work of the previous sections by defining our manifold and metric with respect to global
coordinates. Fix M > 0,Rb ≥ 0, let τc− =
√
2R3b
3M , and consider R
4 = R×R3. Here R is parametrised by τ and
R3 is parametrised by the usual spherical polar coordinates. We then defineM by:
M := R4\{τ ∈ [τc−,∞),r = 0}. (1.21)
In these coordinates, we then have the metric:
gM,Rb =

−
(
1− 2Mr2
r3b
)
dτ2+2
√
2Mr2
r3b
drdτ+dr2+ r2gS2 r < rb(τ)
−(1− 2Mr )dτ2+2√2Mr drdτ+dr2+ r2gS2 r ≥ rb(τ) (1.22)
where rb(τ) is defined by
rb(τ) =
(
R3/2b −
3τ
2
√
2M
)2/3
. (1.23)
Note that choice of Rb is equivalent to choosing when τ = 0. Also note the r = 0 line (as a subset of R4) ceases
to be part of the manifoldM when the singularity “forms" at τc− , where rb = 0. For τ < τc− , r = 0 is included
in the manifold, as the metric extends regularity to this line.
We define our future event horizon by:
H + =
{
r = rb(τ)
(
3−2
√
rb(τ)
2M
)
,τ ∈ [τc−,τc]
}
∪{r = 2M,τ ≥ τc}. (1.24)
Note that geometrically, this family of space-times (H1loc Lorentzian manifolds), (M ,gM,Rb), is a one
parameter family of space-times. The geometry depends only on M, as Rb just corresponds to the coordinate
choice of where τ = 0. Thus constants which only depend on the overall geometry of the space-time only
depend on M.
1.2 Generalising to the RNOS model 11
We can also explicitly calculate ρ in these coordinates for r < rb(τ):
ρ(τ) =
3M
4πr3b(τ)
=
3M
4π
(
R3/2b − 3τ2
√
2M
)2 = R3br3b(τ)ρ0. (1.25)
In the exterior of the space-time, we have one timelike Killing field, ∂t∗ = ∂τ , which is not Killing in the
interior. Throughout the whole space-time, we have 3 angular Killing fields, {Ωi}3i=1, which between them span
all angular derivatives. When given in the usual θ ,ϕ coordinates, these take the form:
Ω1 = ∂ϕ
Ω2 = cosϕ∂θ − sinϕ cotθ∂ϕ (1.26)
Ω3 =−sinϕ∂θ − cosϕ cotθ∂ϕ
1.1.4 Penrose Diagram of (M ,g)
i+
i0
i−
t∗c , 2MH + I +
I
−
r
=
0
r
=
R
∗ (
t∗
)
r = 0
Figure 1.1 Penrose diagram of
Oppenheimer–Snyder space-time
We now look to derive the Penrose diagram for the space-time
(M ,g). Recall that the Penrose diagram corresponds to the range
of globally defined radial double null coordinates. Using the original
R and τ coordinates in (1.9), we obtain that the interior of the dust
cloud has metric
g =−dτ2+
(
1− 3
√
2Mτ
2Rb
3
2
) 4
3 (
dR2+R2gS2
)
(1.27)
for R≤ Rb and τ ≤ τc = 2Rb
3
2
3
√
2M
. We then choose a new time coordin-
ate, η such that
η(τ) =
∫ τ
τ ′=0
(
1− 3
√
2Mτ ′
2Rb
3
2
)− 23
dτ ′. (1.28)
Then we change to coordinates u = η−R, and v = η+R. Thus we
obtain the metric to be of the form
g =
(
1− 3
√
2Mτ(u,v)
2Rb
3
2
) 4
3
(−dudv+R(u,v)2gS2). (1.29)
In this coordinate system, the range of u and v is given by u+ v≤ 2τc and 0≤ v−u≤ 2R. Thus the interior
of the star is conformally flat. Hence the Penrose diagram for the interior is that of Minkowski space-time,
subject to the above ranges of u+ v and v− u. We also note that we have that RabcdRabcd blows up as η
approaches ηc = η(τc), so this corresponds to a singular boundary of space-time.
On the exterior of the dust cloud, our solution is a subregion of Schwarzschild space-time. The boundary of
this region is given by a timelike curve going from past timelike infinity to r = 0. Matching these two diagrams
across the relevant boundary, we obtain the Penrose diagram shown in Figure 1.1. Again, remember the metric
is only a piecewise smooth and H1loc function of u and v.
1.2 Generalising to the RNOS model
In this section, we begin the novel work of this thsis. We will generalise background models of spherically
symmetric dust cloud collapse to include charged matter. In section 1.2.1, we derive the metric of Reissner–
12 RNOS Spacetimes
i+
i−
i0
r
=
0
I
−
I
+
R
=
R
b
η = ηc
i+
i−
i0B
H
+
H
−
I
−
I
+
r = 0
r = 0
r
=
rb
Figure 1.2 Penrose diagram of Minkowski (left) and Schwarzschild (right) space-times, with appropriate
boundaries.
Nordstrom Oppenheimer–Snyder (RNOS) spacetimes in the exterior of our collapsing dust cloud. If the reader
is not interested in this derivation, they may skip straight to section 1.2.4, where the background manifold is
defined, with some interesting and/or useful properties stated.
1.2.1 Derivation
In this section, we derive our metric under the following assumptions: We assume our manifold is a spherically
symmetric solution of the Einstein–Maxwell equations
Rµν − 12Rgµν = 8πT µν (1.30)
T µν =
1
4π
(
FµαFνα − 14F
αβFαβgµν
)
(1.31)
∇νFνµ = 0 (1.32)
∇µFνα +∇νFαµ +∇αFµν = 0, (1.33)
with coordinates t∗ ∈ (−∞,∞), (θ ,ϕ) ∈ S2, r ∈ [r˜b(t∗),∞). We define
r˜b := max{rb(t∗),r+} (1.34)
r+ := M(1+
√
1−q2). (1.35)
i+
i0
(t∗c , r+)
i−
H +
I
+
I
−
r
=
r b
Σt ∗
Figure 1.3 Penrose Diagram
of RNOS Model, with space-
like hyper surface Σt∗ .
Here r = rb(t∗) is a hypersurface generated by a family of timelike, ingoing
radial curves such that, for any fixed θ ,ϕ , the curve {t∗,rb(t∗),θ ,ϕ} describes
the motion of a particle moving only under the electromagnetic force, with
charge to mass ratio matching that of the black hole. That is, we assume that the
surface of the cloud is itself massive and charged, with the same charge density
as the cloud itself. For our results, we will actually only require certain bounds
on rb and r˙b (see already Remarks 2.7.2, 3.4.8), but here we will derive the
behaviour of rb in full. We also note that we are looking solely at the exterior
of the black hole. Thus, we will not be considering the region rb(t∗)< r+. We
will instead use r˜b(t∗) = max{rb(t∗),r+} as the boundary of our manifold. The
topology of our manifold {t∗,r ≥ r˜b(t∗),θ ,ϕ} is that of a cylinder in 3+ 1D
1.2 Generalising to the RNOS model 13
Lorentzian space. As this is simply connected, equation (1.33) means we can
choose an A such that
F = dA. (1.36)
Given an asymptotically flat, spherically symmetric solution of the Einstein–Maxwell equations, we know
that our solution is a subset of a Reissner–Nordström spacetime (see for example [42]). This gives the first two
parameters of our spacetime; M, the mass of the Reissner–Nordström black hole spacetime our manifold is a
subset of, and q = Q/M, the charge density of our underlying Reissner–Nordström spacetime. We will assume
q has modulus less than or equal to 1, as otherwise our dust cloud will either not collapse, or will form a naked
singularity rather than a black hole.
Exterior Reissner–Nordström spacetime has global coordinates:
g =−
(
1− 2M
r
+
q2M2
r2
)
dt∗2+2
(
2M
r
− q
2M2
r2
)
dt∗dr+
(
1+
2M
r
− q
2M2
r2
)
dr2+ r2gS2 (1.37)
A =
qM
r
dt∗, (1.38)
where gS2 is the metric on the unit 2-sphere.
We now proceed to calculate the path moved by a radially moving charged test particle, with charge density
q. The motion of this particle extremises the following action:
S =
∫
r=rb
Ldτ = m
∫
r=rb
1
2
gabvavb−qvaAadτ (1.39)
= m
∫
r=rb
(
1
2
(
−
(
1− 2M
r
+
q2M2
r2
)(
dt∗
dτ
)2
+2
(
2M
r
− q
2M2
r2
)
dt∗
dτ
dr
dτ
+
(
1+
2M
r
− q
2M2
r2
)(
dr
dτ
)2)
− q
2M
r
dt∗
dτ
)
for va the velocity of the particle with respect to τ , A as defined in (1.36), and τ the proper time for the particle,
i.e. normalised such that gabvavb =−1.
We can then use first integrals of the Euler–Lagrange equations to find constants of the motion. Firstly, L is
independent of explicit τ dependence, and so gabvavb is constant. By rescaling τ , we choose gabvavb to be 1.
The second constant we obtain is from L being independent of t∗. Thus
T ∗ =
(
1− 2M
r
+
q2M2
r2
)
dt∗
dτ
−
(
2M
r
− q
2M2
r2
)
dr
dτ
+
q2M
r
(1.40)
is constant.
Using (1.40) to remove dependence of gabvavb on dt
∗
dτ , we obtain(
1− 2M
r
+
q2M2
r2
)−1(
T ∗− q
2M
r
)2
=
(
1− 2M
r
+
q2M2
r2
)(
dt∗
dτ
)2
−2
(
2M
r
− q
2M2
r2
)
dt∗
dτ
dr
dτ
+
(
2M
r − q
2M2
r2
)2
1− 2Mr + q
2M2
r2
(
dr
dτ
)2
=−gabvavb+
1+ 2Mr − q2M2r2 +
(
2M
r − q
2M2
r2
)2
1− 2Mr + q
2M2
r2
 drdτ 2
= 1+
(
1− 2M
r
+
q2M2
r2
)−1( dr
dτ
)2
, (1.41)
14 RNOS Spacetimes
which rearranges to (
dr
dτ
)2
=
(
T ∗− q
2M
r
)2
−
(
1− 2M
r
+
q2M2
r2
)
(1.42)
=
(
T ∗− q
2M
r
)2
−
(
1−M
r
)2
+
(1−q2)M2
r2
.
From (1.40), we can see that if this particle’s velocity is to be future directed, we need T ∗ > 0.
For |q| ≤ 1, T ∗ ≥ 1, and (|q|,T ∗) ̸= (1,1), equation (1.42) tells us that drdτ is positive. As drdτ is a continuous
function of r, this implies that rb must tend to ∞ as t∗→−∞. If q =±1, i.e. the extremal case, and T ∗ = 1, then
we have drdτ ≡ 0. Thus the dust cloud will not collapse, so we will not be considering |q|= 1 = T ∗.
We can also see, from writing out the statement gabvavb =−1, that(
−
(
1− 2M
r
+
q2M2
r2
)
+2
(
2M
r
− q
2M2
r2
)
dr
dt∗
+
(
1+
2M
r
− q
2M2
r2
)(
dr
dt∗
)2)(dt∗
dτ
)2
=−1 (1.43)
which tells us that drdt∗ >−1.
1.2.2 T ∗ < 1
We now look at the behaviour of rb in the case where T ∗ < 1.
If T ∗ < 1, then looking at r → ∞ we can see drdτ vanishes at a finite radius, so the dust cloud will tend to that
radius, either reaching it at a finite time, or as t →−∞.
We therefore look at integrating equation (1.42) to obtain τ(r), which gives us
τ =
((
T ∗r−q2M)2− (r2−2Mr+q2M2)) 12
1−T ∗2 −
2M(1−q2T ∗)
2(1−T ∗2)3/2 sin
−1
(
1−T ∗2
D
r− 2M(1−q
2T ∗)
D
)
(1.44)
where D = M
√
(1−q2T ∗)2−q2(1−q2)(1−T ∗2) is a constant.
Equation (1.44) tells us that in the case T ∗ < 1, the dust cloud’s radius obtains its limit within a finite (and
therefore compact) proper time interval. As t∗ is a continuous increasing function of τ , rb obtains its limit in
finite coordinate (t∗) time. We will call this finite time t∗−. At this point, the curve would collapse back into the
black hole, hitting the past event horizon. Therefore, in order to have a collapsing model, we will, in the T ∗ < 1
case, assume that the radius of the dust cloud, rb(t∗) remains at rb(t∗−) for all t∗ ≤ t∗−.
1.2.3 T ∗ ≥ 1
Here we have that our dust cloud radius tends to ∞, as τ →−∞. Thus the main part we will need to concern
ourselves with is what happens to the surface of the dust cloud as τ→−∞, rb →∞. Equations (1.42) and (1.40)
give us that
r˙b :=
dr
dt∗
=
dr
dτ
dt∗
dτ
→−
√
T ∗2−1
T ∗
=:−a as t∗→−∞ (1.45)
where we will refer to a ∈ [0,1) as the asymptotic speed of the surface of the dust cloud.
1.2 Generalising to the RNOS model 15
1.2.4 Definition of the RNOS Manifold and Global Coordinates
We have a 3 parameter family of collapsing spacetimes, with parameters M ≥ 0, q ∈ [−1,1], and T ∗ ∈ (0,∞).
However, we exclude the points with T ∗ = 1 = q. The topologies of the underlying manifolds are all given by:
M = R× [1,∞)×S2 (1.46)
= {(t∗,x,θ ,ϕ)}.
We scale the second coordinate in (1.46) to define r = xr˜b(t∗), so that the boundary is not at 1, but is at
r˜b(t∗) = max{r+,rb(t∗)}, where t∗ is the first coordinate. Then we have metric
g =−
(
1− 2M
r
+
q2M2
r2
)
dt∗2+2
(
2M
r
− q
2M2
r2
)
dt∗dr+
(
1+
2M
r
− q
2M2
r2
)
dr2+ r2gS2 (1.47)
t∗ ∈ R r ∈ [r˜b(t∗),∞)
where M ≥ 0, q ∈ [−1,1], r+ is given by (1.35), and gS2 is the Euclidean metric on the unit sphere.
Note that r˜b(t∗) is not a smooth function of t∗. Thus our manifold’s smooth structure, as given by (t∗,r)
coordinates, is not the same as the smooth structure given by (t∗,x) coordinates.
We have derived the following statements about rb:
r˙b(t∗) :=
dr
dt∗
∈ (−1,0] (1.48)
∃t∗c s.t. rb(t∗c ) = r+,rb(t∗)> r+ ∀t∗ < t∗c , (1.49)
where r+ is the black hole horizon for the Reissner–Nordström spacetime given by (1.35).
We also have 2 possible past asymptotic behaviours for rb. If T ∗ < 1, we have
rb(t∗) = r0 ∀t∗ ≤ t∗−, (1.50)
and if T ∗ ≥ 1, we have
drb
dt∗
→−a :=
√
T ∗2−1
T ∗
∈ (−1,0] as t∗→−∞. (1.51)
If T ∗ = 1, then rb ∼ (−t∗)2/3 as t∗→−∞.
The RNOS models have the same exterior Penrose diagram as the original Oppenheimer–Snyder model, see
Figure 2.1, derived in [2], for example.
We will also be using the double null coordinates given by:
u = t∗−
∫ r
s=3M
1+ 2Ms − q
2M2
s2
1− 2Ms + q
2M2
s2
ds (1.52)
v = t∗+ r (1.53)
∂u =
1
2
(
1− 2M
r
+
q2M2
r2
)
(∂t∗−∂r) (1.54)
∂v =
1
2
((
1+
2M
r
− q
2M2
r2
)
∂t∗+
(
1− 2M
r
+
q2M2
r2
)
∂r
)
(1.55)
g =−
(
1− 2M
r
+
q2M2
r2
)
dudv+ r(u,v)2gS2. (1.56)
Much of the later discussion will be concerning u and v coordinates. Therefore, we will find it useful to
parameterise the surface of the cloud by u and v. That is, given any u, define vb(u) to be the unique solution to
r(u,vb(u)) = rb(t∗(u,vb(u))). (1.57)
16 RNOS Spacetimes
We will also define ub in the domain v≤ vc as the inverse of vb, i.e.
ub(v) := v−1b (v). (1.58)
We will be making use of the following properties of vb:
vb(u)→ vc := v(t∗c ,r+) as u→ ∞ (1.59)
vc− vb(u) =
Ae−κu+O(e−2κu) |q|< 1A
u +O(u
−3) |q|= 1.
(1.60)
v′b(u) =
Ae−κu+O(e−2κu) |q|< 1A
u2 +O(u
−4) |q|= 1
. (1.61)
These are straightforward calculations, once we note in the extremal case we can choose where u = 0 to remove
the u−2 term to be zero in vc− vb. Here, κ is the surface gravity of the Reissner–Nordström black hole that our
cloud is collapsing to form, given by
ka∇akb = κkb, (1.62)
where ka is the null Killing vector field tangent to the horizon. In Reissner–Nordström, ka = ∂t∗ , and we have
κ =
√
1−q2(
2+2
√
1−q2−q2
)
M
. (1.63)
Finally, we have four linearly independent Killing vector fields in our space time. The timelike Killing field,
∂t∗ does not preserve the boundary {r = rb(t∗)}. However, we have 3 angular Killing fields, {Ωi}3i=1 (as given
in (1.26)) which span all angular derivatives and are tangent to the boundary of the dust cloud.
Chapter 2
The Scattering Map
2.1 Introduction and Overview of Main Theorems
i+
i0
(t∗c , r+)
i−
H +
I
+
I
−
r
=
0
φ
=
0
Σt ∗
Figure 2.1 Penrose diagram of
RNOS space-time with reflect-
ive boundary conditions, with
spacelike hypersurface Σt∗ .
In this chapter we will be studying energy boundedness of solutions to the linear
wave equation
□gφ =
1√−g∂a(
√−ggab∂bφ) = 0 (2.1)
on both Oppenheimer–Snyder space-time (M ,g) [39] and RNOS [1] back-
grounds, as discussed in the previous chapter.
In the case of Oppenheimer–Snyder, we will further consider two different
sets of boundary conditions: reflective, where we will impose the condition
φ = 0 on r = rb(t∗), (2.2)
where this is understood in a trace sense, and permeating, where we will be
solving the linear wave equation throughout the whole space-time, including
the interior of the star. In the case of the RNOS backgrounds, we will only be
considering reflective boundary conditions, as the interior will depend entirely
on one’s choice of matter model. We will then be using these results to define a
scattering theory for these space-times.
i+
i0
(τc , 2M)
i−
H +
I
+
I
−
r
=
0
r
=
r b
(τ
)
=
r b
(t
∗ )
Στ
Figure 2.2 Penrose Diagram
of Oppenheimer–Snyder
space-time with permeating
boundary conditions, with
spacelike hypersurface Στ .
The first main theorem dealing with solutions of (2.1) in the bulk of the
space-time is informally stated below:
Theorem 1 (Non-degenerate Energy (N-energy) boundedness). In both RNOS
space-times (with reflective boundary conditions) and Oppenheimer–Snyder
space-time (with permeating boundary conditions), let the mapF(t∗0 ,t∗1 ) take the
solution of (2.1) on a time slice Σt∗0 (or Στ0), forward to the same solution on
a later time slice, Σt∗1 ∪ (H +∩{t∗ ∈ [t∗0 , t∗1 ]}) (or Στ1 ∪ (H +∩{τ ∈ [τ0,τ1]})).
ThenF(t∗0 ,t∗1 ) is uniformly bounded in time with respect to the non-degenerate
energy. Furthermore, for t∗1 ≤ t∗c (or τ1 ≤ τc), its inverse is also bounded with
respect to this non-degenerate energy.
The contents of this theorem are stated more precisely across Theorems
2.5.3, 2.5.4, 2.5.5.
The sphere (t∗c ,2M) and the time slice Σt∗ (for t∗ < t∗c ) are shown in Figure
2.1. The sphere (τc,2M) and the time slice Στ (τ < τc) are shown in Figure 2.2.
Non-degenerate energy will be defined more accurately later in this chapter,
but it can be defined as the energy with respect to an everywhere timelike vector
field (including on the horizonH +) which coincides with the timelike Killing
18 The Scattering Map
vector in a neighbourhood of null infinity I ±. This energy controls the L2 norm of each 1st derivative of the
field, φ .
In the reflective case of Oppenheimer–Snyder, we also go on to show forwards and backwards boundedness
of higher order derivatives, see Theorem 2.6.1 and 2.6.2. In the permeating case we go on to show forwards and
backwards boundedness of 2nd order derivatives, see Theorem 2.6.3.
We then consider the limiting process to look at the radiation field on past null infinity I −, and obtain the
following result.
Theorem 2 (Existence and Non-degenerate Energy Boundedness of the Past Radiation Field). In both RNOS
space-times (with reflective boundary conditions) and Oppenheimer–Snyder space-time (with permeating
boundary conditions), we define the mapF− as taking the solution of (2.1) on Σt∗0 , t
∗≤ t∗c (or Στ0∪(H +∩{τ ≤
τ0})) to the radiation field on I −. F− is well-defined, and is bounded with respect to the non-degenerate
energy.
This theorem is stated more precisely as Theorem 2.7.1.
On Reissner–Nordström space-times, we know that the future radiation field exists, so the map G+ from
data on Σt∗ toI +∪H + exists (see [35] for example). It is also bounded in terms of the N-energy, [13, 3]. It is,
however, unbounded, going backwards, in terms of the N-energy (see for example [16]). This is stated more
precisely as Proposition 2.7.2. This result immediately applies to Oppenheimer–Snyder space-time. Together
with Theorem 2 and a new result about decay towards the past on asymptotically null foliations (see Lemma
2.7.1), this allows us to define the inverse ofF−,F+ (see Proposition 2.7.1). This combination also gives us
the final theorem:
Theorem 3 (Boundedness but non-surjectivity of the scattering map). We define the scattering map,
S + : E ∂t∗I − → E
∂t∗
I +×E NH + (2.3)
S + := G+ ◦F+
on RNOS space-times (with reflective boundary conditions) and Oppenheimer–Snyder space-time (with permeat-
ing boundary conditions) from data on I − to data on I +∪H +. S + is injective and bounded, with respect
to the non-degenerate energy (L2 norms of ∂v(rφ) on I − andH + and ∂u(rφ) on I +). One can then define
the inverse,S −, of (2.3), going backwards fromS +(E ∂t∗I −) (dense in E
∂t∗
I +×E NH +), in either the reflective or
permeating case. However,S − is not bounded with respect to the non-degenerate energy. It follows thatS + is
not surjective. Moreover, E ∂t∗I +×{0}H + is not a subset of Im(S +).
This Theorem is stated more precisely as Theorem 2.7.2.
In proving Lemma 2.7.1, we obtain a result on the rate at which our solution decays (towards i−, with respect
to this asymptotically null foliation) for data decaying sufficiently quickly towards spatial infinity. However we
do not look at optimising this rate, as only very weak decay is required for Theorem 3.
The non-invertibility ofS + is inherited from that of G+. This ultimately arises from the red-shift effect
alongH +, which for backwards time evolution corresponds to a blue-shift instability. It is the existence of
the mapF+ mapping into the space of non-degenerate energy however, that extends this non-invertibility to
data on I −. Note that for I −, the notion of energy is completely canonical. This is in contrast to the pure
Reissner–Nordström case, where no suchF+ exists.
It remains an open problem to precisely characterise the image of the scattering mapS +.
2.2 Previous Work
There has been a substantial amount of work done concerning the scattering map on Schwarzschild. However
there has been considerably less concerning the scattering map for collapsing space-times such as Oppenheimer–
Snyder. The exterior of the star is a vacuum spherically symmetric space-time and therefore has the Schwarz-
schild metric by Birkhoff’s Theorem, [40]. We will thus be using a couple of results in this region from previous
2.2 Previous Work 19
papers. However, we will not be discussing the scattering map on Schwarzschild very much beyond this. For a
more complete discussion of the wave equation on Schwarzschild, see [14].
Most previous works on scattering in gravitational collapse, such as [8, 26, 6, 34], assume that the star/dust
cloud is at a finite radius from infinite past up to a certain time and then proceed to let this cloud collapse, as in
the RNOS model with T ∗ < 1. Thus these models are stationary in all but a compact region of space-time. This
model allows these previous works to avoid the difficulty of allowing the star to tend to infinite radius towards
the past, as happens in the original Oppenheimer–Snyder model that we will be studying here. Also, dynamics
on the interior of the star have not been examined, and so only the case of reflective boundary conditions has
been studied previously. The energy current techniques we will be using here can, with relatively little difficulty,
also be applied to these finite-radius models. These energy current methods are also more easily generalisable to
other space-time models: for example, to obtain boundedness of the forward scattering map, all that is required
to apply these techniques is that the star is collapsing. Nonetheless, in this thesis, the only interior we will
consider is that of the Oppenheimer–Snyder model.
In this thesis, we look at defining the scattering mapS + geometrically as a map from data on I − to data
onH +∪I + (equation (2.3)). This is treating scattering in terms of the Friedlander radiation formalism (as in
[21]). In the above papers ([8, 26, 6, 34]), their solution is evolved forward a finite time, then evolved back to
t = 0 with respect to either Schwarzschild metric (for the horizon radiation field) or Minkowski metric (for the
null infinity radiation field). Then the authors show that the limit as that time tends to infinity exists. All this is
done using the language of wave operators. For a comparison of these two approaches to scattering theory, the
reader may wish to refer to Section 4 of [38].
Let us discuss two related works in more detail. The work [8] studies the Klein–Gordon equation ((2.1) is
the massless Klein–Gordon equation, thus is studied as a special case) on the finite-radius model discussed
above. In this context, the author obtains what can be viewed as a partial result towards the analogue of Theorem
1 for each individual spherical harmonic. However they do not find a bound independent of angular frequency.
Again in the finite-radius model, [26] studies the Dirac equation for spinors. However, as this has a 0th order
conserved current, this allows a Hilbert space to be defined such that the propagator through time is a unitary
operator. Thus there is no need for the (first order) energy currents we will be using. This also allows questions
of surjectivity to be answered with relative ease.
There have been no mathematical works considering the scattering map in the charged case. There have,
however, been other works considering the underlying models of charged collapse, and there have been other
works considering scattering on Reissner–Nordström backgrounds.
Most papers modelling collapsing models focus on the interior of the collapsing star. This thesis, however,
will not focus on the specifics of interior models such as these in the charged case. There are many such models,
which entirely depend on what equation of state is chosen for the interior of the dust cloud.
For simplicity in the charged case, this thesis will assume only that the surface of the collapsing cloud follows
the motion of charged particles in the exterior spacetime, as discussed in Section 1.2. In fact the techniques in
this chapter do not require an assumption this substantial, and will allow the results to be generalised beyond
this. However, to avoid becoming mired in the details of the interior of the dust cloud, we will restrict to these
RNOS models.
Generally, study of the scattering map in the exterior sub-extremal Reissner–Nordström spacetime is paired
together with that of Schwarzschild, as it has similar behaviour (see [14]). The extremal case has been studied
in detail separately, see [3]. Behaviour in this case differs substantially from the sub-extremal case. Scattering
in the interior has also been studied independently, [32]. The exterior of the RNOS Models (see Section 1.2)
is a spherically symmetric, vacuum solution to the Einstein–Maxwell equations, and thus has the Reissner–
Nordström metric, by uniqueness (see [42], for example). However, this thesis will not be discussing the
scattering map on Reissner–Nordström much beyond this, and instead will quote results from [14] (in the
sub-extremal case) and [3] (in the extremal case). We refer the reader to these for a more complete discussion of
scattering in Reissner–Nordström spacetimes.
20 The Scattering Map
2.3 Notation
In this section, we will be using similar notation to [2, 1].
We will be considering the following hypersurfaces in our manifold, equipped with the stated normals
and volume forms. Note these normals will not neccessarily be unit normals, but have been chosen such that
divergence theorem can be applied without involving additional factors.
Σt∗0 := {(t∗,r,θ ,ϕ) :, t∗ = t∗0} dV = r2drdω dn =−dt∗
(2.4)
Σu0 := {(t∗,r,θ ,ϕ) : u(t∗,r) = u0} dV =
1
2
(
1− 2M
r
+
q2M2
r2
)
r2dvdω dn =−du
(2.5)
Σv0 := {(t∗,r,θ ,ϕ) : v(t∗,r) = v0} dV =
1
2
(
1− 2M
r
+
q2M2
r2
)
r2dudω dn =−dv
(2.6)
S[t∗0 ,t∗1 ] = {(t
∗, r˜b(t∗),θ ,ϕ) s.t. t∗ ∈ [t∗0 , t∗1 ]} dV = r2dt∗dω dn = dρ := dr− ˙˜rbdt∗,
(2.7)
where dω is the Euclidean volume form on the unit sphere i.e.
dω = sinθdθdϕ. (2.8)
Note that dω will not be used as the volume form on the unit sphere in Chapter 3, to avoid confusion with ω
being a frequency.
We will also later be using, for the permeating case, the eventually null foliation, Σ˜τ0 . These are the set of
points with τ = τ0 for r < rb, and v = v0 for r ≥ rb. Here v0 is the value of v at (τ0,rb(τ0)).
Σ˜τ0 = (Στ0 ∩{r < rb(τ0)})∪ (Σv0 ∩{r ≥ rb(τ0)}) . (2.9)
This will have the same volume form as Στ0 for r < rb(τ0) and the same volume form as Σv0 for r ≥ rb(τ0).
We will finally make use of
Σ¯t0,R := (Σu=t0+R∩{r∗ ≤−R})∪ (Σt0 ∩{r∗ ∈ [−R,R]})∪ (Σv=t0+R∩{r∗ ≥ R}) . (2.10)
The volume form of Σ¯t0,R matches that of Σu0 , Σv0 and Σt0 in each section.
Whenever considering the reflective case, we will restrict these surfaces to r ≥ r˜b(t∗). However, in the
permeating case, or when considering pure Reissner–Nordström spacetime, this will not be required.
We define future/past null infinity by:
I + := R×S2 dV = dudω I − := R×S2 dV = dvdω. (2.11)
Past null infinity is viewed as the limit of Σu0 as u0 → ∞. For an appropriate function f (u,v,θ ,ϕ), we will
define the function “evaluated on I +" to be
f (v,θ ,ϕ)|I − := limu→−∞ f (u,v,θ ,ϕ). (2.12)
Similarly, I + is considered to be the limit of Σv0 as v0 → ∞. For an appropriate function f (u,v,θ ,ϕ), we
will define the function “evaluated on I +" to be
f (u,θ ,ϕ) := lim
v→∞ f (u,v,θ ,ϕ). (2.13)
2.3 Notation 21
From here onwards, any surface integral that is left without a volume form will be assumed to have the
relevant volume form listed above, and all space-time integrals will be assumed to have the usual volume form√−det(g).
We will be considering solutions of (2.1) which vanish on the surface r = rb(t∗) (in a trace sense). We will
generally be considering these solutions to arise from initial data on a spacelike surface. Initial data will consist
of imposing the value of the solution and its normal derivative, with both smooth and compactly supported.
We will then consider the following seminorms of a spacetime function f , given by:
∥ f∥2L2(Σ) =
∫
Σ
| f |2dV. (2.14)
We will also define the H˙1 norm as:
∥ f∥2H˙1(Σt∗0 ) :=
∫
Σt∗0
|∂t∗ f |2+ |∂r f |2+ 1r2∥ /˚∇ f∥
2dV (2.15)
∥ f∥2H˙1(Σu0) :=
∫
Σu0
|∂v f |2(
1− 2Mr + q
2M2
r2
)2 + 1r2∥ /˚∇ f∥2dV (2.16)
∥ f∥2H˙1(Σv0) :=
∫
Σv0
|∂u f |2(
1− 2Mr + q
2M2
r2
)2 + 1r2∥ /˚∇ f∥2dV, (2.17)
where /˚∇ is the induced gradient on the unit sphere. This is a tensor on the unit sphere, and we define the norm
of such a tensor by
∥T∥2 =
n
∑
a1,a2,...am=1
|Ta1,a2,...an|2 (2.18)
for T an m tensor on Sn, in any orthonormal basis tangent to the sphere at that point.
Note that we have not yet defined the spaces for which the H˙1 norms will actually be norms.
The generalisation of the H˙1 norm are the H˙n norms, which we will define on Σt∗ by
∥ f∥2H˙n(Σt∗0 ) := ∑n1,n2,n3
1≤n1+n2+n3≤n
n1,n2,n3≥0
∫
Σt∗0
1
r2n3
∥ /˚∇n3∂ n2r ∂ n1t∗ f∥2dV. (2.19)
Let C∞0 (S) be the space of compactly supported functions on surface S, which vanish on {r = rb(t∗)}∩S.
We will define the H˙1(Σt∗0 ) norm on a pair of functions φ0,φ1 ∈C∞0 (Σt∗0 ) as follows:
∥(φ0,φ1)∥H˙1(Σt∗0 ) := ∥φ∥H˙1(Σt∗0 ) for any φ s.t. (φ |Σt∗0 ,∂t∗φ |Σt∗0 ) = (φ0,φ1). (2.20)
We similarly define the H˙1(Σu0) and H˙1(Σv0) on φ0 ∈C∞0 (Σu0,v0) as follows:
∥φ0∥H˙1(Σu0,v0) := ∥φ∥H˙1(Σu0,v0) for any φ s.t. φ |Σu0,v0 = φ0. (2.21)
We will also need to consider what functions we will be working with. For this, we will be using the same
notation as [15, 2]. We first need to look at the notions of energy momentum tensors and energy currents (note
22 The Scattering Map
this energy momentum tensor will be expressed as T , and is different from T in (1.1)).
Tµν(φ) = ∇µφ∇νφ − 12gµν∇
ρφ∇ρφ (2.22)
JXµ = X
νTµν (2.23)
KX = ∇µJXµ (2.24)
JX ,wµ = X
νTµν +w∇µ(φ2)−φ2∇µw (2.25)
KX ,w = ∇νJX ,wν = KX +2w∇µφ∇µφ −φ2□gw (2.26)
X-energy(φ ,S) =
∫
S
dn(JX). (2.27)
Here, dn is the normal to S. It should be noted that applications of divergence theorem do not introduce any
additional factors with our choice of volume form and normal, i.e.∫
t∗∈[t∗0 ,t∗1 ]
KX ,ω =−
∫
Σt∗1
dn(JX ,ω)+
∫
Σt∗0
dn(JX ,ω)−
∫
S[t∗0 ,t
∗
1 ]
dn(JX ,ω), (2.28)
with similar equations holding for Σu,v and in the permeating case.
For any Tµν obeying the dominant energy condition, X future pointing and causal, and S spacelike, then the
X-energy is non-negative.
For any pair of functions, φ0,φ1 ∈C∞0 (Σt∗0 ), and X a causal, future pointing vector, we define the X norm by
∥(φ0,φ1)∥2X ,Σt∗0 := X-energy(φ ,Σt∗0 ) for any φ s.t. (φ |Σt∗0 ,∂t∗φ |Σt∗0 ) = (φ0,φ1). (2.29)
We similarly define for φ0 ∈C∞0 (Σu0,v0)
∥φ0∥2X ,Σu0,v0 := X-energy(φ ,Σu0,v0) for any φ s.t. φ |Σu0,v0 = φ0. (2.30)
Note that for any causal, future pointing X which coincides with the timelike Killing vector field ∂t∗ in a
neighbourhood of I ±, we have that the X norm is Lipschitz equivalent to the H˙1 norm.
For causal timelike vector X , we define the following function spaces
E XΣt∗0
:=ClX ,Σt∗0
(C∞0 (Σt∗0 )×C∞0 (Σt∗0 )) (2.31)
E XΣu0,v0
:=ClX ,Σu0,v0 (C
∞
0 (Σu0,v0)) (2.32)
where these closures are in H1loc with respect to the subscripted norms.
For ψ0 ∈C∞0 (I ±), we define
∥ψ0∥2∂t∗ ,I + :=
∫
I +
|∂vψ0|2dvdω (2.33)
∥ψ0∥2∂t∗ ,I − :=
∫
I −
|∂uψ0|2dudω. (2.34)
We define the energy spaces E ∂t∗I ± by
E
∂t∗
I ± :=Cl∂t∗ ,I ±(C
∞
0 (I
±)). (2.35)
We will finally be considering "well behaved" functions to prove results, and extending our results by density
arguments. If a function onM is smooth and compactly supported on Σt∗ for every t∗ (or compactly supported
on Στ and in H2loc(Στ) for every τ), then we will refer to this function as being in C
∞
0∀t∗ (or H
2
0∀τ ).
2.4 Existence and Uniqueness of Solutions to the Wave Equation 23
2.4 Existence and Uniqueness of Solutions to the Wave Equation
Consider initial data given by φ = φ0 and ∂t∗φ = φ1 on the spacelike hypersurface Σt∗0 (or ∂τφ = φ1 on Στ0). For
the reflective case, we also impose Dirichlet conditions on the surface of the star, φ = 0 on r = R(t). We first
show existence of a solution to the forced wave equation,
□gφ =
1√−g∂a(
√−ggab∂bφ) = F (2.36)
for g as given in (1.47) and (1.22).
These are standard results which can be taken from literature, but there is no elementary reference. For
completeness we will include a proof here.
2.4.1 The Reflective Case
In the reflective case, we initially prove existence and uniqueness for smooth, compactly supported initial data.
We will be proving existence up until the surface of the star passes through the horizon. For later t∗ times, we are
then in exterior Schwarzschild space-time with the usual boundary ar r = 2M, so can refer to standard existing
proofs of existence and uniqueness (see for example proposition 3.1.1 in [14]). The proof below closely follows
that of Theorems 4.6 and 5.3 of Jonathan Luk’s notes on Nonlinear Wave Equations [33], and comes in two
parts:
We proceed by first proving uniqueness via the following lemma:
Lemma 2.4.1 (Uniqueness of Solution to the Forced Wave Equation). Let φ ∈C∞0∀t∗ be a solution to equation
(2.36) in some region r ≥ rb(t∗)≥ 2M, t∗−1 ≤ t∗0 ≤ t∗1 , with
φ = 0 on r = rb(t∗)
φ = φ0
∂t∗φ = φ1
}
on Σt∗0
(2.37)
with gab given by (1.47).
It follows that ∃A,C > 0 s.t.
sup
t∗∈[t∗−1,t∗1 ]
∥φ∥H˙1(Σt∗) ≤C
(
∥(φ0,φ1)∥H˙1(Σt∗0 )+
∫ t∗1
t∗−1
∥F∥L2(Σt∗)(t∗)dt∗
)
exp
(
A|t∗1 − t∗−1|
)
. (2.38)
In particular, if φ ,φ ′ ∈C∞0∀t∗ are both solutions to the above problem, then consider ζ = φ −φ ′. We have
that ζ solves equation (2.36) with F = 0 and has 0 initial data. Thus, ζ = φ −φ ′ = 0 everywhere, and we have
uniqueness.
Proof. We first consider coordinates (t∗,ρ,θ ,ϕ), where ρ = r− rb(t∗)+ r+. This causes ∂t∗ to be tangent to
the boundary r = rb(t∗). The metric then takes the form
g =−
((
1− 2M
r
+
q2M2
r2
)
− 2M
r
(
2− q
2M
r
)
r˙b−
(
1+
2M
r
− q
2M2
r2
)
r˙b2
)
dt∗2 (2.39)
+
(
2M
r
− q
2M2
r2
+
(
1+
2M
r
− q
2M2
r2
)
r˙b
)
2dρdt∗+
(
1+
2M
r
− q
2M2
r2
)
dρ2
+ r(t∗,ρ)2gS2.
24 The Scattering Map
We integrate the following identity:
∂0φ(∂a(g¯ab∂bφ)− F¯)) = 0 (2.40)
for
g¯ab =
√−ggab (2.41)
F¯ =
√−gF. (2.42)
We look at the cases a = b = 0, a = i,b = j, and {a,b}= {0, i} separately, where i, j ∈ {1,2,3}.
∫ t∗1
t∗0
∫
Σt∗
∂0φ∂0(g¯00∂0φ)dρdω2dt∗=
1
2
(∫ t∗1
t∗0
∫
Σt∗
(∂0g¯00)(∂0φ)2dρdω2dt∗+
(∫
Σt∗1
−
∫
Σt∗0
)
g¯00(∂0φ)2dρdω2
)
.
(2.43)∫ t∗1
t∗0
∫
Σt∗
∂0φ∂i(g¯i j∂ jφ)dρdω2dt∗=
1
2
(∫ t∗1
t∗0
∫
Σt∗
(∂0g¯i j)(∂iφ∂ jφ)dρdω2dt∗−
(∫
Σt∗1
−
∫
Σt∗0
)
g¯i j(∂iφ∂ jφ)dρdω2
)
.
(2.44)∫ t∗1
t∗0
∫
Σt∗
∂0φ(∂i(g¯i0∂0φ)+∂0(g¯i0∂iφ))dρdω2dt∗ =
∫ t∗1
t∗0
∫
Σt∗
(∂0g¯i0)(∂0φ∂iφ)dρdω2dt∗. (2.45)
Here we have integrated by parts. Using the fact that as φ = 0 on our boundary and ∂0 is tangent to our boundary,
we can see that ∂0φ = 0 on our boundary. We have used this to simplify the above boundary terms. Using
g¯ab =
 −(1+ 2Mr − q2M2r2 ) 2Mr − q2M2r2 +(1+ 2Mr − q2M2r2 ) r˙b
2M
r − q
2M2
r2 +
(
1+ 2Mr − q
2M2
r2
)
r˙b
(
1− 2Mr + q
2M2
r2
)
− 2Mr
(
2− q2Mr
)
r˙b−
(
1+ 2Mr − q
2M2
r2
)
r˙b2
r2 sinθ
(2.46)
for a,b = 0,1, we see that
|∂0g¯ab|
≤ A′r2 a,b = 0,1= 0 a,b = 2,3. (2.47)
(Note that coordinate singularities have been removed from gab by multiplying by
√−g.)
We then have, by summing (2.43), (2.44) and (2.45) together, that
∫
Σt∗1
g¯i j∂iφ∂ jφ − g¯00(∂0φ)2dρdω2 =
∫
Σt∗0
g¯i j∂iφ∂ jφ − g¯00(∂0φ)2dρdω2+ 12
∫ t∗1
t∗0
(∂0g¯ab)∂aφ∂bφ − F¯∂0φdρdω2dt∗
≤
∫
Σt∗0
g¯i j∂iφ∂ jφ − g¯00(∂0φ)2dρdω2 (2.48)
+
1
2
∫ t∗1
t∗0
∥φ∥H˙1(Σt∗)∥F∥L2(Σt∗)+A
′∥φ∥2H˙1(Σt∗)dt
∗.
Note that the bar is removed from F as the factor of
√−g is absorbed into the volume form in the norms of F
and φ .
Then we define
E(t∗1) =
∫
Σt∗1
g¯i j∂iφ∂ jφ − g¯00(∂0φ)2dρdω2. (2.49)
As the surface of the star is timelike, we have that −g00 = g11 is bounded above and below by positive
constants independent of time (from equation (1.5)). We note that the r2 term from using g¯ instead of g is
identical to the volume form in ∥φ∥2H˙1(Σt∗1 )
. This implies E(t∗1)∼ ∥φ∥2H˙1(Σt∗1 )
. Thus, using the fact that the RHS
2.4 Existence and Uniqueness of Solutions to the Wave Equation 25
of (2.48) is increasing in t∗1 , we have
f (t∗1) : = sup
t∗∈[t∗0 ,t∗1 ]
∥φ∥2H˙1(Σt∗) ≤C supt∗∈[t∗0 ,t∗1 ]
E(t∗)≤C′∥φ∥2H˙1(Σ0)+C
∫ t∗1
t∗0
∥φ∥H˙1(Σt∗)∥F∥L2(Σt∗)+A
′∥φ∥2H˙1(Σt∗)dt
∗
≤C′ f (t∗0)+C
√
f (t∗1)
∫ t∗1
t∗0
∥F∥L2(Σt∗)dt∗+C
∫ t∗1
t∗0
A′ f (t∗)dt∗
≤C′ f (t∗0)+
C2
2
(∫ t∗1
t∗0
∥F∥L2(Σt∗)dt∗
)2
+
f (t∗1)
2
+C
∫ t∗1
t∗0
A′ f (t∗)dt∗. (2.50)
We can then subtract the f (t∗1)/2 term from both sides to end up with an inequality of the form
f (t∗1)≤ A(t∗0 , t∗1)+
∫ t∗1
t∗0
f (t∗)h(t∗)dt∗. (2.51)
An application of Gronwall’s inequality gives our result, but with t∗−1 replaced with t
∗
0 . We then repeat the same
argument with time reversed to obtain the final result.
Note we have written out the above argument explicitly in coordinates. It could be written out using the
energy momentum tensor and a suitable vector field multiplier, as we have done in Section 2.5.
Next we need to deal with existence. To do this, we prove the following theorem:
Theorem 2.4.1 (Existence of Reflective Solutions). Let F ∈Ck([t∗−1, t∗1 ];C∞0 (Σt∗)) ∀k ∈ N and gab as above. Let
also φ0 and φ1 smooth, compactly supported functions on Σt∗0 such that φ0(rb(t
∗
0),θ ,ϕ) = 0, and such that these
initial conditions are not incompatible with the wave equation at r = rb(t∗)*. There exists a C∞0∀t∗ solution to
equation (2.36) subject to (2.37).
*By this, we mean that setting φ |Σt∗0 = φ0 and ∂t∗φ1|Σt∗0 = φ1 imposes the value of all first and second order
derivatives of φ on Σt∗0 , other than ∂
2
t∗φ . As we also require φ to vanish on the surface r = rb(t∗), this boundary
condition imposes that ∂t∗φ+ r˙b∂rφ = 0 and (∂t∗+ r˙b∂r)2φ = 0 on r = rb(t∗). We require that setting φ |Σt∗0 = φ0
and ∂t∗φ1|Σt∗0 = φ1 does not cause a contradiction between this boundary condition and the wave equation at
(t∗0 ,rb(t
∗
0),θ ,ϕ).
Proof. We begin the proof with the case (φ0,φ1) = (0,0). Let the set C0 ⊂C∞0∀t∗ be the image under the map
□g of C∞0 (M ). We define the map W by:
W : C0 → R
□gψ 7→
∫ t∗1
t∗−1
∫
Σt∗
ψF
√−gdρdθdϕdt∗ =: ⟨F,ψ⟩
This is well defined by our previous uniqueness lemma: suppose two functions ψ1,ψ2 ∈C0 have □gψ1 =□gψ2.
Then we can choose t∗0 to be far back enough that Σt∗0 does not intersect the support of either ψ1 or ψ2. Thus
ψ1−ψ2 solves (2.1) with vanishing initial data. Lemma 2.4.1 then gives ψ1−ψ2 = 0 everywhere, i.e. they are
equal.
We then proceed by quoting Lemma 5.2 in [33], which relies on definitions of H−k spaces. The space
H−k(Σt∗) is defined to be the dual of Hk(Σt∗) (the space of bounded linear maps from Hk(Σt∗) to R). Note also
that, as a Hilbert space, Hk(Σt∗) is reflexive, i.e. the dual of H−k(Σt∗) is Hk(Σt∗). In the permeating case, we
define H−k(Στ) in an identical manner.
Lemma 2.4.2. Suppose ψ ∈C∞0 ((−∞, t∗1)×Σt∗), supported away from r = rb(t∗), and g as above. Fix t∗0 ∈
(−∞, t∗1). Then for any m ∈ Z, ∃C =C(m, t∗0 , t∗1 ,g)> 0 s.t.
∥ψ∥Hm(Σt∗2 ) ≤C
∫ t∗1
t∗0
∥□gψ∥Hm−1(Σt∗)(s)dt∗ ∀t∗2 ∈ [t∗0 , t∗1 ]. (2.52)
26 The Scattering Map
Remark 2.4.1. To see this from [33], one must first “Euclideanise”, i.e. replace angular and r coordinates with
some x,y,z in order for these coordinates to be everywhere regular. We can then extend our metric smoothly to
inside the star. Using the result of Lemma 2.4.1 allows the proof to proceed exactly as in [33]. Note that linear
maps on the space extended inside the star are also linear maps when restricted to functions on the outside of
the star.
Lemma 2.4.2 then gives the bound
|W (□gψ)|=
∣∣∣∣∫ t∗1t∗0
∫
Σt∗
ψF
√−gdρdθdϕdt∗
∣∣∣∣
≤C
(∫ t∗1
t∗−1
∥F∥Hk−1(Σt∗)(t
∗)dt∗
)
( sup
t∗∈[t∗−1,t∗1 ]
∥ψ∥H−k+1(Σt∗))≤C
∫ t∗1
t∗−1
∥□gψ∥H−k(Σs)(s)ds, (2.53)
for smooth and compactly supported functions away from the horizon. We then take the closure of such functions
with respect to the Hk norm, for which W is linear and bounded. Thus by Hahn–Banach (Theorem 5.1, [33]),
there exists a function φ ∈ (L1((−∞,T );H−k(Σt∗)))∗ = L∞((−∞,T );Hk(Σt∗)) ∀k, which extends W as a linear
map. This means
⟨F,ψ⟩= ⟨φ ,□gψ⟩ ∀ψ ∈C∞0 (M ). (2.54)
Now □g obeys
⟨□gψ1,ψ2⟩= ⟨ψ1,□gψ2⟩ ∀ψ1,2 ∈C∞0 ((−∞, t∗1)×Σt∗). (2.55)
Thus equation (2.54) means that φ is a solution of (2.36) in the sense of distributions.
We then consider the following equation which ∂t∗φ solves, in a distributional sense:
vµ∇µ(φ˙) = hφ˙ +F1, (2.56)
for
vµ = (−g00,−2g01,−2g02,−2g03) (2.57)
h =
1√−g∂ν
(
gν0
√−g) (2.58)
F1 =
1√−g∂ν
(
gν i
√−g)∂iφ +gi j∂i∂ jφ −F. (2.59)
We explicitly have h and F1, as we have φ and its spacelike derivatives. We can then easily solve this along
integral curves of vµ to obtain that φ˙ exists as a function and is continuous.
We then look at the difference between equation (2.56) and the wave equation (2.36). Here we are considering
everything as distributions rather than functions. This gives us that
vµ∇µ
(
φ˙ −∂t∗φ
)−h(φ˙ −∂t∗φ)= 0. (2.60)
Applying the zero distribution is the same as integrating against the zero function. We also know φ˙ −∂t∗φ is
zero on the initial surface. It is then zero along all integral curves of vµ , and is therefore the zero function
everywhere. Thus ∂t∗φ exists everywhere and is continuous.
Then, by considering equation (2.36) and its derivatives, we can determine further weak derivatives with
respect to time. If F ∈Ck([t∗−1, t∗1 ];C∞0 (Σt∗)) ∀k, then our final solution has finite Hk(Σt∗) norm for all k and
all t∗ ∈ [t∗−1, t∗1 ], This means it is smooth. Due to finite speed of propagation of the wave equation it is also
compactly supported on each Σt∗ .
Finally, we show φ is a classical solution. Let ψ be an arbitrary function in C∞0 (M ), supported away from
the boundary. We can then integrate (2.54) by parts. Using the fact φ is smooth, we can see that □gφ = F .
By choosing k = 1, we note that φ extends W to the closure of C∞0 (M ) under the H
1 norm. In particular,
this includes functions which are smooth with non-vanishing derivative at the horizon. From this set, we can
2.4 Existence and Uniqueness of Solutions to the Wave Equation 27
choose any arbitrary smooth compactly supported function ψ . Let us choose one which is zero at the boundary,
but with non-zero normal derivative at the boundary. The boundary term we obtain when integrating (2.54) by
parts gives that φ = 0 on r = rb(t∗), as required, provided F = 0 on the boundary.
Now let (φ0,φ1) be smooth, as in the statement of the theorem. Let u ∈C∞0 ([0, t∗1 ]×Σt∗) be any function
with (u,∂tu) = (φ0,φ1) on t = t∗0 , and u = 0 on the boundary r = rb(t
∗). Then if we solve
□gν = F2 = F−□gu (2.61)
(ν ,∂tν) = (0,0) on t = t∗0 , (2.62)
then φ := ν+u is our required solution. The existence of such a u for which F2 vanishes at r = rb makes use of
* in the statement of the Theorem.
Remark 2.4.2. Theorem 2.4.1 will allow us to extend other results. Suppose we obtain any result on boundedness
between times slices Σt∗ in the H1(Σt∗) norm (not necessarily uniform in time). We can use a density argument
to obtain that given initial data in H1(Σt∗0 ), there exists an H
1(Σt∗) ∀t∗ solution. Again, this would be a solution
in the sense of distributions (see already Theorem 2.5.2).
2.4.2 The Permeating Case
The proof for the permeating case follows almost identical lines to that of the reflective case. There are fewer
concerns about the boundary, but the solution itself cannot be shown to be smooth for smooth initial data.
We still have Lemma 2.4.1 applying in this case, with almost no change to the proof. Lemma 2.4.2 also
remains the same for all m≤ 2. This can be seen by considering (τ,R,θ ,ϕ) coordinates ((1.27) in chapter 1).
We can then commute with both angular derivatives and ∂τ , and also rearranging (2.36) for ∂ 2Rφ . This just leaves
the analogue of Theorem 2.4.1:
Proposition 2.4.1 (Existence of Permeating Solutions with Initial Data Constraints). Suppose
F ∈C1([τ−1,τ1];H1(Στ)), and gab as above. Suppose also we are given (φ0,φ1) ∈H1(Στ) such that there exists
a function u ∈□−1g (H1(M ))∩H2(M ) with (φ0,φ1) = (u,∂τu) on Στ0 . Then there exists an H2([τ−1,τ1]×Στ)
weak solution to equation (2.36), subject to:
φ = φ0
∂τφ = φ1
}
on Στ0. (2.63)
Proof. Again, we begin with the (φ0,φ1) = (0,0) case. We define the map W exactly as in the reflecting case.
We define it on C′0, the image under the map □g on C∞0 (M ). Note the components of g are H1loc functions, so
have weak derivatives in L2loc. Thus this operator still exists. As before, this operator is well defined, is linear,
and is bounded.
Thus, again by Hahn–Banach, there exists a function φ ∈ (L1((−∞,τ1);H−k(Σt∗)))∗=L∞((−∞,τ1);Hk(Σt∗))
∀k ≤ 2 such that
⟨F,ψ⟩= ⟨φ ,□gψ⟩ ∀ψ ∈C∞0 (M ). (2.64)
As before, we can show φ has a τ derivative by considering the equation obeyed by ∂τφ as a distribution. If F is
in H1, then φ has two weak spatial derivatives. It also has a τ derivative with spacelike weak derivatives. By
integrating (2.64), we obtain that it also has a second weak time derivative. Thus it is H2, and thus our solution
is a weak solution of (2.36).
We then proceed with the final section in exactly the same way. Note that given our function, u, we can take
F2 = F−□gu well behaved. Thus our solution is H2, and therefore is a solution in a weak sense. However, this
sense is sufficient for the applications listed in later sections.
The final thing we need in order to complete existence of solutions is the following: we need to show that
initial data matching our condition (φ0,φ1) = (u,∂τu) on Στ0 is dense in H1(Στ0):
28 The Scattering Map
Proposition 2.4.2. Let (φ0,φ1) be any pair of functions in C∞0 (Στ0)×C∞0 (Στ0). Then there exists a sequence of
globally defined functions un ∈C∞0 (M )∩□−1g (H1(M )) such that
(un|Στ0 ,∂τun|Στ0 )
H1(Στ0)−−−−→ (φ0,φ1). (2.65)
Proof. We first remove the region over which g is not smooth by defining a smooth sequence (φ0,n,φ1,n) such
that
(φ0,n,φ1,n)
H1(Στ0)−−−−→ (φ0,φ1) (2.66)
and ∂rφ0,n = φ1,n = ∂rφ1,n = 0 for the region [rb−1/n,rb+1/n].
Let χ be a smooth cut-off function which is 1 outside [−2,2] and 0 inside [−1,1]. We first construct φ1,n: Let
φ1,n = φ1χ(n(r− rb)). It is clear that this tends to φ1 in the H0 norm. It is also clear that it and its r derivative
are 0 in the required region. Then we choose φ0,n = χ(n(r− rb))φ0+(1−χ(n(r− rb)))φ0(rb).
It is clear to see that the r derivative vanishes while χ = 0. All that remains is to show that
∥((1−χ(n(r− rb)))(φ0−φ0(rb)),0)∥H1(Στ0)→ 0. (2.67)
It is easy to see that the L2 norm of this tends to 0. Similarly the angular derivatives tend to 0 in L2. Then all
that is left to prove is that the r derivative tends to 0.
∥∂r(φ0−φ0,n)∥L2(Στ0) = ∥(1−χ(n(r− rb)))∂rφ0−nχ
′(n(r− rb))(φ0−φ0(rb))∥L2(Στ0)
≤ ∥(1−χ(n(r− rb)))∂rφ0∥L2(Στ0)+∥nχ
′(n(r− rb))(φ0−φ0(rb))∥L2(Στ0) (2.68)
≤ ∥1−χ(n(r− rb))∥L2(Στ0) sup |∂rφ0|
+ sup
r∈[rb−1/n,rb+1/n]
|n(φ0−φ0(rb))|∥χ ′(n(r− rb))∥L2(Στ0).
The first term in the RHS tends to 0, as (1−χ(n(r−rb)))∈ [0,1] is only supported in r ∈ [rb−2/n,rb+2/n].
The supremum in the second terms tends to |∂rφ0(rb)|, so is bounded. The χ ′ in the second term is bounded and
only non-zero in a region whose volume tends to 0. Therefore the whole second term also tends to 0.
Now, given the pair (φ0,n,φ1,n), we define un := (φ0,n+ τφ1,n)(1−χ((2nτ))).
As ∂τrb ≤ 1, we have that ∂run = ∂τun = ∂r∂τun = 0 for all r ∈ [rb− 1/2n,rb + 1/2n]. We also have
(un,∂τun) = (φ0,n,φ1,n) at τ = 0. Thus we can see
□gun =
1√−g∂a(
√−ggab∂bun) = H1-terms+ ∑
a,b∈{r,τ}
(∂agab)∂bun. (2.69)
The only terms in the sum where ∂agab /∈ H1loc is at r = rb. However, in that region we have ∂run = ∂0τun = 0.
Thus □gun ∈ H1loc. As φ0 and φ1 are compactly supported, we then obtain that □gun ∈ H1.
Combining Propositions 2.4.1 and 2.4.2 gives us the following Theorem:
Theorem 2.4.2 (Existence of Permeating Solutions). There exists a dense subset D of H1(Στ0) with the following
property. Given initial data (φ0,φ1) ∈ D, there exists an H20∀τ solution to (2.1) and (2.63) in the permeating
case.
Proof. We use the subset given by
D = {(u,∂τu),u ∈C∞0 (M )∩□−1g (H1(M ))}. (2.70)
This is dense, by Proposition 2.4.2, and has an H20∀τ solution by Proposition 2.4.1.
Remark 2.4.3. As previously, Theorem 2.4.2 allows us to extend other results. Suppose we obtain a result on
boundedness between times slices Στ in the H1(Στ) norm (not necessarily uniform in time). Then we can use a
2.5 Boundedness of the Wave Equation 29
density argument to obtain the following: given initial data in H1(Στ0)), there exists an H1(Στ) ∀τ solution, in
the sense of distributions (see already Theorem 2.5.6).
2.5 Boundedness of the Wave Equation
In this section we work towards proving Theorem 1, as stated in the overview. We will prove the result stated in
that Theorem firstly for the RNOS model with reflective boundary conditions, then we will prove the result in
the Oppenheimer–Snyder case with permeating boundary conditions.
2.5.1 The Reflective Case
We will prove this boundedness result in two sections. We will first prove that going forwards we have a uniform
bound on the H˙1 norm, i.e. there exists a constant C(M,q,T ∗) such that
∥φ∥H˙1(Σt∗1 ) ≤C∥φ∥H˙1(Σt∗0 ) ∀t
∗
1 ≥ t∗0 . (2.71)
In the second section we will prove the analogous statement going backwards in time:
∥φ∥H˙1(Σt∗0 ) ≤C∥φ∥H˙1(Σt∗1 ) ∀t
∗
0 ≤ t∗1 ≤ t∗c . (2.72)
Note the backwards in time version includes a condition on t∗1 ≤ t∗c , as, were t∗1 > t∗c , we can lose arbitrarily
large amounts of energy across the event horizon.
From here on in this thesis, when we say solution, unless stated otherwise, we mean φ which has finite
H˙1(Σt∗) norm for all t∗, and is a solution of (2.1) in a distributional sense, i.e.∫
M
gab∂b f∂aφ = 0 ∀ f ∈C∞0 (M ). (2.73)
Again, note that smooth compactly supported solutions of (2.1) are dense within these functions with respect to
the H˙1(Σt∗) norm. The methods in this section will closely follow [2].
We begin by proving a local in time bound on solutions of (2.1) which are compactly supported on each Σt∗ ,
though this can be extended to all H˙1 functions by a density argument.
Theorem 2.5.1 (Finite in Time Energy Bound). Given an RNOS model given by M,q,T ∗, φ a solution of the
wave equation (2.1) with reflective boundary conditions (2.2), and a time interval t∗0 ≤ t∗1 ≤ t∗c , we have that
there exists a constant C =C(M,q,T ∗, t∗0)> 0 such that
C−1∥φ∥H˙1(Σt∗0 ) ≤ ∥φ∥H˙1(Σt∗1 ) ≤C∥φ∥H˙1(Σt∗0 ) (2.74)
Proof. We start by choosing a vector field in the region t∗ ∈ [t∗0 , t∗1 ] which is everywhere timelike, including on
the surface of the dust cloud. We will also choose this vector field to be tangent to the surface of the dust cloud.
For example
X = ∂t∗+ r˙b(t∗)∂r. (2.75)
Then we have that
−dt∗(JX) = 1
2
((
1+
2M
r
− q
2M2
r2
)
(∂t∗φ)2+2
(
1+
2M
r
− q
2M2
r2
)
r˙b(t∗)∂t∗φ∂rφ
+
(
1− 2M
r
+
q2M2
r2
+
2M
r
(
2− q
2M
r
)
|r˙b(t∗)|
)
(∂rφ)2+
1
r2
| /˚∇φ |2
)
. (2.76)
30 The Scattering Map
Note, in every RNOS model, when the dust cloud crossed r = r+, r˙b(t∗) ̸= 0, and as r˙b(t∗) ∈ (−1,0], we
have that there exists a time independent constant A = A(M,q,T ∗) such that
A−1∥φ∥2H˙1(Σt∗) ≤−
∫
Σt∗
dt∗(JX)≤ A∥φ∥2H˙1(Σt∗) (2.77)
Then we look at the energy current through the surface of the dust cloud
dρ(JX) = 0 (2.78)
once we notice that on the surface of the dust cloud Xν∇νφ = 0 and dρ(X) = 0 for dρ the normal to the surface
of the dust cloud.
If we then calculate KX , we get:
|KX |=
∣∣∣∣∣
(
1+
M
r
)
r˙b(t∗)
r
(∂t∗φ)2−
(
2Mr˙b(t∗)
r2
+
(
1+
2M
r
− q
2M2
r2
)
r¨b(t∗)
)
∂t∗φ∂rφ
−
((
1−M
r
)
r˙b(t∗)
r
−
(
2M
r
− q
2M2
r2
)
r¨b(t∗)
)
(∂rφ)2
∣∣∣∣∣≤ B∥φ∥2H˙1(Σt∗) (2.79)
for B = B(t∗) a continuous function of t∗.
Define
f (t∗) :=−
∫
Σt∗
dt∗(JX) = ∥φ∥2X ,Σt∗ . (2.80)
Now, if we integrate KX in the space t∗ ∈ [t∗0 , t∗1 ] and apply (2.28), we get
f (t∗1)−
∫ t∗1
t∗0
B(t∗) f (t∗)dt∗ ≤ f (t∗0)≤ f (t∗1)+
∫ t∗1
t∗0
B(t∗) f (t∗)dt∗. (2.81)
Then an application of Gronwall’s Inequality to each of the inequalities in (2.81), and applying (2.77) to the
result gives us that(
Ae
∫ t∗1
t∗0
B(t∗)dt∗
)−1
∥φ∥2H˙1(Σt∗0 ) ≤ ∥φ∥
2
H˙1(Σt∗1 )
≤
(
Ae
∫ t∗1
t∗0
B(t∗)dt∗
)
∥φ∥2H˙1(Σt∗0 ). (2.82)
Letting C2 = Ae
∫ t∗c
t∗0
B(t∗)dt∗
gives us the required result.
Theorem 2.5.1 gives us the conditions mentioned in Remark 2.4.2, so we have the following Theorem:
Theorem 2.5.2 (H1 Existence of Reflective Solutions). Let (φ0,φ1) ∈ H1(Σt∗0 ), where t∗0 ≤ t∗c . There exists a
solution φ to the wave equation (2.1) with reflective boundary conditions such that
(φ |Σt∗0 ,∂t∗φ |Σt∗0 ) = (φ0,φ1). (2.83)
Here this restriction holds in a trace sense, and φ is a solution in the sense of distributions. Finally, φ ∈ H˙1(Σt∗)
for all t∗ ≤ t∗c .
Proof. This is a result of Theorem 2.4.1 and a density argument. This density argument relies on the bounds
given by Theorem 2.5.1.
Remark 2.5.1. Note that our existence result, Theorem 2.5.2 allows us to define the forwards map:
F(t∗0 ,t∗1 ) : E
X
Σt∗0
→ E XΣt∗1 (2.84)
F(t∗0 ,t∗1 ) (φ0,φ1) :=
(
φ |Σt∗1 ,∂t∗φ |Σt∗1
)
where φ is the solution to (2.1) and (2.37). (2.85)
2.5 Boundedness of the Wave Equation 31
Then Theorem 2.5.1 gives boundedness ofF(t∗0 ,t∗1 ):
∥F(t∗0 ,t∗1 ) (φ0,φ1)∥X ≤C∥(φ0,φ1)∥X ∀t
∗
0 ≤ t∗1 ≤ t∗c (2.86)
for some C =C(M, t∗0 , t
∗
1)> 0.
Now we wish to obtain a bound for our solution which does not depend on the time interval we are looking
in. We first prove such a uniform boundedness result for the case T ∗ < 1:
Theorem 2.5.3 (Uniform in Time Energy Bound for T ∗ < 1). Given an RNOS model given by M, q and T ∗ < 1,
and φ a solution of the wave equation (2.1) with reflective boundary conditions (2.2), we have that there exists a
constant C =C(M,q,T ∗)> 0 such that
C−1∥φ∥H˙1(Σt∗0 ) ≤ ∥φ∥H˙1(Σt∗1 ) ≤C∥φ∥H˙1(Σt∗0 ) ∀t
∗
0 ≤ t∗1 ≤ t∗c . (2.87)
Proof. This proof is identical to the proof for Theorem 2.5.1. Once we note that for T ∗ < 1 and t∗ < t∗−,
r˙b(t∗) = r¨b(t∗) = 0, we can take B(t∗) = 0 for t∗ ≤ t∗−. Therefore the constant given by Theorem 2.5.1 is actually
a uniform in time bound for all t∗0 < t
∗−, i.e. take
C2 = Ae
∫ t∗c
t∗−
B(t∗)dt∗
. (2.88)
The T ∗ ≥ 1 case is much more difficult than the T ∗ < 1 case, so we will break this boundedness result into
two different Theorems. We will start with the forward bound:
Theorem 2.5.4 (Uniform Forward in Time Energy Bound for T ∗ ≥ 1). Given an RNOS model with parameters
q and T ∗ ≥ 1, and a solution φ of the wave equation (2.1) with reflective boundary conditions (2.2), we have
that there exists a constant C =C(M,q,T ∗)> 0 such that
∥φ∥H˙1(Σt∗1 ) ≤C∥φ∥H˙1(Σt∗0 ) ∀t
∗
0 ≤ t∗1 ≤ t∗c . (2.89)
Proof. We will proceed similarly to Theorem 2.5.1, but we will take the vector field
X = ∂t∗ . (2.90)
Then we obtain the following results:
KX = 0 (2.91)
−dt∗(JX) = 1
2
((
1+
2M
r
− q
2M2
r2
)
(∂t∗φ)2+
(
1− 2M
r
+
q2M2
r2
)
(∂rφ)2+
1
r2
| /˚∇φ |2
)
(2.92)
dρ(JX) =− r˙b(t
∗)
2
(1+ r˙b(t∗))
((
1− 2M
r
+
q2M2
r2
)
−
(
1+
2M
r
− q
2M2
r2
)
r˙b(t∗)
)
(∂rφ)2 ≥ 0, (2.93)
recalling that r˙b ∈ (−1,0].
Now if we take an arbitrary t∗0 < t
∗
c , then in the region t
∗ ≤ t∗0 , r ≥ rb(t∗)≥ rb(t∗0)> r+. Thus there exists an
ε > 0 such that
1− 2M
r
+
q2M2
r2
≥ ε. (2.94)
Therefore in the region t∗ ≤ t∗0
ε∥φ∥2H˙1(Σt∗) ≤−
∫
Σt∗
dt∗(JX)≤ ∥φ∥2H˙1(Σt∗). (2.95)
32 The Scattering Map
Then, as before, we integrate KX in a region t∗ ∈ [t∗1 , t∗2 ] for t∗1 ≤ t∗2 ≤ t∗0 . Once we note that the boundary
term has the correct sign, we have
∥φ∥2H˙1(Σt∗2 ) ≤ ε
−1∥φ∥2H˙1(Σt∗1 ). (2.96)
An application of Theorem 2.5.1 will then allow us to extend our bound over the remaining finite interval [t∗0 , t
∗
c ]
to obtain the required result.
Now we look at obtaining the backward in time bound:
Theorem 2.5.5 (Uniform Backward in Time Energy Bound for T ∗ ≥ 1). Given an RNOS model given by M, q
and T ∗ ≥ 1, and a solution φ of the wave equation (2.1) with reflective boundary conditions (2.2), we have that
there exists a constant C =C(M,q,T ∗)> 0 such that
∥φ∥H˙1(Σt∗0 ) ≤C∥φ∥H˙1(Σt∗1 ) ∀t
∗
0 ≤ t∗1 ≤ t∗c . (2.97)
Proof. For this proof, we will need to use the modified currents, as defined in (2.25). Given asymptotic speed
a < 1 (see (1.45)), let b ∈ (a,1). Looking in the region t∗ < 0, we will use the vector field and modifier
X = f (t∗)∂t∗−b∂r (2.98)
f (t∗) : =
1+ 1
log
( |t∗|
2M
)
 (2.99)
w =− b
2r
. (2.100)
We then calculate
KX ,w =−
 1+ 2Mr − q2M2r2
2|t∗|
(
log
( |t∗|
2M
))2 − bMr2
(
1− q
2M
r
)(∂t∗φ)2
−
 1− 2Mr + q2M2r2
2|t∗|
(
log
( |t∗|
2M
))2 − bMr2
(
1− q
2M
r
)(∂rφ)2− 2bMr2
(
1− q
2M
r
)
∂t∗φ∂rφ
− 1
r2
b
r
+
1
2|t∗|
(
log
( |t∗|
2M
))2
 | /˚∇φ |2− b
r4
(
1− q
2M
r
)
φ2 (2.101)
dρ(JX ,w) =−1
2
(b+ f (t∗)r˙b(t∗))(1+ r˙b(t∗))
((
1− 2M
r
+
q2M2
r2
)
−
(
1+
2M
r
− q
2M2
r2
)
r˙b(t∗)
)
(∂rφ)2
(2.102)
−dt∗(JX ,w) = 1
2
((
1+
2M
r
− q
2M2
r2
)
f (t∗)(∂t∗φ)2−2b
(
1+
2M
r
− q
2M2
r2
)
∂t∗φ∂rφ
+
((
1− 2M
r
+
q2M2
r2
)
f (t∗)+
2bM
r
(
2− q
2M
r
))
(∂rφ)2+
1
r2
f (t∗)| /˚∇φ |2 (2.103)
+
2bM
r
(
2− q
2M
r
)
φ
r
∂rφ −2b
(
1+
2M
r
− q
2M2
r2
)
φ
r
∂t∗φ +
bM
r
(
2− q
2M
r
)
φ2
r2
)
.
We now note that either a ̸= 0, or r˙b(t∗) ∼ √rb as t∗ → −∞. In either of these cases, terms of order(
|t∗|
(
log
( |t∗|
2M
))2)−1
dominate terms of order r−2 as t∗→−∞. Thus for sufficiently negative t∗, KX ,w ≤ 0.
Similarly we have dρ(JX ,w) ≥ 0, for sufficiently negative t∗, since b+ f (t∗)r˙b(t∗)→ b− a > 0 as t∗→ ∞.
2.5 Boundedness of the Wave Equation 33
Finally, we consider −dt∗(JX ,w). Integrating over Σt∗ , we obtain:
−
∫
Σt∗
dt∗(JX ,w) =
1
2
∫
Σt∗
((
1+
2M
r
− q
2M2
r2
)
f (t∗)(∂t∗φ)2−2b
(
1+
2M
r
− q
2M2
r2
)
∂t∗φ∂rφ
+
((
1− 2M
r
+
q2M2
r2
)
f (t∗)+
2bM
r
(
2− q
2M
r
))
(∂rφ)2+
1
r2
f (t∗)| /˚∇φ |2
+
2bM
r
(
2− q
2M
r
)
φ
r
∂rφ −2b
(
1+
2M
r
− q
2M2
r2
)
φ
r
∂t∗φ +
bM
r
(
2− q
2M
r
)
φ2
r2
)
=
1
2
∫
Σt∗
((
1+
2M
r
− q
2M2
r2
)
( f (t∗)−b)(∂t∗φ)2
+b
(
1+
2M
r
− q
2M2
r2
)((
∂t∗φ −∂rφ − φr
)2
−2φ
r
∂rφ
)
(2.104)
+
(
1− 2M
r
+
q2M2
r2
)
( f (t∗)−b)(∂rφ)2+ 1r2 f (t
∗)| /˚∇φ |2+ bM
r
(
2− q
2M
r
)(
φ
r
+∂rφ
)2)
=
1
2
∫
Σt∗
((
1+
2M
r
− q
2M2
r2
)
( f (t∗)−b)(∂t∗φ)2+b
(
1+
2M
r
− q
2M2
r2
)(
∂t∗φ −∂rφ − φr
)2
+b
(
1+
q2M2
r2
)
φ2
r2
+
(
1− 2M
r
+
q2M2
r2
)
( f (t∗)−b)(∂rφ)2+ 1r2 f (t
∗)| /˚∇φ |2
+
bM
r
(
2− q
2M
r
)(
φ
r
+∂rφ
)2)
.
We then note a version of Hardy’s inequality. If h is a differentiable function of one variable, with h(0) = 0,
then
∃C > 0 s.t.
∫
Σt∗
(
h(r)
r
)2
≤C
∫
Σt∗
(∂rh(r))2 , (2.105)
providing the right hand side is finite.
Using (2.105), we have that there exists a t∗ independent constant A such that
0≤
∫
Σt∗
(
b
(
1+
2M
r
− q
2M2
r2
)(
∂t∗φ −∂rφ − φr
)2
+b
(
1+
q2M2
r2
)
φ2
r2
+
1
r2
f (t∗)| /˚∇φ |2 (2.106)
+
bM
r
(
2− q
2M
r
)(
φ
r
+∂rφ
)2)
≤ A∥φ∥2H˙1(Σt∗)
Note f (t∗)→ 1 as t∗→−∞, and b≤ 1. Thus for sufficiently large negative t∗, there exists a t∗ independent
ε such that
ε∥φ∥2H˙1(Σt∗) ≤−
∫
Σt∗
dt∗(JX ,w)≤ ε−1∥φ∥2H˙1(Σt∗). (2.107)
Finally, for negative and large enough t∗, t∗ ≤ t∗2 , say, we have −
∫
Σt∗ dt
∗(JX ,w) non-decreasing in t∗.
Therefore
∥φ∥2H˙1(Σt∗0 ) ≤ A
2∥φ∥2H˙1(Σt∗1 ) ∀t
∗
0 ≤ t∗1 ≤ t∗2 . (2.108)
Combining (2.108) with Theorem 2.5.1 for the interval [t∗2 , t
∗
c ] gives us the final result.
2.5.2 The Permeating Case
We now look at the permeating case. In the Oppenheimer–Snyder model for the interior of the star, we have that
our metric is C0, but piecewise smooth. Thus, as given by Theorem 2.4.2, we are dealing with a weak solution
34 The Scattering Map
rather than a classical solution, i.e. φ ∈ H1(Στ) a solution to∫ ∞
τ=−∞
∫
Στ
√−ggab∂aφ∂b
(
ψ√−g
)
dV dτ = 0 ∀ψ ∈C∞0 (M ). (2.109)
(Note that in the coordinates chosen below, the determinant of
√−g is r2 sinθ )
The metric in the interior of our star has the form (see Section 1.1.2):
ds2 =

−
(
1− 2Mr2
r3b
)
dτ2+2
√
2Mr2
r3b
dτdr+dr2+ r2gS2 r < rb := (R
3
2
b − 3τ2
√
2M)
2
3
−(1− 2Mr )dτ2+2√2Mr dτdr+dr2+ r2gS2 r ≥ rb (2.110)
for constants Rb and M. Here, rb is the boundary of the star.
We note that the null hypersurface given by r = rb
(
3−2
√
rb
2M
)
is part of our event horizon. This means
when we construct the backwards scattering map, we will require data on this as well as the r = 2M, τ > τc
surface.
We begin our study of boundedness by noticing that our usual ∂τ -energy does not give the same bound as
before. This is due to the fact that ∂τ is no longer a Killing vector. We therefore obtain a term arising from K∂τ
inside the star. We can still obtain a bound from integrating K∂τ , however it is now exponentially growing in τ :
Lemma 2.5.1 (Finite in Time Forwards Bound in the Permeating Case). Let φ ∈ H20∀τ be a weak solution to the
wave equation (2.1) with permeating boundary conditions. There exists a constant B = B(M)> 0 such that
∥φ∥2H˙1(Στ1) ≤−2
∫
Στ1
dτ(JX [φ ])≤−2
∫
Στ0
dτ(JX [φ ])eB(τ1−τ0) ≤ 4∥φ∥2H˙1(Στ0)e
B(τ1−τ0) ∀τ1 ≥ τ0 (2.111)
for suitably chosen future directed timelike X.
Proof. Choose f (r) to be a smooth cut off function
f (r)

=−12 r ∈ [3M2 , 5M2 ]
= 0 r /∈ [M,3M]
∈ [−12 ,0] r ∈ (M, 3M2 )∪ (5M2 ,3M)
. (2.112)
Note f has bounded derivatives. Then if we let X = ∂τ + f (r)∂r we have that:
−dτ(JX) =

1
2
(
1
r2 | /˚∇φ |2+(∂τφ)2+2 f∂τφ∂rφ +
(
1− 2Mr2
r3b
−2 f
√
2Mr2
r3b
)
(∂rφ)2
)
r < rb
1
2
(
1
r2 | /˚∇φ |2+(∂τφ)2+2 f∂τφ∂rφ +
(
1− 2Mr −2 f
√
2M
r
)
(∂rφ)2
)
r ≥ rb
. (2.113)
KX =

1
2
(
− f ′r2 | /˚∇φ |2+(
2 f
r + f
′)(∂τφ)2−
(
6Mr
r3b
+ 6
√
2M f√
r3b
)
∂τφ∂rφ
+
(
(1− 2Mr2
r3b
) f ′− (1− 4Mr2
r3b
)2 fr +
3
√
2M
3
r2√
r9b
)
(∂rφ)2
)
r < rb
1
2
(
− f ′r2 | /˚∇φ |2+(
2 f
r + f
′)(∂τφ)2− 3 fr
√
2M
r ∂τφ∂rφ
+
(
(1− 2Mr ) f ′− (1− Mr )2 fr
)
(∂rφ)2
)
r ≥ rb
. (2.114)
Thus KX can always be bounded by multiples of −dτ(JX).
−
∫
Στ0
dτ(JX) =−
∫
Στ1
dτ(JX)+
∫ τ1
τ=τ0
∫
Στ
KX −
∫
H +∩{r<2M}
dn(JX). (2.115)
2.5 Boundedness of the Wave Equation 35
We can also note that the contribution from the part of the horizon in (2.115) is of the form −TabXanb for
future directed normal n. By the dominant energy condition, we have that this term has the correct sign. Thus
letting g(τ) =−∫Στ dτ(JX), we have that
g(τ)≤ g(τ0)+A
∫ τ
s∗=τ0
g(s)ds, (2.116)
which gives us our result by Gronwall’s Inequality.
Remark 2.5.2. For the purposes of the scattering map however, we will not want to disregard the surface term
from the event horizon. Instead we will want to consider a norm on the horizon such that the map from a surface
Στ to Στc ∪ (H ∩{r < 2M}) is bounded in both directions. Letting X = ∂τ + f (r)∂r, then we have
−dn(JX) = 1
2
(
3
√
2M
rb
−1+ f (r)
)(
∂τφ +3
(
1−
√
2M
rb
)
∂rφ
)2
+
(
3
(
1−
√
2M
rb
)
− f
)
1
2r2
| /˚∇φ |2.
(2.117)
If we then use the f from Lemma 2.5.1, we have all these terms being positive definite. Therefore the norm
we will consider on the surface contains only the L2 norms of the angular derivatives and the derivative with
respect to the vector ∂τ +3
(
1−
√
2M
rb
)
∂r.
Lemma 2.5.2 (Finite-in-Time Backward Non-degenerate Energy Boundedness for the Permeating Case). Let
φ ∈ H20∀τ be a weak solution to the wave equation (2.1) with permeating boundary conditions. There exists a
constant B = B(M)> 0 such that for all τ0 ≤ τ1 ≤ τc− , we have
∥φ∥2H˙1(Στ0) ≤ 4∥φ∥
2
H˙1(Στ1)
eB(τ1−τ0). (2.118)
Here (τc−,r = 0) is defined by equation (1.18).
Proof. This is proved identically to the previous lemma. We bound KX below instead of above, and we ignore
the boundary term, asH +∩{τ < τc−}= 0.
Lemmas 2.5.1 and 2.5.2 give us the conditions mentioned in Remark 2.4.3. Thus we have the following
Theorem:
Theorem 2.5.6 (H1 Existence of Permeating Solutions). Let (φ0,φ1) ∈ H1(Στ0), where τ0 ≤ τc− . There exists a
solution φ to the wave equation (2.1) with permeating boundary conditions, such that
(φ |Στ0 ,∂τφ |Στ0 ) = (φ0,φ1) (2.119)
Here this restriction holds in a trace sense, and φ is a solution in the sense of distributions. Finally, φ ∈ H˙1(Στ)
for all τ ≤ τc.
Proof. This is a result of Theorem 2.4.2 and a density argument. This density argument relies on the bounds
given by Lemmas 2.5.1 and 2.5.2.
Remark 2.5.3. As in the permeating case, our existence result Theorem 2.5.6 allows us to define the forwards
map:
F(τ0,τ1) : E
X
Στ0
→ E XΣτ1 (2.120)
F(τ0,τ1) (φ0,φ1) :=
(
φ |Στ1 ,∂t∗φ |Στ1
)
where φ is the solution to (2.1) and (2.63). (2.121)
We can use Lemmas 2.5.1 and 2.5.2 to bound the solution over any finite time interval. Thus we can now
consider only the case where rb ≫ 2M, i.e. 2Mrb < ε for some small, fixed epsilon. Once we have a uniform
36 The Scattering Map
bound for 2Mrb < ε , we can bound solutions of the wave equation for τ ≤ τc using Lemmas 2.5.1 and 2.5.2.
Previous work on the external Schwarzschild space-time gives us the required bounds for τ > τc.
This brings us to our next result:
Proposition 2.5.1 (Forward Non-degenerate Energy Boundedness for the Permeating Case, Sufficiently Far
Back). Let φ be a solution to the wave equation (2.1) with permeating boundary conditions (as given by
Theorem 2.5.6). There exists a constant, A = A(M)> 0, and a time, τ∗ such that
∥φ∥H˙1(Στ0) ≤ A∥φ∥H˙1(Στ1) ∀τ0 < τ1 ≤ τ
∗. (2.122)
Proof. For this proof, we choose a time dependent vector field. Let Y = h(τ)∂τ . Then we have that
−dτ(JY ) =

h
2
(
1
r2 | /˚∇φ |2+(∂τφ)2+
(
1− 2Mr2
r3b
)
(∂rφ)2
)
r < rb
h
2
(
1
r2 | /˚∇φ |2+(∂τφ)2+
(
1− 2Mr
)
(∂rφ)2
)
r ≥ rb
. (2.123)
KY =

−12
(
h′
r2 | /˚∇φ |2+h′(∂τφ)2− 2Mrr3b 3h∂τφ∂rφ
+
(
h′
(
1− 2Mr2
r3b
)
− 2Mr2
r3b
3h
r
)
(∂rφ)2
)
r < rb
h′
2
(
1
r2 | /˚∇φ |2+(∂τφ)2+
(
1− 2Mr
)
(∂rφ)2
)
r ≥ rb
. (2.124)
Now, we would like both of these to be everywhere positive definite. For this, we need to pick h > 0 and
bounded. We also need h′ < 0, with −h′ > 3M
r2b
h. Thus we can choose, for example,
h(τ) = 1−
(
2M
rb
)1/4
∈ [1−
(
2M
rb(τ∗)
)1/4
,1] (2.125)
h′(τ) =−
( rb
2M
)1/4 2M
4r2b
≤−
(
rb(τ∗)
2M
)1/4 2M
4r2b
<−3M
r2b
h (2.126)
where we have chosen τ∗ s.t.
(
rb(τ∗)
2M
)1/4
> 6. This choice also gives us
(
1− 2M
rb(τ∗)
)(
1−
(
2M
rb(τ∗)
) 1
4
)
∥φ∥2H˙1(Στ ) ≤−2
∫
Στ
dτ(JX)≤ ∥φ∥2H˙1(Στ ). (2.127)
Finally, we apply these inequalities to divergence theorem (the permeating equivalent of (2.28)) in the region
τ ∈ [τ0,τ1] to obtain
∥φ∥2H˙1(Στ0) ≥−2
∫
Στ0
dτ(JY ) =−2
∫
Στ1
dτ(JY )+2
∫ τ1
τ=τ0
∫
Στ
KY ≥−2
∫
Στ1
dτ(JY )≥ A∥φ∥2H˙1(Στ1) (2.128)
as required.
Theorem 2.5.7 (Forward Non-degenerate Energy Boundedness for the Permeating Case). Let φ be a solution
to the wave equation (2.1) with permeating boundary conditions (as given by Theorem 2.5.6). There exists a
constant A = A(M) such that
∥φ∥H˙1(Στ1) ≤ A∥φ∥H˙1(Στ0) ∀τ0 ≤ τ1. (2.129)
Proof. Previous works on Schwarzschild exterior space time (e.g. [14]), gives us that (2.129) holds for τc ≤
τ0 ≤ τ1. Thus if we prove the result for the case τ1 ≤ τc, we can combine these results to obtain (2.129) for all
τ0 ≤ τ1.
2.5 Boundedness of the Wave Equation 37
Let A and τ∗ be defined as in Proposition 2.5.1. Let B be as defined in Lemma 2.5.1. Remember τc is defined
to be the time at which the surface of the star crosses r = 2M. We then have that
∥φ∥2H˙1(Στ1) ≤ 4Ae
B(τc−τ∗)∥φ∥2H˙1(Στ0) ∀τ0 ≤ τ1 ≤ τc. (2.130)
In a similar way, we can obtain a boundedness statement for the reverse direction:
Theorem 2.5.8 (Backwards Non-degenerate Energy Boundedness for the Permeating Case). Let φ be a solution
to the wave equation (2.1) with permeating boundary conditions (as given by Theorem 2.5.6). There exists a
constant A = A(M), and a time τ∗− such that
∥φ∥H˙1(Στ0) ≤ A∥φ∥H˙1(Στ1) ∀τ0 ≤ τ1 ≤ τ
∗−. (2.131)
Proof. As before, we consider Y = h(τ)∂τ . However, this time we require the sign of KY to be non-positive.
Thus, we choose
h(τ) = 1+
(
2M
rb
) 1
4
∈ [1,2]. (2.132)
h′(τ) =
( rb
2M
)1/4 2M
4r2b
≥
(
rb(τ∗−)
2M
)1/4 2M
4r2b
>
3M
r2b
h. (2.133)
Therefore, if we choose τ∗− s.t.
(
rb(τ∗−)
2M
)1/4
> 12, we have KY ≤ 0 (see equation (2.124)). Then, as before
we obtain:
∥φ∥2H˙1(Στ1) ≥−
∫
Στ1
dτ(JY ) =−
∫
Στ0
dτ(JY )−
∫ τ1
τ=τ0
∫
Στ
KY ≥−
∫
Στ0
dτ(JY )≥ 1
2
(
1− 2M
rb(τ∗−)
)
∥φ∥2H˙1(Στ0).
(2.134)
Corollary 2.5.1 (Uniform Boundedness for the Permeating Case). Let φ be a solution to the wave equation
(2.1) with permeating boundary conditions (as given by Theorem 2.5.6). There exist constants B = B(M) >
0,b = b(M)> 0 such that
b∥φ∥H˙1(Στ1) ≤ ∥φ∥H˙1(Στ0) ≤ B∥φ∥H˙1(Στ1) ∀τ0,τ1 ≤ τc−. (2.135)
Proof. We have the forward bound due to Theorem 2.5.7. The backwards bound is done by combining Theorem
2.5.8 and Corollary 2.5.2 over the finite time interval [τ∗−,τc−].
Corollary 2.5.2. Let φ be a solution to the wave equation (2.1) with permeating boundary conditions (as given
by Theorem 2.5.6). There exists a constant, C =C(M)> 0, such that
−
∫
Σ˜v0
dn(JY )≤−
∫
Σ˜v1
dn(JY )≤C∥φ∥2H˙1(Στ0) ∀v0 ≤ v1 ≤ τ0+ rb(τ0) (2.136)
where n is the normal to Σ˜v.
Proof. We integrate KY between the relevant surfaces and use Stokes’ theorem to obtain these bounds.
Remark 2.5.4. The first order energy results from this section can be given using the forwards map and the
energy space notation from (2.27) and (2.31). Let X be strictly timelike everywhere (for example, as in Lemma
38 The Scattering Map
2.5.1). We then have that there exist A1 = A1(M)> 0 and A2 = A2(M)> 0 such that
A−11 ∥(φ0,φ1)∥X ≤ ∥Ft∗0 ,t∗1 (φ0,φ1)∥X ≤ A1∥(φ0,φ1)∥X ∀t∗0 ≤ t∗1 ≤ t∗c (2.137)
A−12 ∥(φ0,φ1)∥X ≤ ∥Fτ0,τ1(φ0,φ1)∥X ≤ A2∥(φ0,φ1)∥X ∀τ0 ≤ τ1 ≤ τc− (2.138)
∥Fτ0,τ2(φ0,φ1)∥X ≤ A2∥(φ0,φ1)∥X ∀τ0 ≤ τc− ≤ τ2 ≤ τc. (2.139)
2.6 Higher Order Boundedness
We now try to extend Theorems 2.5.3, 2.5.4 and 2.5.5 to H˙n norms. However, it turns out that this cannot be
done unless the surface of the dust cloud has asymptotic velocity 0, i.e. T ∗ ≤ 1. We begin with the following
3-part Lemma:
Lemma 2.6.1. Given an RNOS model given by M,q,T ∗ ≤ 1, and a solution φ ∈C∞0∀t∗ of the wave equation
(2.1) with reflective boundary conditions (2.2), we have the following results:
1. Let Ωi be the angular Killing vector fields earlier (see (1.26)). Then □g
(
1
r|p|Ω
p∂mr ∂
n−1−m−|p|
t∗ φ
)
only
contains at most nth order derivatives. Furthermore, all coefficients of these derivatives are smooth and
have all their derivatives bounded. Thus there exists a constant D = D(M,n)> 0 such that∥∥∥∥□g( 1r|p|Ωp∂mr ∂ n−1−m−|p|t∗ φ
)∥∥∥∥
L2(Σt∗)
≤ D∥φ∥H˙n(Σt∗). (2.140)
2. There exists a t∗0 ≤ t∗c and a constant C =C(M, t∗0)> 0 such that
C
(
∥∂¯ nt∗φ∥2H˙1(Σt∗)+∥φ∥
2
H˙n−1(Σt∗)
)
≥ ∥φ∥2H˙n(Σt∗) ∀t
∗ ≤ t∗0 . (2.141)
Here ∂¯t∗ is the t∗ derivative with respect to (t∗,ρ = r− rb(t∗)+2M,θ ,ϕ) coordinates, as given in (2.39).
3. Given any finite time t∗0 ≤ t∗1 ≤ t∗c , there exists a constant A = A(n, t∗0 , t∗1 ,M) such that
1
A
∥φ∥2H˙n(Σt∗0 ) ≤ ∥φ∥
2
H˙n(Σt∗1 )
≤ A∥φ∥2H˙n(Σt∗0 ). (2.142)
Remark 2.6.1. Note that when calculating ∥ψ∥2H˙1(Σt∗), we can use the ∂¯t∗ derivative in place of the ∂t∗ derivative.
This is due to the fact that these norms can differ by at most a factor of 2, since
∥∂t∗ψ∥L2(Σt∗)−∥∂rψ∥L2(Σt∗)≤∥(∂t∗+ r˙b∂r)ψ∥L2(Σt∗)= ∥∂¯t∗ψ∥L2(Σt∗)≤∥∂t∗ψ∥L2(Σt∗)+∥∂rψ∥L2(Σt∗). (2.143)
This in turn implies
1
2
(
∥∂t∗ψ∥2L2(Σt∗)+∥∂rψ∥
2
L2(Σt∗)
)
≤ ∥∂¯t∗ψ∥2L2(Σt∗)+∥∂rψ∥
2
L2(Σt∗)
≤ 2
(
∥∂t∗ψ∥2L2(Σt∗)+∥∂rψ∥
2
L2(Σt∗)
)
.
(2.144)
Proof. 1. Note Ωi and ∂t∗ commute with □g. Thus for this part we only need to check □g
(
1
r|p|∂
n−1
r φ
)
explicitly. Using the fact that □gφ = 0, we obtain:
□g
(
1
r|p|
∂ n−1r φ
)
=
n−1
r2+|p|
(
(n−2)∂ 2t∗∂ n−3r φ +2(r−M)∂ 2t∗∂ n−2r φ −4M∂t∗∂ n−1r φ −n∂ n−1r φ −2(r−M)∂ nr φ
)
− |p|
r1+|p|
∇µr∇µ∂ n−1r φ . (2.145)
Given in the above case, |p| ≤ n−1, then we have our result.
2.6 Higher Order Boundedness 39
2. We first look at how the wave operator commutes with ∂¯t∗:
□g
(
1
r|p|
Ωp∂¯ n−|p|t∗ (φ)
)
=□g
(
1
r|p|
Ωp (∂t∗+ r˙b∂r)n−|p|φ
)
(2.146)
=
n−|p|
∑
m=0
(
n−|p|
m
)
r˙bm
r|p|
□g
(
Ωp∂ n−|p|−mt∗ ∂
m
r φ
)
+(Bounded lower order terms) .
As ∂t∗ and Ω commute with □g, we can ignore the m = 0 term in the sum. Then, by the first part of the
Lemma, we can bound the right hand side of (2.146). It is bounded by |r˙b| times a constant multiple of
the H˙n+1 norm, plus lower order terms:∥∥∥∥□g( 1r|p|Ωp∂¯ n−|p|t∗ (φ)
)∥∥∥∥2
L2(Σt∗)
≤ D
(
∥φ∥2H˙n(Σt∗)+ r˙b
2∥φ∥2H˙n+1(Σt∗)
)
. (2.147)
We also have that Ωp∂¯ nt∗(φ) = 0 on the boundary of the star. Thus we can then proceed by using an elliptic
estimate (such as in [37]) on ∂¯ nt∗φ .
We consider the elliptic operator, L, given by
Lψ :=
(
1− 2M
r
− 4Mr˙b
r
−
(
1+
2M
r
)
r˙b2
)
∂ 2r ψ+
1
r2
/˚△ψ = f (t∗,r)∂ 2r ψ+
1
r2
/˚△ψ. (2.148)
Thus we have∫
Σt∗
(Lψ)2 =
∫
Σt∗
f 2(∂rψ)2+
2 f
r2
∂ 2r ψ /˚△ψ+
1
r4
( /˚△ψ)2
=
∫
Σt∗
f 2(∂ 2r ψ)
2+
2 f
r2
|∂r /˚∇ψ|2+ 2∂r fr2 /˚∇ψ.∂r /˚∇ψ+
1
r4
| /˚∇ /˚∇ψ|2−
∫
St∗
2 f
r2
/˚∇ψ.∂r /˚∇ψ (2.149)
≥
∫
Σt∗
1
2
(∂ 2r ψ)
2+
1
2r2
|∂r /˚∇ψ|2− Cr2 | /˚∇ψ|
2+
1
r4
| /˚∇ /˚∇ψ|2−
∫
St∗
2 f
r2
/˚∇ψ.∂r /˚∇ψ
≥ 1
2
∥ψ∥2H˙2(Σt∗)−
1
2
∥∂t∗ψ∥2H˙1(Σt∗)−C∥ψ∥
2
H˙1(Σt∗)
−
∫
St∗
2 f
r2
/˚∇ψ.∂r /˚∇ψ.
By rearranging equation (2.1) in coordinates given by (2.39), we have that
∥Lψ∥2L2(Σt∗) ≤C
(
∥∂¯t∗ψ∥2H˙1(Σt∗)+∥□gψ∥
2
L2(Σt∗)
+∥ψ∥2H˙1(Σt∗)
)
. (2.150)
Combining (2.149) and (2.150) with ψ = ∂¯ nt∗φ (= 0 on St∗), and noting that
∥∂t∗ψ∥2H˙1(Σt∗) ≤C
(
∥∂¯t∗ψ∥2H˙1(Σt∗)+∥ψ∥
2
H˙1(Σt∗)
)
, (2.151)
we obtain
∥∂¯ nt∗φ∥2H˙2(Σt∗) ≤C
(
∥∂¯ n+1t∗ φ∥2H˙1(Σt∗)+∥φ∥
2
H˙n(Σt∗)
+ r˙b2∥φ∥2H˙n+1(Σt∗)
)
. (2.152)
We then look at ψ = 1
r|p|Ω
p∂¯ n−1t∗ (φ), where p is a multi-index of size 1 (as this also vanishes on St∗).
As the L2 norms of ∂¯ 2t∗ψ and ∂¯t∗∂rψ are bounded by the left hand side of (2.152), we repeat the above
argument to get that
∥∂¯ nt∗φ∥2H˙2(Σt∗)+∥
1
r|p|
Ωp∂¯ n−|p|t∗ (φ)∥2H˙2(Σt∗) ≤C
(
∥∂¯ n+1t∗ φ∥2H˙1(Σt∗)+∥φ∥
2
H˙n(Σt∗)
+ r˙b2∥φ∥2H˙n+1(Σt∗)
)
,
(2.153)
for |p|= 1.
We repeat this argument n times to obtain that (2.153) is true for all |p| ≤ n. The coefficient of ∂ 2r in
(2.1) (with respect to the coordinates in (2.39)) is bounded above and away from 0. This means, we can
40 The Scattering Map
rearrange (2.1), to bound all r derivatives to obtain:
∥φ∥2H˙n+2(Σt∗) ≤C
(
∥∂¯ n+1t∗ φ∥2H˙1(Σt∗)+∥φ∥
2
H˙n(Σt∗)
+ r˙b2∥φ∥2H˙n+1(Σt∗)
)
. (2.154)
If we then choose t∗0 such that r˙b
2C < 1, then we can rearrange the above to get the required result.
3. We proceed in a very similar way to our previous results for finite-in-time boundedness; we use energy
currents, Stokes’ theorem, and then Gronwall’s inequality. For this case, our energy currents will be
n=N
∑
n=1
n−1
∑
|p|=0
J∂¯t∗
(
1
r|p|
Ωp∂¯ n−1−|p|t∗ φ
)
, (2.155)
where ∂¯t∗ is timelike, so −
∫
Σt∗ dt
∗(J∂¯t∗ ) ∼ ∥.∥2H˙1(Σt∗). Note here that Ω are our angular Killing vector
fields, and p is a multi-index. Now, as □g 1r|p|Ω
p∂¯ n−1−|p|t∗ φ ̸= 0, we obtain an extra term in our bulk
integral:
∫ t∗1
t∗=t∗0
(
K∂¯t∗ + ∂¯t∗
(
1
r|p|
Ωp∂¯ n−1−|p|t∗ φ
)
□g
(
1
r|p|
Ωp∂¯ n−1−|p|t∗ φ
))
=
∫
Σt∗0
(−dt∗(J∂¯t∗ ))−
∫
Σt∗1
(−dt∗(J∂¯t∗ ))−
∫
S[t∗0 ,t∗1 ]
dρ(J∂¯t∗ ). (2.156)
As in part 2, we have that the coefficients of ∂ 2r in (2.1) are bounded above and away from 0. Suppose we
have bounded the L2 norms of all derivatives up to Nth order that have fewer that 2 r derivatives. We can
then use (2.1) to bound the remaining derivatives up to Nth order.
Now we consider the new second term in (2.156). The first part of this Lemma gives us that the sum of
these additional term can be bounded by∣∣∣∣∣n=N∑n=1
n−1
∑
|p|=0
∂¯t∗
(
1
r|p|
Ωp∂¯ n−1−|p|t∗ φ
)
□g
(
1
r|p|
Ωp∂¯ n−1−|p|t∗ φ
)∣∣∣∣∣≤C
∫ t∗1
t∗=t∗0
∥φ∥2H˙N(Σt∗)dt
∗ (2.157)
≤−C′
n=N
∑
n=1
n−1
∑
|p|=0
∫ t∗1
t∗=t∗0
∫
Σt∗
dt∗
(
J∂¯t∗
(
1
r|p|
Ωp∂¯ n−1−|p|t∗ φ
))
,
where we have used that □g commutes with ∂t∗ and each Ωi. As usual, we can then bound the K∂¯t∗ terms
by a multiple of this.
Finally, we note that Ωp∂¯ n−1−|p|t∗ φ = 0 on S[t∗0 ,t∗1 ], and ∂¯t∗ is tangent to this surface. Therefore dρ(J
∂¯t∗ )
also vanishes. Thus from equation (2.156), we obtain
g(t∗1) :=
n=N
∑
n=1
n−1
∑
|p|=0
∫
Σt∗0
(−dt∗(J∂¯t∗ (Ωp∂¯ n−1−|p|t∗ φ))≤ g(t∗0)+C
∫ t∗1
t∗=t∗0
g(t∗)dt∗, (2.158)
g(t∗0)≤ g(t∗1)+C
∫ t∗1
t∗=t∗0
g(t∗)dt∗. (2.159)
Then, in a similar manner to Gronwall’s inequality, we will show g(t∗)≤ ec(t∗−t∗0 )g(t∗0) for t∗ ≥ t∗0 .
The g(t∗0) = 0 is trivial. We proceed to prove that if g(t
∗
0) is non-zero, then g(t
∗)< (1+δ )ec(t∗−t∗0 )g(t∗0)
for all δ > 0. Suppose that there exists a t∗2 such that g(t
∗
2) = (1+δ )e
C(t∗2−t∗0 )g(t∗0), but up to this point,
2.6 Higher Order Boundedness 41
g(t∗2)< (1+δ )e
C(t∗2−t∗0 )g(t∗0). Then we obtain
g(t∗2)< (1+δ )g(t
∗
0)+C
∫ t∗2
t∗=t∗0
g(t∗)dt∗ ≤ (1+δ )g(t∗0)+C
∫ t∗2
t∗=t∗0
(1+δ )eC(t
∗−t∗0 )g(t∗0)dt
∗
= (1+δ )g(t∗0)+ [(1+δ )e
C(t∗−t∗0 )g(t∗0)]
t∗2
t∗0
(2.160)
= (1+δ )eC(t
∗
2−t∗0 )g(t∗0) = g(t
∗
2),
which gives us a contradiction.
We similarly have g(t∗0)≤ eC(t
∗
1−t∗0 )g(t∗1).
Thus by letting A = eC(t
∗
1−t∗0 ) in the statement of the Lemma, we are done.
The above lemma then allows us to come to our nth energy uniform boundedness results:
Theorem 2.6.1 (Forward nth order Non-degenerate Energy Boundedness for the Reflective Case). Given an
RNOS model given by M,q,T ∗ ≤ 1, and a solution φ ∈C∞0∀t∗ to the wave equation (2.1) with reflective boundary
conditions (2.2), there exists a constant E = E(n,M) such that
∥φ∥H˙n(Σt∗1 ) ≤ E∥φ∥H˙n(Σt∗0 ) ∀φ ∈C
∞
0∀t∗ ∀t∗0 ≤ t∗1 ≤ t∗c . (2.161)
Proof. As with previous uniform boundedness results, we look at bounding the energy uniformly for sufficiently
far back in time. Then we use our local result (part 3 of Lemma 2.6.1) to obtain a uniform bound for all t∗.
If T ∗ < 1, then for sufficiently negative times, r˙b = 0, and therefore ∂ nt∗φ = ∂¯t∗φ is a solution to the
wave equation (2.1) with reflective boundary conditions (2.2). Therefore we have uniform boundedness of
∥∂ nt∗φ∥H˙1(Σt∗). Then by Lemma 2.6.1 part 2, we have H˙n boundedness, as required. Therefore we now focus on
the T ∗ = 1 case.
We proceed inductively, by considering ∂¯ nt∗(φ). Here ∂¯t∗ = ∂t∗+ r˙b∂r is the partial t
∗ derivative with respect
to the coordinates given in (2.39).
□g
(
∂¯ nt∗(φ)
)
=□g ((∂t∗+ r˙b∂r)nφ)
=
n
∑
m=0
(
n
m
)
r˙bm□g
(
∂ n−mt∗ ∂
m
r φ
)
+ r¨b (Lower order terms with bounded coefficients)
=
n
∑
m=0
(
n
m
)
r˙bmm
(
2
r
(
1−M
r
)(
∂ n−m+2t∗ ∂
m−1
r φ −∂ n−mt∗ ∂m+1r φ
)− 4M
r2
∂m−n+1t∗ ∂
m
r φ (2.162)
+
m−1
r2
∂ n−m+2t∗ ∂
m−2
r φ −
m+1
r2
∂ n−mt∗ ∂
m
r φ
)
+ r¨b(Lower order terms with bounded coefficients).
In RNOS models with T ∗ = 1, r¨b < 0 for sufficiently negative t∗. We have, by the induction hypothesis, that
for some A = A(M,n)> 0
∫ t∗1
t∗=t∗0
∫
Σt∗
|r¨b(Lower order terms with bounded coefficients)|2 dt∗ ≤
∫ t∗1
t∗0
A∥φ∥2H˙n(Σt∗0 )|r¨b|
2dt∗ (2.163)
42 The Scattering Map
We also have that if we fix t∗− to be large and negative enough, then rb(t∗)≥A|t∗|2/3≥ 0, 0≤−r˙b≤B|t∗|−1/3
for all t∗ ≤ t∗−. This means that if t∗0 , t∗1 ≤ t∗−,
∫
Σt∗
∣∣∣∣∣ n∑m=0
(
n
m
)
r˙bmm
(
m−1
r2
∂ n−m+2t∗ ∂
m−2
r φ −
m+1
r2
∂ n−mt∗ ∂
m
r φ
)∣∣∣∣∣
2
≤ 1
rb4
C′∥φ∥2H˙n(Σt∗0 ) (2.164)
≤ C
′2
3A4
|t∗|−8/3∥φ∥2H˙n(Σt∗0 ) ≤C|t
∗|−8/3∥φ∥2H˙n(Σt∗0 ),
for some C =C(M,n, t∗−)> 0.
So now we consider the modified current JX ,ε/2r(∂¯ nt∗(φ)), as given by (2.25). Here X = ∂t∗ + ε∂r and
0 < ε ≪ 1 is a fixed, small constant. Given we are already restricting ourselves to t∗0 , t∗1 ≤ t∗−, we can calculate
−
∫
Σt∗
dt∗(JX ,ε/2r(∂¯ nt∗(φ)))≥ c∥∂¯ nt∗φ∥2H˙1(Σt∗) (2.165)
for some positive constant, c = c(M,n, t∗−)> 0. We also have
dρ(JT (Xn(φ)))≥ 0. (2.166)
Thus applying generalised Stokes’ theorem, we obtain
−
∫
Σt∗1
dt∗(JX ,ε/2r(∂¯ nt∗φ))≤−
∫
Σt∗0
dt∗(JX ,ε/2r(∂¯ nt∗φ))−
∫ t∗1
t∗0
∫
Σt∗
∂t∗
(
∂¯ nt∗φ
)
□g(∂¯ nt∗(φ))+KX ,ε/2rdt∗
≤−
∫
Σt∗0
dt∗(JX ,ε/2r(∂¯ nt∗φ)) (2.167)
+
∫ t∗1
t∗0
∥∂t∗ ∂¯ nt∗φ∥L2(Σt∗)
(
A|r¨b|+C|t∗|−4/3
)
∥φ∥H˙n(Σt∗0 )−K
X ,ε/2rdt∗
−
∫ t∗1
t∗0
∫
Σt∗
∂t∗ ∂¯ nt∗(φ)
(
n
∑
m=0
(
n
m
)
r˙bmm(
2
r
(
1−M
r
)(
∂ n−m+2t∗ ∂
m−1
r φ −∂ n−mt∗ ∂m+1r φ
)− 4M
r2
∂m−n+1t∗ ∂
m
r φ
))
dt∗.
We now note that in the case m ≥ 2, every term has a coefficient which can be bounded by Ar˙b2/rb ≤
B|t∗|−4/3. We can similarly bound any terms with a 1/r2 coefficient. Thus we have that
−
∫ t∗1
t∗0
∫
Σt∗
∂t∗ ∂¯ nt∗φ
(
n
∑
m=0
(
n
m
)
r˙bmm
(
2
r
(
1−M
r
)(
∂ n−m+2t∗ ∂
m−1
r φ −∂ n−mt∗ ∂m+1r φ
)− 4M
r2
∂m−n+1t∗ ∂
m
r φ
))
dt∗
≤
∫ t∗1
t∗0
B|t∗|−4/3∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗−
∫ t∗1
t∗0
∫
Σt∗
nr˙b
r
(
∂ n+1t∗ φ −∂ n−1t∗ ∂ 2r
)
∂t∗ ∂¯ nt∗(φ)dt
∗,
(2.168)
Here, we have used part 2 of Lemma 2.6.1 to bound ∂t∗ ∂¯ nt∗(φ) by ∥∂¯ nt∗φ∥H˙1(Σt∗).
In order to bound the final term, it is useful to note that swapping between ∂t∗ and ∂¯t∗ introduces terms with
a factor of r˙b. Any derivative that now has an extra factor of r˙b can be absorbed into the B term in (2.168). Thus
we can freely swap between the two derivatives when bounding this final term. We can similarly ignore any ∂¯t∗r
2.6 Higher Order Boundedness 43
terms.∫ t∗1
t∗0
∫
Σt∗
nr˙b
r
(
∂ n+1t∗ φ −∂ n−1t∗ ∂ 2r
)
∂t∗ ∂¯ nt∗(φ)dt
∗ ≥
∫ t∗1
t∗0
∫
Σt∗
nr˙b
r
∂¯ n−1t∗
(
∂ 2t∗φ −∂ 2r φ
)
∂¯ n+1t∗ φ dt
∗
−
∫ t∗1
t∗0
B|t∗|−4/3∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗ (2.169)
≥
∫ t∗1
t∗0
∫
Σt∗
nr˙b
r3
∂¯ n−1t∗ /˚△φ∂¯ n+1t∗ φ dt∗−
∫ t∗1
t∗0
B|t∗|−4/3∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗
≥−
∫ t∗1
t∗0
∫
Σt∗
nr˙b
r3
∂¯ n−1t∗ /˚∇φ .∂¯
n+1
t∗ /˚∇φ dt
∗−
∫ t∗1
t∗0
B|t∗|−4/3∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗
≥
∫ t∗1
t∗0
∫
Σt∗
nr˙b
r3
|∂¯ nt∗ /˚∇φ |2dt∗−
∫ t∗1
t∗0
B|t∗|−4/3∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗
−
∣∣∣∣nr˙brb
∣∣∣∣
t∗0
∥φ∥2H˙n+1(Σt∗0 )−
∣∣∣∣nr˙brb
∣∣∣∣
t∗1
∥φ∥2H˙n+1(Σt∗1 ).
We then have, that
∫ t∗1
t∗0
∫
Σt∗
KX ,ε/2rdt∗ ≥
∫ t∗1
t∗0
∫
Σt∗
ε
r3
|∂¯ nt∗ /˚∇φ |2dt∗−
∫ t∗1
t∗0
D|t∗|−4/3∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗, (2.170)
for some fixed constant D = D(M,n, t∗−)> 0.
Finally, we note that
∫ t∗1
t∗0
∥∂t∗ ∂¯ nt∗φ∥L2(Σt∗)
(
A|r¨b|+C|t∗|−4/3
)
∥φ∥H˙n(Σt∗0 )dt
∗
≤ A
∫ t∗1
t∗0
(
|r¨b|+ |t∗|−4/3
)
∥∂¯ nt∗φ∥2H˙1(Σt∗)+
(
|r¨b|+ |t∗|−4/3
)
∥φ∥2H˙n(Σt∗0 )dt
∗
−
∫ t∗1
t∗0
∫
Σt∗
ε+nr˙b
r3
|∂¯t∗ /˚∇φ |2dt∗ (2.171)
≤ A
∫ t∗1
t∗0
(
|r¨b|+ |t∗|−4/3
)
∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗+E∥φ∥2H˙n(Σt∗0 ),
for E = E(M,n, t∗−)> 0. This is given t∗− negative enough that |r˙b| ≤ ε/n and that we can apply part 2 of Lemma
2.6.1.
Adding these all together, we get
∥φ∥2H˙n+1(Σt∗1 ) ≤C∥φ∥
2
H˙n+1(Σt∗0 )
+D∥φ∥H˙n(Σt∗0 )+A
∫ t∗1
t∗0
(
|r¨b|+ |t∗|−4/3
)
∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗ (2.172)
≤C∥φ∥2H˙n+1(Σt∗0 )+A
∫ t∗1
t∗0
(
|r¨b|+ |t∗|−4/3
)
∥∂¯ nt∗φ∥2H˙1(Σt∗)dt
∗
where constants C,D,A all only depend on M, n and t∗−.
Thus by Gronwall’s inequality, we have
∥φ∥2H˙n+1(Σt∗1 )≤C∥φ∥
2
H˙n+1(Σt∗0 )
exp
(
A
∫ t∗1
t∗0
(
|r¨b|+ |t∗|−4/3
))
dt∗≤C∥φ∥2H˙n+1(Σt∗0 ) exp
(
A
(
|R˙(t∗−)|+ |t∗−|−1/3
))
,
(2.173)
for all t∗0 ≤ t∗1 ≤ t∗−. We can then proceed to cover the interval [t∗−, t∗c ] by using part 3 of Lemma 2.6.1. Thus we
obtain our result.
The last theorem we then prove in this section is backwards nth order energy boundedness.
Theorem 2.6.2 (Backwards nth order Non-degenerate Energy Boundedness for the Reflective Case). Let
φ ∈C∞0∀t∗ be a solution to the wave equation (2.1) with reflective boundary conditions. There exists a constant
44 The Scattering Map
E = E(n,M) such that
∥φ∥H˙n(Σt∗0 ) ≤ E∥φ∥H˙n(Σt∗1 ) ∀φ ∈C
∞
0∀t∗ ∀t∗1 ∈ [t∗0 , t∗c ]. (2.174)
Proof. This is proved identically to Theorem 2.6.1, but let X = ∂t∗ − ε∂r, and we are done (for positive
definiteness of the surface terms, see Theorem 2.5.5).
We finally look to extend the boundedness result in the permeating case to the H˙2 norm. Note we cannot
extend the result beyond this, as we do not know that solutions with higher order derivatives even exist.
Theorem 2.6.3. Let φ ∈H20∀τ be a solution to the wave equation (2.1) (as given by Theorem 2.4.2). There exists
a constant C =C(M)> 0 such that
∥φ∥2H˙2(Στ1) ≤C∥φ∥
2
H˙2(Στ0)
∀τ0 < τ1 < τc (2.175)
∥φ∥2H˙2(Στ0) ≤C∥φ∥
2
H˙2(Στ1)
∀τ0 < τ1 < τc−. (2.176)
Proof. As in the reflective case, we first prove the local in time case. Let
X = ∂τ +χ
(
r
rb
)
r˙b∂r, (2.177)
where χ is a smooth cut-off function which vanishes outside [1/2,3/2] and is identically 1 inside [2/3,4/3].
Note that X is tangent to the boundary over which irregularities of g occur. This means that derivatives of
components of g in the X direction are still H1loc.
We then proceed to write (2.1) in terms of this X :
□gφ =

−∂τ(Xφ)+
(
2
√
2Mr2
r3b
+χ r˙b
)
∂r(Xφ)+
(
1− 2Mr2
r3b
− r˙b
(
2
√
2Mr2
r3b
+χ r˙b
))
∂ 2r φ + 1r2 /˚△φ
+Lower Order Terms r ≤ rb
−∂τ(Xφ)+
(
2
√
2M
r +χ r˙b
)
∂r(Xφ)+
(
1− 2Mr − r˙b
(
2
√
2M
r +χ r˙b
))
∂ 2r φ + 1r2 /˚△φ
+Lower Order Terms r ≥ rb.
(2.178)
Here note:
1− 2Mr
2
r3b
− r˙b
(
2
√
2Mr2
r3b
+χ r˙b
)
= 1− 2Mr
2
r3b
+
√
2M
rb
(
2
√
2Mr2
r3b
−χ
√
2M
rb
)
(2.179)
= 1− 2Mr
2
r3b
+
2Mr
r2b
(
2−χ rb
r
)
> 0
1− 2M
r
− r˙b
(
2
√
2M
r
+χ r˙b
)
= 1− 2M
r
+
√
2M
rb
(
2
√
2M
r
−χ
√
2M
rb
)
(2.180)
= 1− 2M
r
+
2M√
rrb
(
2−χ
√
r
rb
)
> 0.
We can approximate φ on Στ by smooth functions, and then manipulate (2.178) in an identical way to (2.149) to
obtain: ∫
Στ
(∂τXφ)2 ≥ ε∥φ∥2H˙2(Σt∗)−C∥Xφ∥
2
H˙1(Σt∗)
−C∥φ∥2H˙1(Σt∗)−
∫
Sτ,r=0
2 f /˚∇φ .∂r /˚∇φdω2. (2.181)
Here ε > 0 and C > 0 may depend on the time interval we are considering. One can then show that for smooth
approximations to φ , the final term in (2.181) vanishes.
2.7 The Scattering Map 45
Thus to prove the local in time result, we can consider the following:
f (τ) := ∥φ∥2H˙1(Στ )−
∫
Στ
dτ
(
JX(Xφ)
)∼ ∥φ∥2H˙2(Στ ). (2.182)
Looking at □g(Xφ) as a distribution, we obtain:∫
Στ
|(□g(Xφ))XXφ |=
∫
Στ
|(□g(Xφ)−X(□gφ))XXφ | ≤C∥φ∥2H˙2(Στ ). (2.183)
As in previous cases, we have that |KX(Xφ)| ≤ −Cdτ (JX(Xφ)) for some C > 0. Applying Stokes theorem
and boundedness of H˙1(Σ) norms, we obtain:
1
C
f (τ1)−
∫ τ1
τ0
f (τ)dτ ≤ f (τ0)≤C f (τ1)+
∫ τ1
τ0
C f (τ)dτ (2.184)
for some C = C(M,τ0,τ1) ≥ 0. Then an application of Gronwall’s lemma gives the local result, in either
direction.
Once we are sufficiently far back in time, we can consider
Y± =
(
1±
(
2M
rb
)1/4)
∂τ , (2.185)
as given in the proof of Proposition 2.5.1 and Theorem 2.5.8. Define
g(τ)± := ∥φ∥2H˙1(Στ )−
∫
Στ
dτ
(
JY±(Tφ)
)∼ ∥φ∥2H˙2(Στ ). (2.186)
∫
Στ
|(□g(Tφ))Y±Tφ |=
∫
Στ
|(□g(Tφ)−T (□gφ))Y±Tφ | ≤ Cr2b
∥φ∥2H˙2(Στ ) ≤
C′
r2b
g(τ)±. (2.187)
Now we can choose τ ≤ τ∗ negative enough that ∓KY±(Tφ)≥ 0. Thus
∓KY±(Tφ)+ |(□g(Tφ))Y±Tφ | ≤ C
′
r2b
g(τ)±, (2.188)
In this region we can use the boundedness of H˙1 norms given by Theorems 2.5.7 and 2.5.8 to obtain
g(τ1)+ ≤Cg(τ0)++
∫ τ1
τ0
C
r2b
g(τ)+dτ (2.189)
g(τ0)− ≤Cg(τ1)−+
∫ τ1
τ0
C
r2b
g(τ)−dτ. (2.190)
An application of Gronwall’s lemma completes the proof, on noting that
∫ τ∗
−∞ rb(τ)−2dτ < ∞.
2.7 The Scattering Map
We now consider bounds on the radiation fields. We will be considering the maps G+ and F−, which take
data from Σt∗c to data on I
− and I + respectively. We will also consider their inverses (where defined), G−
and F+, which take data from I + and I − respectively to Σt∗c . We will look at obtaining boundedness or
non-boundedness for these.
Finally, we will define the scattering map,S + := G+ ◦F+, and consider boundedness results for this.
46 The Scattering Map
2.7.1 Existence of Radiation Fields
To look at these maps, we will first need a definition of radiation field. We will then need to show it exists for all
finite energy solutions of the wave equation.
Proposition 2.7.1 (Existence of the Backwards Radiation Field). Given φ a solution to the wave equation (2.1)
with boundary conditions (2.2), there exist ψ+,− such that
r(u,v)φ(u,v,θ ,ϕ)
H1loc−−−−→
u→−∞ ψ−(v,θ ,ϕ) (2.191)
r(u,v)φ(u,v,θ ,ϕ)
H1loc−−−→
v→∞ ψ+(u,θ ,ϕ). (2.192)
Proof. This existence has been done many times before, see for example [35, 2].
2.7.2 Backwards Scattering from Σt∗c
Now we have existence of the radiation field, we define the following map:
F− : E XΣt∗c −→ F
−
(
E XΣt∗c
)
⊂ H1loc(I −) (2.193)
(φ |Σt∗c ,∂t∗φ |Σt∗c ) 7→ ψ−
Dv0
i+
i0
i−
H +
I
+
I
−
r
=
r b
Σt ∗2
Σ
v0
Σ
v1
where the ψ− is as defined in Proposition 2.7.1, and the X is any everywhere
timelike vector field (including on the event horizon) which coincides with
the timelike Killing vector field ∂t∗ for sufficiently large r. An X with these
properties is chosen, so that the X norm is equivalent to the H˙1 norm.
We define the inverse ofF− (once injectivity is established on the image
ofF−) asF+.
• Firstly, we will showF− is bounded. (Proposition 2.7.2)
• Then we will show that F+, if it can be defined, would be bounded,
which gives us thatF− is injective. (Proposition 2.7.3)
• Finally, we show that Im(F−) is dense in E TI − . (Proposition 2.7.4)
We will then combine these results in Theorem 2.7.1 to obtain thatF− is
a linear, bounded bijection with bounded inverse between the spaces E XΣt∗c
and
E
∂t∗
I − .
We will begin with the following:
Proposition 2.7.2 (Boundedness ofF−). There exists a constant A(M,q,T ∗) such that
∥F−(φ)∥2∂t∗ ,I − =
∫
I −
(∂v(F−(φ)))2dvdω ≤ A∥φ∥2H˙1(Σt∗c ). (2.194)
Proof. We will first prove this for compactly supported smooth functions, and then extend to H1 functions using
a density argument.
Let X , w and t∗2 be as in the proof of Theorem 2.5.5. Let φ be smooth and compactly supported on
Σt∗2 . Take v0 large enough such that on Σt∗2 , φ is only supported on v ≤ v1. We integrate KX ,w, in the region
Dv0 = {v ∈ [v0,v1], t∗ ≤ t∗2}, for any v0 ≤ v1.
2.7 The Scattering Map 47
We then apply generalised Stokes’ Theorem in Dv0 to obtain the following boundary terms:
−
∫
Σt∗2
dt∗(JX ,w) =
∫
Dv0
KX ,w−
∫
{v=v0}
dv(JX ,w)+
∫
S[t∗0 ,t∗2 ]
dρ(JX ,w)− lim
u0→−∞
∫
{u=u0}∩[v0,v1]
du(JX ,w)
≥−
∫
{v=v0}
dv(JX ,w)− lim
u0→−∞
∫
{u=u0}∩[v0,v1]
du(JX ,w) (2.195)
where t∗0 is the value of t
∗ at the sphere where {v = v0} intersects {r = rb(t∗)}.
−
∫
{v=v0}
dv(JX ,w) =
∫
{v=v0}
b
2
(
∂t∗φ −∂rφ − φr
)2
(2.196)
+
1
2
((
1− 2M
r
+
q2M2
r2
)
f (t∗)+
(
2M
r
− q
2M2
r2
)
b
)
(∂t∗φ −∂rφ)2 ≥ 0
−
∫
{u=u0}∩[v0,v1]
du(JX ,w)
=
∫
{u=u0}∩[v0,v1]
f (t∗)−b
2
(
1− 2Mr + q
2M2
r2
) ((1+ 2M
r
− q
2M2
r2
)
∂t∗φ −
(
1− 2M
r
+
q2M2
r2
)
∂rφ
)2
+
b
r
φ
1+ 2Mr − q2M2r2
1− 2Mr + q
2M2
r2
∂t∗φ −∂rφ
− bφ2
2r2
(2.197)
+
(
1− 2Mr + q
2M2
r2
)
f (t∗)−
(
1+ 2Mr − q
2M2
r2
)
b
2r2
(
1− 2Mr + q
2M2
r2
) | /˚∇φ |2
However, as we know that rφ tends to an H1loc function, and the volume form on {u = u0} is r2, we can see
that the terms in (2.197) with a factor of φ tend to 0 as u0 → ∞. Similarly, by applying the rotational Killing
fields Ωi (defined in (1.26)) to φ , we can see rΩiφ has an H1loc limit. Thus terms in (2.197) involving /˚∇φ will
also tend to 0 in the limit u0 → ∞.
Thus in the limit u0 → ∞ (and therefore r → ∞, t∗→−∞) we obtain:
− lim
u0→−∞
∫
{u=u0}∩[v0,v1]
du(JX ,w) = lim
u0→−∞
∫
{u=u0}∩[v0,v1]
1−b
4
(
1− 2M
r
+
q2M2
r2
)
(∂t∗φ −∂rφ)2 r2dvdω
= lim
u0→−∞
∫
{u=u0}∩[v0,v1]
1−b
4
(
1− 2M
r
+
q2M2
r2
)
(∂t∗(rφ)−∂r(rφ))2 dvdω
(2.198)
=
∫
I −∩[v0,v1]
1−b
4
(∂t∗ψ−−∂rψ−)2 dvdω ≥ ε
∫
I−∩[v0,v1]
(∂vψ−)2 dvdω
where to get from the first line to the second, we have ignored terms of order φ , as these tend to 0.
Substituting (2.198) and (2.196) into (2.195), and noting that −∫Σt∗2 dt∗(JX ,w) can be bounded by the H˙1
norm, we have that: ∫
I −∩[v0,v1]
(∂v(F−(φ)))2dvdω ≤ A∥φ∥2H˙1(Σt∗2 ), (2.199)
where A is independent of v0. Thus taking a limit as v0 →−∞ and imposing Theorem 2.5.1 in the region [t∗0 , t∗2 ]
gives us the result of the theorem.
We then move on to showingF+, if it exists, would be bounded:
Proposition 2.7.3 (Boundedness ofF+). There exists a constant A such that
∥φ∥2H˙1(Σt∗c ) ≤ A
∫
I −
(∂v(F−(φ)))2dvdω. (2.200)
48 The Scattering Map
To prove this, we will first need to show a decay result:
Lemma 2.7.1 (Decay of Solutions Along a Null Foliation). Let φ be a solution to (2.1) with reflective boundary
conditions. Then
lim
v0→−∞
∫
Σv0
dv(J∂t∗ ) = 0. (2.201)
Similarly, let φ ∈ H20∀τ be a solution to (2.1) with permeating boundary conditions and Ωiφ ∈ H20∀τ . Then
lim
v0→−∞
∫
Σ˜v0
dv(JY ) = 0, (2.202)
for JY as in Theorem 2.5.1.
Proof. We first deal with the reflective case by showing the result for φ compactly supported on some Σv1 , and
then extend the result by a density argument.
Firstly, we calculate −dv(J∂t∗ ) and −du(J∂t∗ ).
−
∫
Σv0
dv(J∂t∗ ) =
1
2
∫
Σv0
(
1− 2M
r
+
q2M2
r2
)
(∂t∗φ −drφ)2+ 1r2 | /˚∇φ |
2
=
∫
Σv0
2
(
1− 2M
r
+
q2M2
r2
)−1
(∂uφ)2+
1
2r2
| /˚∇φ |2 =
∫
Σv0
(∂uφ)2+
1− 2Mr + q
2M2
r2
4r2
| /˚∇φ |2r2dudω
(2.203)
−
∫
Σu0∩[v0,v1]
du(J∂t∗ ) =
1
2
∫
Σu0∩[v0,v1]
(
1− 2M
r
+
q2M2
r2
)1+ 2Mr − q2M2r2
1− 2Mr + q
2M2
r2
∂t∗φ −∂rφ
2+ 1
r2
| /˚∇φ |2
(2.204)
=
∫
Σu0∩[v0,v1]
(∂vφ)2+
1− 2Mr + q
2M2
r2
4r2
| /˚∇φ |2r2dvdω ≥ 0
Integrating K∂t∗ in the area Du0 := {v ∈ [v0,v1]}∩{u≥ u0}, using (2.91), (2.93), and then letting u0 →−∞
gives us that
−
∫
Σv1
dv(J∂t∗ )≤−
∫
Σv0
dv(J∂t∗ ). (2.205)
We then proceed to use the rp method [12]. We consider the wave operator applied to rφ :
4∂u∂v(rφ)
1− 2Mr2 +
q2M2
r2
=
1
r2
/˚△(rφ)− 2M
r3
(
1− q
2M
r
)
rφ . (2.206)
2.7 The Scattering Map 49
We apply (2.206) to the following integral over Du0 :
∫
Σv1
(
2r
1− 2Mr + q
2M2
r2
(∂u(rφ))2
)
dudω ≥
(∫
Σv1
−
∫
Σv0
−
∫
S[v0,v1]
)(
2r
1− 2Mr + q
2M2
r2
(∂u(rφ))2
)
dudω
=
∫
Du0
4r∂u(rφ)∂u∂v(rφ)
1− 2Mr2 +
q2M2
r2
+2(∂u(rφ))2 ∂v
(
1
r
− 2M
r2
+
q2M2
r2
)−1
dudvdω
=
∫
Du0
(
1
r
/˚△(rφ)− 2M
r2
(
1− q
2M
r
)
rφ
)
∂u(rφ)+
(
1− 4Mr + 3q
2M2
r2
)
(∂u(rφ))2
1− 2Mr + q
2M2
r2
dudvdω
(2.207)
=
∫
Du0
(
1− 4Mr + 3q
2M2
r2
)
(∂u(rφ))2
1− 2Mr + q
2M2
r2
− 1
2r
∂u
(
| /˚∇rφ |2
)
−M
r2
(
1− q
2M
r
)
∂u((rφ)2)dudvdω
≥
∫
Du0
(
1− 4Mr + 3q
2M2
r2
)
(∂u(rφ))2
1− 2Mr + q
2M2
r2
+
(
1− 2M
r
+
q2M2
r2
)(
1
4r2
| /˚∇rφ |2+
(
1− 3q
2M
2r
)
M
r3
(rφ)2
)
dudvdω
≥ 1
2
∫
Du0
(∂uφ)2+
1− 2Mr + q
2M2
r2
2r2
| /˚∇φ |2r2dudvdω ≥ 1
2
∫ v1
v0
(
−
∫
Σv0
dv(J∂t∗ )
)
dv
In order to obtain the last line, we have used that for r large enough,
∫
Σv
1− 4Mr + 3q
2M2
r2
1− 2Mr + q
2M2
r2
(∂u(rφ))2dudω ≥ 12
∫
Σv
(∂u(rφ))2dudω =
1
2
∫
Σv
r2(∂uφ)2+∂u
(
r∂urφ2
)−∂ 2u r (rφ)2r dudω
≥ 1
2
∫
Σv
r2(∂uφ)2− 12
(
1− 2M
r
+
q2M2
r2
)(
1− q
2M
r
)
M
r2
φ2dudω
(2.208)
≥ 1
2
∫
Σv
r2(∂uφ)2− M2r2φ
2dudω.
The left hand side of (2.207) is independent of v0, so if we choose φ to be compactly supported on Σv1 (these
functions are dense in the set of H1 functions), then we can let v0 tend to −∞ to obtain
∫ v1
−∞
(
−
∫
Σv0
dv(J∂t∗ )
)
dv≤
∫
Σv1
(
4r
1− 2Mr + q
2M2
r2
(∂u(rφ))2
)
dudω. (2.209)
Thus there exists a sequence vi →−∞ such that
−
∫
Σvi
dv(J∂t∗ )→ 0 as i→ ∞. (2.210)
We then note that given any ε > 0, and a solution φ to (2.1) with finite ∂t∗ energy on Σv0 , there exists a
smooth compactly supported function φε such that
∥φ −φε∥∂t∗ ,Σv0 ≤ ε/2. (2.211)
Furthermore, by (2.205), we know that for all v1 ≤ v0, we have
∥φ −φε∥∂t∗ ,Σv1 ≤ ε/2. (2.212)
50 The Scattering Map
By (2.210), there exists a v1 ≤ v0 such that
∥φε∥∂t∗ ,Σv1 ≤ ε/2. (2.213)
By combining (2.212) (2.213), and (2.205) again, we obtain that
∥φ∥∂t∗ ,Σv ≤ ∥φ∥∂t∗ ,Σv1 ≤ ∥φε∥∂t∗ ,Σv1 +∥φ −φε∥∂t∗ ,Σv1 ≤ ε, (2.214)
for all v≤ v1.
Thus given any solution φ to (2.1) with finite ∂t∗ energy, and given any ε > 0, there exists a v1 such that
∥φ∥∂t∗ ,Σv ≤ ε, (2.215)
for all v≤ v1.
Next we look at the permeating case. For this case, let Y = h(τ)∂τ . We then have
−dn(JY ) =

h
2
(
1
r2 | /˚∇φ |2+
(
1−
√
2M
r
)(
1+
√
2M
r
)−1(
∂τφ −
(
1+
√
2M
r
)
∂rφ
)2)
r ≥ rb
h
2
(
1
r2 | /˚∇φ |2+(∂τφ)2+
(
1− 2Mr2
r3b
)
(∂rφ)2
)
r < rb
(2.216)
where n is the normal to Σ˜τ .
We then perform something similar to the reflective case above. However, as we do not have an explicit
coordinate system u,v, we will use ∂τ and ∂r. Let f be given by
f (τ,r) =

√
2Mr2
r3b
r < rb√
2M
r r ≥ rb
(2.217)
so our metric is of the form
g =−(1− f 2)dτ2+2 f dτdr+dr2+ r2gS2. (2.218)
Again, let ψ := rφ . We obtain
(∂τ +(1− f )∂r)(∂τ − (1+ f )∂r)ψ = f ′(∂τ − (1+ f )∂r)ψ+(2 f f ′− f˙ )ψr +
1
r2
/˚△ψ. (2.219)
Note that (2 f f ′− f˙ ) is not continuous over r = rb.
2.7 The Scattering Map 51
Then, in a similar way to the reflective case, we obtain that:
C
∫
Σv1
r((∂τ − (1+ f )∂r)ψ)2dudω2 ≥
∫∫ v1
v0
(∂τ +(1− f )∂r)(r(∂τ − (1+ f )∂r)ψ)2dudvdω2
=
∫∫ v1
v0
(
− (∂τ − (1+ f )∂r)
(
1
r
| /˚∇ψ|2− (2 f f ′− f˙ )ψ2
)
+(1+ f )
1
r2
| /˚∇ψ|2
− [(∂τ − (1+ f )∂r)(2 f f ′− f˙ )]ψ2
+(1+2r f ′− f )((∂τ − (1+ f )∂r)ψ)2
)
dudvdω2
≥
∫∫ v1
v0
1
r2
| /˚∇ψ|2+ 1
2
((∂τ − (1+ f )∂r)ψ)2dudvdω2
+
∫∫
r<rb
M
2r3b
ψ2dudvdω2+
∫∫
r≥rb
2M
r3
ψ2dudvdω2 (2.220)
+
∫
r=rb
(2 f f ′− f˙ )|rb+
r−b
ψ2dvdω2+
∫
r=0
(2 f f ′− f˙ )ψ2dvdω2
≥
∫∫ v1
v0
1
r2
| /˚∇ψ|2+ 1
2
((∂τ − (1+ f )∂r)ψ)2dudvdω2
+
∫∫
r<rb
M
2r3b
ψ2dudvdω2+
∫∫
r≥rb
2M
r3
ψ2dudvdω2−
∫
r=rb
3M
r2b
ψ2dvdω2.
We will only be using this rp method to bound the Y -energy for the exterior, so we note
∫
Σv∩{r≥rb}
(∂uψ)2dudω2 =
∫
Σv∩{r≥rb}
r2(∂uφ)2+∂u
(
r∂urφ2
)−∂ 2u rψ2r dudω2
≥
∫
Σv∩{r≥rb}
r2(∂uφ)2dudω2−
∫
r=rb
2
rb
ψ2dudω2
=
∫
Σv∩{r≥rb}
r2(∂uφ)2−
∫
r=rb
2
rb
ψ2dudω2. (2.221)
Now, we need to bound the ψ/rb surface term, and therefore also bound the 3Mψ/r2b surface term in (2.220).
We proceed by noting
∫
r=rb
1
r
ψ2dudω2 =−
∫
r<rb
(∂τ − (1+ f )∂r)
(
1
r
χ
(
2Mr2
r3b
)
ψ2
)
dudvdω2
=
∫
r<rb
−2
r
ψ(∂τ − (1+ f )∂r)ψ− (1+ f ) 1r2ψ
2+
(
(1+ f )
4M
r3b
+
2Mr
r4b
r˙b
)
χ ′ψ2dudvdω2
≤
∫
r<rb
((∂τ − (1+ f )∂r)ψ)2−
(
1
r
ψ+(∂τ − (1+ f )∂r)ψ
)2
+
CM
r3b
ψ2dudvdω2
≤
∫
r<rb
((∂τ − (1+ f )∂r)ψ)2+CM
r3b
ψ2dudvdω2. (2.222)
Dividing equations (2.220) and (2.222) by 4C, say, we can absorb the ψ2 term in (2.222) by our ψ2 bulk
term in (2.220). This gives us
∫
Σv1
r((∂τ − (1+ f )∂r)ψ)2dudω2 ≥ c
∫ v1
v0
∫
Σv∩{r≥rb}
(
1
r2
| /˚∇φ |2+(∂uφ)2
)
≥ c
∫ v1
v0
∫
Σv∩{r≥rb}
(−dn(JT )).
(2.223)
52 The Scattering Map
Σ
v1
Σ
v0
v=
v0
Στ0
Figure 2.3 Left: Region for integrating in the rp method as above. Right: Region for integrating for interior
decay see below.
Finally, we need to consider the interior of our star. To do this, we will restrict ourselves to v≤ v1 < 0. In
this region, we will consider
X =−
√
2Mr2
r3b
∂r, w =−12
√
2M
r3b
. (2.224)
We will integrate the modified current JX ,w, as in (2.25).
This has the properties that:
−∇.JX ,w ≥ 0 (2.225)
−∇.JX ,w|r<rb ≥ c
√
2M
r3b
(−dτ(JY )) = ∂τ(logrb)c(−dτ(JY )) (2.226)
|dτ(JX ,w)| ≤ A(−dτ(JY ))+
√
2M
r3b
ψdτ(φ)+
6M
4rb3
φ2. (2.227)
We now integrate ∇.JX ,w in the region Rτ0,v1 := {v≤ v1,τ ≥ τ0} to obtain:
∫
Σv1∩{τ≤τ0}
−dn(JX ,w) −
∫
Στ0∩{v≤v1}
−dτ(JX ,w) = c
∫
Rτ0,v1
−∇.JX ,w ≥
∫
[τ0,τ1]∩{r<rb}
√
2M
r3b
(−dτ(JY )).
(2.228)
Here τ1 is chosen such that (τ1,rb(τ1)) has v≤ v1.
The terms on the left hand side of equation (2.228) can both be bounded by data on Σv1 .
As 2M
r3b
→ 0, by choosing τ1 sufficiently far back, we can combine equations (2.220) and (2.228) to see
C ≥
∫ τ1
τ0
√
2M
r3b
∫
Σ˜τ
(−dn(JY )). (2.229)
This C only depends on our initial data on Σv1 , and is finite for compactly supported data. As
√
2M
r3b
is not
integrable, there must be a sequence of vi → ∞ such that the required result holds. The same logic as before
then gives the required result.
2.7 The Scattering Map 53
Proof of Proposition 2.7.3. Fix t∗1 < t
∗
c , and let φ be a solution of (2.1) with boundary conditions (2.2) such that
φ has finite ∂t∗ energy on Σt∗1 . Note as t
∗
1 < t
∗
c fixed, finite ∂t∗ energy is equivalent to having finite X energy. An
explicit calculation gives(
1− 2M
rb(t∗1)
+
q2M2
rb(t∗1)2
)
∥φ∥2H˙1(Σt∗1 ) ≤−
∫
Σt∗1
dt∗(J∂t∗ )≤ ∥φ∥2H˙1(Σt∗1 ) (2.230)
We prove Proposition 2.7.3 by simply integrating K∂t∗ in the region Du0,v0 = {u > u0,v≥ v0, t∗ < t∗1}. We
will then let u0 →−∞ to get:
−
∫
Σt∗1
dt∗(J∂t∗ ) = lim
u0→−∞
(
−
∫
Σt∗1∩{u≥u0}
dt∗(J∂t∗ )
)
(2.231)
=− lim
u0→−∞
∫
Σu0∩{v≥v0}
du(J∂t∗ )− lim
u0→−∞
∫
Σv0∩{u≥u0}
dv(J∂t∗ )−
∫
S[v0,t∗1 ]
dρ(J∂t∗ )
≤
∫
I −∩{v≥v0}
(∂vψ−)2dvdω−
∫
Σv0
dv(J∂t∗ ).
Here we have used (2.93) to ignore the S[v0, t∗1 ] term. Letting v0 →−∞, and using Lemma 2.7.1, we obtain(
1− 2M
rb(t∗1)
+
q2M2
rb(t∗1)2
)
∥φ∥2H˙1(Σt∗1 ) ≤−
∫
Σt∗1
dt∗(J∂t∗ )≤
∫
I −
(∂vψ−)2dvdω. (2.232)
Theorem 2.5.1 on the interval t∗ ∈ [t∗1 , t∗c ] then gives us our result.
i0
i−
ψ−
r
=
r b
Σ
v0
Σ t∗0
, φ
′ | Σ t∗0
We now have that F− is bounded and injective, so the inverse is well
defined. We also have thatF+ is bounded where it is defined. The final result
needed to define the scattering map on the whole space E ∂t∗I − is that the image
ofF− is dense in E ∂t∗I −:
Proposition 2.7.4 (Density of Im(F−) in E ∂t∗I −). Im(F
−) is dense in E ∂t∗I − .
Proof. We prove this using existing results on the scattering map on the full
exterior of Reissner–Nordström spacetime. We show that compactly supported
smooth functions on I − are in the image ofF−. These are dense in E ∂t∗I − .
Given any smooth compactly supported function ψ− ∈ E ∂t∗I − , supported
in v ≥ v0, we can find a t∗0 such that the sphere (t∗0 ,rb(t∗0)) is in the region
v≤ v0. Using previous results from [25], there exists a solution, φ ′ in Reissner–
Nordström with radiation field ψ and vanishing on the past horizon. By finite speed of propagation, φ ′ will be
supported in v≥ v0. Thus both φ ′ and its derivatives on Σt∗0 vanishes around r = rb.
We then evolve (φ ′|Σt∗0 ,∂t∗φ
′|Σt∗0 ) from Σt∗0 in RNOS, call this solution φ . By finite speed of propagation and
uniqueness of solutions, we must have φ = φ ′ for t∗ ≤ t∗0 . By boundedness ofF+ (Proposition 2.7.3) we have
that φ is in E XΣt∗c
, and so the radiation field, ψ−, is in the image ofF−.
Theorem 2.7.1 (Bijectivity and Boundedness ofF−). F− is a linear, bounded bijection with bounded inverse
between the spaces E XΣt∗c
and E ∂t∗I − .
Proof. F+ is continuous (linear and bounded, by Proposition 2.7.3), and its image, E XΣt∗c
, is a closed set.
Therefore (F+)−1
(
E XΣt∗c
)
is closed. Thus
F−(E XΣt∗c ) =
(
F+
)−1(
E XΣt∗c
)
=Cl
((
F+
)−1(
E XΣt∗c
))
⊃ E ∂t∗I −, (2.233)
asF−(E XΣt∗c
) is dense in E ∂t∗I − (Proposition 2.7.4).
54 The Scattering Map
However, asF− is also bounded, we have
F−(E XΣt∗c )⊂ E
∂t∗
I − (2.234)
Therefore
F−(E XΣt∗c ) = E
∂t∗
I −. (2.235)
Thanks to Propositions 2.7.2 and 2.7.3,F− andF+ are bounded, and thus we have the required result.
2.7.3 Forward Scattering from Σt∗c
In a similar manner to Section 2.7.2, we define the map taking initial data on Σt∗c to radiation fields onH
+∪I +:
G+ : E XΣt∗c
−→ G+
(
E XΣt∗c
)
⊂ H1loc(H +∪I +) (2.236)
(φ |Σt∗c ,∂t∗φ |Σt∗c ) 7→ (φ |H +,ψ+)
where ψ+ is as in Proposition 2.7.1, and X is again any everywhere timelike vector field (including on the event
horizon) which coincides with the timelike Killing vector field ∂t∗ for sufficiently large r. We will define the
inverse of G+ (only defined on the image of G+) as
G− :G+
(
E XΣt∗c
)
−→ E XΣt∗c (2.237)
G+(φ |Σt∗c ,∂t∗φ |Σt∗c ) 7→ (φ |Σt∗c ,∂t∗φ |Σt∗c ).
Remark 2.7.1. Note that G± are defined using scattering in pure Reissner–Nordström. Thus they have been
studied extensively already, see for example [14] for the sub-extremal case (|q|< 1) and [3] for the extremal
case (|q|= 1).
We will be using the following facts about G+:
Lemma 2.7.2. • G+ is injective.
• For the sub-extremal case (|q|< 1), G+ is bounded with respect to the X norm on Σt∗c andH + and the
∂t∗ norm onI +. In the extremal case (|q|= 1), we use the weaker result that G+ is bounded with respect
to the X norm on Σt∗c and the ∂t∗ norm on I
+ andH +.
• G+ is not surjective into E ∂t∗I + , for both sub-extremal and extremal Reissner–Nordström.
• G− is not bounded, again with respect to the X norm on Σt∗c andH
+, and the ∂t∗ norm on I +.
Proof. Thanks to T energy conservation, we have that G+ is injective.
For G+ bounded in the sub-extremal case, we apply the celebrated red-shift vector [13] in order to obtain
boundedness of the X energy onH +.
In the extremal case, we do not have the red-shift effect. In this case, the best we can do is apply conservation
of T energy which immediately gives the weaker extremal result.
For G+ not surjective, we can look at any solution with finite ∂t∗ energy on Σt∗0 , but infinite X energy, such
as φ =
√
r− r+, ∂t∗φ = 0. Let G+(φ) = (φ+,ψ+), which has finite X and ∂t∗ energy respectively (the angular
component vanishes by spherical symmetry). G+ is injective from the ∂t∗ energy space, thus no other finite ∂t∗
energy data on Σt∗c can map to (φ+,ψ+). Therefore no finite X energy solution can map to (φ+,ψ+), and thus
G+
(
E XΣt∗c
)
̸= E XH +×E
∂t∗
I + . For a more detailed discussion of non-surjectivity in the sub-extremal case see [16]
(note this proves non-surjectivity for Kerr, but the proof can be immediately applied to Reissner–Nordström).
For the extremal case, again see [3].
By taking a series of smooth compactly supported functions approximating (φ+,ψ+) in the above paragraph,
we can see that G− is not bounded.
2.7 The Scattering Map 55
2.7.4 The Scattering Map
We are finally able to define the forwards Scattering Map:
S + : E ∂t∗I − −→S +
(
E
∂t∗
I −
)
(2.238)
S + := G+ ◦F+
and similarly with the backwards scattering map:
S − :S +
(
E
∂t∗
I −
)
−→ E ∂t∗I − (2.239)
S − :=F− ◦G−.
NoteS − is defined only on the image ofS +.
Theorem 2.7.2 (The Scattering Map). The sub-extremal (|q| < 1) forward scattering map S + defined by
(2.238) is an injective linear bounded map from E ∂t∗I − to E
X
H + ∪E
∂t∗
I + . The extremal (|q|= 1) forward scattering
mapS +, again defined by (2.238), is an injective linear bounded map from E ∂t∗I − to E
∂t∗
H + ∪E
∂t∗
I + . In both cases,
S + is not surjective, and its image does not even contain 0+E ∂t∗I + . When defined on the image of S
+, its
inverseS − is injective but not bounded.
Proof. This is an easy consequence of Theorem 2.7.1 and Lemma 2.7.2.
Remark 2.7.2. [Generality of Theorem 2.7.2] It should be noted that in the reflective case, proving Theorem
2.7.2 only uses the following two behaviours of rb:
• The tangent vector (1, r˙b(t∗),0,0) at the point (t∗,rb(t∗),θ ,ϕ) is timelike (for all t∗,θ ,ϕ , including at
t∗ = t∗c ).
• There exists a t∗− and an ε > 0 such that for all t∗ ≤ t∗−, r˙b(t∗) ∈ (−1+ ε,0).
Provided rb obeys these points, then Theorem 2.7.2 remains true.
This is in immediate contrast with Reissner–Nordström spacetime. The scattering map in Reissner–
Nordström spacetime is an isometry with respect to the T energy, and this immediately follows from the
fact that T is a global Killing vector field. Moreover, this imposes the canonical choice of energy on I ±.
However, in the RNOS model, if one considers the T energy on I −, thenF+ gives an isometry between
E TI − and E
X
Σt∗c
. Thus, we are forced to consider the non-degenerate X energy, when considering the solution on
Σt∗c . This is the main contrast with Reissner–Nordström spacetime, where we consider T energy throughout the
whole spacetime.
In both Reissner–Nordström and RNOS spacetimes, we can consider the backwards reflection map, which
takes finite energy solutions from I + to I −. On both these surfaces, choice of energy is canonically given by
the existence of Killing vector fields in the region around I ±. In Reissner–Nordström this map is bounded,
however in RNOS, this map does not even exist as a map between finite energy spaces.

Chapter 3
Hawking Radiation
3.1 Overview
i+
i0
(t∗c , r+)
i−
H +
I
+
I
−
r
=
r b
Σ
v
Figure 3.1 Penrose Diagram
of RNOS Model, with space-
like hyper surface Σv.
The problem of Hawking radiation for massless zero-spin bosons can be for-
mulated as follows. We study solutions of the linear wave equation (2.1) on
the exteriors of collapsing spherically symmetric spacetimes. In this exterior,
the spacetime is a subset of Reissner–Nordström spacetime [41]. Therefore we
have coordinates t∗, r, θ , ϕ for which the metric takes the form
g =−
(
1− 2M
r
+
q2M2
r2
)
dt∗2+2
(
2M
r
− q
2M2
r2
)
dt∗dr (3.1)
+
(
1+
2M
r
− q
2M2
r2
)
dr2+ r2gS2 t
∗ ∈ R r ∈ [max(rb(t∗),r+),∞),
where gS2 is the metric on the unit 2-sphere, and r = r+ is the horizon of the
underlying Reissner–Nordström metric, as given by (1.35). Here rb(t∗) is the
area radius of the boundary of the collapsing dust cloud as a function of t∗.
See Section 1.2 for further details of this. We refer to M as the mass of the
underlying Reissner–Nordström spacetime, and q ∈ [−1,1] as the charge to
mass ratio. In particular, we are allowing the extremal case, |q|= 1.
We will be imposing Dirichlet conditions on the boundary of the dust cloud,
i.e. φ = 0 on {r = rb(t∗)}.
These collapsing charged models will include the Oppenheimer–Snyder
Model [39] (for which q = 0), and we will refer to these more general models as Reissner–Nordström
Oppenheimer–Snyder (RNOS) models, as defined in Chapter 1.
The main Theorem of this chapter is informally stated below:
Theorem 4 (Late Time Emission of Hawking Radiation). Let ψ+(u,θ ,ϕ) be a Schwartz function on the 3-
cylinder, with ψˆ+ only supported on positive frequencies. Let φ be the solution of (2.1), as given by Theorem
2.4.1, such that
lim
v→∞r(u,v)φ(u,v,θ ,ϕ) = ψ+(u−u0,θ ,ϕ) (3.2)
lim
u→∞r(u,v)φ(u,v,θ ,ϕ) = 0 ∀v≥ vc, (3.3)
Define the function ψ−,u0 by
lim
u→−∞r(u,v)φ(u,v,θ ,ϕ) = ψ−,u0(v,θ ,ϕ). (3.4)
58 Hawking Radiation
Then for all |q|< 1, n ∈ N, there exist constants An(M,q,T ∗,ψ+) such that∣∣∣∣∣
∫ 0
ω=−∞
∫ 2π
ϕ=0
∫ π
θ=0
|ω||ψˆ−,u0(ω,θ ,ϕ)|2 sinθdωdθdϕ−
∫
H −
|ω||ψˆH −(ω,θ ,ϕ)|2
e
2π|ω|
κ −1
sinθdωdθdϕ
∣∣∣∣∣≤ Anu−n0 ,
(3.5)
for sufficiently large u0.
Here, ψH − is the reflection of ψ+ in pure Reissner–Nordström spacetime (as will be discussed in Section
3.3), and κ is the surface gravity of the Reissner–Nordström black hole.
In the case |q|= 1, there exists a constant A(M,q,T ∗) such that∣∣∣∣∫ 0ω=−∞
∫ 2π
ϕ=0
∫ π
θ=0
|ω||ψˆ−,u0(ω,θ ,ϕ)|2 sinθdωdθdϕ
∣∣∣∣≤ A
u3/20
, (3.6)
for sufficiently large u0.
This result is restated in Theorem 3.4.1 and Corollary 3.4.6, including the precise relationship of An,A on
ψ+.
This proof will rely on certain scattering results for solutions of (2.1), which are proven in our Chapter
2. It will also make use of scattering results in pure Reissner–Nordström spacetime, some high frequency
approximations, and finally will use an r∗p weighted energy estimate, based very closely on the estimates given
in [3].
Physically, in order to be normalised, any massless boson cannot have frequency equal to ω . Instead, we
will be considering a ψ+ which has ψˆ+ peaked sharply about ω . The integral
∫ 0
−∞ |σ ||ψI −|2dσ represents the
number of such particles emitted by the black hole (see Section 0.2, or [28, 40] for a discussion of why this is
the case).
The limits (3.5) and (3.6) demonstrate that a sub-extremal Reissner–Nordström black hole forming in the
collapse of a dust cloud gives off radiation approaching that of a black body with temperature κ2π (in units where
h¯ = G = c = 1). In the extremal case, this limit demonstrates that the amount of radiation emitted by a forming,
extremal Reissner–Nordström black hole tends towards 0. This is thus a rigorous result confirming Hawking’s
original calculation in both extremal and subextremal settings.
Equation (3.5) and (3.6) give estimates for the rate at which the limit is approached. In the case of (3.5), this
rate is very fast. In the case of (3.6), however, we note that the bound for the rate obtained is integrable. This
means that, as the ‘final’ temperature is zero, the total radiation emitted by an extremal Reissner–Nordström
black hole that forms from collapse is finite. Thus, extremal black holes are indeed stable to Hawking radiation.
3.2 Previous Works
Hawking radiation on collapsing spacetimes has been mathematically studied in several other settings, for
example [8, 25, 18, 20]. Each of these papers primarily work in frequency space, and work in different contexts
to this thesis. Let us discuss some of these differences in more detail.
The original mathematical study of Hawking radiation, [6], considered Hawking radiation of massive or
massless non-interacting bosons for a spherically symmetric uncharged collapsing model, and performs this
calculation almost entirely in frequency space. Thus [6] obtains what can be viewed as a partial result towards
Theorem 4, where the surface of the collapsing star is assumed to remaining at a fixed radius for all sufficiently
far back times, and no rate at which the limit is approached is calculated.
In [25], Hawking radiation of fermions is studied for sub-extremal charged, rotating black holes. The Dirac
equation itself has a 0th order conserved current, which avoids many of the difficulties of considering the linear
wave equation, for which no such current exists. The extremal case is not considered in [25].
The paper [18] considers the full Klein–Gordon equation, but on the Schwarzschild de-Sitter metric. This
paper is also the first paper in this setting to obtain a rate at which the limit is obtained, independent of the
3.3 Classical Scattering and Transmission and Reflection Coefficients of
Reissner–Nordström Spacetimes 59
angular mode. The asymptotically flat case we will be studying introduces several of new issues, due to the lack
of a cosmological horizon at a finite radius.
The paper [19] looks at calculating the Hawking radiation of extremal and subextremal Reissner–Nordström
black holes in one fewer dimension, with no rate obtained. This paper also considers Hawking radiation in the
context of the Unruh-type vacuum rather than Hawking radiation generated from a collapsing spacetime.
Hawking radiation on a charged background has been considered in several other papers in the physics
literature, the most relevant being [22, 9]. The second of these, [9], considers Hawking radiation emitted by
extremal black holes in the style of Hawking’s original paper. Many papers also make use of the surprising fact
that the extremal Reissner–Nordström Hawking radiation calculation is very similar to an accelerated mirror in
Minkowski space [24].
A more thorough discussion of the physical derivation of Hawking radiation in general, along with a full
explanation of Hawking’s original method for the calculation, can be found in chapter 14.4 of General Relativity
by Wald, [45].
As mentioned in the introduction, one can also rigorously consider the Hartle–Hawking state in order to
determine the thermal temperature of a black hole [29, 17], but we will be considering the collapsing model
derivation of Hawking radiation in this thesis, as this derivation is more generalisable.
In contrast to many of the above works, the considerations of this thesis are almost entirely in physical space.
We will be using the Friedlander radiation formalism, [21], for the radiation field, and we hope this will make
the proof more transparent to the reader.
3.3 Classical Scattering and Transmission and Reflection Coefficients
of Reissner–Nordström Spacetimes
We will begin this section by first stating a well known result which will be used frequently in this chapter:
Theorem 3.3.1 (Existence of Scattering Solution in pure Reissner–Nordström). Let ψ+(u,θ ,ϕ) be a smooth
function, compactly supported in [u−,u+]×S2 on the 3-cylinder. Then there exists a unique finite energy smooth
solution, φ(u,v,θ ,ϕ) to (2.1) in the region r ≥ r+ such that
lim
v→∞r(u,v)φ(u,v,θ ,ϕ) = ψ+(u,θ ,ϕ) (3.7)
lim
u→∞r(u,v)φ(u,v,θ ,ϕ) = 0. (3.8)
Further, there exist functions ψRN and ψH − such that
lim
v→−∞r(u,v)φ(u,v,θ ,ϕ) = ψH −(u,θ ,ϕ) (3.9)
lim
u→−∞r(u,v)φ(u,v,θ ,ϕ) = ψRN . (3.10)
Finally, φ(u,v,θ ,ϕ) = 0 for all u≥ u+. i+
i0B
i−
H
+ I +
I
−H −
v=
v(t ∗0 ,r0 )
u=
u(
t∗ 0,
r 0
)
u=
u(
t∗ 0,
r 1
)
v=
v(t ∗0 ,r1 )
Figure 3.2 The Domain of De-
pendence
We refer the reader to [15] (sub-extremal) and [3] (extremal) for this result.
Another result we will frequently use is the existence of a domain of de-
pendence:
Theorem 3.3.2 (Domain of Dependence of the wave equation). Let φ(t∗0 ,r,θ ,ϕ)
be a smooth solution of (2.1), such that on surface Σt∗0 , φ(t
∗
0 ,r,θ ,ϕ) is supported
on r ∈ [r0,r1].
Then φ vanishes in the 4 regions {t∗> t∗0}∩{v≤ v(t∗0 ,r0)}, {t∗> t∗0}∩{u≤
u(t∗0 ,r1)}, {t∗ < t∗0}∩{v≥ v(t∗0 ,r1)} and {t∗ < t∗0}∩{u≥ u(t∗0 ,r0)}.
60 Hawking Radiation
This result is a trivial consequence of T -energy conservation.
An important part of the Hawking radiation calculation is the use of transmission and reflection coefficients,
so we will discuss their definitions and useful properties here. For a more full discussion of these, we refer the
reader to [15, 10].
We will define the transmission and reflection coefficients in the same way as [15]. We first change
coordinates to the tortoise radial function, r∗, and then consider fixed frequency solutions of the wave equation,
ψ = eiωtuω,m,l(r∗)Yl,m(θ ,ϕ). The equation obeyed by this uω,l,m(r∗) is
u′′+(ω2−Vl)u = 0, (3.11)
where
Vl(r) =
1
r2
(
l(l+1)+
2M
r
(
1− q
2M
r
))(
1− 2M
r
+
q2M2
r2
)
. (3.12)
Considering asymptotic behaviour of possible solutions, there exist unique solutions Uhor and Uin f [10],
characterised by
Uhor ∼ e−iωr∗ as r∗→−∞ (3.13)
Uin f ∼ eiωr∗ as r∗→ ∞. (3.14)
We can also see that U¯hor and U¯in f are solutions to (3.11). Moreover Uin f and U¯in f are linearly independent,
so we can write Uhor in terms of Uin f and U¯in f :
T˜ω,l,mUhor = R˜ω,l,mUin f +U¯in f . (3.15)
Here R˜ and T˜ are what we refer to as the reflection and transmission coefficients, respectively.
Now we consider Reissner–Nordström spacetime again. For a Schwartz function ψ+(u), we can impose the
future radiation field ψ+(u)Yl,m(θ ,ϕ) on I +, and 0 onH +. Therefore we can consider only one spherical
harmonic, and our solution of the wave equation is of the form φ = Yl,m(θ ,ϕ)r ψ . We then rewrite the wave
equation in terms of ψ , and take a Fourier transform with respect to the timelike coordinate t, where ∂t is our
timelike Killing vector field. We obtain that φˆ(ω,r∗) obeys (3.11) for each fixed value of ω . By considering
r∗→−∞, we know that
ψ(r∗, t) =
1
2π
∫ ∞
ω=−∞
ψˆ(ω,r∗)eiωtdω ∼ 1
2π
∫ ∞
ω=−∞
ψˆH +(ω)eiω(t+r
∗)+ ψˆH −(ω)eiω(t−r
∗)dω as r∗→−∞.
(3.16)
Similarly we can consider r∗→ ∞:
ψ(r∗, t) =
1
2π
∫ ∞
ω=−∞
ψˆ(ω,r∗)eiωtdω ∼ 1
2π
∫ ∞
ω=−∞
ψˆI −(ω)eiω(t+r
∗)+ ψˆI +(ω)eiω(t−r
∗)dω as r∗→ ∞.
(3.17)
Using that ψH + = 0 and ψI + = ψ+(u), we obtain
ψˆ(r∗,ω) = ψˆ+(ω)T˜ω,l,mUhor(r∗) = ψˆ+(ω)
(
R˜ω,l,mUin f (r∗)+U¯in f (r∗)
)
. (3.18)
Therefore we can obtain our solution onH − and I −:
ψH −(v) =
1
2π
∫ ∞
ω=−∞
ψˆ+(ω)T˜ω,l,meiωvdω (3.19)
ψI −(u) =
1
2π
∫ ∞
ω=−∞
ψˆ+(ω)R˜ω,l,meiωudω. (3.20)
We will only be using two properties of R˜ and T˜ . Firstly, as a result of conservation of T energy, we have
|R˜ω,l,m|2+ |T˜ω,l,m|2 = 1. (3.21)
3.4 The Hawking Radiation Calculation 61
Secondly we will consider the decay of the reflection coefficient for large ω . Corollary A.0.1 states there
exists a constant C (independent of M,q, l,m,ω) such that
|R˜ω,l,m|2 ≤ C(l+1)
2
1+M2ω2
. (3.22)
The final Theorem we will be using is (part of) Proposition 7.4 in [3], which we will restate here:
Proposition 3.3.1. Let φ be a solution to (2.1) in an extremal Reissner–Nordström spacetime, with radiation
field ψ = rφ . Let M < r0 < 2M. Then there exists a constant, C =C(M,r0)> 0 such that:∫
Σu1∩{r≤r0}
(
1− r
M
)−2 |∂vψ|2 sinθdθdϕdv+∫
Σv=u1+r∗0∩{r
∗>0}
r2|∂uψ|2 sinθdθdϕdu (3.23)
≤C
∫
H −∩{u≥u1}
(
M2+(u−u1)2
) |∂uψ|2+ ∣∣∣ /˚∇ψ∣∣∣2 sinθdθdϕdu
+C
∫
I −∩{v≤u1}
(M2+(v−u1− r∗0)2)|∂vψ|2+
∣∣∣ /˚∇ψ∣∣∣2 sinθdθdϕdv.
Proof. To prove this, we have taken the rI + in the original statement of the theorem to be where r∗ = 0.
3.4 The Hawking Radiation Calculation
In this section we will be proving Theorem 4 from the overview, which is a combination of Theorem 3.4.1 and
Corollary 3.4.6.
Theorem 3.4.1 (Hawking Radiation). Let ψ+(u,θ ,ϕ) be a Schwartz function on the 3−cylinder. Let φ be the
solution of (2.1), as given by Theorem 2.4.1, such that
lim
v→∞r(u,v)φ(u,v,θ ,ϕ) = ψ+(u−u0,θ ,ϕ) (3.24)
lim
u→∞r(u,v)φ(u,v,θ ,ϕ) = 0 ∀v≥ vc, (3.25)
Define the function ψI −,u0 by
lim
u→−∞r(u,v)φ(u,v,θ ,ϕ) = ψI −,u0(v,θ ,ϕ). (3.26)
Then for |q|< 1, there exists a constant A(M,q,T ∗) such that∣∣∣∣∣
∫ ∞
σ=−∞
∫
S2
|σ ||ψˆI −,u0(σ ,θ ,ϕ)|2 sinθdσdθdϕ (3.27)
−
∫ ∞
σ=−∞
∫
S2
|σ |
(
coth
(π
κ
|σ |
)
|ψH −(σ ,θ ,ϕ)|2+ |ψRN(σ ,θ ,ϕ)|2
)
2sinθdσdθdϕ
∣∣∣∣∣
≤ A(e−κu1I.T.[ψ+]+ e2κu1I.E[ψ+,vc,u1,u0]) ,
for sufficiently large u0.
Here, ψH − and ψRN are the transmission and reflection of ψ+ in pure Reissner–Nordström spacetime, as
defined by Theorem 3.3.1, κ is the surface gravity of the Reissner–Nordström black hole. Finally, I.T. and I.E.
62 Hawking Radiation
are given by
I.E.[ψ+,vc,u1,u0] =
∫ u1
−∞
∫
S2
[
(M+u0− vc)(M+u0−u1)|∂uψH −|2+ |ψH −|2
]
sinθdθdϕdu
+(M+u0−u1)
∫ vc
−∞
∫
S2
[
(M2+(vc−u)2)|∂uψH −|2
]
sinθdθdϕdu (3.28)
+(M+u0−u1)
∫ vc
−∞
∫
S2
[
(M2+M(u0− vc)+(vc− v)2)|∂vψRN |2+ | /˚∇|2|ψRN |2
]
sinθdθdϕdv.
+
[∫ ∞
u=u0−u1
∫
S2
(M2+(u−u0+u1)2)|∂uψ+(u)|2 sinθdθdϕdu
]∗
I.T.[ψ+] =
∫ ∞
−∞
∫
S2
(M2+u2)(1+ | /˚∇|4)(|∂uψ+(u)|2+ |ψ+(u)|2)sinθdθdϕdu. (3.29)
Here, /˚∇ is the derivative on the unit sphere, and we write | /˚∇|4| f |2 to mean
∣∣∣∣ /˚∇2 f ∣∣∣∣2. Note that I.T.[ψ+] controls
similarly weighted norms of ψH − and ψRN , thanks to reflection and transmission coefficients being bounded
above by 1 (see Section 3.3). Finally, it should be noted that the final term in I.E. (marked by [ ]∗), is only
required in the extremal (|q|= 1) case.
In the case |q|= 1, there exists a constant A(M,q,T ∗) such that∣∣∣∣∫
I −
|σ ||ψˆI −,u0|2dσdθdϕ−
∫ ∞
−∞
|σ ||ψˆ+|2dσ
∣∣∣∣≤ A
(
I.T.[ψ+]
u3/21
+u5/20 I.E[ψ+,vc,u0−
√
Mu0,u0]
)
, (3.30)
for sufficiently large u0.
Furthermore, let us fix δ > 0. If we suppose that |q| < 1, and that ψ+ be such that all I.E.[ψ+,vc,(1−
δ )u0,u0] terms decay faster than e−3κ(1−δ )u0 . Then there exists a constant B(M,q,T ∗,ψ+,δ ) such that∣∣∣∣∫
I −
|σ ||ψˆ|2dσ −
∫ ∞
−∞
|σ |coth
(π
κ
|σ |
)
|ψˆH −|2dσ −
∫ ∞
−∞
|σ ||ψˆRN |2dσ
∣∣∣∣≤ Be−κ(1−δ )u0 , (3.31)
for sufficiently large u0.
3.4.1 The Set-up and Reduction to Fixed l
i+
i0
(t∗c , r+)
i−
H
+ ,φ
=
0 I
+
,rφ →
ψ
+Y
l,m
I
− ,r
φ →
ψ I
−Y l
,m
r
=
r b
,φ
=
0
Figure 3.3 The set-up for the
Hawking radiation calculation
We will prove Theorem 3.4.1 by first restricting to a fixed spherical harmonic,
as these are orthogonal. We further restrict ψ+(x) to be a smooth compactly
supported function in one variable. Let φ be the solution to (2.1), subject to
ψ = 0 on r = rb(t∗), with future radiation field Yl,m(θ ,ϕ)ψ+(u−u0), and φ = 0
onH +, as given by Theorem 2.4.1. Here Yl,m is spherical harmonic, see for
example [36]. Note that this result will then immediately generalise to Schwartz
functions by an easy density argument.
We will generally be considering ψ(u,v), given by
ψ(u,v)Yl,m(θ ,ϕ) := r(u,v)φ(u,v,θ ,ϕ) (3.32)
rather than φ itself. Note ψ(u,v) is independent of θ ,ϕ , as spherical symmetry
of our system implies that if we restrict scattering data in Theorems 3.3.1 and
2.4.1 to one spherical harmonic, the solution will also be restricted to that harmonic.
Re-writing the wave equation for fixed l,m in terms of ψ , we obtain:
4∂u∂vψ =− 1r2
(
1− 2M
r2
+
q2M2
r2
)(
l(l+1)+
2M
r
(
1− q
2M
r
))
ψ =:−4V (r)ψ. (3.33)
ψ(u,vb(u)) = 0, (3.34)
3.4 The Hawking Radiation Calculation 63
where vb is as given in (1.57).
3.4.2 Summary of the Proof
i+
i0
R2
i−
R1
R3
H
+ ,φ
=
0 I
+
,rφ →
ψ
+Y
l,m
I
− ,r
φ →
ψ I
−Y l
,mr =
r b
v=
vc
Figure 3.4 The regions we will
consider in the Hawking radi-
ation calculation
This proof will be broken up into 5 sections.
1. Firstly we will consider the evolution of the solution determined by
scattering data (0,ψ+Yl,m) onH ++I + through the region R1 := {v ∈
[vc,∞)}, where vc is the value of the v coordinate at (t∗c ,2M) given by
(1.53). Note that evolution in R1 is entirely within a region of Reissner–
Nordström spacetime, so is relatively easy to compute.
We will obtain that
ψˆ(σ ,v)≈ T˜σ ,l,mψˆ+(σ), (3.35)
where T˜ω,l,m is the transmission coefficient (section 3.3) from I + back
toH − in Reissner–Nordström for the spherical harmonic Yl,m (again see
[36]). Here, when we say "≈", we mean to leading order for large u0, and
the exact nature of these error terms will be covered in more detail in the
full statement of Corollary 3.4.2.
2. Secondly we will consider the reflection of the solution off the surface of
the dust cloud. This will occur in the region R2 := {v≤ vc,u≤ u1}∩{r≥
rb} for the same u0 in the definition of ψ+.
We will obtain that, for v≤ vc,
ψ(u1,v)≈ ψ(ub(v),vc), (3.36)
where here u = ub(v) is parametrising the surface r = rb(t∗) in terms of
the null coordinates. See Corollary 3.4.3 for a precise statement of this.
3. Thirdly we will consider the high frequency transmission of the solution from near the surface of the dust
cloud to I −. This will occur in a region we will call R3 := {v≤ vc,u≥ u1}.
In a very similar manner to (3.35), we will obtain
ψˆI −(σ)≈ Tσ ,l,mψˆ(u1,σ), (3.37)
where Tσ ,l,m is the transmission coefficient fromH + to I − (or equivalently fromH − to I +).
However, as ψ|Σu1∩{v≤vc} is supported in a small region, we will also obtain that
ψˆI −(σ)≈ Tσ ,l,mψˆ(u1,σ)≈ ψˆ(u1,σ). (3.38)
See Corollary 3.4.4 for the precise statement of this.
4. Before the final calculation, we will consider the integrated error terms, I.E., and how they behave,
depending on our future radiation field, ψ+. We will look to show that, provided φˆ+ is only supported on
ω ≥ 0,
I.E.[ψ+,vc,u1,u0]≤ An(u0−u1)−n, (3.39)
for all n, provided M ≤ u1 ≤ u0. Here An depends on n,M,q and ψ+ itself.
5. Finally, we will consider the actual calculation of I[ψ+, l,u0] on I −. We will use a conserved current to
show that ∫
σ∈R
σ |ψˆI −|2dσ =
∫
σ∈R
σ |ψˆ+|2dσ . (3.40)
64 Hawking Radiation
Using a useful Lemma by Bachelot (Lemma II.6 in [7], Lemma 3.4.2 here), we obtain for the sub-extremal
case ∫ ∞
−∞
|σ ||ψˆI −|2dσ ≈
∫ ∞
−∞
|σ ||R˜σ l,m|2|ψˆ+|2+ |T˜σ ,l,m|2|σ |coth
(π
κ
|σ |
)
|ψˆ+|2dσ . (3.41)
Here, κ is the surface gravity of the Reissner–Nordström black hole, as given in (1.63). We obtain the
equivalent result on the extremal case, using Lemma 3.4.3.
Thus combining (3.40) and (3.41), we will obtain the final result:
lim
u0→∞
I[ψ+, l,u0]≈
∫ ∞
−∞
|T˜σ ,l,m|2
2
(
coth
(πω
κ
)
−1
)
|σ ||ψˆ+|2dσ , (3.42)
subject to an extra condition on the support of ψˆ+.
We also obtain the extremal equivalent:
lim
u0→∞
I[ψ+, l,u0]≈
∫ ∞
−∞
|T˜σ ,l,m|2
2
(1−1) |σ ||ψˆ+|2dσ = 0. (3.43)
See Theorem 3.4.1 and Corollary 3.4.6 for the precise statement of this.
We will prove this result almost entirely in physical space rather than frequency space, which will hopefully
be a more transparent proof.
3.4.3 Evolution in Pure Reissner–Nordström
In this section we will be considering the following problem: In Reissner–Nordström spacetime, if we im-
pose radiation field ψ+(u)Yl,m(θ ,ϕ) on I + and that our solution vanishes on H +, what happens to the
solution on a surface of constant v as we let v →−∞? By transporting our solution along the Killing vec-
tor, T , this is equivalent to considering a solution with radiation field ψ+(u− u0)Yl,m(θ ,ϕ) on I + on a
surface of fixed v, and allowing u0 → ∞. We obtain the following result:
i+
i0B
i−
R1
H
+ ,φ
=
0
I
+
,rφ →
ψ
+Y
l,m
I
−H −
v=
vc ,rφ
=
ψY
l,m
Figure 3.5 The first region we
will consider in the Hawking
radiation calculation
Proposition 3.4.1 (Reissner-Nordström Transmission). Let ψ+ : R→ C be a
smooth, compactly supported function. Let ψ be the solution of (3.33), as given
by Theorem 3.3.1, with radiation field on I + equal to ψ+, and which vanishes
onH +. Let vc,u1 ∈ R. Define
ψˆH −(σ) := T˜σ ,l,mψˆ+(σ), (3.44)
Then there exists a constant A(M,q) such that∫ ∞
u=u1
|∂uψ(u,vc)−∂uψH −(u)|2du≤ AI.T.[ψ+](r(u1,vc)− r+)2. (3.45)
(1+ l)4 sup
v≤vc
(∫ ∞
u1
|ψ(u,v)|2du
)
≤ AI.T.[ψ+], (3.46)
provided r(u1,vc)≤ 3M.
Moreover, if |q|< 1, then we also have a constant B(M,q) such that∫ ∞
u=u1
|ψ(u,vc)−ψH −(u)|2du≤ AI.T.[ψ+](r(u1,vc)− r+)2+A|ψ(u1,vc)|2, (3.47)
again provided r(u1,vc)≤ 3M.
3.4 The Hawking Radiation Calculation 65
In the case |q|= 1, we have∫ ∞
u=u1
|ψ(u,vc)−ψH −(u)|2du≤ AI.T.[ψ+](u0−u1)2(r(u1,vc)− r+)2
+A(M2+(u0−u1)2)|ψ(u1,vc)−ψH −(u1)|2 (3.48)
+4I.E.[ψ+,vc,u1,u0].
Here I.T. and I.E. are as defined in Theorem 3.4.1.
Remark 3.4.1. We can also define the past radiation field in pure RN by ψRN , which is given by
ψˆRN := R˜σ ,l,mψˆ+, (3.49)
where R˜σ ,l,m are the reflection coefficients from I + to I − in Reissner–Nordström for the spherical harmonic
Yl,m (again see [36]).
Proof. We know from many previous works on Reissner-Nordström (see [35] for example) that
lim
v→−∞ψ(u,v) = ψH −(u), (3.50)
for ψH − as in the statement of the Proposition.
The proof of (3.45) is fairly straightforward:
∫ ∞
u=u1
|∂uψ(u,vc)−∂uψH −(u)|2du =
∫ ∞
u=u1
∣∣∣∣∫ vc−∞ ∂v∂uψdv
∣∣∣∣2 du = ∫ ∞u=u1
∣∣∣∣∫ vc−∞Vψdv
∣∣∣∣2 du
≤
(∫ vc
−∞
(∫ ∞
u1
V 2|ψ|2du
)1/2
dv
)2
(3.51)
≤ sup
v≤vc
(∫ ∞
u1
|ψ(u,v)|2du
)(∫ vc
−∞
V (u1,v)dv
)2
≤ A(l+1)4(r(u1,vc)− r+)2 sup
v≤vc
(∫ ∞
u1
|ψ(u,v)|2du
)
.
Here, we have used Minkowski’s integral inequality to reach the second line.
In the case of |q|< 1, we first show that to prove (3.47), it is sufficient to bound ∫ ∞u=u1(u−u1)2|∂uψ(u,vc)−
∂uψH −(u)|2du.
Let χ be a smooth cut off function such that
χ(x)

= 0 x≥ 1
∈ [0,1] x ∈ [0,1]
= 1 x≤ 0
. (3.52)
66 Hawking Radiation
Then for any function φ ,
∫ ∞
u=u1
|φ(u,v)|2du≤ 2
∫ ∞
u=u1
∣∣∣∣φ(u,v)−φ(u1,vc)χ(u−u1M
)∣∣∣∣2+ ∣∣∣∣φ(u1,vc)χ(u−u1M
)∣∣∣∣2 du (3.53)
≤ 8
∫ ∞
u=u1
(u−u1)2
∣∣∣∣∂uφ(u,v)−φ(u1,vc)M−1χ ′(u−u1M
)∣∣∣∣2 du+2M|φ(u1,vc)|2 ∫ ∞x=0 |χ(x)|2 dx
≤ 16
∫ ∞
u=u1
(u−u1)2 |∂uφ(u,v)|2+(u−u1)2
∣∣∣∣φ(u1,vc)M−1χ ′(u−u1M
)∣∣∣∣2 du
+2M|φ(u1,vc)|2
∫ ∞
x=0
|χ(x)|2 dx
≤ 16
∫ ∞
u=u1
(u−u1)2 |∂uφ(u,v)|2 du+2M|φ(u1,vc)|2
∫ ∞
x=0
|χ(x)|2+8x2 ∣∣χ ′(x)∣∣2 dx
≤ 16
∫ ∞
u=u1
(u−u1)2 |∂uφ(u,v)|2 du+A(M)|φ(u1,vc)|2,
Here, we have used Hardy’s inequality.
Now we prove (3.47), in a similar way to (3.45):
∫ ∞
u=u1
(u−u1)2|∂uψ(u,vc)−∂uψH −(u)|2du =
∫ ∞
u=u1
(u−u1)2
∣∣∣∣∫ vc−∞ ∂v∂uψdv
∣∣∣∣2 du = ∫ ∞u=u1(u−u1)2
∣∣∣∣∫ vc−∞Vψdv
∣∣∣∣2 du
≤
(∫ vc
−∞
(∫ ∞
u1
(u−u1)2V 2|ψ|2du
)1/2
dv
)2
(3.54)
≤ sup
v≤vc
(∫ ∞
u1
|ψ(u,v)|2du
)(∫ vc
−∞
sup
u≥u1
{(u−u1)V (u,v)}dv
)2
≤ B(l+1)4(r(u1,vc)− r+)2 sup
v≤vc
(∫ ∞
u1
|ψ(u,v)|2du
)
.
Here we have used that there exists a constant C(M,q) such that
C−1(l+1)2eκ(v−u) ≤V (u,v)≤C(l+1)2eκ(v−u), (3.55)
for r ≤ 3M.
In order to prove (3.46), we use similar logic to (3.53) to show∫ ∞
u=−∞
|φ(u,v)|2du≤ 16
∫ ∞
u=−∞
(u−u0)2 |∂uφ(u,v)|2 du+A(M)|φ(u0,vc)|2, (3.56)
And we then consider
φ =
ψ u≥ u1u0−u1
u0−u ψ(u1,vc) u < u1
, (3.57)
to obtain∫ ∞
u1
|ψ(u,v)|2du≤
∫ ∞
u=−∞
|φ(u,v)|2du≤ 16
∫ ∞
u=−∞
(u−u0)2 |∂uφ(u,v)|2 du+A(M)|φ(u0,vc)|2 (3.58)
≤ 16
∫ ∞
u=u1
(u−u0)2 |∂uψ(u,v)|2 du+A(M)|ψ(u0,vc)|2+(u0−u1)|ψ(u1,vc)|2.
3.4 The Hawking Radiation Calculation 67
In order to bound ψ(u0,vc), we look at
|ψ(u0,vc)|2 ≤
∣∣∣∣∫ ∞u0 ∂uψ(u,vc)du
∣∣∣∣2 (3.59)
≤
(∫ ∞
u0
1
M2+(u0−u)2 du
)(∫ ∞
u0
(
M2+(u0−u)2
) |∂uψ(u,vc)|2du)
≤ π
2M
∫ ∞
u0
(
M2+(u0−u)2
) |∂uψ(u,vc)|2du.
We then consider the conserved T -energy. In u,v coordinates, this is given by:
T-energy(φ ,Σv) =
∫ ∞
−∞
|∂uψ(u,v)|2+V (r)|ψ(u,v)|2du. (3.60)
We apply (3.33) to a weighted version of the T -energy in the region u≥ u0:∫ ∞
u0
(
M2+(u−u0)2
)(|∂uψ(u,v)|2+V |ψ(u,v)|2)du = ∫ ∞
u0
(
M2+(u−u0)2
) |∂uψ+|2du (3.61)
−
∫
u≥u0,v′≥v
2(u−u0)
(|∂uψ(u,v′)|2+V |ψ(u,v′)|2)dv′du
≤
∫ ∞
u0
(
M2+(u−u0)2
) |∂uψ+|2du.
We bound the u≤ u0 in a similar way:∫ u0
u1
(
M2+(u−u0)2
)(|∂uψ(u,v)|2+V |ψ(u,v)|2)du (3.62)
=
∫ u0
u1
(
M2+(u−u0)2
) |∂uψH −|2du
+
∫ vc
−∞
(
M2+(u0−u1)2
)(|∂uψ(u1,v)|2+V |ψ(u1,v)|2)dv
−
∫
u∈[u1,u0],v′≥v
2(u0−u)
(|∂uψ(u,v′)|2+V |ψ(u,v′)|2)dv′du
≤
∫ u0
−∞
(
M2+(u−u0)2
) |∂uψH −|2du+∫ vc−∞ (M2+(u0−u1)2) |∂vψRN |2.
For the extremal case, we simply use Poincaré’s inequality to bound
∫ 2u0−u1
u=u1
|ψ−ψH −|2du≤ A(u0−u1)2
∫ 2u0−u1
u=u1
|∂uψ−∂uψH −|2du+(M2+(u0−u1)2)|ψ(u1,vc)−ψH −(u1)|2
≤ AI.T.[ψ+](u0−u1)2(r(u1)− r+)2+(M2+(u0−u1)2)|ψ(u1,vc)−ψH −(u1)|2.
(3.63)
We are then left to bound∫ ∞
u=2u0−u1
|ψ−ψH −|2du =
∫ ∞
x=0
1
x2
|ψ−ψH −|2dx≤ 4
∫ ∞
x=0
|∂xψ−∂xψH −|2dx (3.64)
≤ 4
∫ ∞
u=2u0−u1
(u−2u0+u1)2|∂uψ−∂uψH −|2du.
This can then be bounded in exactly the same way as (3.61) to obtain∫ ∞
u=2u0−u1
(u−2u0+u1)2|∂uψ−∂uψH −|2du≤ 2
∫ ∞
u=2u0−u1
(u−2u0+u1)2
(|∂uψH −|2+ |∂uψH −|2)du
≤ 4
∫ ∞
u=2u0−u1
(u−2u0+u1)2|∂uψ+|2du, (3.65)
as required.
68 Hawking Radiation
We will also need to calculate the r-weighted energy of our solution on Σvc .
Proposition 3.4.2 (Reissner–Nordström Weighted Bounds). Let ψ+ : R→ C be a smooth, compactly supported
function. Let ψ be the solution of (3.33), as given by Theorem 3.3.1, with radiation field on I + equal to ψ+,
and which vanishes onH +. Let χ be a smooth function such that
χ(x)

= 1 x≥ 1
∈ [0,1] x ∈ [0,1]
= 0 x≤ 0
, (3.66)
Let r0 > r+ and vc be fixed. Define ψH − and ψRN as in Theorem 3.3.1.
Then there exists constants A(M,q,r0,χ) and B(M,q,r0,χ) such that
∫
Σvc
χ
(
r− r0
M
)
r2|∂uψ|2du≤ A
(∫ vc−2r∗0
u=−∞
(
1+(vc−2r∗0−u)2
) |∂uψH −|2du (3.67)
+
∫ vc
v=−∞
(
1+(vc− v)2
) |∂vψRN |2+ l(l+1)|ψRN |2dv),
for l ̸= 0, and
∫
Σvc
χ
(
r− r0
M
)
r3|∂uψ|2du≤ B
(∫ vc−2r∗0
u=−∞
(
1+(vc−2r∗0−u)3
) |∂uψH −|2du (3.68)
+
∫ vc
v=−∞
(
1+(vc− v)3
) |∂vψRN |3+2M|ψRN |2dv),
for l = 0. Here, r∗0 =
1
2(v−u) when r = r0.
Remark 3.4.2. In the extremal (q = 1) case, (3.67) follows easily from Proposition 3.3.1, but the proof we will
offer below will not distinguish between the extremal and sub-extremal cases.
Proof. We first write the conserved T−energy flux through in terms of ψ , though the surface
Σ¯vc,0 := (Σu=vc−2r∗0 ∩{r ≤ r0})∪ (Σvc ∩{r > r0}) (3.69)
An explicit calculation gives:
T − energy(φ , Σ¯vc,0) =
∫
Σu=vc−2r∗0∩{r≤r0}
2|∂vφ |2+ 1− 2Mr + q2M2r22r2 l(l+1)|φ |2
sinθdθdϕdv (3.70)
+
∫
Σvc∩{r>r0}
2|∂uφ |2+ 1− 2Mr + q2M2r22r2 l(l+1)|φ |2
sinθdθdϕdv
=
∫
Σu=vc−2r∗0∩{r≤r0}
2
[|∂vψ|2+V (r)|ψ|2]dv+∫
Σvc∩{r>r0}
2
[|∂uψ|2+V (r)ψ2]du
=
∫
Σ¯vc,0
2
[
|∂˜r∗ψ|2+V (r)|ψ2|
]
dr∗,
where here we define ∂˜r∗ to be the r∗ derivative along Σ¯v,0, i.e.
∂˜r∗ =
∂v r ≤ r0∂u r > r0 . (3.71)
We define ∂˜r :=
(
1− 2Mr + q
2M2
r2
)
∂˜r∗ .
3.4 The Hawking Radiation Calculation 69
We note here that even if l = 0, the T−energy bounds (χ ( r−r0M )ψ)2/r2 for any r0 > 2M:
∫
Σ¯v,0
χ
( r−r0
M
)2 |ψ|2
r2
dr∗ ≤
(
1− 2M
r0
+
q2M2
r20
)−1 ∫
Σ¯v,0
χ
( r−r0
M
)2 |ψ|2
r2
dr
≤ A(r0)
∫
Σ¯v,0
|∂˜r(χψ)|2dr = A(r0)
∫
Σ¯v,0
|χ∂˜rψ|2−χχ ′′ |ψ|
2
M2
dr (3.72)
≤ B(M,r0,χ)
∫
Σ¯v,0
|∂˜r∗ψ|2+V (r)|ψ|2dr∗.
Here we have used Hardy’s inequality.
We look at the integral of χ
( r−r0
M
)
rp(∂uψ)2 on Σvc:∫
Σvc
χ
(
r− r0
2M
)
rp|∂uψ|2du =
∫
v≤vc
[
1
2
(
1− 2M
r2
+
q2M2
r2
)(
prp−1χ+
rp
M
χ ′
)
|∂uψ|2+2χrpV (r)∂u(|ψ|2)
]
dvdu
=
∫
v≤vc
[(
1− 2M
r2
+
q2M2
r2
)(
1
2
(
prp−1χ+
rp
M
χ ′
)
|∂uψ|2+∂r(χrpV (r))|ψ|2
)]
dvdu
+
∫
I −∩{v≤vc}
2rp−2l(l+1)|ψ|2dv (3.73)
≤
∫
v≤vc
prp−1
2
χ|∂uψ|2dvdu+A
∫
v≤vc,r≥r0
|∂r∗ψ|2+V (r)|ψ|2dvdu
+
∫
I −∩{v≤vc}
2rp−2l(l+1)|ψ|2dv.
For p = 1, we obtain
∫
Σvc
χ
(
r− r0
M
)
r|∂uψ|2du≤ A
∫
v≤vc,r≥r0
|∂r∗ψ|2+V (r)|ψ|2dvdu. (3.74)
Then for p = 2, we obtain
∫
Σvc
χ
(
r− r0
M
)
r2|∂uψ|2du≤ A
∫ vc
v=−∞
(
A
∫
v′≤v,r≥r0
|∂r∗ψ|2+V (r)|ψ|2dv′du
)
dv (3.75)
+
∫
v≤vc,r≥r0
|∂r∗ψ|2+V (r)|ψ|2dvdu+
∫
I −∩{v≤vc}
2l(l+1)|ψ|2dv.
By T - energy conservation, we have that
∫ vc
v=−∞
(∫
v′≤vc,r≥r0
|∂r∗ψ|2+V (r)|ψ|2dv′du
)
dv =
∫ vc
v=−∞
(∫ v
v′=−∞
T − energy(φ , Σ¯v′,0)dv′
)
dv (3.76)
=
∫
H −,u≤vc−2r∗0
∫
u′≤u
∫
u′′≤u′
|∂uψ|2du′′du′du+
∫
I −,v≤vc
∫
v′≤v
∫
v′′≤v′
|∂vψ|2dv′′dv′dv
=
∫
H −,u≤vc−2r∗0
(vc−2r∗0−u)2|∂uψ|2du+
∫
I −,v≤vc
(vc− v)2|∂vψ|2dv,
where we have integrated by parts to obtain the last line. (Note ψ on I − or H − is Schwartz, so we have
arbitrarily large polynomial decay.)
Combining (3.75) and (3.76), we obtain:
∫
Σvc
χ
(
r− r0
M
)
r2|∂uψ|2du≤ A
(∫
H −,u≤vc−2r∗0
(
1+(vc−2r∗0−u)2
) |∂uψ|2du (3.77)
+
∫
I −,v≤vc
(
1+(vc− v)2
) |∂vψ|2+ l(l+1)|ψ|2dv).
70 Hawking Radiation
for l ̸= 0, and
∫
Σvc
χ
(
r− r0
M
)
r3|∂uψ|2du≤ A
(∫
H −,u≤vc−2r∗0
(
1+(vc−2r∗0−u)3
) |∂uψ|2du (3.78)
+
∫
I −,v≤vc
(
1+(vc− v)3
) |∂vψ|3+2M|ψ|2dv),
for l = 0, as required.
Corollary 3.4.1 (Pointwise Bounds). Let ψ+ : R→ C be a smooth, compactly supported function. Let ψ be the
solution of (3.33), as given by Theorem 3.3.1, with radiation field on I + equal to ψ+, and which vanishes on
H +. Let r0 > r+ and vc be fixed. Then there exists a constant A(M,q,r0) such that
|ψ(u1,vc)|2 ≤ AI.E[ψ+,vc,u1,u0], (3.79)
for any u1 > vc− r∗0. Here, r∗0 = 12(v−u) on r = r0. Here we define ψH − and ψRN as in Theorem 3.3.1.
Proof. This is a fairly straight forward consequence of Proposition 3.4.2.
|ψ(u1,vc)|2 ≤ 2
∣∣∣∣∫ vc−∞ ∂vψRN(v)dv
∣∣∣∣2+2 ∣∣∣∣∫ vc−r∗0−∞ ∂uψ(u,vc)du
∣∣∣∣2+2 ∣∣∣∣∫ u1vc−r∗0 ∂uψ(u,vc)du
∣∣∣∣2 (3.80)
≤ 2
(∫ vc
−∞
1
M2+(vc− v)2 dv
)(∫ vc
−∞
(
M2+(vc− v)2
) |∂vψRN(v)|2dv)
+2
(∫ vc−r∗0−M
−∞
r−2dv
)(∫ vc
−∞
r2|∂uψ(u,vc)|2dv
)
+2
(∫ u1
vc−r∗0−M
dv
)(∫ u1
vc−r∗0−M
|∂uψ(u,vc)|2dv
)
≤ A
(∫ vc−2r∗0
u=−∞
(
M2+(vc−2r∗0−u)2
) |∂uψH −|2du+∫ u1
u=−∞
(u1− vc+ r∗0)|∂uψH −|2du
+
∫ vc
v=−∞
(
M2+M(u1− vc+ r∗0)+(vc− v)2
) |∂vψRN |2+ l(l+1)|ψRN |2dv),
as required.
Proposition 3.4.3 (Extremal Weighted Energy Bounds). Let ψ+ : R→ C be a smooth, compactly supported
function. Let ψ be the solution of (3.33), as given by Theorem 3.3.1, on an Extremal Reisnner–Nordström
background, with radiation field on I + equal to ψ+, and which vanishes on H +. Let u1,vc be such that
r(u1,vc)< 32M. Then there exists a constant C =C(M)> 0 such that
∫
Σu1∩{v≤vc}
(
1−M
r
)−2
|∂vψ|2 sinθdθdϕdv≤C(u1− vc)2I.E.[ψ+,vc,u1,u0]. (3.81)
Proof. We will be considering the solution to the wave equation (2.1), φ˜ , with radiation field ψ˜ , given by
ψ˜|H − = ψH − (3.82)
ψ˜|I − =
χ
(v−vc
M
)
ψRN(vc) v > vc
ψRN v≤ vc
, (3.83)
where χ is as defined in Proposition 3.4.2.
By a standard domain of dependence argument, we can see that
∫
Σu1∩{v≤vc}
(
1−M
r
)−2
|∂vψ|2dv =
∫
Σu1∩{v≤vc}
(
1−M
r
)−2
|∂vψ˜|2dv. (3.84)
3.4 The Hawking Radiation Calculation 71
Here we will be making use of Proposition 3.3.1. By choosing r0 = 32M, we can then bound
∫
Σu1∩{v≤vc}
(
1−M
r
)−2
|∂vψ˜|2 sinθdθdϕdv≤C
∫
H −∩{u≥u1}
(
M2+(u−u1)2
) |∂uψ˜|2+ ∣∣∣ /˚∇ψ˜∣∣∣2 sinθdθdϕdu
+C
∫
I −∩{v≤u1}
(M2+(v−u1)2)|∂vψ˜|2+
∣∣∣ /˚∇ψ˜∣∣∣2 sinθdθdϕdv
≤C(u1− vc)2I.E.[ψ+,vc,u1,u0]. (3.85)
Corollary 3.4.2 (Hawking Radiation Error from Reissner–Nordström Transmission). Let f be a smooth
compactly, supported function with f (0) = 1. Let ψ+ : R→ C be a Schwartz function. Let ψ be the solution of
(3.33), as given by Theorem 3.3.1, with radiation field on I + equal to ψ+, and which vanishes onH +. Let
vc,u1 ∈ R, both tending to −∞, with vc ≤ u1. Let u0 be fixed. Define
ψ0(u) :=
ψ(u,vc)− f (u−u1)ψ(u1,vc) u≥ u10 u < u1 . (3.86)
In the extremal case, we also restrict ψˆ+ to be supported on positive frequencies, and u0−u1 ≤ u1− vc. Then
there exists a constant A(M,q, f )> 0 such that
∣∣∣∣∫σ∈R (κ+ |σ |)(|ψˆH −|2−|ψˆ0|2)dσ
∣∣∣∣≤

A
(
eκ(vc−u1)I.T.[ψ+]+ I.E.[ψ+,vc,u1,u0]1/2I.T.[ψ+]1/2
)
|q|< 1
A
((
ln
(
u1−vc
M
)
+u0−u1
)
I.T.[ψ+]
(u1−vc)2
+(u1− vc)1/2I.E.[ψ+,vc,u1,u0]1/2I.T.[ψ+]1/2
)
|q|= 1
,
(3.87)
where κ is given by (1.63). Here I.E.[ψ+,vc,u1,u0] are “Integrated Errors" due to the tail of ψ+, ψH − and
ψRN , and I.T.[ψ+,vc,u1,u0] are “Integrated Terms", both given in the statement of Theorem 3.4.1. where
ψRN ,ψH − are as defined in Proposition 3.4.1 and Remark 3.4.1.
Remark 3.4.3. We have chosen the above form of ψ0 to ensure that it is a weakly differentiable function. If ψ0
were less well behaved, then we do not know for certain that the integral in (3.87) would converge.
Remark 3.4.4. Wherever vc,u1,u0 occur in Corollary 3.4.2, they occur as a difference. Thus when we propagate
our solution along T , these differences remain the same.
Proof. We will consider |q|< 1 first:
∫
σ∈R
(κ+ |σ |) ∣∣|ψˆH −|2−|ψˆ0|2∣∣dσ ≤ A(∫
σ∈R
|ψˆH −− ψˆ0|2 dσ
)1/2(∫
σ∈R
(
κ2+σ2
)(|ψˆH −|2+ |ψˆ0|2)dσ)1/2
≤ A
(∫ ∞
u1
|ψH −−ψ(u,vc)|2 du+
∫ u1
−∞
|ψH −|2du+ |ψ(u1,vc)|2
)1/2
(∫ ∞
u1
κ2
(|ψH −|2+ |ψ0|2)+ |∂uψH −|2+ |∂uψ0|2du+∫ u1−∞κ2|ψH −|2+ |∂uψH −|2du
)1/2
≤ A(I.T.[ψ+](r(u1,vc)− r+)2+ I.E.[ψ+,u0])1/2 (3.88)(∫ ∞
u1
κ2|ψ+|2+ |∂uψ+|2du+
∫ u1
−∞
κ2|ψH −|2+ |∂uψH −|2du+ |ψ(u1,vc)|2
)1/2
≤ A((r(u1,vc)− r+)2I.T.[ψ+]+ I.E.[ψ+,vc,u1,u0])1/2 (I.T.[ψ+]))1/2 ,
as required. We have used Proposition 3.4.1 to bound
∫ ∞
u1 |ψH −−ψ0|
2 du, and Corollary 3.4.1 to bound
|ψ1(u1,v)|2.
72 Hawking Radiation
For the |q|= 1 case, we have κ = 0. We then proceed slightly differently to obtain our result, by first noting:
− iσ∂vψˆ = V̂ψ. (3.89)
Here, ψˆ is the Fourier transform of ψ with respect to u. While this transform may not exist in an L2 sense, as
Vψ is an L2 function on Σu, this implies that ∂vψˆ exists in a distributional sense.
We will write
ψ0(u,v) :=
ψ(u,v)− f (u−u1)ψ(u1,v) u≥ u10 u < u1 . (3.90)
Substituting (3.90) into (3.89), we obtain
− iσ∂̂vψ0 =−V̂ψ0− f̂ ′∂vψ−ψ(u1,v)V̂ f Iu≥u1, (3.91)
where
Iu≥u1 =
1 u≥ u10 u≤ u1 . (3.92)
We therefore obtain:∣∣∣∣∫σ∈R |σ |(|ψˆH −|2−|ψˆ0|2)dσ
∣∣∣∣≤ 2 ∣∣∣∣∫σ∈R σ|σ |
∫ vc
v=−∞
R
(
i ¯ˆ0ψ
(
V̂ψ0+ f̂ ′∂vψ+ψ(u1,v)V̂ f Iu≥u1
))
dvdσ
∣∣∣∣
+
∣∣∣∣∫σ∈R |σ |(|ψˆH −|2−|ψˆ0(u,−∞)|2)dσ
∣∣∣∣
≤ 2
∣∣∣∣∫σ∈R σ|σ |
∫ vc
v=−∞
R
(
i ¯ˆψ0V̂ψ0
)
dvdσ
∣∣∣∣ (3.93)
+2
(∫
u
∫
v≤vc
(
1−M
r
)2
|ψ0|2dudv
)1/2(∫
u
∫
v≤vc
f ′2
|∂vψ|2(
1− Mr
)2 dudv
)1/2
+2 sup
v≤vc
|ψ(u1,v)|
(∫
u
∫
v≤vc
(
1−M
r
)2
|ψ0|2dudv
)1/2
(∫
u≥u1
∫
v≤vc
V 2(
1− Mr
)2 f 2dudv
)1/2
+AI.E.[ψ+,vc,u1,u0].
We note that ∫
u
∫
v≤vc
(
1−M
r
)2
|ψ0|2dudv≤ A(r(u1,vc)−M)
(l+1)4
I.T.[ψ+], (3.94)
using Proposition 3.4.1, and given f is a compactly supported function, we can bound
∫
u≥u1
∫
v≤vc
V 2(
1− Mr
)2 f 2dudv≤ A(l+1)4(r(u1,vc)−M). (3.95)
We can also bound∫
u
∫
v≤vc
f ′2
|∂vψ|2(
1− Mr
)2 dudv≤ A sup
f ′ ̸=0
∫
v≤vc
|∂vψ|2(
1− Mr
)2 dv≤C(u1− vc)2I.E.[ψ+,vc,u1,u0], (3.96)
using Proposition 3.4.3.
Thus we have∣∣∣∣∫σ∈R |σ |(|ψˆH −|2−|ψˆ0|2)dσ
∣∣∣∣≤ 2 ∣∣∣∣∫σ∈R σ|σ |
∫ vc
v=−∞
R
(
i ¯ˆψ0V̂ψ0
)
dvdσ
∣∣∣∣ (3.97)
+((u1− vc)I.T.[ψ+]I.E.[ψ+,vc,u1,u0])1/2 .
3.4 The Hawking Radiation Calculation 73
Given that we know in some sense that φ → φH − , and V ∼ l(l+1)r∗2 as v → −∞, we will replace φ0 =
φ0H −+δφ(u,v) and V =
l(l+1)
r∗2 +δV .
Then we obtain∣∣∣∣∫σ∈R σ|σ |
∫ vc
v=−∞
R
(
i ¯ˆψ0V̂ψ0
)
dvdσ
∣∣∣∣≤ l(l+1)
∣∣∣∣∣
∫
σ∈R
σ
|σ |
∫ vc
v=−∞
R
(
i ¯ˆψ0H −
̂( ψ0H −
(u− v)2
))
dvdσ
∣∣∣∣∣ (3.98)
+
∫ vc
v=−∞
|δV (u1,v)|∥ψ0∥2Σvdv
+2
∫ vc
v=−∞
V (u1,v)∥ψ0∥Σv∥δψ0∥Σvdv
≤ l(l+1)
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i ¯ˆψ0H −
̂(ψ0H −
u− vc
))
dσ
∣∣∣∣∣
+AI.T.[ψ+]
(
(r(u1,vc)−M)2 ln
(
r(u1,vc)
M
−1
)
+
u0−u1
u1− vc (r(u1,vc)−M)
)
+
M+u0−u1
u1− vc I.T.[ψ+]
1/2I.E.[ψ+,vc,u1,u0]1/2.
We have bounded ∥δψ0∥Σv using Proposition 3.4.1, and |δV (u1,v)≤ A(l+1)2
(
1− Mr
)3 ln( rM −1), by an
explicit calculation. Also, as ψ0H − is compactly supported, we can bring the integral over v inside the Fourier
transform. Denoting δψH − = ψH −−ψ0H − , we then obtain
74 Hawking Radiation
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i ¯ˆψ0H −
̂(ψ0H −
u− vc
))
dσ
∣∣∣∣∣≤
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i ¯ˆψH −
̂(ψH −
u− vc
))
dσ
∣∣∣∣∣
+
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i ¯ˆδψH −
̂(ψH −−ψH −(vc)
u− vc
))
dσ
∣∣∣∣∣
+
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i ¯δψˆH −
̂(ψH −(vc)
u− vc
))
dσ
∣∣∣∣∣
+
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i
( ¯ˆψH −+ ¯δψˆH −) ̂(δψH −−δψH −(vc)u− vc
))
dσ
∣∣∣∣∣
+
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i
( ¯ˆψH −+ ¯δψˆH −) ̂(δψH −(vc)u− vc
))
dσ
∣∣∣∣∣
≤
∣∣∣∣∣
∫
σ∈R
σ
|σ |R
(
i ¯ˆψH −
̂(ψH −
u− vc
))
dσ
∣∣∣∣∣+∥δψH −∥L2
∥∥∥∥ψH −−ψH −(vc)u− vc
∥∥∥∥
L2
+
√
π
2
∣∣∣∣∫σ∈RR
( ̂¯δψH −(−σ)ψH −(vc)e−ivcσ)dσ ∣∣∣∣ (3.99)
+∥ψ¯H −+ ¯δψH −∥L2
∥∥∥∥δψH −−δψH −(vc)u− vc
∥∥∥∥
L2
+
√
π
2
∣∣∣∣∫σ∈RR(( ¯ˆψH −+ ¯δψˆH −)(−σ)δψH −(vc)e−ivcσ)dσ
∣∣∣∣
≤
√
π
2
∣∣∣∣∫σ∈R
∫
σ ′∈R
σ
|σ |
σ ′
|σ ′|R
(
¯ˆψH −(σ)ψˆH −(σ −σ ′)e−ivcσ
′)
dσ ′dσ
∣∣∣∣
+2∥δψH −∥L2∥∂uψH −∥L2
+ | ¯δψH −(vc)ψH −(vc)|+2∥ψ¯H −+ ¯δψH −∥L2∥∂uδψH −∥L2
+ |(ψ¯H −(vc)+ ¯δψH −(vc))δψH −(vc)|
≤
√
π
2
∣∣∣∣∫σ∈R
∫
σ ′∈R
σ
|σ |
σ ′
|σ ′|R
(
ˆ¯ψH −(−σ)ψˆH −(σ −σ ′)e−ivcσ
′)
dσ ′dσ
∣∣∣∣
+
AI.T.[ψ+]1/2I.E.[ψ+,vc,u1,u0]1/2
(l+1)2
.
Here we have used that the Fourier transform of u−1 is
√
π/2 σ/|σ |, and we have used Hardy’s Inequality
to bound ∥∥∥∥ f (u)− f (vc)u− vc
∥∥∥∥
L2
≤ 2∥∂u f∥L2. (3.100)
Thus we have∣∣∣∣∣
∫
σ∈R
|σ |(|ψˆH −|2−|ψˆ0|2)dσ
∣∣∣∣∣
≤
√
π
2
∣∣∣∣∫σ∈R
∫
σ ′∈R
σ
|σ |
σ ′
|σ ′|R
(
ˆ¯ψH −(−σ)ψˆH −(σ −σ ′)e−ivcσ
′)
dσ ′dσ
∣∣∣∣ (3.101)
+A
(
ln
(
u1− vc
M
)
+u0−u1
)
I.T.[ψ+]
(u1− vc)2
+((u1− vc)I.T.[ψ+]I.E.[ψ+,vc,u1,u0])1/2 .
3.4 The Hawking Radiation Calculation 75
If ψˆ is only supported on positive frequencies, then we can simplify the following√
π
2
∣∣∣∣∣
∫
σ∈R
∫
σ ′∈R
σ
|σ |
σ ′
|σ ′|R
(
ˆ¯ψH −(−σ)ψˆH −(σ −σ ′)e−ivcσ
′)
dσ ′dσ
∣∣∣∣∣
=
√
π
2
∣∣∣∣∫σ∈R
∫
σ ′∈R
σ ′
|σ ′|R
(
ˆ¯ψH −(−σ)ψˆH −(σ −σ ′)e−ivcσ
′)
dσ ′dσ
∣∣∣∣
=
√
π
2
∣∣∣∣∫σ ′∈R σ ′|σ ′|R
(
̂|ψH −|2(−σ ′)e−ivcσ
′)
dσ ′
∣∣∣∣ (3.102)
=
√
π
2
∣∣∣∣∫σ ′∈RR
(
̂|ψH −|2(σ ′)eivcσ
′)
dσ ′
∣∣∣∣= |ψH −(vc)|2
≤ I.E.[ψ+,vc,u1,u0].
3.4.4 The Reflection
In this section we will consider evolving our solution in the small compact (in r, t∗ coordinates) region, given by
R2 := {v≤ vc,u≤ u1}∩{r ≥ rb}.
We will consider the surface r = rb(t∗) to be instead parametrised by v = vb(u), or equivalently by u =
ub(v) = v−1b (v), as in (1.57).
Proposition 3.4.4 (Reflection Energy Bounds). Let ψ be a smooth solution to (3.33) subject to (3.34). Define
the function ψ0 by
ψ0 := ψ(u,vc)−ψ(ub(v),vc). (3.103)
Then there exists a constant A(M,q,T ∗) such that
∫ vc
vb(u1)
∣∣∣∣dubdv
∣∣∣∣−1 |∂vψ(u1,v)−∂vψ0(u1,v)|2dv≤
Ae
−3κu1I.T.[ψ+] |q|< 1
Au20I.T.[ψ+]
u81
|q|= 1
, (3.104)
for any sufficiently large u1.
Furthermore, there exists a constant B(M,q,T ∗) such that
∫ vc
vb(u1)
∣∣∣∣dubdv
∣∣∣∣ |ψ(u1,v)−ψ0(u1,v)|2dv≤
Bu1e
−3κu1I.T.[ψ+] |q|< 1
Bu20I.T.[ψ+]
u61
|q|= 1 . (3.105)
Finally, there exists a constant C(M,q,T ∗) such that
∫ vc
vb(u1)
|ψ(u1,v)|2dv≤
C
I.T.[ψ+]
(l+1)4 e
−κu1 |q|< 1
C I.T.[ψ+]u
2
0
(l+1)4u41
|q|= 1.
(3.106)
If ψ+ decays quickly enough as u→ ∞, then∫ vc
vb(u1)
|ψ(u1,v)|2dv≤ eκ(u0−3u1)
∫ ∞
u=−∞
eκu|∂uψ+|2du. (3.107)
Remark 3.4.5. Note the form of ψ0 here is the solution to the equation
∂u∂vψ0 = 0, (3.108)
with initial conditions
ψ0(u,vc) = ψ(u,vc). (3.109)
76 Hawking Radiation
Therefore, ψ0 is reflected as if it were in Minkowski spacetime. Thus, Proposition 3.4.4 gives a bound on how
much the solution differs from a reflection in 1+1 dimensional Minkowski.
This solution ψ0 takes the form:
ψ0(u,v) = ψ(u,vc)−ψ(ub(v),vc). (3.110)
Proof. We begin by considering how the derivatives of ψ and ψ0 vary on the surface of the dust cloud, and
applying (3.33):
∫
S[u1,∞)
(u−u1)p|∂uψ−∂uψ0|2du≤
∫
S[u1,∞)
(u−u1)p
∣∣∣∣∫ vcvb(u1) ∂u∂vψdv
∣∣∣∣2 du = ∫S[u1,∞)(u−u1)p
∣∣∣∣∫ vcvb(u1)Vψdv
∣∣∣∣2 du
≤
∫
S[u1,∞)
(u−u1)p∥1∥2L2(Σu) ∥Vψ∥
2
L2(Σu) du (3.111)
≤
∫ ∞
u=u1
∫ vc
v=vb(u)
(u−u1)p(v− vb(u))V 2|ψ|2dvdu.
We then proceed to do the same to compare the derivatives of ψ on the surface of the dust cloud to the
derivatives of ψ on Σu1:∫
Σu1
|u′b|−1(ub(v)−u1)p|∂uψ(u1,v)−∂uψ(ub(v),v)|2du
≤
∫
Σu1
|u′b|−1(ub(v)−u1)p
∣∣∣∣∫ vcvb(u1) ∂u∂vψdv
∣∣∣∣2 dv
≤
∫
Σu1
|u′b|−1(ub(v)−u1)p
∣∣∣∣∫ vcvb(u1)Vψdv
∣∣∣∣2 dv (3.112)
≤
∫
Σu1
|u′b|−1(ub(v)−u1)p∥
√
V∥2L2(Σv)
∥∥∥√Vψ∥∥∥2
L2(Σv)
dv
≤ A
∫ ∞
u=u1
∫ vc
v=vb(u)
|u′b|−1(ub(v)−u1)p
(∫ ∞
u1
V (u′,vc)du′
)
V |ψ|2dvdu.
Combining equations (3.111) and (3.112), we obtain∫
Σu
|u′b|−1(ub(v)−u1)p|∂vψ−∂vψ0|2dv (3.113)
≤ 2
∫ ∞
u=u1
∫ vc
v=vb(u)
(
(v− vb(u))(u−u1)pV + |u′b|−1(ub(v)−u1)p
(∫ ∞
u1
V (u′,vc)du′
))
V |ψ|2dvdu
≤
A(l+1)
2 ∫ ∞
u=u1
∫ vc
v=vb(u)
(
ub(v)p−2+ub(v)p−1e−κu1
)
V |ψ|2dvdu |q|< 1
A(l+1)2
∫ ∞
u=u1
∫ vc
v=vb(u)
(
(v− vb(u))up−2+ ub(v)
p−2
u1−vc
)
V |ψ|2dvdu |q|= 1
,
where we have used the behaviour of ub and vb for late times to obtain the final line.
We first consider the extremal case, and the non-extremal case for sufficiently good decay. We will be using
energy boundedness results from [1] (Theorem 1).
This tells us that the non-degenerate energy of φ on Σvc bounds the non-degenerate energy of φ on Σu,
i.e. there exists a constant A(M,q,T ∗) such that for all u′ > u1∫ ∞
u1
|∂uφ(u,vc)|2
1− 2Mr + q
2M2
r2
+l(l+1)
(
1− 2M
r
+
q2M2
r2
)
|φ(u,vc)|2du (3.114)
≥ A
∫ vc
vb(u′)
|∂vφ(u′,v)|2
1− 2Mr + q
2M2
r2
+ l(l+1)
(
1− 2M
r
+
q2M2
r2
)
|φ(u′,v)|2dv
≥ A(
1− 2Mr(u′,vc) +
q2M2
r(u′,vc)2
)
(vc− vb(u′))2
∫ vc
vb(u′)
|φ(u′,v)|2dv.
3.4 The Hawking Radiation Calculation 77
Here we have used that φ vanishes on the surface of the dust cloud in order to apply Poincaré’s inequality. In
the extremal case, we can bound this non-degenerate energy as follows:
∫ ∞
u1
|∂uφ(u,vc)|2
1− 2Mr + q
2M2
r2
+l(l+1)
(
1− 2M
r
+
q2M2
r2
)
|φ(u,vc)|2du (3.115)
≤ A
∫ ∞
u1
(M2+u2)|∂uφ(u,vc)|2+ l(l+1)
(
1− 2M
r
+
q2M2
r2
)
|φ(u,vc)|2du
≤ A
∫ ∞
u1
(M2+(u−u0)2+u20)|∂uφ(u,vc)|2+ l(l+1)
(
1− 2M
r
+
q2M2
r2
)
|φ(u,vc)|2du.
We can use (3.61) and (3.62) to bound this by I.T.[ψ+] plus u20 times T energy. Cambining (3.114) and (3.115),
we obtain (3.106) for the extremal case. We then proceed by combining this with (3.113):
A(l+1)2
∫ ∞
u=u1
∫ vc
v=vb(u)
(
(v− vb(u))up−2+ ub(v)
p−2
u1− vc
)
V |ψ|2dvdu (3.116)
≤ A(l+1)2 sup
u≥u1,v<vc
((
(v− vb(u))up−2+ ub(v)
p−2
u1− vc
)
V
)∫ ∞
u1
∫ vc
vb(u)
|φ(u,v)|2
≤ A(l+1)−2 sup
u≥u1,v<vc
((
(v− vb(u))up−2+ ub(v)
p−2
u1− vc
)
V
)
∫ ∞
u1
((
1− 2M
r(u′,vc)
+
q2M2
r(u′,vc)2
)
(vc− vb(u′))2
)
u20I.T.[ψ+]du
≤ A(l+1)
−2u20I.T.[ψ+]
u31
sup
u≥u1,v<vc
((
(v− vb(u))up−2+ ub(v)
p−2
u1− vc
)
V
)
.
The two cases we will be considering are the cases p = 0 and p = 2. These give the following results:
∫
Σu
|u′b|−1|∂vψ−∂vψ0|2dv≤
Au20I.T.[ψ+]
u81
(3.117)
∫
Σu
|u′b|−1(ub(v)−u1)2|∂vψ−∂vψ0|2dv≤
Au20I.T.[ψ+]
u61
. (3.118)
In the less well behaved sub-extremal case, we instead consider energy currents in order to bound∫ ∞
u=u1
∫ vc
v=vb(u)
V |ψ|2dvdu≤ A
∫ ∞
u=u1
∫ vc
v=vb(u)
V |φ |2dvdu. (3.119)
We apply divergence theorem to the following vector field
J :=M∂u+α(v−vc)∂v−
(
1− 2M
r
+
q2M2
r2
)
(M−α(v− vc)) φ∇φ2r +φ
2∇
(
M−α(v− vc)
4r
(
1− 2M
r
+
q2M2
r2
))
,
(3.120)
in the region u ≥ u1,v ∈ [vb(u),vc]. Here, α = α(M,q,T ∗) > 0 is chosen such thatα−1 ≥ 8(1− q2)−1 and
α−1 ≥ (v− vc)u′b(v) for v− vc sufficiently small.
78 Hawking Radiation
A simple calculation reveals the following results:
∇.J =
(
α
2r2
+
M−α(vc− v)
2r3
(
M
r
(
1− q
2M
r
)
−
(
1− 2M
r
+
q2M2
r2
)))
l(l+1)|φ |2 (3.121)
+
(
M4(1−q2)
r6
(
2−Mq
2
r
)
−
(
q2M4(8−q2)
r6
+
q2M5(4−3q2)
r7
)
α
)
|φ |2
+
(
O
(
1− 2M
r
+
q2M2
r2
)
+O(vc− v)
)
|φ |2 ≥ 0
−du(J) = 2α(vc− v)|∂vφ |
2
1− 2Mr + q
2M2
r2
+
Ml(l+1)|φ |2
2r2
−∂v
(
(M−α(vc− v))|φ |2
2r
)
+
(
M2
2r3
(
1− q
2M
r
)
+
α
2r
+
αM2
2r3
(
4−q2− q
2M
r
))
|φ |2
+
(
O
(
1− 2M
r
+
q2M2
r2
)
+O(vc− v)
)
|φ |2
−dv(J)|v=vc =
2M|∂uφ |2
1− 2Mr + q
2M2
r2
−Mφ∂uφ
r
+
M
4r2
(
2M
r
(
1− q
2M
r
)
+
(
1− 2M
r
+
q2M2
r2
))
|φ |2 (3.122)
≤ 4M|∂uφ |
2
1− 2Mr + q
2M2
r2
(du−u′bdv)(J)|S =
2u′b
(
1−α(v− vc)u′b
) |∂uφ |2
1− 2Mr + q
2M2
r2
≥ 0, (3.123)
for vc− v sufficiently small. Here ‘sufficiently small’ only depends on m,q,T ∗.
We then apply divergence theorem:∫
∇.J+
∫
Σu0∩{v≤vc}
(−du(J))+
∫
((du− v′bdv)(J)) =
∫
Σvc
(−dv(J)), (3.124)
to obtain ∫
Σu0∩{v≤vc}
(vc− v)|∂vφ |2dv≤ 2Mα
∫
Σvc
|∂uφ |2du. (3.125)
An application of Hardy’s inequality to the function f
(vc−v
M
)
φ , for f a smooth function which vanishes at 0
yields
∫ vc
vb
f 2|φ |2
(vc− v)2 dv≤ 4
∫ vc
vb
f 2|∂vφ |2− f f
′
M
∂v
(|φ |2)+ f ′2
M2
|φ |2dv (3.126)
≤ 4
∫ vc
vb
f 2|∂vφ |2− f f
′′
M2
|φ |2dv. (3.127)
Choosing f (x) = x
√− log(x) gives
∫ vc
vb
− log
(
vc− v
M
)
|φ |2dv≤ 4
∫ vc
vb
− log
(
vc− v
M
)
(vc− v)2|∂vφ |2+
(
1
2
+
1
4(− log(vc−vM ))
)
|φ |2
M2
dv.
(3.128)
Provided vc− v≤ Me , then this can be rearranged for∫ vc
vb
− log
(
vc− v
M
)
|φ |2dv≤ 16
∫ vc
vb
− log
(
vc− v
M
)
(vc− v)2|∂vφ |2, (3.129)
3.4 The Hawking Radiation Calculation 79
and as we know the form u′b takes for v close to vb, we have that∫
Σu∩{v≤vc}
u′b(v)|φ |2dv≤ A
∫
Σu∩{v≤vc}
− log
(
vc− v
M
)
(vc− v)2|∂vφ |2 (3.130)
≤−A log
(
vc− vb(u1)
M
)
(vc− vb(u1))
∫
Σvc
|∂uφ |2du.
This can be used to immediately obtain (3.106), and we can also apply it to (3.113) to obtain∫
Σu
|u′b|−1(ub(v)−u1)p|∂vψ−∂vψ0|2dv
≤ A(l+1)2e−κu1
∣∣∣∣log(vc− vb(u)M
)∣∣∣∣p−1 (vc− vb(u1))∫Σvc |∂uφ |2du
∫ ∞
u1
V (u,vc)du
≤
Ae−3κu1I.T.[ψ+] p = 0Au1e−3κu1I.T.[ψ+] p = 2. . (3.131)
Finally, we can use Hardy’s inequality on the function ψ(u1,vb(u))−ψ0(u1,vb(u)), as this vanises on
u = u1, to get∫ vc
vb(u1)
u′b|ψ(u1,v)−ψ0(u1,v)|2dv =
∫ ∞
u1
|ψ(u1,vb(u))−ψ0(u1,vb(u))|2du (3.132)
≤ 4
∫ ∞
u1
(u−u1)2|∂uψ(u1,vb(u))−∂vψ0(u1,vb(u))|2du
= 4
∫ vc
vb(u1)
(u′b(v))
−1(ub(u)−u1)2|∂vψ(u1,v)−∂vψ0(u1,v)|2du.
The proof of (3.107) requires using a weighted T−energy estimate to obtain∫
u=−∞
eκ(u−u0)|∂uψ+(u−u0)|2du≥
∫
u=−∞
eκ(u−u0)|∂uψ(u,vc)|2du (3.133)
≥ ae−κu0
∫ ∞
u=u1
|∂uψ(u,vc)|2
1− 2Mr + q
2M2
r2
du
≥ ae−κu0
∫ vc
v=vb(u1)
|∂vψ(u1,v)|2
1− 2Mr + q
2M2
r2
dv
≥ ae−κ(u0−u1)|∂vψ(u1,v)|2dv≥ ae−κ(u0−3u1)|ψ(u1,v)|2dv.
As required. Here we have again used Theorem 1 from [1], followed by Poincaré’s inequality.
Note we still have other error occurring across the rest of Σvc:
∫ u1
u=−∞
|∂uψ(u,vc)−∂uψH −(u)|2du≤
Aeκu1I.E.[ψ+,vc,u1,u0] |q|< 1AI.E.[ψ+,vc,u1,u0] |q|= 1 . (3.134)
Corollary 3.4.3 (Hawking Radiation Error from the Reflection). Let ψ+ : R→ C be a smooth, compactly
supported function. Let ψ be the solution of (3.33), as given by Theorem 2.4.1, with radiation field on I +
equal to ψ+, and which vanishes onH +. Let f be a smooth compactly supported function such that f (0) = 1,
and define
ψ0(u) :=
ψ(u,vc)− f (u−u1)ψ(u1,vc) u≥ u10 u < u1 . (3.135)
ψ1(v) :=
ψ(u1,v)− (1− f (ub(v)−u1))ψ(u1,vc) v ∈ [vb(u1),vc]0 v /∈ [vb(u1),vc] . (3.136)
80 Hawking Radiation
Then there exists a constant A(M,q,T ∗) such that
∫
σ∈R
(κ+ |σ |) ∣∣|ψˆ0|2−|ψ̂1 ◦ vb|2∣∣dσ ≤
A
√
u1e−
3κ
2 u1I.T.[ψ+] |q|< 1
A
u41
I.T.[ψ+] |q|= 1
, (3.137)
as u0,u1 →∞ with u1 < u0. Here κ is the surface gravity, as in (1.63) and I.T.[ψ+] is as defined in the statement
of Theorem 3.4.1
Proof. This proof is similar to that of Corollary 3.4.2. We consider |q|< 1 first
∫
σ∈R
(κ+ |σ |) ∣∣|ψˆ0|2−|ψ̂1 ◦ vb|2∣∣dσ ≤ A(∫
σ∈R
∣∣ψˆ0− ψ̂1 ◦ vb∣∣2 dσ)1/2(∫
σ∈R
(
κ2+σ2
)(|ψ̂1 ◦ vb|2+ |ψˆ0|2)dσ)1/2
≤ A
(∫ ∞
u1
|ψ0−ψ1 ◦ vb|2 du
)1/2
(3.138)(∫ ∞
u1
|ψ1 ◦ vb|2+ |ψ0|2+ |∂u(ψ1 ◦ vb)|2+ |∂uψ0|2du
)1/2
≤ A
(∫ ∞
u1
|ψ(u1,vc)−ψ(u,vc)−ψ(u1,vb(u))|2 du
)1/2
(∫ ∞
u1
|ψ0−ψ1 ◦ vb|2+ |ψ0|2+ |∂u(ψ0−ψ1 ◦ vb)|2+ |∂uψ0|2du
)1/2
≤ A(I.T.[ψ+]u1e−3κu1)1/2 (I.T.[ψ+])1/2
≤ A√u1e− 3κ2 u1I.T.[ψ+].
As required. Here, we have used Proposition 3.4.4 to reach the penultimate line.
We next consider |q|= 1, where κ = 0
∫
σ∈R
|σ | ∣∣|ψˆ0|2−|ψ̂1 ◦ vb|2∣∣dσ ≤ A(∫
σ∈R
σ2
∣∣ψˆ0− ψ̂1 ◦ vb∣∣2 dσ)1/2(∫
σ∈R
(|ψ̂1 ◦ vb|2+ |ψˆ0|2)dσ)1/2
≤ A
(∫ ∞
u1
|∂uψ0−∂u(ψ1 ◦ vb)|2 du
)1/2(∫ ∞
u1
|ψ1 ◦ vb|2+ |ψ0|2du
)1/2
(3.139)
≤ A
(
I.T.[ψ+]
u81
)1/2
(I.T.[ψ+])1/2
≤ A
u41
I.T.[ψ+].
3.4.5 High Frequency Transmission
We now consider how our solution on Σu1 is transmitted to I −. We first look to bound the energy through
the surface Σvb(u1), as all other energy is transmitted to I
−. However, the map taking solutions on space-like
surfaces back to their past radiation fields is bounded with respect to the non-degenerate energy (Theorem 7.1
in [1]). Thus we need to look at non-degenerate energy through Σvb(u1). The non-degenerate energy on a surface
Σv takes the form ∫
Σv
|∂uψ|2
1− 2Mr + M
2q2
r2
du, (3.140)
3.4 The Hawking Radiation Calculation 81
where we can absorb the ψ term using Hardy’s inequality:
∫
Σv
(
1− 2M
r
+
M2q2
r2
) |ψ2|
(r− rb)2 du = 2
∫
Σv
|ψ|2
(r− rb)2 dr ≤ 8
∫
Σv
|∂rψ|2dr = 4
∫
Σv
|∂uψ|2
1− 2Mr + M
2q2
r2
du. (3.141)
We have the following proposition:
Proposition 3.4.5 (High Frequency Reflection in Pure Reissner–Nordström). Let ψ be a smooth solution to
(3.33), and let v2 ∈ R. Then there exists a constant A(M,q,T ∗) such that∫
Σvb(u1)
1
V
|∂uψ|2du≤ A
(∫
Σvc∩{u≤u1}
1
V
|∂uψ|2du+
∫ vc
v=vb(u1)
∣∣|ψ(u1,v)|2−|ψI −(v)|2∣∣dv) . (3.142)
There also exists a constant B(M,q,T ∗) such that
∫
Σvb(u1)
|∂uψ|2
1− 2Mr + q
2M2
r2
du≤ B
(∫
Σvc∩{u≤u1}
|∂uψ|2
1− 2Mr + q
2M2
r2
du+(l+1)2
∫
Σu1∩{v≤vc}
|ψ|2dv
)
. (3.143)
Furthermore, there exists a constant C(M,q,T ∗) such that
∫
I −∩{v≤vb(u1)}
|ψ|2dv≤C
(∫
Σvc∩{u≤u1}
1
V
|∂uψ|2du+
∫ vc
v=vb(u1)
∣∣|ψ(u1,v)|2−|ψI −(v)|2∣∣dv) . (3.144)
Proof. Given that rb → ∞ as v→−∞, there exists some v∗ such that V ′ ≤ 0 for all v ≤ v∗. We then proceed
using (3.33):
EV (v) : =
∫
Σv∩{u≤u1}
1
V
|∂uψ|2du = EV (vc)+
∫ vc
v
∫
Σv′∩{u≤u1}
∂u(|ψ|2)−∂v
(
1
V
)
|∂uψ|2dudv′
≤ EV (vc)+
∫ vc
v
(
1− 2Mr + q
2M2
r2
)
V ′
V 2
|∂uψ|2dv′+
∫
Σu1∩{v′∈[v,vc]}
|ψ|2dv′−
∫
I −∩{v′∈[v,vc]}
|ψ|2dv′
(3.145)
≤ EV (vc)+A
∫ vc
max{v,v∗}
EV (v′)dv′+
∫
Σu1∩{v′∈[v,vc]}
|ψ|2dv′−
∫
I −∩{v′∈[v,vc]}
|ψ|2dv′
≤ EV (vc)+A
∫ vc
max{v,v∗}
EV (v′)dv′+
∫ vc
v′=vb(u1)
∣∣|ψ(u1,v′)|2−|ψI −(v′)|2∣∣dv′−∫
I −∩{v≤vb(u1)}
|ψI −(v′)|2dv
To reach the penultimate line, we have used (3.141).
We then apply Gronwall’s Inequality to obtain
∫
Σv2
1
V
|∂uψ|2du≤
(∫
Σvc
1
V
|∂uψ|2du+
∫ vc
v=vb(u1)
∣∣|ψ(u1,v)|2−|ψI −(v)|2∣∣dv)e∫ vcv∗ Adv, (3.146)
for all v2 ≤ vb(u1).
By keeping the I − term in (3.145), we can then bound
∫
I −∩{v≤vb(u1)}
|ψ|2dv≤ B
(∫
Σvc∩{u≤u1}
1
V
|∂uψ|2du+
∫ vc
v=vb(u1)
∣∣|ψ(u1,v)|2−|ψI −(v)|2∣∣dv) , (3.147)
as required.
82 Hawking Radiation
We perform a similar calculation for the remaining result, (3.143)
E(v) : =
∫
Σv∩{u≤u1}
|∂uψ|2
1− 2Mr + q
2M2
r2
du = E(vc)+
∫ vc
v
∫
Σv′∩{u≤u1}
V∂u(|ψ|2)
1− 2Mr + q
2M2
r2
−∂v
(
1
1− 2Mr + q
2M2
r2
)
|∂uψ|2dudv′
≤ E(vc)+A
∫ vc
v′=v
E(v′)dv′+
∫
Σu1∩{v≤vc}
V |ψ|2
1− 2Mr + q
2M2
r2
dv−
∫ vc
v
∫
Σv′∩{u≤u1}
∂u
(
V
1− 2Mr + q
2M2
r2
)
|ψ|2dudv′
(3.148)
≤ E(vc)+A
∫ vc
v′=v
E(v′)dv′+A(l+1)2
∫
Σu1∩{v≤vc}
|ψ|2dv+
∫ vc
v
∫
Σv′∩{u≤u1,r≤2M}
(
1− 2M
r
+
q2M2
r2
)
|ψ|2dudv′
≤ E(vc)+A
∫ vc
v′=v
E(v′)dv′+A(l+1)2
∫
Σu1∩{v≤vc}
|ψ|2dv,
to which another application of Gronwall’s Inequality obtains the result.
The final Proposition in this section is:
Proposition 3.4.6 (High Frequency Transmission). Let ψ be a smooth solution to (3.33), (3.34) such that
∫
Σvc∩{u≤u1}
|∂uψ|2
1− 2Mr + q
2M2
r2
du =: Evc < ∞ (3.149)
∫
Σu1
|ψ|2du =: Lu1 < ∞. (3.150)
Then we have that there exists a constant A(M,q,T ∗) such that∫
v≤vc
∣∣∣∂vψ|I −−∂vψ|Σu1 ∣∣∣2 dv≤ A((l+1)6Lu1 +(l+1)4Evc) . (3.151)
Proof. We will start this result by considering the interval [vb(u1),vc]. This section is done in a similar manner
to Proposition 3.4.4.
∫ vc
vb(u1)
|∂vψI +(v)−∂vψ(u1,v)|2dv =
∫ vc
v=vb(u1)
∣∣∣∣∫ u1u=−∞ ∂u∂vψ(u,v)du
∣∣∣∣2 dv = ∫ vcv=vb(u1)
∣∣∣∣∫ u1u=−∞Vψ(u,v)du
∣∣∣∣2 dv
(3.152)
≤
∥∥∥√V∥∥∥2
L2(Σvc)
∫ vc
v=vb(u1)
∫ u1
u=−∞
V |ψ(u,v)|2dudv
≤ A(l+1)4
∫ vc
v=vb(u1)
∫ u1
u=−∞
(
1− 2M
r
+
q2M2
r2
) |ψ(u,v)|2
r2
dudv
≤ A(l+1)4
∫ vc
v=vb(u1)
|ψ(u1,v)|2+
(∫ u1
u=∞
|∂uψ|2
1− 2Mr + q
2M2
r2
du
)
dv
≤ A(l+1)4 (Lu1 +(vc− vb(u1))(Evc +(l+1)2Lu1))
≤ A
(
(l+1)6Lu1 +(l+1)
4Evc
)
.
For the region v≤ vc, we will use Theorem 2 from [1], which gives us energy boundedness of the scattering
map, that is
∫ vb(u1)
v=−∞
|∂vψI −|2dv≤ A
∫ u=u1
u=−∞
|∂uψ(u,vb(u1))|2
1− 2Mr + q
2M2
r2
+V |ψ|2du≤ A(l+1)2 (Evc +(l+1)2Lu1) , (3.153)
which gives us our result.
Propositions 3.4.5 and 3.4.6 give rise to the following corollary.
3.4 The Hawking Radiation Calculation 83
Corollary 3.4.4 (Hawking Radiation Error from High Frequency Transmission). Let ψ+ : R→ C be a smooth,
compactly supported function. Let ψ be the solution of (3.33), as given by Theorem 2.4.1, with radiation field
onI + equal to ψ+(u−u0), and which vanishes onH +. Let f be a smooth compactly supported function such
that f (0) = 1, and define
ψ1(v) :=
ψ(u1,v)−
(
1− f ((l+1)2(ub(v)−u1)))ψ(u1,vc) u≥ u1
0 u < u1
. (3.154)
Let ψI − be the past radiation field. Then there exists a constant A(M,q,T ∗) such that
∫
σ∈R
|σ | ∣∣|ψˆI −|2−|ψˆ1|2−|ψˆRN |2∣∣dσ ≤

A
(
I.T.[ψ+]e−κu1 + e2κu1I.E.[ψ+,vc,u1,u0]
) |q|< 1
A
(
I.T.[ψ+]u0
u5/21
+u7/21 I.E.[ψ+,vc,u1,u0]
)
|q|= 1
, (3.155)
as u0,u1 → ∞ with u1 < u0. Here κ is the surface gravity, as in (1.63) and I.T.[ψ+], I.E.[ψ+,vc,u1,u0] are as
defined in the statement of Theorem 3.4.1. In the extremal case, we will also required u1 > u0/2.
Suppose further that |q|< 1, and that ψ+, ψH − and ψRN decay sufficiently fast that all I.E.[ψ+,vc,(1−
δ )u0,u0] terms decay faster than e−3κ(1−δ )u0 . Then for all δ > 0, there exists a constant B(M,q,T ∗,δ ,ψ+)
such that ∫
σ∈R
|σ | ∣∣|ψˆI −|2−|ψ1|2−|ψˆRN |2∣∣dσ ≤ Be−κ(1−δ )u0 . (3.156)
Proof. Define the following
ψ2 := ψI −−ψ1−ψRN . (3.157)
Note that ψ2 is only supported in v≤ vc, as v > vc is out of the past light cone of the collapsing cloud. Thus, the
solution in v > vc coincides with that of Reissner–Nordström.
We can expand (3.155) to get:∫ ∞
−∞
|σ | ∣∣|ψˆI −|2−|ψˆ1|2−|ψˆRN |2∣∣dσ = ∫ ∞−∞ |σ | ∣∣|ψˆ2|2+2ℜ((ψˆ1+ ψˆ2) ¯ˆψRN + ψˆ1 ¯ˆψ2)∣∣dσ . (3.158)
We can then bound∫ ∞
∞
|σ |Re(ψˆ2 ¯ˆψRN)dσ ≤ ∥ψ2∥L2(I −)∥ψRN∥H˙1(I −) (3.159)∫ ∞
∞
|σ |Re(ψˆ1 ¯ˆψRN)dσ ≤ ∥∥∥∥ ˆσψ11+M2σ2
∥∥∥∥
L2(I −)
∥∥(1+M2σ2) ψˆRN∥∥L2(I −) (3.160)∫ ∞
∞
|σ |Re(ψˆ1 ¯ˆψ2)dσ ≤ ∥ψ1∥L2(I −)∥ψ2∥H˙1(I −) (3.161)∫ ∞
−∞
|σ ||ψˆ2|2dσ ≤ ∥ψ2∥L2(I −)∥ψ2∥H˙1(I −). (3.162)
We already have a bounds on ∥ψ1∥L2(I −), given by Proposition 3.4.4:
∥ψ1∥2L2(I −) ≤ ∥ψ(u1,v)∥2L2({v∈[vb(u1),vc]})+ |ψ(u1,vc)|
2∥ f∥2L2(I −) (3.163)
≤
A
(
I.T.[ψ+]
(l+1)4 e
−κu1 + I.E.[ψ+,vc,u1,u0]
(l+1)2 e
−κu1
)
|q|< 1
A
(
I.T.[ψ+]u20
(l+1)4u41
+ I.E.[ψ+,vc,u1,u0]
(l+1)2u21
)
|q|= 1
.
84 Hawking Radiation
We also have a bound on ∥ψ2∥H˙1(I −), thanks to Proposition 3.4.6:
∥ψ2∥2H˙1(I −) ≤ 2∥ψ−ψ1∥2H˙1(I −)∩{v≤vc}+2∥ψRN∥
2
H˙1(I −)∩{v≤vc}
≤ 4∥ fψ(u1,vc)∥2H˙1(I −)∩{v≤vc}+
∫
v≤vc
∣∣∣∂vψ|I −−∂vψ|Σu1 ∣∣∣2 dv+2∥ψRN∥2H˙1(I −)∩{v≤vc}
≤ A
(
(l+1)2|ψ(u1,vc)|2
v′b(u1)
+
(l+1)4
V (u1,vc)
∫
Σvc∩{u≤u1}
|∂uψ|2du (3.164)
+(l+1)6
∫
Σu1∩{v≤vc}
|ψ|2dv+ I.E.[ψ+,vc,u1,u0]
)
≤
A
(
(l+1)2I.T.[ψ+]e−κu1 +(l+1)2eκu1I.E.[ψ+,vc,u1,u0]
) |q|< 1
A
(
(l+1)2u20
u41
I.T.[ψ+]+ (l+1)2u21I.E.[ψ+,vc,u1,u0]
)
|q|= 1
.
We bound ∥ψ2∥L2(I −) as follows:
∥ψ2∥2L2(I −) ≤ 2∥ψ−ψ1∥2L2(I −)∩{v≤vc}+2∥ψRN∥
2
L2(I −)∩{v≤vc} (3.165)
≤ 2(vc− vb(u1))2∥∂vψ(u1,v)−∂vψI −(v)∥2L2({v∈[vb(u1),vc]}
+2∥ψ∥2L2(I −)∩{v≤vb(u1)}+2I.E.[ψ+,vc,u1,u0]
≤ 2(vc− vb(u1))2
(∫
Σvc∩{u≤u1}
(l+1)4
V
|∂uψ|2du+(l+1)6
∫
Σu1∩{v≤vc}
|ψ|2dv
)
+B
(∫
Σvc∩{u≤u1}
1
V
|∂uψ|2du+
∫ vc
v=vb(u1)
∣∣|ψ(u1,v)|2−|ψI −(v)|2∣∣dv)+2I.E.[ψ+,vc,u1,u0]
≤ A
(
(l+1)6(vc− vb(u1))2∥ψ(u1,v)∥2L2({v∈[vb(u1),vc]})+
(l+1)4
V (u1,vc)
I.E[ψ+,vc,u1,u0]
+ (vc− vb(u1))∥ψ(u1,v)−ψI −∥H˙1({v∈[vb(u1),vc]})(
∥ψ(u1,v)∥L2({v∈[vb(u1),vc]})+∥ψI −(v)∥L2({v∈[vb(u1),vc]})
))
≤ A
(
(l+1)6(vc− vb(u1))∥ψ(u1,v)∥2L2({v∈[vb(u1),vc]})+
(l+1)4
V (u1,vc)
I.E[ψ+,vc,u1,u0]
)
≤
A(l+1)
2 (I.T.[ψ+]e−2κu1 + I.E.[ψ+,vc,u1,u0]eκu1) |q|< 1
A(l+1)2
(
I.T.[ψ+]u20
u51
+ I.E.[ψ+,vc,u1,u0]u21
)
|q|= 1 .
Finally, we consider (1+M2σ2)ψˆ1, for which we will use the following Lemma:
Lemma 3.4.1. Let f be a smooth function supported in the interval [0,ε], for ε < 1. Then there exists a constant
A such that: ∥∥∥∥σ fˆ (σ)1+σ2
∥∥∥∥2
L2
≤ Aε∥ f∥2L2. (3.166)
Proof. Let f−1 be defined by fˆ−1(σ) = σ(1+σ2)−1 fˆ (σ). Then f−1 is an L2 solution to the equation:
− f ′′−1(x)+ f−1(x) = f ′(x). (3.167)
For x < 0 and x > ε , we have that f−1 = Aex+Be−x. In order for f−1 to be L2, this means that
f−1 =
Aex−ε x < 0Beε−x x > ε . (3.168)
3.4 The Hawking Radiation Calculation 85
The solution to (3.167) in the interval [0,ε] is therefore
f−1(x) = Aex−ε −g(x), (3.169)
where
g(x) =
∫ x
x1=0
f (x1)dx1+
∫ x
x1=0
∫ x1
x2=0
∫ x2
x3=0
f (x3)dx3dx2dx1+ ..., (3.170)
assuming such a sequence converges. To show that such a sequence converges, we write:
|g(x)| ≤
∫ x
x1=0
| f (x1)|dx1+
∫ x
x1=0
∫ x1
x2=0
∫ x2
x3=0
| f (x3)|dx3dx2dx1+ ... (3.171)
≤ ∥1∥L2([0,ε])∥ f∥L2([0,ε])+
∫ x
x1=0
∫ x1
x2=0
∥1∥L2([0,ε])∥ f∥L2([0,ε])dx2dx1+ ...
≤√ε∥ f∥L2([0,ε])
(
1+
ε2
2!
+
ε4
4!
+ ...
)
≤ cosh(ε)√ε∥ f∥L2([0,ε]).
We can similarly bound the derivative of g(x):
|g′(x)| ≤ | f (x)|dx2+
∫ x
x2=0
∫ x2
x3=0
| f (x3)|dx3dx2+ ... (3.172)
≤ | f (x)|+
∫ x1
x2=0
∥1∥L2([0,ε])∥ f∥L2([0,ε])dx2+ ...
≤ | f (x)|+√ε∥ f∥L2([0,ε])
(
ε+
ε3
3!
+
ε5
5!
+ ...
)
≤ | f (x)|+ sinh(ε)√ε∥ f∥L2([0,ε]).
We then need to consider the values of A and B which allow this function to be twice weakly differentiable:
f−1(ε) = A+g(ε) = B (3.173)
f ′−1(ε) = A+g
′(ε) =−B. (3.174)
Solving these gives
A =−g(ε)+g
′(ε)
2
(3.175)
B =
g(ε)−g′(ε)
2
. (3.176)
Then the L2 norm of f−1 can be bounded by:
∥ f−1∥2L2(R) ≤
∫ ∞
ε
|B|2e2(ε−x)dx+2
∫ ε
−∞
|A|2e2(x−ε)dx+2
∫ ε
0
|g(x)|2dx (3.177)
≤ A
(
|g(ε)|2+ |g′(ε)|2+
∫ ε
0
|g(x)|2dx
)
≤ A(cosh(ε)2+ sinh(ε)2+ ε cosh(ε)2)ε∥ f∥2L2 ,
as required.
We can apply this Lemma to σ(1+M2σ2)−1ψˆ1 to get that∥∥∥∥ σψˆ11+M2σ2
∥∥∥∥2
L2(I −)
≤ vc− vb(u1)
M
∥ψ1∥2L2(I −). (3.178)
86 Hawking Radiation
We then use (3.22) to obtain
∥(1+M2σ2)ψˆRN∥2L2(I −) =
∫ ∞
−∞
(1+M2σ2)2|R˜σ ,l,m|2|ψˆ+|2dσ ≤ A
∫ ∞
−∞
(1+M2σ2)2
(l+1)2
1+M2σ2
|ψˆ+|2dσ
≤ A
∫ ∞
−∞
(l+1)2(1+M2σ2)|ψˆ+|2dσ ≤ A(l+1)−2I.T.[ψ+] (3.179)
Substituting into (3.158) gives the required results.
For the result with sufficiently fast decay of ψ+, we can use (3.107) to bound
∫ vc
vb(u1)
|ψ(u1,v)|2dv more
accurately. Setting u1 = (1− 34δ )u1, all I.E. terms will decay sufficiently fast to obtain our result.
3.4.6 Treatment of the I.E. Terms
In this section we show the arbitrary polynomial decay of the I.E. terms, provided that ψˆ+ vanishes and has all
derivatives vanishing at ω = 0. This result has been largely done in the extremal (|q| = 1) case, in [3]. This
gives our first Theorem:
Theorem 3.4.2. [Decay of the I.E. terms in the |q|= 1 case] Let ψ+ be a Schwartz function on the cylinder,
with ψˆ+ compactly supported on σ ≥ 0. Then for each n, there exists an An(M,ψ+) such that
I.E.[ψ+,vc,u1,u0]≤ An(u0−u1)−n, (3.180)
as u0− vc,u1− vc → ∞, with u0 ≥ u1. Here, I.E. is as defined in Theorem 3.4.1, in the case of an extremal
(|q|= 1) RNOS model.
Proof. As φˆ+ and all it’s ω derivatives vanish at ω = 0, then ψˆ−n :=ω−nψ+ is also a Schwartz function. Instead
of imposing ψ+ as our radiation field on I +, we can use ψ−n. The resulting solution has the property
∂ nt∗ψ−n = ψ. (3.181)
We then apply Theorem 4.2 (with u0 as the origin) from [3] to ψ−n, to see∫
H −
(1+(u−u0)2)n|∂uψH −|2+(1+(u−u0)2)n| /˚∇ψH −|2 sinθdθdϕdu (3.182)
+
∫
I −
(1+(v−u0−R)2)n|∂uψRN |2+(1+(v−u0−R)2)n| /˚∇ψRN |2 sinθdθdϕdu
≤ An[ψ+].
Restricting the integral to u≤ u1, we can see that
I.E.[ψ+,vc,u1,u0]≤ A
∫ u1
u=−∞
(1+(u−u0)2)3/2|∂uψH −|2+(1+(u−u0)2)3/2| /˚∇ψH −|2 sinθdθdϕdu
+
∫
I −
(1+(v−u0−R)2)3/2|∂uψRN |2+(1+(v−u0−R)2)3/2| /˚∇ψRN |2 sinθdθdϕdu
≤ An[ψ+](1+(u1−u0)2)−n+3/2, (3.183)
giving our result.
We now look to extend this result to the sub-extremal case. The following section will closely follow that of
the extremal case [3].
The next ingredient needed for the r∗p method is integrated local energy decay, or ILED. This will be done
in a manner similar to [13].
Proposition 3.4.7 (ILED for sub-extremal Reissner–Nordström). Let φ be a solution of (2.1) on a sub-extremal
(|q|< 1) Reissner–Nordström background. Let t0 be a fixed value of t, and let R be a large fixed constant. Then
3.4 The Hawking Radiation Calculation 87
there exists a constant A = A(M,q,R,n) such that
∫ t0
−∞
(∫
Σt∩{|r∗|≤R}
|∂rφ |2
)
dt ≤ A
∫
Σ¯t0,R
dn(J∂t )≤ A
∫
Σt0
−dt(J∂t ) (3.184)∫ t0
−∞
(∫
Σt∩{|r∗|≤R}
−dt(J∂t )
)
dt ≤ A ∑
|α|+ j≤1
∫
Σ¯t0,R
dn(J∂t [∂ jt Ωαφ ])≤ A ∑
|α|+ j≤1
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ]). (3.185)
Proof. Consider Reisnner–Nordström spacetime in r∗, t,θ ,ϕ coordinates. For ease of writing, we will denote
D(r∗) = 1− 2M
r
+
M2q2
r2
. (3.186)
As done so far in this chapter, we will restrict to a spherical harmonic. We will first consider the case l ≥ 1.
We choose ω = h′/4+hD/2r, h′(r∗) = (A2+(r∗−R)2)−1, and consider divergence theorem applied to
JX := JX ,ω +
h′
D
βφ2∂r∗ (3.187)
β =
D
r
− r
∗−R
A2+(r∗−R)2 ,
for A and R yet to be chosen.
Note that the flux of this current through any t = const surface is bounded by the ∂r∗φ and ∂tφ terms of the
T energy. The bulk term of this is given by
KX = ∇νJXν =
h′
D
(∂rφ +βφ)2+
(
(r∗−R)2−A2
2D((r∗−R)2+A2)3 +
(
l(l+1)
r2
(
D
r
− D
′
2D
)
+
D′
2r2
− D
′′
2Dr
)
h
)
φ2.
(3.188)
Calculating the coefficient of hφ2 gives us
l(l+1)
r2
(
D
r
− D
′
2D
)
+
D′
2r2
− D
′′
2Dr
=
M4
r7
(
l(l+1)
( r
M
)4−3(l(l+1)−1)( r
M
)3
+(2q2l(l+1)−4q2−8)
( r
M
)2
+15q2
( r
M
)
−6q4
)
(3.189)
=
M4
r7
(
(x−1)(x−2)(l(l+1)x2+3(x−1))− (1−q2)((2l(l+1)−4)x2+15x−6(1+q2)))
= l(l+1)(x−3)x3+(3x−8)x2+q2((2l(l+1)−4)x2+15x−6q2),
where x = r/M. Searching for roots of this, we can see there is a root at r = M, but this is strictly less than 0 for
r < 2M, and strictly greater than 0 for r > 3M. In this interval, we consider the function
f (x) = l(l+1)x−3(l(l+1)−1)+(2q2l(l+1)−4q2−8)x−1+15q2x−2−6q4x−3 (3.190)
f ′(x) = l(l+1)(1−2q2x−2)+(8+4q2)x−2−30q2x−3+18q4x−4 > 0, (3.191)
for x > 2. Therefore the coefficient of hφ2 in (3.188) has exactly one root, in a bounded region of r∗. We label
this point r∗0, and we let
h(r∗0) = 0. (3.192)
As h has a positive gradient, this means that f (x)h≥ 0, with a single quadratic root at r∗0. Provided R > r∗0, we
also know h > π/2A for sufficiently large values of r∗. Thus to ensure KX is positive definite, it is sufficient to
88 Hawking Radiation
show that R and A can be chosen such that
(r∗−R)2−A2
2D((r∗−R)2+A2)3 +
M
r4
f
( r
M
)
h > 0. (3.193)
We only need to consider the region |r∗−R|< A. By choosing R− r∗0−A >> M, we can ensure that in this
region, D > 1− ε , Mr f
( r
M
)≥ l(l+1)(1− ε), and r ≤ r∗(1− ε). Thus it is sufficient to choose R and A, with
R− r∗0−A >> M, such that
l(l+1)π(1− ε)− A
(
A2− (r∗−R)2)r∗3
((r∗−R)2+A2)3 > 0. (3.194)
Let y = r
∗−R
A , then we are looking for the maximum of
(1− y2)(y+ RA)3
(1+ y2)3
. (3.195)
If we choose R− r∗0 = 1.001A, and choose 0.001A >> M, then
sup
−1≤y≤1
(1− y2)(y+1.001)3
(1+ y2)3
<
π
2
, (3.196)
and we have KX is positive definite.
KX ≥ ε|∂rφ +βφ |
2
D(M2+ r∗2)
+ ε
 l(l+1)D tanh
(
r∗−r∗0
M
)2
r3
+
1
D(M2+ r∗2)2
 |φ |2 (3.197)
≥ ε|∂rφ |
2
D(M2+ r∗2)
+ ε
 l(l+1)D tanh
(
r∗−r∗0
M
)2
r3
+
1
D(M2+ r∗2)2
 |φ |2.
To bound the T -energy locally, we can thus consider
A ∑
|α|+ j≤1
KX[∂ jt Ωαφ ]≥
−dt(J∂t )
M2+ r∗2
(3.198)
A ∑
|α|+ j≤1
KX[∂ jt Ωαφ ]≥ A(−dt(J∂t )) ∀|r∗| ≤ R, (3.199)
where Ω are the angular Killing Fields, as given by (1.26).
For the l = 0 case, we again follow the example of [13] and take X = ∂r∗ . Given that all angular derivatives
vanish, applying divergence theorem to JX in the interval r∗ ∈ (−∞,r∗0), we obtain∫ t0
−∞
(∂tφ(r∗0))
2+(∂r∗φ(r∗0))
2r2 sinθdθdϕdt+
∫ r∗0
r∗=−∞
2D
r
∫ t0
−∞
(−(∂tφ)2+(∂r∗φ)2)r2 sinθdθdϕdr∗dt
≤ 4T -energy(Σt∗0 ). (3.200)
Let
F(r∗) :=
∫ r∗0
r∗=−∞
2D
r
∫ t0
−∞
(∂tφ)2r2 sinθdθdϕdr∗dt. (3.201)
Then (3.200) implies
F ′(r∗)≤ 2D
r
F(r∗)+
8D
r
T -energy(Σt∗0 ). (3.202)
3.4 The Hawking Radiation Calculation 89
Noting that
∫ r∗0
r∗=−∞
2D
r dr
∗ = 2log
( r
r+
)
, an application of Gronwall’s inequality yields
F(r∗)≤ A
(
r2
r2+
)
T -energy(Σt∗0 ). (3.203)
By applying this to (3.200), we can obtain(
r2+
r20
)∫ ∞
t0
∫ r∗0
r∗=−∞
2D
r
(
∂tφ)2+(∂r∗φ)2
)
sinθdθdϕdr∗dt ≤ AT -energy (3.204)∫ t0
−∞
(∫
Σt∩{|r∗|≤R}
−dt(J∂t )
)
dt ≤ AT -energy (3.205)
We now have the result for all l using Σt0 .
Once we note that the region {t ≤ t0, |r∗| ≤ R} is entirely in the domain of dependence of Σ¯t0,R, we can
consider the alternative solution, φ˜ , given by the data of φ on Σ¯t0,R, but vanishing onH − and I − to the future
of Σ¯t0,R. We evolve this forward to Σt0 , we can apply the above result. As φ˜ = φ to the past of Σ¯t0,R, we have the
result.
Remark 3.4.6 (Degeneracy at the Photon Sphere). For the l ≥ 1 case, as l → ∞, the root of the h function
chosen tends towards the root of
1− 3M
r
+
2M2q2
r2
= 0, (3.206)
known as the photon sphere, r = rp. If we do not require control of the T -energy at this particular value, then
we do not need to include angular derivatives
∫ ∞
t0
(∫
Σt∩{ε≤|r∗−r∗p|≤R}
−dt(J∂t )
)
dt ≤ A
1
∑
j=0
∫
Σt0
−dt(J∂t [∂ jt φ ]). (3.207)
Remark 3.4.7 (Forward and higher order ILED). By sending t →−t, Proposition 3.4.7 immediately gives us
the result in the forward direction:
∫ ∞
t0
(∫
Σt∩{|r∗|≤R}
−dt(J∂t )
)
dt ≤ A ∑
j+|α|≤1
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ]). (3.208)
We can also apply the Proposition 3.4.7 to ∂ jt Ωαφ to obtain∫ ∞
t0
(∫
Σt∩{|r∗|≤R}
|∇nφ |2
)
dt ≤ A ∑
j+|α|≤n
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ]), (3.209)
where we have rewritten terms in ∇nφ involving more than one r∗ derivative using (2.1).
Proposition 3.4.8 (Boundedness of r∗ Weighted Energy). Let ψ+ be a Schwartz function. Let ψ be the solution
to (3.33) on a sub-extremal Reissner–Nordström background, with radiation field on I + equal to ψ+, and
which vanishes onH +. Let R be a constant, and let t0 be a fixed value of t. Then for each n ∈ N0, we have the
following bounds:
∑
j+|α|≤n
∫
Σ¯t0,R
(M2+2 j + |r∗|2+2 j)dn(J∂t [Ωα∂ jt φ ])≤ An ∑
1≤ j+|m|≤n+1
∫ ∞
−∞
(
M2 j +u2 j
)
(l+1)2m
∣∣∂ juψ+∣∣2 du,
(3.210)
where An = an(M,q, t0,R,n).
90 Hawking Radiation
Proof. We start by bounding an rp weighted norm on Σu0 ∩ r∗ ≤−R for some u0 ∈ R and R large.∫
Σu0∩r∗≤−R
(−R− r∗)p|∂vψ|2dv =
∫
u≥u0,r∗≤−R
−p(−r∗−R)p−1|∂vψ|2+(−R− r∗)pV∂v(|ψ|2)dudv (3.211)
≤−
∫
u≥u0,r∗≤−R
∂v((−R− r∗)pV )|ψ|2dudv
≤ A
∫ ∞
u=u0
∫
Σv
V |ψ|2dvdu
≤ A
∫ ∞
u=u0
∫ ∞
u′=u
|∂uψ+|2du′du = A
∫ ∞
u=u0
(u−u0)|∂uψ+|2du.
Here we have used T energy boundedness to reach the last line, along with an explicit calculation to show that
−∂v((−R− r∗)pV ) ≤ AV . A is a constant which depends on M,q, and the choice of R. Note this calculation
applies for all p ∈ N for sub-extremal Reissner–Nordström, but in the extremal case this only applies up to
p = 2. By applying this result to ∂ jt Ωαφ , we obtain the required bound for Σ¯t0,R∩{r∗ ≤−R}.
For r∗ ∈ [−R,R], we note that T -energy boundedness of ∂ jt Ωαφ is sufficient for our result, as the constant
An may depend on our choice of R.
For the equivalent result on Σv0 ∩ r∗ ≥ R, a similar approach does not work, as the T energy on Σv does not
approach 0 as v→∞. Instead, we will make use of the vector field multiplier u2∂u+v2∂v. Let u0 ≤ v0−R. This
will closely follow the proof of Proposition 8.1 in [3].
∫
Σv0∩u≤v0−R
u2|∂uψ|2+ v2V |ψ|2du+
∫
Σu0∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2du
=
∫
I +∩{u≤v0−R}
u2|∂uψ|2+ v2V |ψ|2du
+
∫
Σu=v0−R∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2du (3.212)
+
∫
u∈[u0,v0−R],v≥v0
(
∂v(v2V )+∂u(u2V )
) |ψ|2dudv.
We then note
∂v(v2V )+∂u(u2V ) = 2tV + tr∗V ′ = t(2V + r∗V ′)≤

A|t|
r3 ≤ Ar−2 l = 0
AV |t| log( rM )
r ≤ AV log
( r
M
)
l ̸= 0
, (3.213)
using that |t| ≤ r∗+max(v0−R,−v0) in the region we are considering. Here A depends on the choice of v0 and
R. We can then take a supremum of (3.212) over u0 ≤ v0−R and v≥ v0 to obtain
sup
v≥v0
∫
Σv∩u≤v0−R
u2|∂uψ|2+ v2V |ψ|2du+ sup
u≤v0−R
∫
Σu∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv
≤
∫
I +∩{u≤v0−R}
u2|∂uψ|2+ v2V |ψ|2du
+
∫
Σu=v0−R∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv (3.214)
+A
∫
u≤v0−R,v≥v0
(
V log
( r
M
)
+ r−2
)
|ψ|2dudv.
3.4 The Hawking Radiation Calculation 91
We can bound the final integral using the following:
∫ v0−R
u=−∞
∫ ∞
v=v0
(
V log
( r
M
)
+ r−2
)
|ψ|2dvdu≤
∫ v0−R
u=−∞
∫ ∞
v=v0
(
V log
(−u
M
)
+V log
( v
M
)
+u−2
)
|ψ|2dvdu
≤ A
∫ v0−R
u=−∞
u−2 log
(−u
M
)∫ ∞
v=v0
u2V |ψ|2dvdu
+A
∫ ∞
v=v0
v−2 log
( v
M
)∫ v0−R
u=−∞
v2V |ψ|2dudv (3.215)
+A
∫ v0−R
u=−∞
u−2
∫ ∞
v=v0
|ψ|2dudv
≤ ε sup
u≤v0−R
∫
Σu∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2du
+ ε sup
v≥v0
∫
Σv∩u≤v0−R
u2|∂uψ|2+ v2V |ψ|2du
+ ε sup
u≤v0−R
∫
Σu∩{v≥v0}
|ψ|2dv,
where v0 and R are sufficiently large.
We can then apply Hardy’s inequality to χ(1+R/M)ψ (χ as in Proposition 3.4.2) to get
sup
u≤v0−R
∫
Σu∩{v≥v0}
|ψ|2dv≤ A sup
u≤v0−R
∫
Σu∩{v≥v0}
V |ψ|2dv+ sup
u≤v0−R
∫
Σu∩{v≥v0}
v2|∂vψ|2dv. (3.216)
We can then rearrange (3.214) to see
sup
v≥v0
∫
Σv∩u≤v0−R
u2|∂uψ|2+ v2V |ψ|2du+ sup
u≤v0−R
∫
Σu∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv
≤ A
∫
I +∩{u≤v0−R}
u2|∂uψ|2+ v2V |ψ|2du (3.217)
+A
∫
Σu=v0−R∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv.
By taking an appropriate limit of this, we can see that∫
Σv0∩u≤v0−R
u2|∂uψ|2+ v2V |ψ|2du+
∫
I −∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv
≤ A
∫
I +∩{u≤v0−R}
u2|∂uψ|2+ v2V |ψ|2du (3.218)
+A
∫
Σu=v0−R∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv.
We can also consider a time reversal of this statement to get∫
I +∩{u≤v0−R}
u2|∂uψ|2+ v2V |ψ|2du+
∫
Σu=v0−R∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv
≤ A
∫
Σv0∩u≤v0−R
u2|∂uψ|2+ v2V |ψ|2du (3.219)
+A
∫
I −∩{v≥v0}
v2|∂vψ|2+u2V |ψ|2dv.
In order to add more u and v weighting to this, we commute with the vector field S = u∂ −U + v∂v.
(∂u∂v+V )S( f ) = S[(∂u∂v+V ) f ]+ (2V − r∗∂r∗V ) f +2(∂u∂v+V ) f . (3.220)
92 Hawking Radiation
Thus an easy induction argument gives
|(∂u∂v+V )Snψ| ≤ A|V − r∗∂r∗V |
n−1
∑
k=0
Skψ, (3.221)
noting that
|∂ nr∗(V − r∗∂r∗V )| ≤ A|V − r∗∂r∗V | ≤
A(l+1)2 log
( r
M
)
r3
. (3.222)
Repeating (3.214), but applied to Snψ , we obtain
Fn := sup
v≥v0
∫
Σv∩u≤v0−R
u2|∂uSnψ|2+ v2V |Snψ|2du+ sup
u≤v0−R
∫
Σu∩{v≥v0}
v2|∂vSnψ|2+u2V |Snψ|2du (3.223)
≤
∫
I +∩{u≤v0−R}
u2|∂uSnψ|2+ v2V |Snψ|2du+
∫
Σu=v0−R∩{v≥v0}
v2|∂vSnψ|2+u2V |Snψ|2du
+A
∫
u≤v0−R,v≥v0
(
V log
( r
M
)
+ r−2
)
|Snψ|2dudv
+A
n−1
∑
k=0
∫
u≤v0−R,v≥v0
(l+1)2 log
( r
M
)
r3
|Skψ| ∣∣u2∂uSnψ+ v2∂vSnψ∣∣dudv
≤
∫
I +∩{u≤v0−R}
u2|∂uSnψ|2+ v2V |Snψ|2du+
∫
Σu=v0−R∩{v≥v0}
v2|∂vSnψ|2+u2V |Snψ|2du
+AεFn+AF
1/2
n
n−1
∑
k=0
∫ v0−R
u=−∞
(l+1)4 log
(
r(u,v0)
M
)2
u2V (u,v0)r(u,v0)6
du

1/2
F1/2k
≤ A
∫
I +∩{u≤v0−R}
u2|∂uSnψ|2+ v2V |Snψ|2du+A
∫
Σu=v0−R∩{v≥v0}
v2|∂vSnψ|2+u2V |Snψ|2du
+AF1/2n
n−1
∑
k=0
∫ v0−R
u=−∞
(l+1)4 log
(
r(u,v0)
M
)2
u2V (u,v0)r(u,v0)6
du

1/2
F1/2k
≤ A
∫
I +∩{u≤v0−R}
u2|∂uSnψ|2+ v2V |Snψ|2du+A
∫
Σu=v0−R∩{v≥v0}
v2|∂vSnψ|2+u2V |Snψ|2du
+Aε(l+1)F1/2n
n−1
∑
k=0
F1/2k
≤ A
∫
I +∩{u≤v0−R}
u2|∂uSnψ|2+ v2V |Snψ|2du+A
∫
Σu=v0−R∩{v≥v0}
v2|∂vSnψ|2+u2V |Snψ|2du
+A(l+1)2
n−1
∑
k=0
Fk.
As F0 is bounded by (3.218), we can inductively obtain
∑
k+m≤n
(∫
Σv0∩u≤v0−R
u2(l+1)2m|∂u(Skψ)|2+ v2V (l+1)m|Skψ|2du (3.224)
+
∫
I −∩{v≥v0}
v2(l+1)2m|∂v((v∂v)kψ)|2+u2V (l+1)2m|((v∂v)kψ)|2dv
)
≤ A ∑
k+m≤n
(∫
I +∩{u≤v0−R}
u2(l+1)2m|∂u((u∂u)kψ)|2+ v2V (l+1)2m|((u∂u)kψ)|2du
+
∫
Σu=v0−R∩{v≥v0}
v2(l+1)2m|∂v(Skψ)|2+u2V (l+1)2m|(Skψ)|2dv
)
,
3.4 The Hawking Radiation Calculation 93
along with the time reversed result
∑
k+m≤n
(∫
I +∩{u≤v0−R}
u2(l+1)2m|∂u((u∂u)kψ)|2+ v2V (l+1)2m|((u∂u)kψ)|2du
+
∫
Σu=v0−R∩{v≥v0}
v2(l+1)2m|∂v(Skψ)|2+u2V (l+1)2m|(Skψ)|2dv
)
≤ A ∑
k+m≤n
(∫
Σv0∩u≤v0−R
u2(l+1)2m|∂u(Skψ)|2+ v2V (l+1)2m|(Skψ)|2du
(3.225)
+
∫
I −∩{v≥v0}
v2(l+1)2m|∂v((v∂v)kψ)|2+u2V (l+1)2m|((v∂v)kψ)|2dv
)
.
All that is now left for the result is to bound
∑
k+m≤n
∫
Σu=v0−R∩{v≥v0}
v2(l+1)2m|∂v(Skψ)|2+u2V (l+1)2m|(Skψ)|2dv
≤ ∑
k+m+ j≤n
∫
Σu=v0−R∩{v≥v0}
v2+2k(l+1)2m|∂ k+1v ∂ jt ψ|2 (3.226)
+ v2kV (l+1)2m|(∂ kv ∂ jt ψ)|2dv,
for fixed and arbitrarily large R,v0. We have used (3.33) to remove any ∂u∂v derivatives, and have replaced any
∂u derivatives with ∂t +∂v derivatives. As ∂t and Ω are Killing fields, it is sufficient to bound∫
Σu=v0−R∩{v≥v0}
v2k+2|∂ k+1v ψ|2dv≤ A
∫
Σu=v0−R
χ
(
r−R
M
−1
)
r2(k+1)|∂ k+1v ψ|2dv, (3.227)
for k ≥ 0.
We can immediately apply Proposition 3.4.2 (with time reversed) to obtain the k = 0 case∫
Σu=v0−R
χr2|∂vψ|2du≤ A
∫
I +
(M2+u2)|∂uψ+|2+ l(l+1)|ψ+|2du. (3.228)
Here the constant A depends on choice of v0 and R. We would now like to generalise this to the following result
(closely based on Proposition 7.7 in [3]).
∫
Σu=v0−R
χ
(
r−R
M
−1
)
r2k|∂ kvψ|2dv≤ A ∑
1≤m+ j≤k
∫ ∞
u=v0−R
(
M2m+(u−uR)2m
)
(l+1)2 j|∂mu ψ+|2du, (3.229)
where A depends on M,q,n,R. From here, we will denote v0−R = uR.
We will prove this inductively. First, we consider commuting (3.33) with ∂v to obtain
∂u∂v(∂ nv ψ)+V∂
n
v ψ =−∂ nv (Vψ)+V∂ nv ψ =−
n−1
∑
j=0
(
n
j
)
∂ n− jr∗ V∂
j
vψ ≤ A
n−1
∑
j=0
(l+1)2
r2+n− j
|∂ jvψ|. (3.230)
94 Hawking Radiation
We then look at applying this to the following generalisation of the right hand side of (3.229)
∫
Σu=uR
χrp|∂ kvψ|2dv =
∫
u≥uR
D(χ ′rp+ pχrp−1)|∂ kvψ|2+2R
(
χrp∂ kv ψ¯
k−1
∑
j=0
(
k
j
)
∂ k− jv V∂
j
vψ
)
−D∂r∗ (χrpV ) |∂ k−1v ψ|dudv+
∫
I +
rp−2l(l+1)|∂ k−1v ψ+|2du (3.231)
≤ A
∫
u≥uR,r∗∈[R,R+M]
∑
m+ j≤k−1
−dt((l+1)2 jJ∂t [∂mt ψ])dudv+A
∫
u≥uR
χ
k
∑
j=0
(l+1)2 j
r1+2 j−p
|∂ k− jv ψ|2dudv
≤ A
∫
u=uR
∑
m+ j≤k−1
(l+1)2 j|∂m+1u ψ+|2du+A
∫
u≥uR
χ
k
∑
j=0
(l+1)2 j
r1+2 j−p
|∂ k− jv ψ|2dudv,
where we have used Proposition 3.4.7.
For our induction argument, we will assume we have proved (3.229) for k ≤ n, where n ≥ 1. We first
consider 3.231, with k = n+1 and p = 1+2n.
∫
Σu=uR
χr1+2n|∂ n+1v ψ|2dv≤
∫
u=uR
∑
m+ j≤n
(l+1)2 j|∂m+1u ψ+|2du+
∫
u≥uR
χ
n+1
∑
j=0
(l+1)2 j
r2( j−n)
|∂ 1+n− jv ψ|2dudv
(3.232)
≤ A
∫
u=uR
∑
m+ j≤n
(l+1)2 j|∂m+1u ψ+|2du
+A
∫
u≥uR
χ
n
∑
j=0
(l+1)2 j
r2( j−n)
|∂ 1+n− jv ψ|2+(l+1)2nχ|∂vψ|2dudv
≤ A
∫
u=uR
∑
m+ j≤n
(l+1)2 j|∂m+1u ψ+|2du+A
∫
u≥uR
χ
n
∑
j=0
(l+1)2(n− j)r2 j|∂ jv (∂t +∂u)ψ|2dudv
≤ A
∫
u=uR
∑
m+ j≤n
(l+1)2 j|∂m+1u ψ+|2du+A
∫
u≥uR
χ
n
∑
j=0
(l+1)2(n− j)r2 j|∂ j−1v (Vψ)|2dudv
+A
∫
u=uR
∑
1≤m+ j≤n
∫ ∞
u′=u
(
M2m+(u−uR)2m
)
(l+1)2 j|∂mu ∂tψ+|2du′du
≤ A ∑
0≤m+ j≤n
∫ ∞
u=uR
(
M2m+(u−uR)2m+1
)
(l+1)2 j|∂m+1u ψ+|2du
+A
∫
u≥uR
χ
n
∑
j=0
(l+1)2(n− j)+2r2 j|∂ j−1v (Vψ)|2dudv
≤ A
∫
u=uR
∑
1≤m+ j≤n+1
(l+1)2 j|∂mu ψ+|2du,
where we have used that ∂t is a Killing field along with our induction hypothesis in the final three lines.
We then proceed to prove (3.229):∫
Σu=uR
χr2+2n|∂ n+1v ψ|2dv≤ A
∫
u=uR
∑
m+ j≤n
(l+1)2 j|∂m+1u ψ+|2du (3.233)
+A
∫
u≥uR
χ
n
∑
j=0
(l+1)2 j
r2( j−n)−1
|∂ 1+n− jv ψ|2+(l+1)2nrχ|∂vψ|2dudv
≤ A
∫
u=uR
∑
m+ j≤n
(M+(u−uR))(l+1)2 j|∂m+1u ψ+|2du
+A
∫
u≥uR
χ
n
∑
j=0
(l+1)2(n− j)r2 j+1|∂ jv (∂t +∂u)ψ|2dudv
≤ A ∑
1≤m+ j≤n+1
∫ ∞
u=v0−R
(
M2m+(u−uR)2m
)
(l+1)2 j|∂mu ψ+|2du,
applying (3.232), along with identical reasoning as used in (3.232).
3.4 The Hawking Radiation Calculation 95
Proposition 3.4.9 (Integrated Decay of Higher Order Energy). Let ψ+ be a Schwartz function. Let ψ be the
solution of (3.33) on a sub-extremal Reissner–Nordström background, as given by Theorem 3.3.1, with radiation
field on I + equal to ψ+, and which vanishes onH +. Let R be a constant, and let t0 be a fixed value of t. Then
for each n ∈ N0, we have the following bounds:∫ t0
t2n+1=−∞
∫ tn
t2n=−∞
...
∫ t2
t1=−∞
∫ t1
t=−∞
(∫
Σ¯t,R
−dt(J∂t [∂ nt φ ])
)
dtdt1dt2..dt2n+1 (3.234)
+ ∑
j+|α|+m≤n
∫
v=t0+R,r∗≥R
∫
v≤t0+R,r∗≥R
r1+2 j
(|∂ 1+ ju ∂mt Ωαψ|2+ jV |∂ ju∂mt Ωαψ|2)dudv
+ ∑
j+|α|+m≤n
∫
u=t0+R,r∗≤−R
(−r∗)1+2 j (|∂ 1+ jv ∂mt Ωαψ|2+(−r∗)V |∂ jv ∂mt Ωαψ|2)dudv
≤ An ∑
j+|α|+m≤n
∫
v=t0+R,r∗≥R
r2+2 j|∂ 1+ ju ∂mt Ωαψ|2du
+An ∑
j+|α|+m≤n
∫
u=t0+R,r∗≤−R
(−r∗)2+2 j|∂ 1+ jv ∂mt Ωαψ|2dv
+An ∑
j+|α|≤2n+2
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ]),
where An = An(M,q,n,R).
Proof. This proof again closely follows that of [3]. We will consider T energy through a null foliation, Σ¯t,R (see
(2.10)).
We first look at how the wave operator commutes with both ∂u and ∂v:
∂u∂v(∂ nuψ)+V∂
n
uψ =−∂ nu (Vψ)+V∂ nuψ =
n−1
∑
j=0
(
n
j
)
(−1)n− j+1∂ n− jr∗ V∂ juψ ≤ A
n−1
∑
j=0
V
rn− j
|∂ juψ| (3.235)
∂u∂v(∂ nv ψ)+V∂
n
v ψ =−∂ nv (Vψ)+V∂ nv ψ =−
n−1
∑
j=0
(
n
j
)
∂ n− jr∗ V∂
j
vψ ≤ A
n−1
∑
j=0
Vκn− j|∂ jvψ| (3.236)
We apply the rp and r∗p methods to the null segments of Σ¯t0,R to obtain:∫
v=t0+R,r∗≥R
rpχ
(
r∗−R
M
)
|∂ kuψ|2du (3.237)
=
∫
v≤t0+R,r∗≥R
(
prp−1Dχ+
rp
M
χ ′
)
|∂ kuψ|2− rpχV∂u
(
|∂ k−1u ψ|2
)
dvdu
+
∫
v≤t0+R,r∗≥R
2rpχR
(
∂ ku ψ¯
k−2
∑
j=0
(
k−1
j
)
(−1)k− j∂ k−1− jr∗ V∂ juψ
)
dudv
≥
∫
v≤t0+R,r∗≥R
(
prp−1Dχ+
rp
M
χ ′
)
|∂ kuψ|2−∂r∗ (rpχV )
(
|∂ k−1u ψ|2
)
dvdu
−A
∫
v≤t0+R,r∗≥R
rpχ|∂ kuψ|
k−2
∑
j=0
V r1−k+ j|∂ juψ|dudv+
∫
I −
rpV |∂ k−1u ψ|2dv
≥ a
∫
v≤t0+R,r∗≥R
χrp−1
(
p|∂ kuψ|2+(p−2)V |∂ k−1u ψ|2
)
dudv
−A
∫
R≤r∗≤M+R,t≤t0
|∂ kuψ|2d+V |∂ k−1u ψ|2dr∗dt
−A
k−2
∑
j=0
∫
v≤t0+R,r∗≥R
χV 2r3−2k+2 j+p|∂ juψ|2dudv.
96 Hawking Radiation
∫
u=t0+R
(−r∗)pχ
(−r∗−R
M
)
|∂ kvψ|2dv
=
∫
u≤t0+R
(
p(−r∗)p−1χ+ (−r
∗)p
M
χ ′
)
|∂ kvψ|2
− (−r∗)pχV∂v
(
|∂ k−1v ψ|2
)
−2χ(−r∗)pR
(
∂ kv ψ¯
k−2
∑
j=0
(
k−1
j
)
∂ k−1− jr∗ V∂
j
vψ
)
dvdu
≥
∫
u≤t0+R
(
p(−r∗)p−1χ+ (−r
∗)p
M
χ ′
)
|∂ kvψ|2+∂r∗ ((−r∗)pχV )
(
|∂ k−1v ψ|2
)
−Aχ(−r∗)p|∂ kvψ|
k−2
∑
j=0
Vκk−1− j|∂ jvψ|dvdu (3.238)
≥ a
∫
u≤t0+R
χ(−r∗)p−1
(
p|∂ kvψ|2+(−r∗κ− p)V |∂ k−1v ψ|2
)
dudv
−A
∫
−r∗≥M+R,t≤t0
|∂ kvψ|2+V |∂ k−1v ψ|2dr∗dt
−A
k−2
∑
j=0
∫
u≤t0+R
χ(−r∗)p+1κ2k−2 j−2V 2|∂ jvψ|2dudv.
By summing (3.237) and (3.238) when p = 1, k = 1 (as then the two summations vanish), we obtain:
∫
v=t0+R,r∗≥R
rχ
(
r∗−R
M
)
|∂uψ|2du+
∫
u=t0+R
(−r∗)χ
(−r∗−R
M
)
|∂vψ|2dv+
∫
Σt0
1
∑
j=0
−(l+1)2−2 jdt(J∂t [∂ jt φ ])
≥ a
∫ t0
t=−∞
T -energy(Σ¯t,R)dt (3.239)
Here we have used Proposition 3.4.7.
We then consider the p = 2, k = 1 case to obtain
∫
v=t0+R,r∗≥R
r2χ
(
r∗−R
M
)
|∂uψ|2du+
∫
u=t0+R
(−r∗)2χ
(−r∗−R
M
)
|∂vψ|2dv+
∫
Σt0
2
∑
j=0
−(l+1)4−2 jdt(J∂t [∂ jt Ωαφ ])
≥ a
∫ t0
t=−∞
(∫
v=t+R
rχ
(
r∗−R
M
)
|∂uψ|2du (3.240)
+
∫
u=t+R
(−r∗)χ
(−r∗−R
M
)(|∂vψ|2+((−r∗)κ−2)V |ψ|2)dv
+
∫
Σt
1
∑
j=0
−(l+1)2−2 jdt(J∂t [∂ jt φ ])
)
≥ a
∫ t0
t=−∞
∫ t
t ′=−∞
T -energy(Σ¯t ′,R)dt ′dt.
By using mean value theorem and T -energy boundedness (see [12] for an example of this), one can thus obtain∫
Σ¯t,R
T -energy≤ A(−t)−2
∫
I +
(M2+u2)|∂uψ+|2+ l(l+1)|ψ+|2du. (3.241)
By considering T -energy boundedness between Σ¯t,R andH −∪I−, we can also obtain:
∫ t0
t=−∞
∫ t
t ′=−∞
(∫ t ′+R
u=−∞
|∂uψH −|2du+
∫ t ′+R
v=−∞
|∂vψI −|2dv
)
(3.242)
=
∫ t0+R
u=−∞
(u− t0−R)2|∂uψH −|2du+
∫ t0+R
v=−∞
(v− t0−R)2|∂vψRN |2dv
≤ A
∫
I +
(M2+u2)|∂uψ+|2+ l(l+1)|ψ+|2du.
3.4 The Hawking Radiation Calculation 97
We now proceed to prove the result inductively, given the case n = 0 is (3.240) (Provided R > 3κ). We first
look to bound the r and r∗ weighted summations. We take p = 2+2n,k = n+1 in (3.237)∫
v≤t0+R
χr1+2n
(|∂ 1+nu ψ|2+V |∂ nuψ|2)dudv (3.243)
≤ A
∫
v=t0+R
χr2+2n|∂ 1+nu ψ|2du
+A
∫
R≤r∗≤M+R,t≥t0
|∂ n+1u ψ|2+V |∂ nuψ|2dr∗dt
+A
n−1
∑
m=1
∫
v≤t0+R
χ(l+1)2r1+2mV |∂mu ψ|2dudv
+
∫
v≤t0+R
2χr2+2nR
(
∂ n+1u ψ¯(−1)n∂ nr∗Vψ
)
dudv
≤ A
∫
v=t0+R
χr2+2n|∂ 1+nu ψ|2du+A ∑
m+|α|≤n+1
(l+1)2
∫
Σt0
−dt(J∂t [∂mt ψ])
+A ∑
j+k+m≤n
∫
v=t0+R,r∗≥R
r2+2 j(l+1)2k|∂ 1+ ju ∂mt ψ|2du
+A ∑
j+k+m≤n
∫
u=t0+R,r∗≤−R
(−r∗)2+2 j(l+1)2k|∂ 1+ jv ∂mt ψ|2dv
+A ∑
j+m≤2n+2
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ])+
∫
v≥t0+R,r∗≥R
χ(l+1)2rV |ψ|2dudv
+
∫
v≤t0+R
2χr2+2nR
(
∂ n+1u ψ¯(−1)n∂ nr∗Vψ
)
dudv
≤ A ∑
j+k+m≤n
∫
v=t0+R,r∗≥R
r2+2 j(l+1)2k|∂ 1+ ju ∂mt ψ|2du
+A ∑
j+k+m≤n
∫
u=t0+R,r∗≤−R
(−r∗)2+2 j(l+1)2k|∂ 1+ jv ∂mt ψ|2dv
+A ∑
j+m≤2n+2
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ])
+
∫
v≤t0+R
2χr2+2nR
(
∂ n+1u ψ¯(−1)n∂ nr∗Vψ
)
dudv
In order to bound the final term in (3.243), we first note that the usual method of separating does not work:∫
v=t+R
χr2+2nR
(
∂ n+1u ψ¯(−1)n∂ nr∗Vψ
)
du≤ A
∫
v=t+R
χr2n+1|∂ n+1u ψ|2+ r(l+1)2V |ψ|2du. (3.244)
Unfortunately, we have no way to bound rV |ψ|2. If we consider lower order terms in r, we can use Hardy’s
inequality. ∫
v=t+R
V |ψ|2 ≤ A
∫
v=t+R
(l+1)2|∂uψ|2+A
∫
v=t+R,|r∗|≤R
V |ψ|2du. (3.245)
98 Hawking Radiation
Thus, the only term we need to be concerned about in (3.243) is the leading order in r behaviour of the final
term. This behaves as follows:
∫
v=t+R
2l(l+1)χrnR
(
∂ n+1u ψ¯ψ
)
du =
∫
v=t+R
2l(l+1)R
(
∂uψ¯
n
∑
j=0
(
n
j
)
∂ n− jr∗ (−1) j(χrn)∂ juψ
)
du (3.246)
≤ A
∫
v=t+R
l(l+1)r|∂uψ|2+ l(l+1)
n−1
∑
j=1
r2 j−1|∂ juψ|2du
+
∫
v=t+R
l(l+1)∂ nr∗(χr
n)∂u
(|ψ|2)du
≤ A
∫
v=t+R
l(l+1)r|∂uψ|2+ l(l+1)
n−1
∑
j=1
r2 j−1|∂ juψ|2du
+
∫
v=t+R
l(l+1)∂ n+1r∗ (χr
n)|ψ|2du− l(l+1)n!(−1)n|ψ|2I −.
As ∂ n+1r∗ (r
n)≤ Ar−2, we can use (3.245) to bound this. Combining (3.243), (3.245) and (3.246), we obtain
∫
v≤t0+R
χr1+2n
(|∂ 1+nu ψ|2+V |∂ nuψ|2)dudv≤ A ∑
j+k+m≤n
∫
v=t0+R,r∗≥R
r2+2 j(l+1)2k|∂ 1+ ju ∂mt ψ|2du (3.247)
+A ∑
j+k+m≤n
∫
u=t0+R,r∗≤−R
(−r∗)2+2 j(l+1)2k|∂ 1+ jv ∂mt ψ|2dv
+A ∑
j+m≤2n+2
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ]),
as required.
The (−r∗)p section is made much more easy by the exponential behaviour of the potential. We take
p = 2+2n,k = n+1 in (3.238):∫
u≤t0+R
χ(−r∗)1+2n (|∂ n+1v ψ|2+(−r∗)V |∂ nv ψ|2)dudv
≤ A
∫
u=t0+R
χ(−r∗)2+2n|∂ n+1v ψ|2dv+A
∫
u≤t0+R,−r∗≤2(2+2n)/κ
χ(−r∗)2+2nV |∂ nv ψ|2dudv
+A
∫
R≤−r∗≤R+M,t≤t0
|∂ n+1v ψ|2+V |∂ kvψ|2dr∗dt
+A
n−1
∑
j=0
∫
u≤t0+R
χ(−r∗)3+2nκ2n−2 jV 2|∂ jvψ|2dudv
≤ A
∫
u=t0+R
χ(−r∗)2+2n|∂ n+1v ψ|2dv
+A
∫
R≤−r∗≤max{R+M,2(2+2n)/κ},t≤t0
|∂ n+1v ψ|2+V |∂ nv ψ|2dr∗dt (3.248)
+A
(l+1)2
M2
n−1
∑
j=0
∫
u≤t0+R
χ(−r∗)1+2 jκ2n−2 jV |∂ jvψ|2dudv
≤ A ∑
j+|α|+m≤n
∫
v=t0+R,r∗≥R
r2+2 j|∂ 1+ ju ∂mt Ωαψ|2du
+A ∑
j+|α|+m≤n
∫
u=t0+R,r∗≤−R
(−r∗)2+2 j|∂ 1+ jv ∂mt Ωαψ|2dv
+A ∑
j+|α|≤2n+2
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ]),
as required. In the final inequality, we have used the induction hypothesis.
3.4 The Hawking Radiation Calculation 99
By repeating the above argument, we can also show that∫
u≤t0+R
χ(−r∗)2n (|∂ n+1v ψ|2+(−r∗)V |∂ nv ψ|2)dudv+∫
v≤t0+R
χr2n
(|∂ 1+nu ψ|2+V |∂ nuψ|2)dudv (3.249)
≤ A ∑
j+|α|+m≤n
∫
v=t0+R,r∗≥R
r1+2 j|∂ 1+ ju ∂mt Ωαψ|2du
+A ∑
j+|α|+m≤n
∫
u=t0+R,r∗≤−R
(−r∗)1+2 j|∂ 1+ ju ∂mt Ωαψ|2dv
+A ∑
j+|α|≤2n+1
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ])
We now look to prove the final part. Assuming the result is true in the n case, apply (3.234) to ∂tψ . We then
integrate twice with respect to t0 to obtain∫ t0
t2n+3=−∞
∫ t2n+3
t2n+2=−∞
...
∫ t2
t1=−∞
∫ t1
t=−∞
(∫
Σ¯t,R
−dt(J∂t [∂ n+1t φ ])
)
dtdt1dt2..dt2n+3 (3.250)
≤ A ∑
j+k+m≤n
∫ t0
−∞
∫ t2n+3
−∞
∫
v=t+R,r∗≥R
r2+2k(l+1)2k|∂ 1+ku ∂m+1t ψ|2dudt2n+2dt2n+3
+A ∑
j+k+m≤n
∫ t0
−∞
∫ t2n+3
−∞
∫
u=t+R,r∗≤−R
(−r∗)2+2k(l+1)2k|∂ 1+kv ∂m+1t |2dvdt2n+2dt2n+3
+A ∑
j+|α|≤2n+2
∫ t0
−∞
∫ t2n+3
−∞
(∫
Σt
−dt(J∂t [∂ j+1t Ωαφ ])
)
dt2n+2dt2n+3.
The final term here can be immediately bounded using Proposition 3.4.7. To bound the earlier terms, we
note that ∂u+∂v = ∂t , and we can use (3.33) to remove any mixed u,v derivatives.
∑
j+k+m≤n
∫ t0
−∞
∫ t2n+3
−∞
∫
v=t+R,r∗≥R
r2+2 j(l+1)2k|∂ 1+ ju ∂m+1t ψ|2dudt2n+2dt2n+3 (3.251)
≤ A ∑
j+k+m≤n
∫ t0
−∞
∫ t2n+3
−∞
∫
v=t+R,r∗≥R
r2+2 j(l+1)2k|∂ 2+ ju ∂mt ψ|2dudt2n+2dt2n+3
+A ∑
j+k+m≤n
∫ t0
−∞
∫ t2n+3
−∞
∫
v=t+R,r∗≥R
r2+2 j(l+1)2k|∂ ju∂mt (Vψ)|2dudt2n+2dt2n+3
≤ A ∑
j+k+m≤n
∫ t0
−∞
∫ t2n+3
−∞
∫
v=t+R,r∗≥R
r2+2 j(l+1)2k|∂ 2+ ju ∂mt ψ|2dudt2n+2dt2n+3
+A ∑
j+k+m≤n
∫ t0
−∞
∫ t2n+3
−∞
∫
v=t+R,r∗≥R
r2 j(l+1)2k+2V |∂ ju∂mt ψ|2dudt2n+2dt2n+3
≤ A ∑
j+k+m≤n+1
∫ t0
−∞
∫
v=t+R,r∗≥R
r1+2 j(l+1)2k|∂ 1+ ju ∂mt ψ|2dudt2n+3
+A ∑
j+k+m≤n+1
∫ t0
−∞
∫
u=t+R,r∗≤R
(−r∗)1+2 j(l+1)2k|∂ 1+ jv ∂mt ψ|2dudt2n+3
+A ∑
j+|α|≤2n+3
∫ t0
−∞
(∫
Σt0
−dt(J∂t [∂ jt Ωαφ ])
)
dt2n+3
≤ A ∑
j+k+m≤n+1
∫
v=t+R,r∗≥R
r2+2 j(l+1)2k|∂ 1+ ju ∂mt ψ|2du
+A ∑
j+k+m≤n+1
∫
u=t+R,r∗≤R
(−r∗)2+2 j(l+1)2k|∂ 1+ jv ∂mt ψ|2dv
+A ∑
j+|α|≤2n+4
∫
Σt0
−dt(J∂t [∂ jt Ωαφ ]),
100 Hawking Radiation
as required. An identical argument follows for the −r∗ ≥ R region.
Theorem 3.4.3 (Boundedness of the u and v Weighted Energy). Let ψ+ be a Schwartz function on the cylinder.
Let φ be a solution to (2.1) on a sub-extremal Reissner–Nordström background. Further, let φ vanish onH +
and have future radiation field equal to ψ+. Then there exists a constant An = An(M,q,n) (which also depends
on the choice of origin of u,v) such that
2
∑
k=0
∑
2 j+m+2|α|≤2n
∫
H −∩{u≤0}
(M2( j+1)−k +u2( j+1)−k)|∂ j+m+k+1u ΩαψH −|2du (3.252)
+
2
∑
k=0
∑
1≤ j+|α|,2 j+2|α|+m≤2n+2
∫
I −∩{v≤0}
(M2 j−k + v2 j−k)|∂ j+k+mv ΩαψI −|2dv
≤ A
2
∑
k=0
∑
1≤ j+|α|,2 j+2|α|+m≤2n+2
∫
I −
(M2 j−k +u2 j−k)|∂ j+k+mu Ωαψ+|2du
Proof. This result again follows closely that of [3]. It is an easy combination of Propositions 3.4.8 and 3.4.9,
applied to T mΩα , for α ≤ n− j and m≤ 2n−2k−2α . All that remains is to note
∫ t0
t2n+1=−∞
∫ tn
t2n=−∞
...
∫ t2
t1=−∞
∫ t1
t=−∞
(∫
Σ¯t,R
−dt(J∂t [∂ nt φ ])
)
dtdt1dt2..dt2n+1 (3.253)
=
∫ t0
t2n+1=−∞
∫ tn
t2n=−∞
...
∫ t2
t1=−∞
∫ t1
t=−∞
(∫ t+R
−∞
|∂uψH −|2 sinθdθdϕdu+
∫ t+R
−∞
|∂vψI −|2 sinθdθdϕdv
)
dtdt1dt2..dt2n+1
=
1
(2n+2)!
(∫ t+R
−∞
(u− t0−R)2n+2|∂uψH −|2 sinθdθdϕdu+
∫ t+R
−∞
(v− t0−R)2n+2|∂vψI −|2 sinθdθdϕdv
)
,
by repeated integration by parts.
Corollary 3.4.5 (Arbitrary polynomial decay of I.E. Terms). Let ψ+ be a Schwartz function on the cylinder,
with ψˆ+ supported on ω ≥ 0. Then for each n, there exists an An(M,q,ψ+) such that
I.E.[ψ+,vc,u1,u0]≤ An(u0−u1)−n, (3.254)
as u0− vc,u1− vc → ∞, with u0 ≥ u1. Here I.E. is as defined in Theorem 3.4.1.
Proof. This proof is identical to that of Theorem 3.4.2.
3.4.7 Final Calculation
We now have all the tools we need to calculate the final result. We wish to calculate:
I[ψ+, l,u0] :=
∫ ∞
−∞
|σ ||ψˆI −|2dσ , (3.255)
where ψI − is the radiation field on I −.
Proof of Theorem 3.4.1. We will define ψ0 and ψ1 as in Corollary 3.4.3, that is
ψ0(u) :=
ψ(u,vc)− f (u−u1)ψ(u1,vc) u≥ u10 u < u1 , (3.256)
ψ1(v) :=
ψ(u1,v)− (1− f (ub(v)−u1))ψ(u1,vc) v ∈ [vb(u1),vc]0 v /∈ [vb(u1),vc] , (3.257)
where f is a smooth compactly supported function with f (0) = 1.
Note this coincides with the definition of ψ0 in Corollary 3.4.2 and the definition of ψ1 in Corollary 3.4.4.
3.4 The Hawking Radiation Calculation 101
For this final calculation, we will be using Lemma II.6 from [7]:
Lemma 3.4.2. For β > 0, u ∈C∞0 (R), we define
F(ξ ) =
∫
R
eiξe
βx
u′(x)dx. (3.258)
Then we have ∫
R
|ξ |−1|F(ξ )|2dξ =
∫
R
|ξ |coth
(
π
β
|ξ |
)
|uˆ(ξ )|2dξ . (3.259)
We also have a similar Lemma for the extremal case:
Lemma 3.4.3. Let A ∈ R>0, vc ∈ R be constants. Define
p(v) =
A
vc− v . (3.260)
Then for all u ∈C∞0 (R), we have ∫
R
|σ ||uˆ|2dσ =
∫
R
|σ ||û◦ p|2dσ . (3.261)
Proof. This proof proceeds in an almost identical way to the proof of Lemma 3.4.2 (see [7]).
∫
σ∈R
|σ ||û◦ p|2dσ = lim
ε→0
(∫
σ∈R
|σ |e−ε|σ ||û◦ p|2dσ
)
= lim
ε→0
(∫∫∫
x,x′,σ∈R
|σ |e−ε|σ |eiσ(x′−x)u◦ p(x)u◦ p(x′)dxdx′dσ
)
= lim
ε→0
∫∫
u,u′∈R
(∫
σ∈R
|σ |e−ε|σ |eiσ(Ay− Ay′ )
)
u(y)u¯(y′)
A2
y2y′2
dydy′ (3.262)
= lim
ε→0
∫∫
y,y′∈R

2
(
ε2−
(
A
y − Ay′
)2)
(
ε2+
(
A
y − Ay′
)2)2
u(y)u¯(y′) A2y2y′2 dydy′
= lim
ε→0
∫∫
y,w∈R
2
(
ε2y2(y−w)2
A2 −w2
)
(
ε2y2(y−w)2
A2 +w
2
)2
u(y)u¯(y−w)dydw
=
∫∫
y,w∈R
lim
α→0
(
2
(
α2−w2)
(α2+w2)2
)
u(y)u¯(y−w)dydw =
∫
w∈R
lim
α→0
( ̂|σ |e−α|σ |)(̂|uˆ|2)dw
=
∫
σ∈R
|σ ||uˆ|2dσ ,
as required.
In order to use Lemmas 3.4.2 and 3.4.3, we take a sequence of functions in C∞0 (R) which approximate ψ1
with respect to the L2 and H˙1 norms.
By considering ψˆ1, we can see
−ieiσvcσψˆ1 :=−i
∫
Σ−u1∩{v≤vc}
σeiσ(vc−v)ψ1dv=−i
∫
R
σeiσ(vc−vb(u))ψ1(vb(u))
dvb
du
du=
∫
R
eiσ(vc−vb(u)) (ψ1 ◦ vb)′ du.
(3.263)
In the extremal case, this is similar to the form of F in (3.258), once we note that vc−vb(u) = Ae−κu+O(e−2κu)
for large u.
Thus, we define
γSE(u) :=− 1κ log
(
vc− vb(u)
A
)
= u− 1
κ
log
(
1+O(e−κu)
)
= u+O(e−κu), (3.264)
102 Hawking Radiation
as u→ ∞. Then combining (3.263) and (3.264) gives
− ieiσvcσψˆ1 =
∫
R
eiσ(vc−vb(u)) (ψ1 ◦ vb)′ du =
∫
R
eiσAe
−κγSE
(ψ1 ◦ vb)′ du =
∫
R
eiσAe
−κw (
ψ1 ◦ vb ◦ γ−1SE
)′
dw.
(3.265)
Then we can apply Lemma 3.4.2 to obtain:∫
R
|σ ||ψˆ1|2dσ =
∫
R
|σ |coth
(π
κ
|σ |
)
| ̂ψ1 ◦ vb ◦ γ−1SE |2dσ . (3.266)
Note that |σ |coth(πκ |σ |)≤ κπ + |σ |.
In the extremal case, vc− vb(u) = A0u−1+O(u−3), so we define
γE(u) :=
A0
vc− vb(u) =
A0
A0u−1+O(u−3)
= u+O(u−2), (3.267)
as u→ ∞. For p as in Lemma 3.4.3, we have
p(v) =
A
vc− v =
A
vc− vb(ub(v)) = γE ◦ub. (3.268)
Thus we can apply Lemma 3.4.3 to ψ1 ◦ vb ◦ γ−1E to obtain∫
σ∈R
|σ || ̂ψ1 ◦ vb ◦ γ−1E |2dσ =
∫
σ∈R
|σ || ̂ψ1 ◦ vb ◦ γ−1E ◦ γE ◦ub|2dσ =
∫
σ∈R
|σ ||ψˆ1|2dσ , (3.269)
as in the sub-extremal case.
We now note that in both the extremal and sub-extremal cases, we have:∫
σ∈R
(κ+ |σ |)
∣∣∣| ̂ψ1 ◦ vb ◦ γ−1|2−|ψ̂1 ◦ vb|2∣∣∣dσ
≤ A
(∫ ∞
u1
|∂u(ψ1 ◦ vb)|2+κ2|ψ1 ◦ vb|2du
)1/2(∫ ∞
u1
|ψ1 ◦ vb ◦ γ−1−ψ1 ◦ vb|2du
)1/2
≤ A(I.T.[ψ+])1/2
(∫ ∞
u1
∣∣∣∣∫ γ(u)u ∂u(ψ1 ◦ vb)du
∣∣∣∣2 du
)1/2
(3.270)
≤ A(I.T.[ψ+])1/2
(∫ ∞
u1
∣∣∣∣∫ γ(u1)u1 |∂u(ψ1 ◦ vb)|du′
∣∣∣∣2 du
)1/2
≤ A(I.T.[ψ+])1/2
∫ γ(u1)
u1
(∫ ∞
u1
|∂u(ψ1 ◦ vb)|2du
)1/2
du′
≤ A|γ(u1)−u1|I.T.[ψ+],
where we have dropped the subscript from γSE , γE . We have used Minkowski’s integral inequality to reach the
penultimate line.
By using (3.266) and (3.269), we obtain∣∣∣∣∣
∫
σ∈R
|σ |
(
|ψˆI −|2−coth
(π
κ
|σ |
)
|ψˆH −|2−|ψˆRN |2
)
dσ
∣∣∣∣∣ (3.271)
≤ A
∫
σ∈R
|σ | ∣∣|ψˆI −|2−|ψˆ1|2−|ψˆRN |2∣∣dσ
+A
∫
σ∈R
|σ |coth
(π
κ
|σ |
)∣∣∣| ̂ψ1 ◦ vb ◦ γ−1|2−|ψ̂1 ◦ vb|2∣∣∣dσ
+A
∫
σ∈R
|σ |coth
(π
κ
|σ |
)∣∣|ψ̂1 ◦ vb|2−|ψˆ0|2∣∣dσ
+A
∫
σ∈R
|σ |coth
(π
κ
|σ |
)∣∣|ψˆ0|2−|ψˆH −|2∣∣dσ .
3.4 The Hawking Radiation Calculation 103
In the extremal case, we set u1 = u0−
√
Mu0. Then we can apply (3.270) and Corollaries 3.4.2, 3.4.3 and 3.4.4
to obtain the required result, again noting |σ |coth(πκ |σ |)≤ κπ + |σ |.
If we have sufficient decay in the non-extremal case, then we set u1 =
(
1− 23δ
)
, and again use Corollaries
3.4.2, 3.4.3 and 3.4.4.
Corollary 3.4.6 (Particle Emission by the RNOS Models). Let f be a Schwartz function on the cylinder, with fˆ
supported in [−1,1]×S2, and such that ∫ ∞
−∞
| f (x,θ ,ϕ)|2dx = 1. (3.272)
Then let φ be the solution to (2.1), (3.34) with data onH + vanishing, and radiation field on I + given by
lim
v→∞r(u,v)φ(u,v,θ ,ϕ) = f (ω(u−u0))e
iω(u−u0), (3.273)
as given by Theorem 2.4.1.
Let ψI − be the past radiation field, and let n ∈ N. Then there exist constants An(M,q,T ∗, f ,ω) and
A(M,T ∗, f ,ω) such that∣∣∣∣∣
∫ 0
σ=−∞
|σ ||ψˆI −|2 sinθdθdϕdσ −
∫ ∞
−∞
(
e
2πω|x|
κ −1
)−1
∑
l,m
|T˜ωx,l,m|2
∣∣ fˆl,m (x−1)∣∣2 dx
∣∣∣∣∣≤ Anu−n0 |q| ̸= 1
(3.274)∫ 0
σ=−∞
|σ ||ψˆI −|2dσ ≤
A
u3/20
|q|= 1,
where
fˆl,m =
∫
S2
fˆYl,m sinθdθdϕ, (3.275)
are the projection of fˆ onto spherical harmonics, Yl,m.
Proof. This result follows easily from Theorem 3.4.1 and Corollary 3.4.5. In the sub-extremal case, one can
choose u1 such that e−κu1 = u−n0 to obtain∣∣∣∣∣
∫
I −
|σ ||ψˆ|2 sinθdθdϕdσ −
∫ ∞
−∞
|σ |coth
(π
κ
|σ |
)
∑
l,m
|T˜σ ,l,m|2|ψˆ+l,m|2dσ −
∫ ∞
−∞
|σ |∑
l,m
|R˜σ ,l,m|2|ψˆ+l,m|2dσ
∣∣∣∣∣
≤ Anu−n0 +u2n0 I.E.[ψ+,vc,
n
κ
ln(u0),u0]
≤ Anu−n0 . (3.276)
Note also that
i
2
∫
S
φ¯∇φ −φ∇φ¯dn (3.277)
is a conserved quantity, therefore by taking appropriate limits, we obtain∫ ∞
σ=−∞
σ |ψˆ+|2dσ = i2
∫
I +
ψ¯∇ψ−ψ∇ψ¯du = i
2
∫
I −
ψ¯∇ψ−ψ∇ψ¯dv =
∫ ∞
σ=−∞
σ |ψˆI −|2dσ . (3.278)
104 Hawking Radiation
Thus, we have∣∣∣∣∣
∫
I −
|σ ||ψˆ|2 sinθdθdϕdσ −
∫ ∞
−∞
|σ |coth
(π
κ
|σ |
)
∑
l,m
|T˜σ ,l,m|2|ψˆ+l,m|2dσ −
∫ ∞
−∞
|σ |∑
l,m
|R˜σ ,l,m|2|ψˆ+l,m|2dσ
∣∣∣∣∣
=
∣∣∣∣∣∑l,m
(∫
I −
(|σ |−σ)|ψˆ+l,m|2dσ −
∫ ∞
−∞
|σ |coth
(π
κ
|σ |
)
|T˜σ ,l,m|2|ψˆ+l,m|2dσ (3.279)
−
∫ ∞
−∞
(|σ ||R˜σ ,l,m|2−σ) |ψˆ+l,m|2dσ
)∣∣∣∣∣
=
∣∣∣∣∣∑l,m
(∫ ∞
−∞
(
σ −|σ ||R˜σ ,l,m|2−|σ |coth
(π
κ
|σ |
)
|T˜σ ,l,m|2
)
|ψˆ+l,m|2dσ −2
∫ 0
σ=−∞
|σ ||ψˆ+l,m|2dσ
)∣∣∣∣∣ .
Finally, we note that |T˜σ ,l,m|2 = 1−|R˜σ ,l,m|2, which allows us to simplify:∫ ∞
−∞
(
σ −|σ |+ |σ |− |σ ||R˜σ ,l,m|2−|σ |coth
(π
κ
|σ |
)
|T˜σ ,l,m|2
)
|ψˆ+l,m|2dσ (3.280)
=
∫ ∞
−∞
(
σ −|σ |+ |σ |
(
1− coth
(π
κ
|σ |
))
|T˜σ ,l,m|2
)
|ψˆ+l,m|2dσ
= 2
∫ ∞
−∞
(
x−|x|
2
+
(
e
2πω|x|
κ −1
)−1
|T˜ωx,l,m|2
)∣∣ fˆl,m (x−1)∣∣2 dx
= 2
∫ ∞
−∞
(
e
2πω|x|
κ −1
)−1
|T˜ωx,l,m|2
∣∣ fˆl,m (x−1)∣∣2 dx,
as required. We have used that fˆ (x−1) is only supported on σ ≥ 0. The calculation follows identically for the
extremal case.
Remark 3.4.8 (Generality of Theorem 3.4.1 and Corollary 3.4.6). Similarly to Remark 2.7.2, the only behaviour
of rb required to prove Theorem 3.4.1 and Corollary 3.4.6 are
• The tangent vector (1, r˙b(t∗),0,0) at the point (t∗,rb(t∗),θ ,ϕ) is timelike (for all t∗,θ ,ϕ , including at
t∗ = t∗c ).
• There exists a t∗− and an ε > 0 such that for all t∗ ≤ t∗−, r˙b(t∗) ∈ (−1+ ε,0).
This allows Theorem 3.4.1 and Corollary 3.4.6 to be generalised more easily using these methods.
As discussed in the introduction, Corollary 3.4.6 is the calculation of radiation of frequency ω given off by
the RNOS model of a collapsing black hole, see [28] for a full discussion of this. We will however, comment
that the quantity of particles emitted by Extremal RNOS models is integrable. This means that the total number
of particles given off by the forming extremal black hole is finite, and thus the black hole itself may never
evaporate.
3.5 Future Work
In this thesis, we have considered Hawking radiation in collapsing Reissner–Nordström, with reflective boundary
conditions on the surface of the collapse. There are several ways in which we would like to generalise this result
going forward.
Firstly, we could change the background for the collapse to a Kerr black hole. This result would be of special
significance for two reasons. On the one hand, this has physical application - all black holes that have been
observed have some angular momentum. On the other hand, other methods for calculating black hole emission,
like the Hartle–Hawking–Israel state mentioned in the introduction, do not generalise to this rotating case. We
would then proceed to considering the collapsing Kerr–Newman case, where we allow both rotation and charge.
Secondly, we could consider other equations of motion beyond the wave equation. The most obvious
example of this would be the Klein–Gordon equation (of which the wave equation is a special case). Further
3.5 Future Work 105
generalising this to the charged Klein–Gordon case would allow us to consider what charges are emitted by the
black hole, and thus consider whether extremal black holes could become sub-extremal, or whether sub-extremal
black holes become closer to extremal through this emission of charge.
Finally, extending this result beyond Dirichlet boundary conditions to include the interior of the collapse
would allow us to understand whether different matter models could possibly have any influence in the emission
of particles from the black hole, or whether this emission is, as theorised, independent of the matter collapsing.

Bibliography
[1] F Alford. The scattering map on collapsing charged spherically symmertic spacetimes. In Preparation.
[2] F Alford. The scattering map on Oppenheimer–Snyder space-time. Annales Henri Poincaré, 21(6):2031–
2092, 2020.
[3] Y Angelopoulos, S Aretakis, and D Gajic. A non-degenerate scattering theory for the wave equation on
extremal Reissner–Nordström. https://arxiv.org/abs/1910.07975, 2019.
[4] S Aretakis. Stability and instability of extreme reissner-nordström black hole spacetimes for linear scalar
perturbations i. Communications in mathematical physics, 307(1):17–63, 2011.
[5] S Aretakis. Stability and instability of extreme reissner–nordström black hole spacetimes for linear scalar
perturbations ii. Annales Henri Poincaré, 12(8):1491–1538, 2011.
[6] A Bachelot. The Hawking effect. Comptes Rendus de l’Académie des Sciences - Series I - Mathematics,
325(2):1229–1234, December 1997.
[7] A Bachelot. Quantum vacuum polarization at the black-hole horizon. Annales de l’I.H.P. Physique
théorique, 67(2):181–222, 1997.
[8] A Bachelot. Scattering of scalar fields by spherical gravitational collapse. Journal de Mathématiques
Pures et Appliquées, 76(2):155–210, February 1997.
[9] R Balbinot, A Fabbri, S Farese, and R Parentani. Hawking radiation from extremal and nonextremal black
holes. Physical Review D, 76(12), Dec 2007.
[10] S Chandrasekhar. The mathematical theory of black holes / S. Chandrasekhar. International series of
monographs on physics ; 69. Clarendon Press, Oxford, 1992.
[11] D Christodoulou. Violation of cosmic censorship in the gravitational collapse of a dust cloud. Communic-
ations in Mathematical Physics, 93(2):171–195, Jun 1984.
[12] Mihalis D and Igor R. A new physical-space approach to decay for the wave equation with applications
to black hole spacetimes. XVIth International Congress on Mathematical Physics, P. Exner (ed.), World
Scientific, London, pages 421–433, 2009.
[13] M Dafermos and I Rodnianski. The red-shift effect and radiation decay on black hole spacetimes.
Communications on Pure and Applied Mathematics, 62(7):859–919, 2009.
[14] M Dafermos and I Rodnianski. Lectures on black holes and linear waves. Clay Mathematics Proceedings,
Amer. Math. Soc., Providence, RI, 17:97–205, 2013.
[15] M Dafermos, I Rodnianski, and Y Shlapentokh-Rothman. A scattering theory for the wave equation on
Kerr black hole exteriors. Annales scientifiques de l’ENS, 51, Issue 2:371–486, 2018.
[16] M Dafermos and Y Shlapentokh-Rothman. Time-translation invariance of scattering maps and blue-shift
instabilities on Kerr black hole spacetimes. Communications in Mathematical Physics, 350(3):985–1016,
Mar 2017.
[17] J Dimock and B S Kay. Classical and Quantum Scattering Theory for Linear Scalar Fields on the
Schwarzschild Metric. 1. Annals Phys., 175:366, 1987.
[18] A Drouot. A quantitative version of Hawking radiation. Annales Henri Poincaré, 18(3):757–806, 2017.
[19] G Eskin. Hawking radiation from not-extremal and extremal Reissner–Nordström black holes. 2019.
[20] K Fredenhagen and R Haag. On the derivation of Hawking radiation associated with the formation of a
black hole. Comm. Math. Phys., 127(2):273–284, 1990.
108 Bibliography
[21] F G Friedlander. Radiation fields and hyperbolic scattering theory. Mathematical Proceedings of the
Cambridge Philosophical Society, 88(3):483–515, 1980.
[22] S Gao. Late-time particle creation from gravitational collapse to an extremal Reissner–Nordström black
hole. Physical Review D, 68(4), 2003.
[23] C Gerard. The hartle–hawking–israel state on spacetimes with stationary bifurcate killing horizons.
Reviews in mathematical physics, 33(8):2150028, 2021.
[24] M R R Good. Extremal Hawking radiation. Phys. Rev. D, 101:104050, May 2020.
[25] D Häfner. Asymptotic completeness for the wave equation in a class of stationary and asymptotically flat
space-times. ANNALES DE L INSTITUT FOURIER, 51(3):779–779, 2001.
[26] D Häfner. Quantum Field Theory and Gravity, chapter Some Mathematical Aspects of the Hawking Effect
for Rotating Black Holes, pages 121–136. Springer Basel, Basel, 2012.
[27] S W Hawking. Black hole explosions? Nature, 248(5443):30–31, March 1974.
[28] S W Hawking. Particle creation by black holes. Comm. Math. Phys., 43(3):199–220, 1975.
[29] B S Kay. The Double Wedge Algebra for Quantum Fields on Schwarzschild and Minkowski Space-times.
Commun. Math. Phys., 100:57, 1985.
[30] B S Kay. Quantum field theory in curved spacetime. Encyclopedia of Mathematical Physics, 02 2006.
[31] B S Kay and R M Wald. Theorems on the Uniqueness and Thermal Properties of Stationary, Nonsingular,
Quasifree States on Space-Times with a Bifurcate Killing Horizon. Phys. Rept., 207:49–136, 1991.
[32] C Kehle and Y Shlapentokh-Rothman. A scattering theory for linear waves on the interior of Reissner–
Nordström black holes. Annales Henri Poincaré, 20(5):1583–1650, Feb 2019.
[33] J Luk. Introduction to nonlinear wave equations. https://web.stanford.edu/~jluk/NWnotes.pdf.
[34] F Melnyk. The Hawking effect for spin 1/2 fields. Communications in Mathematical Physics, 244(3):483–
525, Feb 2004.
[35] G Moschidis. The rp-weighted energy method of Dafermos and Rodnianski in general asymptotically flat
spacetimes and applications. Annals of PDE, 2(1):6, May 2016.
[36] C Müller. Spherical harmonics. Lecture notes in mathematics (Springer-Verlag) ; 17. Springer-Verlag,
Berlin, New York, 1966.
[37] L I Nicolaescu. Lectures on the geometry of manifolds / Liviu I. Nicolaescu (University of Michigan, USA).
1999.
[38] J Nicolas. Conformal scattering on the Schwarzschild metric. Annales de l’Institut Fourier, 66(3):1175–
1216, 2016.
[39] J R Oppenheimer and H Snyder. On continued gravitational contraction. Phys. Rev, 56:455–459, 1939.
[40] H Reall. Lectures on black holes. www.damtp.cam.ac.uk/user/hsr1000/black_holes_lectures_2016.pdf,
2016.
[41] H Reissner. Über die eigengravitation des elektrischen feldes nach der einsteinschen theorie. Annalen der
Physik (1900), 355(9):106–120, 1916.
[42] P Ruback. A new uniqueness theorem for charged black holes. Classical and Quantum Gravity, 5(10):L155–
L159, oct 1988.
[43] K Sanders. On the construction of hartle–hawking–israel states across a static bifurcate killing horizon.
Letters in mathematical physics, 105(4):575–640, 2015.
[44] W G Unruh. Notes on black-hole evaporation. Physical review. D, Particles and fields, 14(4):870–892,
1976.
[45] R M Wald. General relativity. University of Chicago Press, London, 1984.
[46] R M Wald. Quantum field theory in curved spacetime and black hole thermodynamics / Robert M. Wald.
Chicago lectures in physics. University of Chicago Press, Chicago ; London, 1994.
Appendix A
High Frequency Behaviour of the Reflection
Coefficient
In this appendix, we will be considering solutions to the equation
u′′+(ω2−Vl)u = 0, (A.1)
where
Vl(r) =
1
r2
(
l(l+1)+
2M
r
(
1− q
2M
r
))(
1− 2M
r
+
q2M2
r2
)
. (A.2)
Throughout this appendix, we will be denoting ′ = ddr∗ .
We define solutions Uhor and Uin f as follows:
Uhor ∼ e−iωr∗ r∗→−∞ (A.3)
Uin f ∼ riωr∗ r∗→ ∞. (A.4)
We define the coefficients UI − and UI + as follows
Uhor = UI +Uin f +UI −U¯in f . (A.5)
Note that these coefficients are related to the reflection and transmission coefficients mentioned in 3.3 by the
following
R˜ω,l,m =
UI +
UI −
(A.6)
T˜ω,l,m =
1
UI −
(A.7)
We then prove the following result:
Theorem A.0.1. There exist constants A,C, independent of M,q, l,m,ω such that for all ω ≥C(l+1)2M−2,
|UI +|2 ≤
A(l+1)2
M2ω2
. (A.8)
Proof. We first consider the red-shift energy current
Qred := |u′+ iωu|2−V |u|2 (A.9)
Qred ′ =−V ′|u|2.
by integrating Qred[Uhor], we obtain that
4ω2|UI +|2 =−
∫ ∞
−∞
V ′|Uhor|2dr∗. (A.10)
110 High Frequency Behaviour of the Reflection Coefficient
Therefore, all that is left to show is
−
∫ ∞
−∞
V ′
(l+1)2
|Uhor|2dr∗ ≤ AM−2, (A.11)
for some constant A. To show this, we consider the Morawetz energy current given by
Qymor := y(|u′|2+(ω2−V )|u|2) (A.12)
Qymor
′ = y′|u′|2+ ((ω2−V )y)′ |u|2,
with
y =−exp
(
−
∫ r∗
s=−∞
M2|V ′|
(l+1)2
ds
)
. (A.13)
Integrating Qymor
′, we obtain
2ω2 ≥ 2ω2− e−
∫ ∞
−∞
M2|V ′|
(l+1)2
dr∗ (
4ω2|UI +|2+2ω2
)
(A.14)
=
∫ ∞
−∞
e
−∫ ∞s=−∞ M2|V ′|(l+1)2 ds(M2|V ′|
(l+1)2
|U ′hor|2+
(
M2|V ′|
(l+1)2
(ω2−V )−V ′
)
|Uhor|2
)
dr∗
≥
∫ ∞
−∞
e
−∫ ∞s=−∞ M2|V ′|(l+1)2 ds( M2ω2
(l+1)2
− M
2V
(l+1)2
−1
)
|V ′||Uhor|2dr∗
≥ e−
∫ ∞
−∞
M2|V ′|
(l+1)2
dr∗
(
M2ω2
(l+1)2
−M
2Vmax
(l+1)2
−1
)∫ ∞
−∞
|V ′||Uhor|2dr∗,
provided that
M2ω2
(l+1)2
≥ M
2Vmax
(l+1)2
+1. (A.15)
Now, if we assume that M
2ω2
(l+1)2 ≥ M
2Vmax
(l+1)2 +1+δ , for any fixed δ > 0, we have
−
∫ ∞
−∞
V ′|Uhor|2dr∗ ≤ 2M2δ e
M2Vmax
(l+1)2 , (A.16)
as required. Note that
sup
|q|≤1
M2Vmax
(l+1)2
< ∞. (A.17)
Corollary A.0.1. Let R˜ω,l,m be the reflection coefficient of a Reissner–Nordström spacetime, as defined above.
Then there exists a constant B, independent of M,q, l,m such that
|R˜ω,l,m|2 ≤ B(l+1)
2
1+M2ω2
. (A.18)
Proof. Let A,C be as given by Theorem A.0.1, and without loss of generality, assume A≥ 1. From T -energy
conservation (in this formalism, this is given by I(u′u¯) = const), we know
|UI −|2 = |UI +|2+1≥ 1. (A.19)
Therefore, in the region ω2 ≥C(l+1)2M−2,
|R˜ω,l,m|2 = |UI +|
2
|UI −|2
=
|UI +|2
|UI +|2+1
≤ A(l+1)
2
A(l+1)2+M2ω2
≤ A(l+1)
2
1+M2ω2
(A.20)
111
Then in the region ω2 ≥C(l+1)2M−2 ≥M−2, we have
|R˜ω,l,m|2 ≤ A(l+1)
2
M2ω2
≤ 2A(l+1)
2
1+M2ω2
, (A.21)
and in the region ω2 <C(l+1)2M−2, we have
|R˜ω,l,m|2 ≤ 1≤ (1+C)(l+1)
2
1+C(l+1)2
≤ (1+C)(l+1)
2
1+M2ω2
. (A.22)
Therefore letting B = max{1+C,2A} gives the result.