JULIANO CHOI RODRIGUES
IMPLICAÇÕES COSMOLÓGICAS DA RADIAÇÃO HAWKING DO
HORIZONTE DE DE SITTER
COSMOLOGICAL IMPLICATIONS OF HAWKING RADIATION FROM THE
DE SITTER HORIZON
CAMPINAS
COSMOLOGICAL IMPLICATIONS OF HAWKING RADIATION FROM THE
DE SITTER HORIZON
IMPLICAÇÕES COSMOLÓGICAS DA RADIAÇÃO HAWKING DO
HORIZONTE DE DE SITTER
Dissertation presented to the "Gleb Wataghin" Institute of Physics of the University of Camp-inas in partial fullfilment of the requirements for the degree of Master, in the area of Physics.
Dissertação apresentada ao Instituto de Física "Gleb Wataghin" da Universidade Estadual de Campinas como parte dos requisitos exigidos para a obtenção do título de Mestre em Física, na área de Física.
Supervisor/Orientador: Dr. Donato Giorgio Torrieri
Este exemplar corresponde à versão final da dissertação defendida pelo aluno Juliano Choi Rodrigues, e orientado pelo Prof. Dr. Donato Giorgio Torrieri
CAMPINAS
Biblioteca do Instituto de Física Gleb Wataghin Lucimeire de Oliveira Silva da Rocha - CRB 8/9174
Rodrigues, Juliano Choi,
R618c RodCosmological implications of Hawking radiation from the De Sitter horizon / Juliano Choi Rodrigues. – Campinas, SP : [s.n.], 2019.
RodOrientador: Donato Giorgio Torrieri.
RodDissertação (mestrado) – Universidade Estadual de Campinas, Instituto de Física Gleb Wataghin.
Rod1. Cosmologia. 2. Gravitação. 3. Teoria quântica de campos. 4. Hawking, Radiação de. 5. De Sitter, Espaço de. I. Torrieri, Donato Giorgio, 1975-. II. Universidade Estadual de Campinas. Instituto de Física Gleb Wataghin. III. Título.
Informações para Biblioteca Digital
Título em outro idioma: Implicações cosmológicas da radiação Hawking do horizonte de De Sitter
Palavras-chave em inglês: Cosmology
Gravitation
Quantum field theory Hawking radiation De Sitter space
Área de concentração: Física Titulação: Mestre em Física Banca examinadora:
Donato Giorgio Torrieri [Orientador] José Ademir Sales de Lima
Pedro Cunha de Holanda Data de defesa: 26-06-2019
Programa de Pós-Graduação: Física
Identificação e informações acadêmicas do(a) aluno(a)
- ORCID do autor: https://orcid.org/0000-0003-1834-6440 - Currículo Lattes do autor: http://lattes.cnpq.br/0059614458281070
MEMBROS DA COMISSÃO JULGADORA DA DISSERTAÇÃO DE MESTRADO DE
JULIANO CHOI RODRIGUES – RA 136381 APRESENTADA E APROVADA AO
INSTITUTO DE FÍSICA “GLEB WATAGHIN”, DA UNIVERSIDADE ESTADUAL DE CAMPINAS, EM 26 / 06 / 2019.
COMISSÃO JULGADORA:
- Prof. Dr. Donato Giorgio Torrieri – Orientador – DRCC/IFGW/UNICAMP
- Prof. Dr. José Ademir Sales de Lima – IAG/USP
- Prof. Dr. Pedro Cunha de Holanda – DRCC/IFGW/UNICAMP
OBS.: Informo que as assinaturas dos respectivos professores membros da banca
constam na ata de defesa já juntada no processo vida acadêmica do aluno.
CAMPINAS
2019
I dedicate this work to my family, friends and
profes-sors who inspired me in this humble path of searching
for knowledge.
I would like to begin this by saying that I’m forever grateful to my advisor, Dr
Giorgio Torrieri, for his patience during my undergrad courses lectured by him,
which were filled with questions and discussions after classes that fueled my
curiosity for learning Physics and certainly led me to where I am today. Also I
would like to thank him for his support and motivation for me to keep pressing on
the research and learning from the previous mistakes along the way.
In the same spirit, I would like to thank some of the professors that inspired
me along the way: Dr Pedro Cunha de Holanda for introducing me to the
wonder-ful world of cosmology, Drs Eduardo Miranda, Marcelo Moraes Guzzo and
Mar-cus Aloizio Martinez de Aguiar for their excellent classes and thought-provoking
discussions which caused me to dive deep into the world of Theoretical Physics.
In special, I would like to thank my girlfriend Ísis, for her motivation to keep me
on track, and for her incredible patience during the times I was working on this
dissertation. Words may never fully describe what it means to feel your support
during these troubling times.
Some people say that great friendships are formed in the heat of battle, and
learning Physics was certainly a battle I was honored to face with my friends, to
whom I can only thank for their support: Danilo, Hugo and William, thank you for
all the great times we spent talking about the most absurd things, from solitons
in pools to philosophical discussions filled with mind blows. Those are memories
I’ll certainly cherish and hold on to when looking back.
I would also like to thank CNPq (process number 132436/2017-5) for their
financial support.
O problema da constante cosmológica é um dos maiores mistérios da física teórica atualmente, onde há uma aparente discordância entre a densidade de energia do vácuo observada como uma constante cosmológica Λ e o valor teórico da energia de ponto zero inferido pela teoria quântica de campos. Aqui neste trabalho introduzimos sobre a radiação Hawking de horizontes aparentes, em específico o horizonte cosmológico de de Sitter e estudamos a dinâmica entre Λe a radiação Hawking. Propomos que o backreaction da radiação Hawking no espaço de de Sitter é o decaimento deΛcom a produção de radiação homogênea e isotrópica.
The cosmological constant problem is one the biggest mysteries of theoretical physics today, where there is an apparent disagreement between the vacuum energy density observed as a cosmological constantΛand the theoretical value of the zero-point energy inferred by quantum field theory. In this work we introduce about Hawking radiation of apparent horizons, more specifically about the cosmological horizon of de Sitter and study the dynamics betweenΛand Hawking radiation. We propose that the backreaction of Hawking radiation in de Sitter space is the decay ofΛwith the production of homogeneous and isotropic radiation.
1 1 Introduction 11
1.1 Cosmological Constant Problem . . . 11
1.2 Some Inconsistencies in Big Bang Cosmology . . . 11
1.2.1 Theory of Inflation . . . 12
2 2 General Relativity/Standard Cosmology 14 2.1 General Relativity . . . 14
2.1.1 Λ ↔ ρvacCorrespondence . . . 15
2.1.2 Energy-Momentum Tensor Conservation . . . 16
2.2 Standard Cosmology: theΛCDMModel . . . 17
2.2.1 Friedmann-Lemaître-Robertson-Walker(FLRW) Metric . . . 17
2.2.2 Friedmann equations . . . 17
2.2.3 de Sitter Space . . . 18
2.2.4 ΛCDM . . . 18
3 3 Hawking Radiation-Black Holes 19 3.1 Introduction . . . 19
3.2 Tunneling . . . 20
3.2.1 Painlevé-Gullstrand Coordinates . . . 20
3.2.2 Derivation . . . 21
3.2.3 Entropy . . . 22
4 4 Hawking Radiation-de Sitter 23 4.1 Birkhoff’s Theorem and generalization . . . 24
4.1.1 Generalized case . . . 24
4.2 Action for s-waves . . . 24
4.2.1 Relativistic Hamilton-Jacobi Equation . . . 24
4.2.2 Imaginary part ofW+ . . . 26
4.2.3 Evolution of Schwarzschild . . . 28
4.2.4 Evolution of dS . . . 28
4.2.5 Temperature of SdS . . . 29
A Some calculations about GR 35 A.1 Bianchi’s Identities and other symmetries . . . 35
A.2 Einstein tensor’s covariant conservation . . . 35
B.1.1 Contra/covariant metric . . . 38 B.1.2 Christoffel Symbols . . . 39 B.1.3 Geodesics . . . 40 B.2 P-G Coordinates . . . 43 B.2.1 Contra/covariant metric . . . 43 B.2.2 Christoffel Symbols . . . 44 B.2.3 Geodesics . . . 46
1
Introduction
1.1
Cosmological Constant Problem
One of the open problems in theoretical physics that connects the small scale of elementary particle physics to the largest scale in cosmology is known as the cosmological constant problem. The problem is a discrepancy between the expected value of the vacuum energy density from Quantum Field Theory(QFT) and the observed value in cosmology. Vacuum energy density should be the same throughout space, and therefore from Lorentz invariance, we see thatTµνvac∝ gµν(shown in section2.1.1).
From QFT, we can calculate some of the contributions to the vacuum energy density from the zero modes,hρvaci ∝ h0|u0u0T00|0i, and one being from the Higgs vacuum expectation value[1]:
ρvac= − 1 4m 2 Hhνi2= − √ 2 16 m2H GF ≈ −1.2 × 10 8GeV4
where mH ≈ 125 GeV is the Higgs mass, hνi is the vacuum expectation value of the Higgs, GF = 1.16 × 10−5GeV−2 is the Fermi Constant.
The standard model of cosmology, known asΛCDM(Section2.2), uses the observations of Supernovae and anisotropies of the cosmic microwave background, among other observations, to the parameter fitting where, according to this model, we live in a universe where there are currently four relevant components of the energy-momentum tensor according to their equations of state(P= wρ), baryonic matter (P≈ 0), dark matter (unknown, but the model assumes P≈ 0), radiation (P= ρ3) and dark energy (P= −wρ), and the last one is dominating the "energy budget", which, has the same characteristics of a cosmological constant(w≈ −1), and we obtain from the data that the vacuum energy density from the cosmological constant is[2](in natural units):
ρvac= ΩΛρcrit1= ΩΛ 3H02
8π G≈ 2.5 × 10
−47GeV4
and this clearly is a tension of over 50 orders of magnitude between theory and observation.
1.2
Some Inconsistencies in Big Bang Cosmology
Despite the successes that the current model ofΛCDMhas as an explanation of our cosmology, there are still open questions that remain, and three of them, known as the flatness problem, horizon problem and magnetic monopole problem.
1
ρcr=3H
2
•The flatness problem is a problem of fine-tuning, where the observable of the density parameter of matter and energy in the universe,Ω = ρρ
cr has to be extremely close to a critical value of 1, which, by the current observations, the current value is|Ω − 1| < 0.01, so, at Planck epoch, the deviation from 1 should be at most10−62.
•The horizon problem is the observed homogeneity of causally disconnected regions of space without any mechanism to make this plausible by setting similar initial conditions everywhere.
Note: It is important to remark that both these problems are not inconsistencies of the model
but are also outside the predictive power of the theory, which is why theorists propose solutions to both cases, in an attempt to calculate, a priori, why we observe a flat universe with such homogeneity.
•The magnetic monopole problem is actually a result of some types of extensions of the stan-dard model of particle physics, where the idea is that the fundamental forces are actually a single gauge theory that unify for sufficiently high energies/temperatures such as the ones in the early universe. As such, these are known as Grand Unified Theories(GUT), which predict that, in the early universe, there is a huge production of heavy and stable "magnetic charges" called mag-netic monopoles, and the fact that we do not observe such exotic relics of the early universe is seen as a problem by some theorists.
These problems are believed to be solved by a theory known as the inflation theory.
1.2.1
Theory of Inflation
The inflation theory was developed in 1979 by Alan Guth to attempt a solution to these open problems in cosmology. While investigating the problem of the exotic relics, he found that if the universe has an scalar field in a positive-energy false vacuum state, usually called inflaton, then the fluctuations of such field around the false vacuum generate, according to General Relativity, an exponential expansion of the space, which is mathematically similar to the effect of the cos-mological constant. We start from the Einstein-Hilbert action summed to an action of a scalar field with a canonical kinetic term and a potencial describing the self-interactions of the field[3]:
S= Z d4x√−g R 2+ 1 2g µν ∂µφ∂νφ − V (φ) = SEH+ Sφ
From which we derive
Tµν(φ)≡ −√2 −g δSφ δgµν = ∂µφ∂νφ − gµν 1 2g µν ∂σφ∂σφ + V (φ)
2.2), which gives an equation of state parameter: w= pφ ρφ = 1 2˙φ2−V 1 2˙φ2+V
which shows us that for high potentials, w can be negative, possibly leading to accelerating expansion (wφ< −13), and in the limit of 21˙φ2 V, we findwφ→ −1, effectively functioning as a
cosmological constant.
As this field goes from a metastable vacuum to the true vacuum, there is an exponential expansion of the metric and also particle production, which happens until some mechanism (model-dependent part) makes the potential "turn off" and then the inflationary phase ends. The details and predictions of this mechanism also depend on the type of field, its potential and the interaction of such field with other particle fields, which makes this highly model-dependent, and also why the criticism that goes into inflation theory is that the potential is inferred ad hoc to adequately fit the data, so the type of interaction of such scalar field would remain a mystery, and also using a scalar field is problematic due to the lack of elementary scalar fields that have been found in our search of elementary particles, which is basically one, the Higgs Boson, but the data concordance of cosmological observations constrain severely any type of Higgs Inflation due to its high mass, which would produce different distortions in the CMB, so we would need a new and lighter scalar field to be inflaton.
Our Approach
We believe there is a possible solution to these problems that use Hawking radiation, which is a phenomenon discovered for the curved spacetime of a black hole, that can evolve an initial vacuum state to a mixed state with particle production from the apparent horizon. Since the definition of a particle depends on asymptotic conditions and the apparent horizon breaks en-tanglement between points in spacetime, this is common to all spacetimes with an apparent horizon, such as de Sitter(dS) with its cosmological horizon.
DS space has been the subject of extensive study in the literature and many studies point towards its vacuum’s instability due to different effects, such as thermodynamic instability[4], infrared effects screening the cosmological constant[5], loss of degrees of freedom across the horizon[6]. This makes it likely that de Sitter space evaporates like black holes do, due to the backreaction.
Our approach differs from [6] as we are trying a calculation of the backreaction that is not an adiabatic approximation, not depending on smallΛor slowly varyingH,−H˙
H2 1.
Such combination has shown some qualitative and quantitative aspects that lead us to be-lieve that the cosmological constant changes in time due to the horizon radiation and this has implications in the problems described earlier.
2
General Relativity/Standard Cosmology
First, we present some of the important points in the theory of General Relativity that are related to the cosmological constant and the vacuum energy density that it is equivalent to, so we can understand the problem at hand.2.1
General Relativity
The theory of General Relativity is a theory of gravity that was postulated by Einstein in 1915 as a more accurate description than the Newton’s law of Universal Gravitation, proposing a unification of the concepts of spacetime, gravity and geometry. In this theory, gravity is a result of the energy and momentum changing the surrounding geometry of spacetime, which is summarized in the words of John Archibald Wheeler:
"Spacetime tells matter how to move; matter tells spacetime how to curve." The theory of General Relativity can be summarized as two postulates:
Postulate 1:
Spacetime is a 4-D Lorentzian (semi-Riemannian
| {z }
a
) manifold with a Levi-Civita | {z } b connection. a) ds2= p
∑
j=1 (dxj)2− n∑
j=p+1 (dxj)2, n = p + q, signature(p, q)ds2=−(dx0)2+(dx1)2+(dx2)2+(dx3)2, signature(3,1) ←(we use this one) ds2=+(dx0)2−(dx1)2−(dx2)2−(dx3)2, signature(1,3)
b)The affine connection∇is a Levi-Civita connection if:
b1)metric compatible:∇g = 0 b2)without torsion:∇XY− ∇YX = [X ,Y ]
Postulate 2:
The dynamics of spacetime is given by the principle of least actionδS = 0applied to the Einstein-Hilbert action: SEH = Z 1 2κ(R − 2Λ) +
L
m √ −g d4x, κ =8 π G c4 (2.1)Given a matter Lagrangian
L
mand the principle of least actionδ S = 0: 0 = δS = Z 1 2κ δ(√−g(R − 2Λ)) δgµν + δ(√−gL
m) δgµν δgµνd4x = Z 1 2κ δ(R − 2Λ) δgµν + (R − 2Λ) √ −g δ√−g δgµν +√1 −g δ(√−gL
m) δgµν δgµν √ −g d4x where the variation of the metricδgµνis generic, so[ ] = 0δR δgµν− 2 δΛ δgµν+ (R − 2Λ) √ −g δ√−g δgµν = κ√−2 −g δ(√−g
L
m) δgµν Rµν− 2 7 0 δΛ δgµν+ (R − 2Λ) √ −g − √ −g 2 gµν = κTµνwhich results in the Einstein’s Field Equations(EFE): Rµν−1
2R gµν+ Λ gµν= κTµν (2.2)
Note
It is important to note that Λ is a scalar that can be added to the Lagrangian density and is usually assumed to be independent of the geometry and, therefore, have no dependence on the metric, which justifies that last line of δΛ
δgµν = 0. But that is an assumption which has its origin in
the first cosmological model, such as Einstein’s first model of cosmology, where he introducedΛ as a way to provide a repulsive force to stop the attraction of astronomical bodies and reproduce a static universe. But this model was unstable under perturbations, and so Einstein abandoned the idea of a cosmological constant.
2.1.1
Λ ↔ ρ
vacCorrespondence
If we assume that the vacuum energy must be homogeneous and isotropic, then it is a perfect fluid:
Tµν= (ρ + p)uµuν+ p gµν (2.3)
but there should be no preferred direction, therefore the first term should be zero:
ρ + p = 0 =⇒ p = −ρ
Tµνvac= pvacgµν= −ρvacgµν
and rearranging EFE(2.2), we have:
Gµν= κ
Tµν−Λ κgµν
Therefore: ρvac= Λ c4 8 π G (2.4) And now we see that either the cosmological constant is vacuum energy density, or that we can have two parts that give the same effect:
Λeff= Λb+ Λvac= Λb+
8πGρvac
c4 (2.5)
whereΛbis the constant in the Einstein-Hilbert action, andΛvacis the vacuum energy density of
fields.
2.1.2
Energy-Momentum Tensor Conservation
Applying the covariant derivative in the EFE(2.2):
* 0 ∇µGµν+ (∇µΛ) gµν+ Λ: 0 (∇µgµν) = ∇µ(κTµν) (2.6)
Where the first term cancels due to the Bianchi Identity(shown in the sectionA.2), and the third one because of the metric-compatible connection. Therefore:
∇µ(κTµν) = (∂µΛ) gµν ∂µ 8πG c4 Tµν+ κ∇µ(Tµν) = (∂µΛ) gµν ∇µTµν= (∂µΛ) gµν− ∂µ 8πG c4 Tµν
Which shows us one of the underlying assumptions of General Relativity, in order to have a covariant conservation of the energy-momentum tensor:
•The scalar functionsΛ,Gandcare constants.
We assume thatc and G are constants, but a question for the future remains whether these assumptions are justified.
There are theoretical reasons to believe that it is more natural to assume a time-dependentΛ(t), instead of restricting it to a fixed constant [7]. Also under these assumptions we see that a time changingΛ(t)produces a current of matter-energy:
∇µTµν= 1
κ(∂µΛ(t)) g
µν= ∂
µρvac(t) gµν
To simplify the language for reference purposes, the time changing functionΛ(t)will still be called cosmological constant(CC) throughout this work.
2.2
Standard Cosmology: the
ΛCDM
Model
2.2.1
Friedmann-Lemaître-Robertson-Walker(FLRW) Metric
One of the first solutions found for the EFE(2.2), the FLRW metric describes a homogenenous and isotropic spacetime and it was proven by Robertson and Walker to be unique with those properties (Note:c= 1in this section):
ds2= −dt2+ a(t)2 dr2 1 − k r2+ r 2 dΩ2 (2.7)
where a(t) is the scale factor, k is the curvature of 3D space anddΩ2= dθ2+ sin2θ dφ2.
2.2.2
Friedmann equations
From this metric, one can calculate the Christoffel symbols and then obtain the Ricci tensor and Ricci scalar, which are shown in the Appendices. With those, we return to the EFE(2.2) and examine the(00)component with observeruµ= (1, 0, 0, 0):
R00− 1 2Rg00+ Λg00= 8πGT00 −3a¨ a+ 3 ¨ a a+ ˙ a2 a2+ k a2 ! − Λ = 8πGρ
which results in the first Friedmann equation: ˙ a2+ k a2 − Λ 3 = 8πG 3 ρ (2.8)
The mixed components(0i, i j, fori6= j)all vanish and now we examine the(ii)component:
Rii−1 2Rgii+ Λgii = 8πGTii ¨a a+ 2 ˙ a2+ k a2 − 3a¨ a+ 3 ˙ a2+ k a2 + Λ g>ii= 8πGpg>ii −2a¨ a− ˙ a2+ k a2 + Λ = 8πGp −2a¨ a− 8πG 3 ρ − Λ 3 + Λ = 8πGp which results in the second Friedmann equation:
¨ a a− Λ 3 = − 4πG 3 (ρ + 3p) (2.9)
2.2.3
de Sitter Space
From (2.8), we can see the evolution of the scale factor for the simplest case we will study here, which is the case of no matter density in flat space with a positive cosmological constant (ρ = 0, k = 0, Λ 6= 0), also known as de Sitter(dS) space:
˙ a a = 1 a da dt = r Λ 3 =⇒ Z da a = Z r Λ 3dt =⇒ a(t) = e c1et/`, ` = r 3 Λ wherec1is an integration constant which we can redefine if we seta(0) = a0, so:
a(t) = a0et/`.
We see that dS space has an exponentially expanding spatial section in this comoving frame. Now from (2.9), we see that the acceleration of dS space is positive:
¨ a(t) = a0 Λ 3e t/`
2.2.4
ΛCDM
The currect standard model of cosmology, ΛCDM uses several observations of astrophysical phenomena, such as supernovae, cosmic microwave background, large scale structure, baryon acoustic oscillation, to constrain the observed parameters of the Big Bang cosmological model, whereΛstands for the dark energy, possibly being a cosmological constant, and CDM stands for cold dark matter, which we have no definite candidate nor observed its properties conclusively, but we believe to be massive enough to cool down fast as the universe expanded and have very small or even no interaction with the particles known in the Standard Model of Particle Phyiscs. From the data of Planck(2018)[2], we see that the distribution of the energy content is the following:
Baryonic Matter: P≈ 0 Ωb= 0.04897(31)
Cold Dark Matter: P≈ 0 Ωc= 0.2607(20)
Dark Energy: P= −ωΛρ ΩΛ= 0.6889(56) ωΛ= −1.03(3)
3
Hawking Radiation-Black Holes
3.1
Introduction
The study of black holes done in the early 70’s by Wheeler, Bekenstein, Hawking, among others, have shown many properties which looked very much like the laws of thermodynamics, which brought many interest as to why there were such similarity. One of those is a theorem proven by Hawking that shows that assuming a weak energy condition, the area of black holes is a non-decreasing function of time:dAdt ≥ 0. This inspired the idea that maybe the area of the black hole is connected to entropy, which is also a non-decreasing function of time.
Bekenstein showed, in 1973, [8] that from the standpoint of information theory, black holes have entropy based on the fact that the horizons hide information from observers, and should be proportional to the event horizon areaS ∝ A.
In 1974, Hawking showed that quantum fields in a Schwarzschild background had a new phe-nomena of particle production due to the curved background, which was the transition of an initial vacuum state to a mixed state of thermal equilibrium proportional to the surface gravity of the Schwarzschild radius of a black hole which is at the event horizon, and this phenomena is known today as Hawking radiation.[9]
This discovery naturally led to other questions that if we were to conserve energy locally, this particle emission produces a flow of energy from the black hole region to the radial infinity, which implies that the black hole loses energy and therefore mass, and this is a mechanism for black hole evaporation, which started a new field of study usually called Quantum Black Hole Thermodynamics.
Since then, several studies have shown that the particle production and the appearance of a thermal spectrum is not limited to Schwarzschild metric and happens in the presence of apparent horizons[10], which indicated that even a simple solution of the EFE(2.2) such as the dS space could have this particle production, but some other questions arise from this:
•What is the analogue phenomena of "black hole evaporation" in dS space? •Does the existence of the horizon radiation imply that dS space is unstable? •How do we define the energy and entropy outside the dS horizon?
3.2
Tunneling
3.2.1
Painlevé-Gullstrand Coordinates
Before we begin the derivation, let’s start by showing a different coordinate system used by Parikh and Wilzcek called Painlevé-Gullstrand coordinates for black holes.
The Schawarzschild metric in static form: ds2= − 1 −2M r dts2+ dr 2 1 −2Mr + r 2 dΩ2, rs= 2M
to which we apply the coordinate changet = ts− f (r):
ds2= −1 −rs r dt2+ 21 −rs r f0dt dr+ 1 1 −rs r +1 −rs r f02 | {z } g(r) dr2+ r2dΩ2
We make the metric regular at the horizon by requiring that the transformation makesg(r) = 1: 1 1 −rs r +1 −rs r f02= 1 1 −rs r f02= 1 − 1 1 −rs r f02= rs r 1 −rs r 2 =⇒ f 0= prs r 1 −rs r
therefore the metric becomes:
ds2= −1 −rs r dt2+ 2r rs rdt dr+ dr 2+ r2 dΩ2.
As is shown in the Appendix, we can derive the radial null geodesics from this metric, which we will use in following derivations:
˙r = ±1 −r rs
r (3.1)
which we can see in the following graph:
We can see in the Figure3.1, that even though the metric is regular in the horizonr= rs, the
outgoing geodesics has a sign change inr= rs, so classically, there is no outgoing waves that
0 1 2 3 4 5 −2 −1 0 1 −2 r rs ˙r
Figure 3.1: blue=outgoing solution, red=ingoing solution
3.2.2
Derivation
Now, I’ll show the tunneling method first derived by Parikh and Wilzcek[11,12]:
The tunneling rate is given by the imaginary part of the action as seen in the WKB method of approximation(We use natural units~ = kb= G = c = 1):
Γ ∼ e−2 Im S≈ e−βE =⇒ T = 1 β=
E 2Im S Therefore we start from the action:
S= Z p dq− Z
H
dt Im S=Im Z p dq− Im ZH
dt= Im Z p dqwhere we use Hamilton’s equations to change from momentum to energy in a radial geodesic:
Im S=Im Z rout rin Z pr 0 d prdr, ˙r = dH d pr r Im S=Im Z rout rin Z E 0 dH ˙r dr= Im Z E 0 dH Z rout rin dr ˙r (3.2)
Now using3.1in3.2we obtain:
Im S= E Z rout rin dr ±1 −prs r
does not have a pole, and therefore only the outgoing solution has an imaginary part: Im S= E Z rout rin dr 1 −prs r = 2πrsE= 4πME
and we obtain the result derived by Hawking: TSch= E
2Im S = 1 8πM
Note
One important point to make is that these temperatures derived for black holes or de Sitter space are not expected to be exactly temperature, because it is not expected to be in thermal equilibrium with matter, and therefore it is more of a parameter to describe an adiabatic evolution of the space. As we expect the black hole evaporation to increase the temperature over time due to its backreaction, we see that this "temperature" is not in equilibrium, and the same is expected of dS space.
3.2.3
Entropy
The entropy change related to an infinitesimal mass being dropped to the black hole is: dS≡ dQ
T = 8πMdQ = 8πMdM = d(4πM
2) = d(πr2 s)
where the area of the black hole is
A= 4πrs2 so we see that
S= A 4
4
Hawking Radiation-de Sitter
As we have seen in the previous chapter, the original Hawking radiation derived for black holes can be obtained with a tunneling approach, but as was mentioned before, there are many rea-sons and arguments as to why the phenomenon of Hawking radiation is not limited to black holes. In [10], it is hypothesized that every solution of the EFE(2.2) with apparent horizons should have a temperature associated with it, and it is mentioned that dS space is included among those, and they calculate the "temperature" of dS space(as said in the previous chapter, not exactly a temperature):
TdS= 1 2π`
This brings a whole series of questions as to what exactly is this temperature in dS space: • In dS space, there is no asymptotically flat region to "prescribe" energy, so it is observer-dependent and not easily defined.
•As the region of the black hole is contained by the Schwarzschild radius, the idea of black hole evaporation is clear from the flow of energy from the inside region to the outside. Even if it is classically forbidden.
In dS space, the only parameter of the metric is the cosmological constant Λ, which was shown to be equivalent to a vacuum energy. Which begs the question as to whether the Hawking radiation should "extract" energy from the cosmological constant.
•dS space does not have a global timelike Killing Vector, which means that there is no agreement of particle content between observers, similar to Unruh effect detectors in different frames, where an inertial one sees vacuum, the accelerated one sees a thermal bath.
•In contrast to Minkowski space, that every observer can "reach" any other point in space, given enough time, In dS space, observers that know the initial conditions, do not know the final ones, and vice-versa, due to the causal horizon.
These points are indications that dS space have a process of quantum decoherence across the horizon. The loss of information due to inacessible degrees of freedom by the observer should be part of the origin of the Hawking radiation.
Now we intend to show what is the expected qualitative aspects of the backreaction of Hawk-ing radiation comHawk-ing from the causal horizon.
4.1
Birkhoff’s Theorem and generalization
This theorem will not be proven here, but can be found in many GR textbooks. This theorem says the following:
"Any spherically symmetric solution of the Vacuum Einstein’s Equations (Tµν= 0) must be static
and asymptotically flat."
This result works ifΛ = 0, and can also be interpreted as:
Around any spherically symmetric body, locally, the metric is uniquely given by a Schwarzschild metric, determined by the body’s mass.
4.1.1
Generalized case
The generalized case withΛimplies neither staticity nor an asymptotically flat region, but only the condition of uniqueness of the metric, locally, being the Schwarzschild-de Sitter(SdS) metric:
ds2= − 1 −rs r − r2 `2 dt2+ 1 1 −rs r − r2 `2 d r 2+ r2 dΩ2
4.2
Action for s-waves
A different derivation of the Hawking radiation temperature can be done with Hamilton-Jacobi Equation in the following way:
4.2.1
Relativistic Hamilton-Jacobi Equation
(Derivation follows [13])
Given a spherically symmetric spacetime with a locally static timelike Killing Vectorkµand norm kµkµ= − f and spatial Killing Vectorsξµi, which are the generators of SO(3), we find a frame of static observers with 4-velocity proportional to the Killing vector:
etµ= √1 fk
µ (4.1)
as the timelike vector field of a spherically symmetric orthonormal frame,eµr, eµj chosen such that
L
ker=L
kej=L
ξier = 0, whereL
is the Lie derivative. In this case we can write the metric as:gµν= ηabeµaeν
Given the energy-momentum relation gµνPµPν+ (mc) 2= 0 and P µ= ∂S ∂xµ we have: gµ∂µS∂νS+ (mc)2= 0 (4.3)
Then using4.2and4.3we have:
ηabeµa∂µSeνb∂νS+ (mc)
2= 0 (4.4)
If we assume that the dominant contribution to the amplitude is the s-wave, then S is invariant under rotations:
L
ξiS= ξi∂µS= 0and thereforeeµj∂µS= 0S must also be covariantly constantunder the action ofkµ, so
L
kS= kµ∂µS= −α. Thisαis usually associated to energy in frames where the observer is in a region which is asymptotically flat(away from Black holes as in r→ inf), or in other words, regions where the norm of the Killing Vector is unity. So we see the following relation:ε = −etµ∂µS= α √
f (4.5)
Now using4.1and4.4we get:
ηabeµa∂µSeνb∂νS+ (mc)2= 0 ηtt(eµt∂µS)2+ ηrr(eµr∂µS)2+ ηj j(eµj∂µS)2+ (mc)2= 0 ηtt k µ √ f∂µS 2 + ηrr(eµr∂µS)2+ ηj j(0)2+ (mc)2= 0 (−1)α 2 f + (e µ r∂µS)2+ (mc)2= 0 −α 2 f + ∂S ∂σ 2 + (mc)2= 0 becauseeµr∂µ= ∂
∂σ is the derivative with respect to the proper distance. =⇒ S = ± Z dσ s α2 f − (mc) 2+ Θ (4.6)
whereΘis a function of integration independent ofσ.
Also, sincekµ∂µ=∂λ∂ is the derivative with respect to the parametrization coordinate of the Killing
Vector, andkµ∂µS= −α, then
S= −αλ + Θ0, (4.7)
Therefore, S= −αλ +W +C (4.8) W = ± Z dσ s α2 f − (mc) 2. (4.9)
In 4.8 we see that an imaginary part of S comes from a pole in the integral W and/or from a complex part of C. A pole in W corresponds to a Killing horizon. Since the Killing Vector has a vanishing norm on the horizon, the integral will have a pole for incoming and outgoing particles, and therefore we require that the probability of absorption to be unity in the classical limit, so it is required that C be complex and chosen. The requirement is thatImC= −ImW−, and in any
static patch we haveImW+ = −ImW−, so we have that
Im S= Im (W++C) = 2ImW+
From the WKB approximation, we’ve seen that the temperature is related to the imaginary part of the action: T = ε 2 Im S ~ kb = α 4√f ImW+ ~ kb (4.10)
4.2.2
Imaginary part of
W
+Spherically symmetric metric in static form:
ds2= − f (r) c2dt2+ 1 g(r)dr
2+ h2(r)d Ω2
To calculateImW±, we use a Taylor expansion of f and g around the pole:
f(r) = *0 f(r0) + f 0 (r0)(r − r0) + 1 2f 00 (r0)(r − r0)2) + ... g(r) = g(r0) + g 0 (r0)(r − r0) + 1 2g 00 (r0)(r − r0)2+ ...
and ifg(r0) = 0, then the measuredσ =√dr
g(r) gives us, in first order,σ = 2 √ g0(r0) √ r− r0 Then f(r) = 14 f0(r0) g 0
(r0)σ2, and for the casem= 0:
ImW+= p 2 α f0(r0) g0(r0) c
Im Z dσ
σ
Now to calculate this contour integral, we see that there an important detail in the contour of integration chosen.
A common mistake in the application of the method is the choice of contour of integration:
C3 C1
C4 C2
Correct contour for proper distance.
C1
C2
C3
C4
Wrong contour generates extra factors of 2.
We can see the reason for this choice of contour as we are integrating over two points along the proper distance, which crosses the pole(horizon), and proper distance is given bydσ =pd r
g(r), so the sign change across the pole changes the proper distance from real to imaginary, so we have to use the left contour, and we obtain:
Im Z C dσ σ = π 2 For example, for Schwarzschild, f(r) = g(r) =1 −2GM
c2r , so f0(rs) = c 2 2MG andImW+ = 2π αMG
c3 , which results in the usual black hole temperature: TBH = 1 8π M ~ c3 G kb 1 √ f (4.11)
We can see that in the Schwarzschild, the redshift factor f(r)is equal to1forr− > inf, so the static observer at infinity sees the usual BH temperature 8πM1 , but as you approach the black hole, f(r)becomes smaller, and therefore you observe a larger temperature.
For dS space, f(r) = g(r) =1 −r`22 =⇒ f0(`) = −2`, soImW+= π ` α 2 c , which results: TdS= 1 2 π ` ~ c kb 1 √ f (4.12)
In dS, we can see that the redshift factor is equal to1atr= 0, so the static observer at the origin sees the dS temperature2π`1 , which increases for static observers closer to the horizon.
4.2.3
Evolution of Schwarzschild
Now, to illustrate the picture of the Birkhoff’s theorem applied to theses cases:
r Shell rb EH M− E M
We see that an observer outside the shell of energyE of the s-wave still observes a black hole with massM, and between the shell and the Event Horizon (EH), we observe a black hole with MassM− E, as expected to preserve causality.
4.2.4
Evolution of dS
r Shell CH rb rc= ` F− EH dS SdSWe see in dS space, that the emission of a shell from the cosmological horizon(CH), changes the metric from dS to SdS, but there is an important detail to be noticed. In this coordinate system, the observer is atr= 0, and the change in the metric happens fromr= `, so the change in the metric is how to interpret incoming signalsafter the shell reachesr= 0, or in other words, how the observer inr= 0interprets the events for other observers outside the shell.
It is important to note that in this case, the massless shell eventually reaches a critical point of density, forming a future event horizon(F-EH), which indicates that in this coordinate system, the radiation from the horizon nucleates into a future black hole, but we believe this is a feature of the coordinate system, which is spherically symmetric.
An important feature of this analysis is that once the shell starts to propagate, the external region to the shell becomes a SdS space, which we will show that has a lower temperature.
4.2.5
Temperature of SdS
In SdS, we have f(r) =1 −rs r − r2 `2which has the following roots:
rb= 2 √ 3` 3 cos π 3+ ψ 3 , ψ = arccos 3 √ 3rs 2` ! rc= 2 √ 3` 3 cos π 3− ψ 3 rn= −2 √ 3` 3 cos ψ 3
rbis the black hole radius,rcis the cosmological horizon radius,rnis negative, so it is neglected. In our case,Mis the mass-energy of the shell, sors `, and we can taylor expand around the x=3 √ 3rs 2` ≈ 0: rb≈ 2 √ 3` 3 x 3+ 4x3 81 + O(x 5) = 2 √ 3` 3 " √ 3rs 2` + 4 81 81√3r3s 8`3 # = rs 1 +r 2 s `2 ≈ rs rc≈ 2√3` 3 " √ 3 2 − x 6+ O(x 2) # = 2 √ 3` 3 " √ 3 2 − √ 3rs 4` # = `h1 − rs 2` i ≈ `h1 − rs 2` i ≈ `
It is enough to use this first order approximation to see a lower temperature in SdS, so we will userb= rs andrc= `.
Now we can calculate the temperature in SdS, approximately, using (4.10) and f0(r) = rs r2−
2r `2:
Black Hole Temperature TSdS[BH]≈ | f 0 (rs)| 4π√f ~ c kb = 1 4πrs 1 −2r 2 s `2 1 √ f ~ c kb = TSch 1 −2r 2 s `2
Cosmological Horizon Temperature
TSdS[CH]≈| f 0 (`)| 4π√f ~ c kb = 1 2π` h 1 − rs 2` i 1 √ f ~ c kb = TdS h 1 − rs 2` i = TdS " 1 −M √ Λ √ 3 G c2 # < TdS
and from this we conclude that Hawking radiation from the horizon lowers the temperature of the next "iteration".
Conclusions
In this work, we explored the phenomenon of Hawking radiation as a quantum tunneling process across the apparent horizon, and concluded that Hawking radiation in dS space has a backre-action that results in the following interpretation:
Interpretation
We started from a dS space, and a vacuum state evolved to radiation due to the horizon, which propagates from the horizon to the observer atr= 0. Every region outside this shell changes to SdS, as seen from the Birkhoff’s Theorem. But in this scenario, the observer atr= 0sees the formation of a black hole.
The question that arises: "Is the black hole just an artifact of the coordinate symmetries?" Given the isotropy and homogeneity of dS space, every point in dS space should have the same properties, given that every point has its own horizon, but this logic gives us the conclusion that every point in dS space nucleates to a black hole.
We have seen that the effect of the radiation is to have a decrease in temperature due to the radiation.
A reinterpretation of the phenomenon of temperature decay of Hawking radiation in dS space, is for a time-varying, decaying cosmological "constant", which can be seen as an approximate thermal bath in every point in dS space, which is the same for inertial frames, so this should be a more consistent scenario as to what is the backreaction of the Hawking radiation in dS space.
Challenges and some questions for future projects
•It was shown by Robinson and Wilczek that there is relationship between Hawking radiation and the cancelation of gravitational anomalies to preserve general covariance at a quantum level in a black hole background[14].
Anomalies in QFT are caused by IR boundary conditions of quantum fluctuations, which do not possess the symmetries of the classical action after the quantization procedure, and these can indicate some new, interesting physics. We hope that, in this case, the quantum fields included dynamically could restore the symmetries[15], like how Hawking radiation can restore general covariance, and that would, hopefully, lead us to generally covariant equations of motion compatible with the backreaction of quantum fields. The gravitational anomaly breaks Lorentz invariance, such as the equation 1 of Robinson and Wilczek’s article:
∇µTµν= 1 96√−gε
βδ
•From GR,Λ is a fixed parameter that dictates the solution, so the evolution of a system with varyingΛimplies an explicitly time-dependent Lagrangian, so we have a violation of the energy conservation.
• One unaswered question is how we can find the energy-momentum tensor of Hawking radiation in dS space. We believe the approach would be:
TµνHR= −√2 −g
δ(ImS) δgµν but this haven’t been done by us yet.
•Is there a way to find a time-dependent solution with this tunneling approach?
This is being investigated, but the symmetries required by the assumptions on the tunneling method, usually imply a static metric, which makes this a difficult task.
• There are no well established criteria on how one obtains the backreaction on the metric due to these effects, so our guide was physical intuition and paradox resolution to remove this conundrum of black holes everywhere.
References
[1] Martin, J. Everything you always wanted to know about the cosmological constant problem (but were afraid to ask). Comptes Rendus Physique 13, 566 – 665 (2012). URL http://www. sciencedirect.com/science/article/pii/S1631070512000497.
[2] Planck Collaboration: Aghanim, N et al. Planck 2018 results. VI. Cosmological parameters. arXiv e-prints (2018). URLhttps://arxiv.org/abs/1807.06209.
[3] Baumann, D. TASI Lectures on Inflation. arXiv e-prints (2009). URLhttps://arxiv.org/abs/ 0907.5424.
[4] Mottola, E. Thermodynamic Instability of de Sitter Space. Phys. Rev.D33, 1616–1621 (1986). URL
https://doi.org/10.1103/PhysRevD.33.1616.
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https://arxiv.org/abs/0709.2899.
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URLhttps://arxiv.org/abs/1703.06898.
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012015 (2013). URLhttps://arxiv.org/abs/1306.1527.
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[9] Hawking, S. W. Particle creation by black holes. Comm. Math. Phys.43, 199–220 (1975). URL
https://projecteuclid.org:443/euclid.cmp/1103899181.
[10] Gibbons, G. W. & Hawking, S. W. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D 15, 2738–2751 (1977). URL https://link.aps.org/doi/10.1103/ PhysRevD.15.2738.
[11] Parikh, M. K. & Wilczek, F. Hawking radiation as tunneling. Phys. Rev. Lett.85, 5042–5045 (2000).
URLhttps://link.aps.org/doi/10.1103/PhysRevLett.85.5042.
[12] Parikh, M. K. New coordinates for de Sitter space and de Sitter radiation. Physics Letters B546,
189–195 (2002). URLhttps://arxiv.org/abs/hep-th/0204107.
[13] Stotyn, S., Schleich, K. & Witt, D. M. Observer-dependent horizon temperatures: a coordinate-free formulation of Hawking radiation as tunneling. Classical and Quantum Gravity 26, 065010 (2009).
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Appendix A
Some calculations about GR
A.1
Bianchi’s Identities and other symmetries
Riemann tensor: Rabcd = gaeRebcd = gae h ∂cΓebd− ∂dΓebc+ Γec fΓ f bd− Γ e d fΓ f bc i
where the Christoffel SymbolsΓγ
αβare given by: Γγ
αβ= gγδ
2 gδα,β+ gδβ,α− gαβ,δ
The Riemann tensor exhibits several symmetries which can simplify many tensor manipula-tions when working with GR, which are the following:
The Skew Symmetry
Rabcd = −Rbacd = −Rabdc (A.1) The interchange symmetry
Rabcd = Rcdab (A.2)
The first Bianchi Identity(algebraic)
Ra[bcd]≡ Rabcd+ Racdb+ Radbc= 0 (A.3)
The second Bianchi Identity(differential)
Rab[cd;e]≡ Rabcd;e+ Racde;c+ Radec;d = 0 (A.4)
A.2
Einstein tensor’s covariant conservation
We start by contracting theA.4withgab, with the metric compatibilitygab;c = 0:
Rcbcd;e+ Rcbec;d+ Rcbde;c= 0 where we useA.1to change the second term:
and now we use the definition of the Ricci tensorRab= Rcacb:
Rbd;e− Rbe;d+ Rcbde;c= 0
we contract it with the metricgbd:
Rdd;e− Rde;d+ Rcdde;c= 0 which we contract to the Ricci ScalarR= Rdd:
R;e− 2Rce;c= 0 which we rewrite in the form:
(Rce−1 2g
c
eR);c= 0
and we contract with the metricged:
(Rcd−1 2g
cdR) ;c= 0
and finally, we identify the Einstein TensorGµν= Rµν−R2gµν:
∇µGµν= 0.
A.3
Comments on Semiclassical and Quantum Gravity
In this approach, we assume a fixed classical background, with quantized fields, that can be resumed in quantizing the RHS of Einstein’s Equations, expressed as expectation values, but preserving the LHS as a classical object.
Gµν+ Λgµν= κTµν
(A.5) From the conservation law2.6,
* 0 ∇µGµν+ (∇µΛ) gµν+ Λ: 0 (∇µgµν) = ∇µ(κTµν)
we see that a nonzero value of the RHS implies a variation ofΛ, because we assume that the geometry is fixed, so the Bianchi Identity cancels the first term and the third one from the metric compatibility.
in a quantum gravity, where the geometry is quantized? Then the same equation
Gµν + Λgµν = κ Tµν
applying the covariant derivative, we would have:
∇µhGµνi + ∇µhΛ gµνi = κ∇µhTµνi
Then a non-zero RHS resulting from a gravitational anomaly, which breaks general covari-ance, makes the LHS be nonzero and then we would have:
∇µhGµνi = κ∇µhTµνi − ∇µhΛ gµνi
then the two options would be that∇µhGµνiis either zero, which is a very fine-tuned
cancel-lation, or non-zero which is a quantum correction breaking Bianchi’s Identities to the geometry of spacetie.
Another possibility, without these assumptions, is that the Einstein’s equations should receive extra terms for a correct account of these effects.
Appendix B
de Sitter metrics
B.1
Static Coordinates
B.1.1
Contra/covariant metric
Given the line element: ds2= − 1 −r 2 `2 c2dt2+ 1 1 −r2 `2 d r 2+ r2 dθ2+ sin2θ dφ2 g00= − 1 −r 2 `2 c2 g11= 1 −r 2 `2 −1 g22= r2 g33= r2sin2θ
Using the relationgµνgµν= δµµwe obtain the following system:
g00g00= 1 g11g11= 1 g22g22= 1 g33g33= 1 =⇒ g00 = 1 g00 = − 1 −r 2 `2 −1 c−2 g11 = 1 g11 = 1 −r 2 `2 g22 = 1 g22 = 1 r2 g33 = 1 g33 = 1 r2sin2 θ gµν= −1 −r`22 c2 0 0 0 0 1 −r`22 −1 0 0 0 0 r2 0 0 0 0 r2sin2θ
gµν= −1 −r`22 −1 c−2 0 0 0 0 1 −r2 `2 0 0 0 0 r12 0 0 0 0 1 r2sin2 θ
B.1.2
Christoffel Symbols
Γγ αβ= gγδ 2 gδα,β+ gδβ,α− gαβ,δ•in 4-D There are 64 symbols to be computed:
Components conditions # of components
Γγ00= −g γ1 2 g00,1 nonzero forγ = 1 4 Γγ01= Γγ10= +g γ0 2 g00,1 nonzero forγ = 0 8 Γγ02= Γγ20= 0 ∀γ 8 Γγ03= Γγ30= 0 ∀γ 8 Γγ11= +gγ1g11,1 nonzero forγ = 1 4 Γγ12= Γγ21= +g γ2 2 g22,1 nonzero forγ = 2 8 Γγ13= Γγ31= +g γ3 2 g33,1 nonzero forγ = 3 8 Γγ22= −g γ1 2 g22,1 nonzero forγ = 1 4 Γγ23= Γγ32= +g γ3 2 g33,2 nonzero forγ = 3 8 Γγ33= −g γ1 2 g33,1− gγ2 2 g33,2 nonzero forγ = 1, 2 4 g00,1=2rc 2 `2 g11,1= 2r `2 1 −r 2 `2 −2 g22,1= 2r g33,1= 2r sin2θ g33,2= r22 sin θ cos θ g00= − 1 −r 2 `2 −1 c−2 g11= 1 g11 = 1 −r 2 `2 g22= 1 r2 g33= 1 r2sin2 θ
Now we compute their value: γ = 0 ( Γ001= +g 00 2 g00,1= − r `2 1 −r 2 `2 −1 γ = 1 Γ100= −g 11 2 g00,1= − r c2 `2 1 −r 2 `2 Γ111= +g 11 2 g11,1= 2r `2 1 −r 2 `2 −1 Γ122= −g 11 2 g22,1= −r 1 −r 2 `2 Γ133= −g 11 2 g33,1= −r sin 2 θ 1 −r 2 `2 γ = 2 Γ221= +g 22 2 g22,1= 1 r Γ233= −g 22 2 g33,2= − sin θ cos θ γ = 3 Γ331= +g 33 2 g33,1= 1 r Γ332= +g 33 2 g33,2= cot θ
B.1.3
Geodesics
The geodesic equation is given by: d2xγ ds2 + Γ γ αβ dxα ds dxβ ds = 0 (B.1)
where s is an affine parameter of the curveφ(s)with coordinatesxα(s), and in particular it could
be the proper time for particle geodesics(timelike) or the coordinate time x0 for null trajecto-ries(lightlike) with a change of variables.
Such a change of variables from s to arbitraryt(s)gives: d ds= dt ds d dt d2 ds2 = dt ds 2 d2 dt2+ d2t ds2 d dt (B.2)
Which we use in equation (B.1): dt ds 2" d2xγ dt2 + Γ γ αβ dxα dt dxβ dt # = −d 2t ds2 dxγ dt " d2xγ dt2 + Γ γ αβ dxα dt dxβ dt # = − " d2t ds2 dt ds 2 # dxγ dt (B.3)
And the right side is equal to zero if d2t
ds2 = 0 =⇒ t(s) = a + bswhich is the condition for t to be affine.
From equation (B.1) we have a system of four equations: d2t ds2+ Γ 0 00(t0)2+ 2Γ010t0r0+ Γ011(r0)2+ Γ022(θ0)2+ Γ033(φ0)2= 0 (B.4) d2r ds2+ Γ 1 00(t0)2+ 2Γ110t0r0+ Γ111(r0)2+ Γ122(θ0)2+ Γ133(φ0)2= 0 (B.5) d2θ ds2 + 2Γ 2 21r0θ0+ 2Γ233(φ0)2= 0 (B.6) d2φ ds2 + 2Γ 3 31r0φ0+ 2Γ332θ0φ0= 0 (B.7)
Where a function primed f0(s)is the derivative of the function in s. And we can see from the third and fourth equations that if we have at any point of the geodesicφ0= 0then it remains so because the acceleration onφ is also zero. Assumingφ0= 0and applying the same reasoning toθ0= 0, we see that the acceleration ofθis also zero. Which is expected from the conservation of angular momentum in the absence of external forces, so we focus on radial geodesics (φ0= θ0= 0).
Radial Null Geodesics
• Approach 1- Choosing the non-affine parameter to be the time coordinate we use equation (B.3) to obtain radial lightlike geodesics (Note:˙t ≡ 1):
¨t+ Γ0 00(˙t)2+ 2Γ010˙t˙r + Γ011˙r2= − " d2t ds2 dt ds 2 # ˙t = − f (t(s))˙t 0 Γ000+ 2Γ010˙r + 0 Γ011˙r2+ f (t(s)) = 0 (B.8) ˙r = −f(t(s)) 2Γ010 = `2 f(t(s)) 2r 1 −r 2 `2
•Approach 2- From the line element we have:
˙r = ± r −g00 g11 = ± v u u u u t − h −1 −r`22 c2i 1 −r`22 −1 = ± 1 −r 2 `2 c =⇒ dr r2− `2 = ∓ c dt `2 =⇒ − tanh−1 r` ` = ∓ ct `2+ k =⇒ r(t) = ` tanh ±ct ` + k` r(0) = 0 =⇒ tanh (k`) = 0 =⇒ k = 0, t≥ 0, r(t) ≥ 0 so we have r(t) = tanhct `
for radial lightlike geodesics that begin atr= 0and go tor= `fort→ ∞ Comparing both approaches seems to imply that
`2f(t(s)) 2r = ±c =⇒ f (t) = ± 2rc ` =⇒ d2t ds2 = ± 2rc ` dt ds 2 = ±2c ` tanh ct ` dt ds 2 dt ds= v(t(s)), d2t ds2 = dv ds = v(t) dv dt =⇒ dv dt = ± 2c(v(t)) ` tanh ct ` , ct ` = x
we choose the + sign because time and proper time should go in the same direction, so: ln(v(t)) = 2 Z tanh(x)dx =⇒ ln(v(t)) = 2 ln[cosh (x)] v(t) = ` c dx ds = cosh 2(x) =⇒ ` c Z sech2(x)dx = Z ds s= ` ctanh ct ` =⇒ t(s) = ` ctanh −1cs ` .
B.2
P-G Coordinates
B.2.1
Contra/covariant metric
Given the line element: ds2= − 1 −r 2 `2 c2dt2− 2r `c dt dr+ dr 2+ r2dΩ2 g00= − 1 −r 2 `2 c2 g01= g10= −rc ` g11= 1 g22= r2 g33= r2sin2θ Using the relationgµνgµν= δµµwe obtain the following system:
g00g00+ g01g10= 1 g00g01+ g01g11= 0 g10g00+ g11g10= 0 g10g01+ g11g11= 1 g22g22= 1 g33g33= 1 =⇒ g00= 1 g00− (g01)2 = −1 c2 g10= −g10g00= g10= − r `c g11= 1 − g10g01= 1 − r2 `2 g22= 1 g22 = 1 r2 g33= 1 g33 = 1 r2sin2 θ gµν= −1 −r`22 c2 −rc ` 0 0 −rc` 1 0 0 0 0 r2 0 0 0 0 r2sin2θ gµν= −1 c2 −`cr 0 0 −r `c 1 −r`22 0 0 0 0 1 r2 0 0 0 0 1 r2sin2 θ
B.2.2
Christoffel Symbols
Γγαβ= gγδ
2 gδα,β+ gδβ,α− gαβ,δ
•in 4-D There are 64 symbols to be computed:
Components conditions # of components
Γγ00= −g γ1 2 g00,1 nonzero forγ = 0, 1 4 Γγ01= Γγ10= +g γ0 2 g00,1 nonzero forγ = 0, 1 8 Γγ02= Γγ20= 0 ∀γ 8 Γγ03= Γγ30= 0 ∀γ 8 Γγ11= +gγ0g01,1 nonzero forγ = 0, 1 4 Γγ12= Γγ21= +g γ2 2 g22,1 nonzero forγ = 2 8 Γγ13= Γγ31= +g γ3 2 g33,1 nonzero forγ = 3 8 Γγ22= −g γ1 2 g22,1 nonzero forγ = 0, 1 4 Γγ23= Γγ32= +g γ3 2 g33,2 nonzero forγ = 3 8 Γγ33= −g γ1 2 g33,1− gγ2 2 g33,2 nonzero forγ = 0, 1, 2 4 g00,1= 2rc2 `2 g01,1= − c ` g22,1= 2r g33,1= 2r sin2θ g33,2= r22 sin θ cos θ g00= −1 c2 g01= − r `c g11= 1 −r 2 `2 g22= 1 r2 g33= 1 r2sin2θ
Now we compute their value: γ = 0 Γ000= − g01 2 g00,1= r2c `3 Γ010= +g 00 2 g00,1= − r `2 Γ011= +g00g01,1= 1 ` c Γ022= −g 01 2 g22,1= r2 ` c Γ033= −g 01 2 g33,1= r2 ` csin 2 θ γ = 1 Γ100= −g 11 2 g00,1= − r c2 `2 1 −r 2 `2 Γ110= +g 10 2 g00,1= − r2c `3 Γ111= +g10g01,1= r `2 Γ122= −g 11 2 g22,1= −r 1 −r 2 `2 Γ133= −g 11 2 g33,1= −r sin 2 θ 1 −r 2 `2 γ = 2 Γ221= +g 22 2 g22,1= 1 r Γ233= −g 22 2 g33,2= − sin θ cos θ γ = 3 Γ331= +g 33 2 g33,1= 1 r Γ332= +g 33 2 g33,2= cot θ
B.2.3
Geodesics
Radial Null Geodesics• Approach 1- Choosing the non-affine parameter to be the time coordinate we use equation (B.3) to obtain radial lightlike geodesics (Note: ˙t ≡ 1):
¨t+ Γ000(˙t)2+ 2Γ010˙t˙r + Γ011˙r2= − " d2t ds2 dt ds 2 # ˙t = − f (t(s))˙t Γ000+ 2Γ010˙r + Γ011˙r2+ f (t(s)) = 0 (B.9) From (B.9) we have: ˙r = −Γ 0 10 Γ011 ± s (Γ010)2 (Γ011)2− Γ000 Γ011 − f(t(s)) Γ011 = r ` c± r r2 `4`2c2− r2c `3 ` c − ` c f (t(s)) = c r `± r −` cf(t(s)) !
•Approach 2- From the line element we have:
ds2= gµνdxµdxν= 0 =⇒ g11˙r2+ 2g10˙r + g00= 0 g11=1 ===⇒ ˙r = −g10± q (g10)2− g00= r c ` ± r r2c2 `2 + c 2−r2c2 `2 = c r `± 1 =⇒ dr r± ` = c dt ` =⇒ ln(r ± `) = ct ` + c3 =⇒ r(t) = e ct `+c3∓ ` r(0) = 0 =⇒ ec3 = ±` = ` =⇒ r(t) = ` ect` − 1
where the constantc3was chosen to guarantee positivity of r for positive t.
Comparing both approaches seems to imply that r −` cf(t) = 1 =⇒ f (t) = −c ` =⇒ d2t ds2 = −c ` dt ds 2 dt ds= v(s) =⇒ dv ds = −c(v(s))2 ` =⇒ − −1 v(s) = c s+ ` c1 ` v(s) = dt ds = ` c 1 s+` c1 c =⇒ t(s) = ` cln s+` c1 c + c2.
Now we connect their initial values t(0)=0:
t(0) =` cln ` c1 c + c2= 0 =⇒ t(s) = ` cln c s ` c1 + 1
Given that s is proper time, we want that coordinate time goes in the same "direction" of it and restricting it to positive s, and we can see from the function v(s) that, independently of the sign chosen, it has a sign change ats= −` c1
c , so to make sure there is no sign change for positive
s,c1> 0.
So the coordinate time along a radial lightlike geodesic fors= τis given by:
t(τ) = ` cln c τ ` c1 + 1 , c1> 0
which implies that proper time is:
τ(t) = ` c1 c ect` − 1 (B.10)
but going back to the radial equation we see that the time it takes for the massless particle to reach the horizon at`is given by:
r(th) = `ecth` − 1 = ` =⇒ ecth` = 2 =⇒ t h= ` cln(2) and plugging this time in equation (B.10) we have:
τ(th) = ` c1
c
and this should be lcin this freely falling frame of reference given by this proper time, so we have c1= 1.
