Quantum Field Theory in Curved SpacetimeLecturer: Ben Freivogel(benfreivogel@gmail.com)

Assistant: Joris Kattemölle(j.j.kattemolle@uva.nl)

December 13, 2017

Contents

I General info

II Lectures

1 Lecture I 1.1 Rindler space . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 The Unruh effect . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Euclidean time . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Euclidean time in Rindler space and the entanglement structure of the

Minkowski vacuum . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

2 Lecture II 2.1 Canonical technique . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.2 Canonical technique in Rindler space . . . . . . . . . . . . . . . . . . . . . . 2.3 Black holes . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

3 Lecture III 3.1 Quantum state near black hole . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Hawking radiation . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Black hole information problem . . . . . . . . . . . . . . . . . . . . . . . . .

4 Lecture IV

III Exercises

5 Exercise class I 5.1 Temperature of a Black Hole . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2 Quantum fields in Rindler Space . . . . . . . . . . . . . . . . . . . . . . . .

5.2.1 Preliminaries . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 5.2.2 Bogolyubov Transformations . . . . . . . . . . . . . . . . . . . . . . 5.2.3 Particle Density in the Modes . . . . . . . . . . . . . . . . . . . . . .

6 Exercise class II

6.1 Euclidean Harmonic Oscillator . . . . . . . . . . . . . . . . . . . . . . . . . 6.2 Black Hole Thermodynamics . . . . . . . . . . . . . . . . . . . . . . . . . . 6.3 The Laws . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 6.4 Black Hole Scattering and the Consequences of the Area Law . . . . . . . . 6.5 The Schwarzschild-AdS Black Hole . . . . . . . . . . . . . . . . . . . . . . .

7 Exercise class III 7.1 Klein-Gordon Inner Product . . . . . . . . . . . . . . . . . . . . . . . . . . . 7.2 Stress Energy Tensor in Rindler Space . . . . . . . . . . . . . . . . . . . . . 7.3 Firewalls . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

8 Exercise class IV 8.1 Scalar field in de Sitter . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

Part I

General info

This module focuses on techniques and results from quantum field theory in curved space-times, such as the real world. Important applications we cover include

• Hawking radiation from black holes and the black hole information paradox.

• Quantum fluctuations during inflation that generate the temperature fluctuations inthe CMB.

Exam. Problems 6.5, 7.2 and 8.1, to be handed in by the end of Friday 22/12/2017.Please email your solutions to both Joris and Ben (email addresses at the very beginningof this document).

You may use whatever references you can find in order to do the exam problems, butyou must write up the solutions in a coherent manner. You must complete the examproblems individually. It is of course no problem if you have collaborated with fellowstudents on the exercises so far, but from this point onward you should not collaborate onthe exam questions.

Note that problem 7.2 is mostly the same as what was already assigned in the exerciseclass, but we have added a couple of further questions at the end.

References.

• D. Harlow, Jerusalem Lectures on Black Holes and Quantum Information, [arXiv:1409.1231[hep-th]].

• L. H. Ford, Quantum field theory in curved space-time, [gr-qc/9707062].

• N. D. Birrell and P. C. W. Davies, Quantum Fields in Curved Space.

• L. Susskind and J. Lindesay, An Introduction to Black Holes, Information and theString Theory Revolution: The Holographic Universe.

Notes. The lecture notes were written down by Ward Vleeshouwers, and he has kindlyagreed to share them with the class. They are schematic, and should be seen as anoverview of the topics covered in the course. The instructors have not edited the notes,which means we can’t guaranty their correctness, but we expect they will be quite usefulfor many students. See the references for more reliability.

Part II

Lectures

1 Lecture I

1.1 Rindler space

ds2 “ ´ρ2dτ2 ` dρ2 ` d~y2

A wedge of Minkowski space, accelerating observers: x “ ` cosh s`, t “ ` sinh s`. s “proper time, and acceleration is 1`.

xMink “ ρ cosh τ , tMink “ ρ sinh τ . Rindler coordinates cover a single wedge. Rindler timeis a boost parameter, which leaves Minkowski invariant: Bτ is a Killing vector.

QFT of Rindler observers:

Take xφφy for φ free massless scalar in 3` 1-dimensional space.

xφpxµ1 qφpxµ2 qy “

1

px1 ´ x2q2 ` iε

“1

px1 ´ x2q2 ` py1 ´ y2q

2 ´ pt1 ´ t2q2 ` iε

Plug in x1 “ ρ1 cosh τ1 etc, gives:

xφpxµ1 qφpxµ2 qy “

1

p∆yq2 ` ρ21 ` ρ

22 ´ 2ρ1ρ2 coshpτ1 ´ τ2q

1.2 The Unruh effect

Time dependence of xφφy:

In Minkowski we get power law decay. In Rindler, we get exponential decay: e´|τ1´τ2|, thisis reminiscent of thermal behavior. Rindler correlator is invariant under τ Ñ τ`2πin, n PZ i.e. correln fct is periodic in imaginary/Euclidean time, which is also indicative of thermalbehavior

To show this, consider some thermal expectation value xOyβ , β “ 1T .

xOyβ “ tr pρβOq “ Z´1tr´

e´βHO¯

“ Z´1ř

i 〈i| e´βHO |i〉, where |i〉 is the basis where Ois diagonal.

Typically: |ψptq〉 “ e´iHt |ψp0q〉, Uptq “ e´iHT . Now: e´tEH is the Euclidean timeevolution operator. Then

ř

i 〈i| e´βHO |i〉 tells us that we evolve |i〉 by e´βH after whichit returns to its original state. Hence |i〉 is periodic in imaginary time with period tE “ β.

Rindler observer sees a thermal spectrum with β “ 1T “ 2π (dimensionless). Dimension-lessness comes from the fact that τ in Rindler metric is dimensionless. Physical temperatureis then given in terms of proper time τproper “ ρτ , so that Tτ “ 1

2π , Tprop “1

2πρ , whereρ is proper distance form horizon. This is called the Unruh effect: accelerating observerinside Minkowski vacuum sees thermal radiation with temperature T “ 1

2πρ “a

2π , wherea is acceleration.

Is Rindler really thermal?

1.3 Euclidean time

Here: Euclidean path integral

Exercises: Canonical

Why talk about Euclidean time? Because xφφyZ´1ş

DrφseiS is not well-defined. LorentzianQFT: many states possible.

Choose state: |0〉M „ limtEÑ8 e´HtE |χ〉. Calculate vacuum correlation function.

Take SHO: S “ 12

ş

dt´

9φ2 ´ ω2φ2¯

. We take ψGSpφ0q “şφptE“0q“φ0φp´8q rDφs e´SEucrφs, SEuc “

12

ş

dtE

´

9φ2 ` ω2φ2¯

Use saddle point approx: ψGS pφ0qFree„ e´SEucrφcls, with φclp´8q “ 0, φclp0q “ φ0

Calculate ψGS pφ0q for SHO. SEucl “ ω2φ2o

ş0´8

dte2ωt “ 12ωφ

20. Then: ψGSpφ0q “ exp

`

´ω2 φ

20

˘

Generalizing:

ψGS pφpxqq “ xφpxq|0y “ #

ż φpx,t“0q“φ0pxq

Drφse´SEucpφpxqq Free“ #e´SEucpφclq

Slogan: Euclidean evolution selects state. This state is called the “Euclidean vacuum”.

Correlators:

x0|φφ|0y “@

χˇ

ˇe´HTEφφe´HTEˇ

ˇχD

.Then:

x0|φpx, t “ 0qφpy, t “ 0q|0y “ Z´1

ż

Drφse´SEucrφsφpx, 0qφpy, 0q

At finite temperature:

Z “ trρ “ÿ

i

A

iˇ

ˇ

ˇe´βH

ˇ

ˇ

ˇiE

“

ż

φptE`β,xq“φptE ,xqDrφse´SEucpφq

I.e. we integrate over field configurations that are periodic in Euclidean time. Hence:

xφpx, t “ 0qφpy, t “ 0qy “ Z´1

ż

periodicDrφse´SEucpφqφpx, 0qφpy, 0q

1.4 Euclidean time in Rindler space and the entanglement structure ofthe Minkowski vacuum

Rindler

ds2 “ ´ρ2dτ2 ` dρ2 ` d~y2

Euclidean: τ “ iτEuc

ds2E “ ρ2dτ2

E ` dρ2 ` d~y2

Metric smooth at ρ “ 0 : τE has to have period 2π. Quick and dirty way to see thermalityis to check periodicity of Euclidean time.

A cleaner way to see this is from the fact that |0〉M is prepared by Euclidean time evolution.

We have ψ pφpxqq “@

φpxqˇ

ˇe´HTEˇ

ˇχD

τE Ñ τE ` π takes us from right to left Rindler wedge. We have:

ψGS pφLpxqφRpxqq “

ż φpx,tE“0q“φRpxq

φpx,tE“πq“φLpxqe´SEucrφs “

@

φRpxqˇ

ˇe´πHRˇ

ˇφ˚LpxqD

This is the Minkowski vacuum in Rindler basis. Go to energy basis of Rindler Hamiltonian:

ψGS pEL, ERq “@

ELˇ

ˇe´πHRˇ

ˇERD

Hence EL “ ER. We thus have:

|0〉Mÿ

E eigenstates

e´piEi |i〉˚L |i〉R (1)

This is the thermofield double state. It is pure now in the form of entangled states of leftand right wedges. Trace over left wedge:

ρR “ trL |0〉M 〈0|M “ÿ

i

e´2πEi |i〉R 〈i|R “ e´βH

2 Lecture II

Last time: Rindler: ds2 “ ´ρ2dτ2 ` dρ2, gives finite T .

Let τ Ñ ßτE : ds2 “ dρ2 ` ρ2dτ2E , τE „ τE ` 2π. Hence: Tτ “ 1

2π so that Tproper “ 12πρ “

acceleration2π .

This derivation is a bit slick, Euclidean continuation is not always ‘available’ for generalcurved spaces.

2.1 Canonical technique

Canonical technique: Valid in general, but requires a lot of assumptions, e.g. free masslessscalar in 1` 1 dimensions.

In flat space:

φpx, tq “

ż

dk

2πd´1

ˆ

1?ωkake

ipkx´ωtq ` h.c.

˙

,”

ak, a:

k1

ı

“ δkk1

I.e. we have one SHO for each k.

In curved space:

φ pxµq “

ż

dλ´

uλ pxµq aλ ` vλ px

µq a:λ

¯

, luλ “ 0 “ lvλ

Nice choice:

puλ, uλ1qKG “ δλλ1 , pu, vq “ 0

Andpuλ, uλ1qKG “ ´δλλ1

Where:pf, gqKG “ i

ż

?´g nµ pf˚Bµg ´ pBµf

˚qgq

Here: g is induced metric, nµ is normal vector to spatial slice.

uλ are called positive frequency modes.

Flat: |0〉M defined by a |0〉M “ 0.

For curved space: aλ |0〉 “ 0 for all λ: this state |0〉 is not unique in general. Choice ofstate is in the choice of which solutions we call positive frequency.

Time-like Killing vector Ø Hamiltonian, we can then define |0〉 by H |0〉 “ 0.

2.2 Canonical technique in Rindler space

For 1` 1 free massless scalar:

1

ρ2B2τφ “

1

ρBρ pρBρφq

up “ ρipe´iωpτ R , up “ 0 L

With ωp “ |p|

And vRp “ u˚p

φ “

ż

dp

ˆ

uRp

´

bR¯:

` vRp bR ` uLp b

L ` vp

´

bL¯:˙

How does ak |0〉M “ 0 look in terms of Rindler modes b?

Define fk “ eipkx´ωtq?ωk

:

ak “´

fk, φ¯

, bp “´

up, φ¯

KG

bp “

ˆ

up,

ż

dk pfkak ` c.c.q

˙

“

ż

dk´

αpkak ` βpka:

k

¯

Whereαpk “ pup, fkq , βpk “ pup, f

˚k q

i.e. they are the inner product of Minkowski fk and Rindler up modes.

Calculate number of particles in each mode in Minkowski vacuum:

xNy “@

0Mˇ

ˇb:bˇ

ˇ0MD

“

B

0M

ˇ

ˇ

ˇ

ˇ

ż

dk1´

α˚k1pa:

k1 ` β˚k1pak1

¯

ż

dk´

αkpak ` βkpa:

k

¯

ˇ

ˇ

ˇ

ˇ

0M

F

“

ż

dk|βkp|2

Trick: Find combination of Rindler modes which is positive frequency from Minkowskipoint of view.

eipkx´ωktq is well-behaved in lower half of complex t-plane. Let t “ iτ : eωτ : blows upat infinity. If we take a function which is analytic (well-behaved) in lower half complext-plane, this is some superposition of positive frequency modes We combine Rindler modesin such a way that they are well-behaved in lower half t-plane.

uRp “`

ρe´τ˘ip“ px´ tqip , uLp “ 0

With x “ ρ cosh τ , t “ ρ sinh τ . up is not analytic since it jumps to zero when we crossfrom R to L.

Write: u “ px ´ tqip “ eipplnpx´tqq, gives branch cut at x ´ t “ 0 in upper half t-plane.Hence one side x ´ t is positive and other side x ´ t is negative. Hence on right handside we have eip lnpx´tq, on left hand side we have same function but with x ´ t ă 0 i.e.lnpx´ tq “ lnpt´xq` lnp´1q “ lnpt´xq˘ iπ, ˘ depending on where we have branch cut.

Hence: uR “ eip lnpx´tq , uL “ eippln|x´t|`iπq. Hence u “ uRp ` e´pπvLp has positiveMinkowski frequency. Choosing branch cut gives unique analytic continuation. We thusget positive frequency on the right and negative on the left from Rindler point of view,which is positive frequency for Minkowski.

I.e. uRp `e´ωpπvLp „ positive in right wedge and negative in left wedge has positive Minkowski frequency.Similarly: uLp ` e´ωpπvRp has pos freq.

Negative Minkowski frequency is given by vRp ` e´ωpπuLp , vLp ` e´ωpπuRp .

This gives an annihilation operator for Minkowski vacuum for each Rindler mode:

´

bRp ` e´ωpπ

`

bLp˘:¯

|0〉M “ 0

*break*

We have:

ˆ

bRp ´ e´πωp

´

bLP

¯:˙

|0〉M “ 0

ˆ

bLp ´ e´πωp

´

bRP

¯:˙

|0〉M “ 0

Ex: Show that |0〉M “ 1Z

ś

p

`ř

n e´βωpn |n〉L |n〉R

˘

, using b |n〉 “?n |n´ 1〉 , b: |n〉 “

?n` 1 |n` 1〉

Easy exercise. |0〉M is unique!

2.3 Black holes

Schwarzschild black hole in 3` 1 dimensions.

ds2 “ ´fprqdt2 `dr2

fprq` r2dΩ2

2

Where fprq “ 1 ´ rsr , rs “ 2GM . Horizon a r “ rs. Near horizon, change of coordinates

to give:

ds2 “ ´ρ2

4r2s

dt2 ` dρ2 ` r2sdΩ2

2

I.e. we get Rindler coordinates close to horizon (except for spherical part).

*conf diagram*

If you are much smaller than the Schwarzschild radius, you’re safe. Proper time to r “ 0„ rs.

We look at eternal black holes „ non-eternal black holes at late times, which behaves asRindler.

Euclidean near-horizon geometry:

ds2 “ r2sdΩ2

2 ` dρ2 ` ρ2d

ˆ

t

2rs

˙2

Hence: tE2rs„

tE2rs`2π i.e. tE „ tE`4πrs. Observers at r “constant see thermal radiation

at temperature T “ 14πrs

, such that the proper temperature is:

Tproper “1

4πrs

1a

fprq“

1

4πrs

1b

`

1´ rsr

˘

Euclidean black hole:

ds2 “ fprqdt2E `dr2

fprq` r2dΩ2

2

I.e. tE „ tE ` 4πrs

Physics:

An observer at infinity sees thermal radiation: observer in locally inertial frame sees ob-server at infinity emit radiation.

xTtry „ energy flux.Everything is static i.e. invariant under tÑ ´t. However, energy fluxis not static, hence it is equal to zero. I.e. things look thermal without energy flux, hencewe have a black hole in thermal equilibrium with Hawking radiation.

State = Euclidean vacuum = Hartle-Hawking state.

Consider a free massless scalar in black hole background:

Exercises: Tortoise coordinate, spherical harmonics.

Modes:

uk`m “eiωkt?ωkY`m pΩ2qΨ`

kprq

Where Ψprq solves a Schrödinger type equation.

3 Lecture III

We look at real black hole i.e. collapsing matter. After relaxation time: black hole no hairtheorem.

In Euclidean vacuum (Hartle-Hawking state) T “ 14πrs

, respects symmetries such astime-like translation. This means that everything has to be static i.e. xT0r „ flux“ 0. Hence, H.H. state describes black hole in equilibrium with radiation, which is anunstable equilibrium. This means that H.H. is not the appropriate state to describe realblack holes in asymptotically flat space.

3.1 Quantum state near black hole

lφ “ 0

uk`m “1eiωkt

r?ωkY`mpΩqΨ

`mk prq ,

dr

fprq” dr

Then, ‘effective Schrödinger equation’:

`

B2r ` V`prq

˘

Ψ`kprq “ k2Ψ`

k , V` “fprq

r2

”

`p`` 1q `rsr

ı

Notice the mixed use of r and r, look at exercises for week 2 and picture on phone .

For large `, we can ignore the rsr term. The maximum value of the angular momentumpotential barrier is achieved at r “ 3

2rs, where the potential is roughly `p``1qr2s

.

These wave-functions make our problem a quantum mechanics problem. We consideringoing modes (from the left) „ eikr with transmitted and reflected pieces:

eikr Ñ Rpkqe´ikr ` T pkqeikr

Similar for ingoing modes from the right „ eikr. Note that the horizon is at r “ ´8 andwe are looking at scattering from angular momentum barrier outside horizon.

Quantum mechanics energy: E “ k2 “ ω2. For ω !`p``1qr2s

, T pkq ! 1, i.e. lowtransmission for low energy, compare to ’tH.

We have T “ 14πrs

, modes with ω " T „ 1rs

are not excited. We want `p``1qr2s

ď ω2

for the modes to get through the barrier, and ω ď 1r2s, hence we want `p``1q

r2sď ω ď 1

r2s.

Hence: high ` is boring. ωk is the frequency seen by the asymptotic observer. Note thatthe frequencies that we are interested in are of order ω „ T „ 1

rsfor asymptotic observer.

Focus on ` “ 0 “s-wave”. s-Wave has barrier Vmax „1r2s. Radiation has λ „ rs.

We now need to choose a quantum state. First choice: lowest energy state. |B〉 annihilatedby a`mk . We also have Hartle Hawking state |HH〉, which is thermal: xa:ayHH “ 1

eβω´1.

Near horizon

dr “ drfprq , rr ´ rs ` rs ln

´

rrs´ 1

¯

. Near r “ 8: r « r, hence outgoing modes are

approximately plane waves. Near the horizon r “ rs, r “ rs log´

rrs´ 1

¯

.

Hence:

eikrnear horizon

Ñ

ˆ

r

rs´ 1

˙ikrsproper distance

Ñ ρ2ikrs

Recall: For Rindler metric ds2 “ dρ2 ´ ρ2dτ2, u „ ρipeiωpτ

I.e. Rindler modes close to horizon, Minkowski plane wave for r Ñ 8. If we take |B〉close to the horizon, it becomes the Rindler vacuum |0R〉, which is singular at the horizon.This state |B〉 is called the Boulware vacuum. A quick way to see that it is singular at thehorizon is that |B〉 is a pure state outside the horizon i.e. |B〉 is pure in ‘region I’. Hencethere is no entanglement across horizon. We have a stress energy term pBxφq

2, hence weget a singular stress tensor when there is no entanglement between fields since fields aren’tcorrelated.

3.2 Hawking radiation

Consider Unruh state. We put in-moving modes from the right in a vacuum state, and out-moving modes (in-moving modes from the left) in a thermal state. Tracing back Hawkingparticles to very close to the horizon, it looks like a Rindler wave packet. Proper wavelengthredshifts as λprop „

a

fprq , λproprÑ8Ñ rs, λprop

rÑrsÑ 0.

We have three states |HH〉, |B〉, and |U〉: |U〉 is the relevant state, which has thermallypopulated outgoing modes and ingoing modes in the vacuum. |U〉 is thus a mixed state.Look at outgoing mode H with λ „ rs. There is a non-zero energy flux out to infinity i.e.black hole is evaporating via Hawking radiation. The energy out-flux goes as T 4 ˚Area “r´4s r2

s “1r2s. Recall: interesting modes have ω „ 1

rsand ` « 0. These have occupation

number „ Op1q and spacing between Hawking photons of order rS .

We have:

xNωy “1

eβω ´ 1|T pkq|2loomoon

greybody factor

Lifetime of black hole:

dM

dt“ Op1q ˚ ´1

r2s

,1

G

drsdt“ ´

1

rs

Hence

tlifetime „ r3sGN

3.3 Black hole information problem

Suppose we make a black hole in a pure state. It seems to evaporate to produce a mixed,thermal state. But unitarity demands that pure states evolve into pure states, hence wehave a apparent violation of unitarity.

We analyzed near-horizon region + exterior. This is weakly curved, so we can use QFT incurved spaces. We are basically using an effective field theory. This gives thermal Hawkingradiation, which is (of course) independent of thermal state. Estimate amount of Hawkingparticles: ω „ 1rs, M „ rsG. Hence N „ M

ω „r2s4G “ SBH , i.e. the number of Hawking

particles is of the order of the entropy.

Schematically: black hole evaporation gives S photons, which, by unitarity, should be in apure state. We (thus) expect that these photons are highly entangled. A cartoon pictureis given by S qubits. Page:calculate entanglement entropy SE as a function of the numberof photons. Page then calculated that SE roughly increases linearly with the number ofqubits, since they are roughly randomly entangled. We thus get SE „ Nγ for Nγ ă S2,and SE “ S ´ Nγ for Nγ ą S2. Page: let half of a black hole evaporate. If outgoingHawking photon is entangled with early radiation, we know that black hole was (likely) ina pure state.

Unitarty demands: SH RH « 0 (Hawking particle and previous Hawking radiation), SH P «

0, SH « 1.

Strong subadditivity:

SH RH ` SH P ě SH ` SH RH

But this is in conflict with values for the entropy as demanded by unitarity.

pure Ñ mixed via transmission/reflection

Partner particle P goes to r “ 0 in finite time.

ñ 32rs “ 3M : location of potential maximum for Schwarzschild as in Carroll analysis.

4 Lecture IV

...

Part III

Exercises

5 Exercise class I

5.1 Temperature of a Black Hole

Consider a metric of the form

ds2 “ ´fprqdt2 `dr2

fprq` r2dΩ2

d´2, (2)

where, for example, fprq “ 1´ 2GMr for a d “ 4 Schwarzschild black hole.

a) Analytically continue to Euclidean signature by sending tÑ ´iτ . Assume that fprqchanges sign at a certain horizon radius rh. Now zoom in at the near-horizon regionby defining a coordinate ξ ” r ´ rh ě 0, and show that the metric can be broughtinto the form

ds2 “ f 1prhqξdτ2 `

dξ2

f 1prhqξ, (3)

where we’ve suppressed any angular dependence.

b) Figure out a coordinate transformation which brings the the metric into

ds2 “ ρ2

ˆ

d

„

f 1prhqτ

2

˙2

` dρ2. (4)

c) Does this metric look familiar? Explain why this metric makes sense.

d) In order for the metric to be well-behaved near the origin, what condition do you needto impose on the τ coordinate? Argue that this leads to a black hole temperature of

T “f 1prhq

4π. (5)

e) Compute the temperature of a Schwarzschild black hole in S.I. units. What can yousay about the various constants that appear?

5.2 Quantum fields in Rindler Space

In this exercise you will do quantum field theory in Rindler space, and derive the spectrumof the particles that an accelerating observer detects in the Minkowski vacuum.

5.2.1 Preliminaries

Consider the action of a massless scalar field φ in d dimensions,

Srφs “

ż

ddx?´g

1

2gµνBµφBνφ. (6)

a) Show that this action is invariant under conformal transformations of the metricgµν Ñ Ω2pxqgµν when d “ 1` 1.

The Rindler metric is conformally flat, i.e. it can be written as ds2 “ e2aξp´dτ2 ` dξ2q

where a is related to the acceleration of the observer. Since the action is invariant underconformal transformation of the metric, we know that the equation of motion in bothcoordinate systems can be written as

B2φ

Bt2“B2φ

Bx2,

B2φ

Bτ2“B2φ

Bξ2. (7)

Remember that, using Minkowski coordinates, the free field operator can be expanded overthe plane waves as

φpt, xq “

ż `8

´8

dk

p2πq121

a

2|k|

”

ake´i|k|t`ikx ` a:ke

i|k|t´ikxı

, (8)

where standard commutation relations for a and a: have been imposed. Since the fieldsatisfies the same eom in the Rindler coordinates, writing down the mode expansion of thefield φ in the Rindler frame becomes trivial1,

φpτ, ξq “

ż `8

´8

dp

p2πq121

a

2|p|

”

bpe´i|p|τ`ipξ ` b:pe

i|p|τ´ipξı

. (9)

Note that we now have expanded the field operator φ into linear combinations of twodifferent sets of mode functions with operator valued coefficients ak and bk. Both setsof operators define vacuum and particle states, which are different quantum states of thesame field. The Minkowski vacuum |0My is defined as the state for which ak|0My “ 0 forall k. Similarly, the Rindler vacuum is defined as the state for which bp|0Ry “ 0 for all p.

For further analysis, it is convenient to introduce lightcone coordinates

u ” t´ x v ” t` x and u ” τ ´ ξ v ” τ ` ξ. (10)

They satisfy#

u “ ´a´1e´au

v “ a´1eav. (11)

1This mode expansion is only valid within the Rindler wedge |x| ą t. The Rindler coordinates are notdefined outside this domain, and in this case the two mode expansions cannot be compared with eachother.

In these coordinates the metric becomes

ds2 “ dudv “ eapu´vqdudv. (12)

b) Substitute the lightcone coordinates in the obtained mode expansions while denotingω “ |k| in the Minkowski frame and Ω “ |p| in the Rindler frame, and obtain thefollowing lightcone mode expansions

φpu, vq “

ż `8

0

dω

p2πq121?

2ω

”

aωe´iωu ` a:ωe

iωu ` a´ωe´iωv ` a:´ωe

iωvı

(13)

φpu, vq “

ż `8

0

dΩ

p2πq121

?2Ω

”

bΩe´iΩu ` b:Ωe

iΩu ` b´Ωe´iΩv ` b:

´ΩeiΩv

ı

. (14)

Notice that the uu and vv coordinates are separated from each other within the modeexpansions,

φpu, vq “ Apuq ` Bpvq, (15)

φpu, vq “ P puq ` Qpvq. (16)

According to the transformations (11), we can now extract two separate equations for thefield expansion,

Apupuqq “ P puq, Bpvpvqq “ P pvq. (17)

Writing out the first relation yieldsż `8

0

dω

p2πq121?

2ω

”

aωe´iωupuq ` a:ωe

iωupuqı

“

ż `8

0

dΩ

p2πq121

?2Ω

”

bΩe´iΩu ` b:Ωe

iΩuı

,

(18)which shows that there is no mixing between operators with positive and negative momen-tum. This means we can do a separate analysis for the positive and negative momentumoperators: in everything that follows, the results derived for positive modes will be equallyvalid for negative modes.

5.2.2 Bogolyubov Transformations

a) We now Fourier-transform eq. (18)2. We start with the RHS. Show that

ż `8

´8

du?

2πeiΩuP puq “

1a

2|Ω|

#

bΩ if Ω ą 0

b:|Ω| if Ω ă 0

. (19)

b) Do exactly the same thing to the LHS of (18), and find the following Bogolyubovtransformation,

bΩ “

ż `8

0dω

“

αωΩaω ` βωΩa:ω

‰

where

$

&

%

αωΩ “

b

ΩωF pω,Ωq

βωΩ “

b

ΩωF p´ω,Ωq

ω ą 0,Ω ą 0,

(20)2As the result of this calculation clearly shows, the case Ω “ 0 is not covered, since the mode expansions

(8) and (9) ignore the k “ 0p “ 0 solution, also called the zero mode. In 1+1 dimensions, quantizationof the zero mode is a technical issue, but it can be shown that it doesn’t contribute in four-dimensionaltheory, so we’ll ignore it.

with

F pω,Ωq ”

ż `8

´8

du

2πeiΩu´iωu “

ż `8

´8

du

2πexp

”

iΩu` iω

ae´au

ı

. (21)

This Bogolyubov transformation expresses the annihilation operator of the Rindler modesin terms of annihilation and creation operators of the Minkowski modes, with various ω.Hermitian conjugation of equation (20) gives a Bogolyubov transformation for the creationoperator b:Ω. The construction of the Bogolyubov transformations for the negative momen-tum modes is completely analogous.

5.2.3 Particle Density in the Modes

We are interested in computing what particle excitations an accelerated observer (usingthe Rindler frame) will see in the Minkowski vacuum. Of course,

Rx0|NΩ|0yR “R x0|b:

ΩbΩ|0yR “ 0. (22)

a) Now show that

Mx0|NΩ|0yM “

ż

dω|βωΩ|2. (23)

This is the number of particles in the Minkowski vacuum the Rindler observer is expectedto see. We will now compute this integral explicitly.

b) Bonus question. Prove the following essential property

F pω,Ωq “ F p´ω,Ωq exp

ˆ

πΩ

a

˙

. (24)

Hint: extend u to the complex plane.

c) Compute the expected Rindler particle density in the Minkowski vacuum

Mx0|NΩ|0yM “

„

exp

ˆ

2πΩ

a

˙

´ 1

´1

δp0q. (25)

Hint: The commutation relation”

bΩ, b:

Ω1

ı

“ δpΩ´ Ω1q (26)

imposes a normalization condition on the Bogolyubov coefficients. Write out thisnormalization condition and set Ω “ Ω1 in the end. Use the property of the functionF above.

d) Interpret your result. Is there a non-trivial particle spectrum? What temperaturecan you associate to it? Why is the divergent factor δp0q there? Where do theseparticles come from, and who is providing them with the necessary energy?

e) Reinstate the correct factors of ~, c and kB, and compute the temperature you’dmeasure when accelerating at 6.5m

s2with your Aston Martin in the Minkowski vac-

uum.

6 Exercise class II

6.1 Euclidean Harmonic Oscillator

“The career of a young theoretical physicist consists of treating the harmonic oscillator inever-increasing levels of abstraction.”– Sidney Coleman

In this exercise, you will compute the partition function of the harmonic oscillator bytaking a detour through euclidean time. This is very inefficient, but because you are al-ready familiar with the answer you should get, you can focus more on the techniques. So,the goal is not to obtain the partition function, but rather to interpret the steps in betweencarefully.

a) From memory or with a small derivation, write down the partition function of theHarmonic Oscillator Z “ trpe´βHq in closed form. (Without using path integrals.)

It is a quite trivial observation that

Z “ tr e´βH “ tr e´itH |tÑ´iβ.

This result, however, is the reason that by going to Euclidean time, we go from unitarytime-evolution to (thermal) statistics.

b) Without making the substitution, write out the trace in the last expression in theposition basis, and write the propagator you find in your expression as a path integral.Keep the boundary conditions explicit.

c) Now make the substitution.The substitution changes the contour of the integral over the Lagrangian (

şt0 dt1 LÑ

ş´iβ0 dt1L). Instead, change the integrand so that you get an expression involving theintegral

şβ0 dtELE , where LE is the Euclidean Lagrangian.

d) Can you see why ‘Euclidean time is cyclic’?

We could proceed by finding the Euclidean action and solving the subsequent path integral.However, to save some work, we’ll just take the non-Euclidean propagator for the HarmonicOscillator from a book, and substitute to get the Euclidean propagator. In closed form, itreads

Kpx2, t2;x1, t1q “

c

mω

2πi sinrωpt2 ´ t1qs

ˆ exp

"

imω

2 sinrωpt2 ´ t1qs

“

px21 ` x

22q cosrωpt2 ´ t1qs ´ 2x1x2

‰

*

.

e) Find the Euclidean propagator for the Harmonic oscillator.

f) Using what you’ve learned in questions b and e, compute the partition function ofthe Harmonic Oscillator. Compare your answer with a.

The ground state wave-function 〈x|Ω〉 of a quantum mechanical system can be computedby

〈x|Ω〉 „ limβÑ8

〈x| e´βH |χ〉 .

Here |χ〉 can be any state that is not orthogonal to |Ω〉.

g) Compute the ground state of the Harmonic Oscillator using the Euclidean propagatoryou found in e. (If you already solved this in class, you can skip this exercise.)

6.2 Black Hole Thermodynamics

6.3 The Laws

Last session we showed that a d “ 3` 1 Schwarzschild black hole has a Hawking temper-ature T “ 1

8πGM .

a) Using the first law of thermodynamics, show that you can associate an entropy tothe black hole

SBH “A

4G, (27)

where A is the surface area of the black hole.

b) What does the second law of thermodynamics say about A? For now, assume youcan ignore evaporation. Your result is called the second law of black hole thermody-namics.

b) Compute the heat capacity of this black hole, and interpret. If you place a black holein an infinite heat bath with T ‰ TBH , is a stable equilibrium possible? Why (not)?

6.4 Black Hole Scattering and the Consequences of the Area Law

Consider two well-separated Schwarzschild black holes at rest. Let their masses be M1

and M2. Assume that they coalesce to form a Schwarzschild black hole of mass M3, withgravitational waves carrying off energy E in the process.

a) Use conservation of energy and the second law of black hole thermodynamics (asderived in the previous exercise) to prove that the efficiency of the creation of gravi-tational waves obeys η ď 1´ 1?

2.

The radiated energy could be used to do work, hence the second law of black hole thermo-dynamics restricts the useful energy that can be extracted from merging black holes.

b) On 14 September 2015, LIGO detected the gravitational waves created by the coa-lescence of two black holes with M1 « 36Md, M2 « 29Md, and E “ 3Md. Forthe sake of simplicity, assume the black holes 1 and 2 were once two well-separatedSchwarzschild black holes at rest, and that black hole 3 is also a Schwarzschild blackhole.3 Did the efficiency of this process come close to bound that was shown in a?

Now consider an initial Schwarzschild black hole of mass M that evolves to form twowell-separated approximately Schwarzschild black holes at rest.

c) Use conservation of energy and the second law of black hole thermodynamics to showthat this cannot happen. (This is a special case of a completely general result thatblack holes cannot bifurcate.)

3This isn’t true, but according to LIGO, M3 « 62Md, so that indeed M1 `M2 «M3 ` E.

6.5 The Schwarzschild-AdS Black Hole

The unique spherically-symmetric solution of Einstein’s field equations with a negativecosmological constant is known as the Schwarzschild-AdS metric. In d “ 3` 1 dimensionsit is given by

ds2 “ ´fprqdt2 `dr2

fprq` r2dΩ2

2, fprq “ 1´2MG

r`r2

b2. (28)

Here b is related to the cosmological constant by Λ “ ´3b2ă 0. The event horizon of this

black hole is at r`, the largest root of fprq.

a) The classical equation of motion of the massless scalar field reads

lφ “ 0.

Here the d’Alembertian (‘box’) is, of course, the generalized d’Alembertian for curvedspacetimes. As an ansatz, take φ to be a spherical harmonic,

φ “ e´iωtψωlprq

rYlmpθ, ϕq.

Show that this indeed solves the equation of motion if the radial part ψωl satisfiesthe effective Schrödinger equation

´B2r˚ψωlprq ` V prqψωlprq “ ω2ψωlprq,

V prq “fprq

r2rlpl ` 1q ` rBrfprqs .

Here r is always to be understood as rpr˚q, where r˚ is the coordinate that takes themetric (28) to the form

ds2 “ fprqp´dt2 ` dr˚2q ` r2dΩ2

2.

b) Plug in the Schwarzschild-AdS fprq, rewrite, and sketch the effective potential. Inter-pret your result. In particular, comment on the physical differences with the bÑ8

limit, which is the usual Schwarzschild effective potential discussed in class.

c) Finally, compute the temperature of the Schwarzschild-AdS black hole as a functionof r` and b only. Plot the temperature as a function of its size and compare with theSchwarzschild case. Interpret carefully. Are there thermodynamically stable blackholes in AdS?

7 Exercise class III

7.1 Klein-Gordon Inner Product

The space of solutions to the Klein-Gordon equation is a vector space, endowed with theKlein-Gordon inner product

pφ1, φ2q “ ´i

ż

Σpφ1∇µφ˚2 ´ φ˚2∇µφ1qn

µ?γdd´1Σ. (29)

Here Σ is a spacelike hypersurface with unit normal nµ, and γµν is the (induced) metricon Σ.

a) Show that the Klein-Gordon product does not depend on the choice of Σ.Hint. You may use Stoke’s theorem, which relates the integral over a manifold M toan integral over the boundary of that manifold, BM ,

ż

Mdnx

a

|g|∇µV µ “

ż

BMdn´1y

a

|γ|nµVµ.

7.2 Stress Energy Tensor in Rindler Space

The action for a free, massless scalar field in flat spacetime is given by

S “ ´

ż

ddx?´g

1

2gµνBµφBνφ. (30)

The stress-energy tensor is defined as

Tµν “ ´2

?´g

δS

δgµν. (31)

a) Bonus. Derive that

δ?´g “ ´

1

2

?´g gµνδg

µν .

b) Show that Tµν “ BµφBνφ´ 12gµνBσφB

σφ.

For further analysis it is convenient to introduce light cone coordinates in d “ 1 ` 1Minkowski space, U “ t ´ x and V “ t ` x. The right Rindler wedge is then given byU ă 0 ă V .

c) Write down the metric of flat, 1+1 dimensional spacetime in lightcone coordinates.

A complete basis for the right-moving modes (i.e. functions of t ´ x “ U that solve theeom.) is formed by

φω,R “ Θp´Uq1

?4πω

p´aUqiωa , φω,L “ ΘpUq

1?

4πωpaUq

´iωa .

We can now expand the field φ as

φpUq “

ż 8

0dω

´

bω,Rφω,R ` b:

ω,Rφ˚ω,R

¯

` pLØ Rq ,

where we have neglected the left moving modes. (Later in the exercise, you will see whywe do not have to consider the left-moving modes.)

d) Interpret the quantity TUU in (classical) GR.

e) Compute the expectation value of the UU component of the stress-energy tensor inthe Rindler vacuum.(Note: Your answer to the previous part should give TUU “ BUφBUφ.)To get rid of a UV-divergence, the expectation value is defined as

xTUUyR ” x0L|x0R|TUU |0Ly|0Ry ´ x0M |TUU |0My.

Hint. First show that

x0|TUU |0y “

ż 8

0dω

ω

4πa2U2p2nω ` 1q, (32)

where |0y could be either the Minkowski vacuum or the Rindler vacuum, and wherenω ” xb

:ωbωy is the expected number density of particles in the respective state. Also,

you can use thatż 8

0

dω

a

ω

a

1

e2πωa ´ 1

“1

24. (33)

As discussed in class, for real black holes that form in a collapse, the outgoing modes arethermal while the ingoing modes are in the vacuum. So roughly, when we zoom in nearthe horizon and ignore the angular directions, it looks like 1+1 Rindler spacetime in the‘Unruh vacuum’ state |0Uy defined by putting the left-moving modes in the Minkowskivacuum and the right-moving modes in the Rindler vacuum.

f) Calculate all components of the stress tensor in the Unruh vacuum,

xTµνyU ” x0U |Tµν |0Uy ´ x0M |Tµν |0My (34)

g) Calculate the energy flux xTtxy in the both the Unruh vacuum and the Rindlervacuum. As above, these quantities are defined relative to the Minkowski vacuum.Give a physical interpretation of your result.

7.3 Firewalls

We have seen that the Minkowski vacuum |0yM can be expressed as a sum of Rindler modes,which are entangled in very particular way across the surface x “ 0. In this exercise wewill show that this entanglement is crucial in order to have a smooth transition from theleft or right Rindler wedge to the future interior.

Instead of putting system in the ground state |0yM , let us put it in the mixed state ρ “ρL b ρR where ρL,R is the thermal density matrix obtained by tracing out the R,L-wedgein the Minkowski vacuum. Note that this state is fully separable: there is no entanglement.However, for any observer who lives in the L,R wedge, the state is indistinguishable fromthe state |0yM .

Given is the fact that the Minkowski vacuum |0yM is annihilated by the operator

cωk “1

?1´ e´2πω

´

bRωk ´ e´πωb:Lωp´kq

¯

, (35)

where the bL,R’s are the Rindler creation and annihilation operators, and

|0yM “â

ω,k

«

a

1´ e´2πωÿ

n

e´πωn|nyLωp´kq b |nyRωk

ff

. (36)

a) Compute the expectation value of the Minkowski number operator in this state andshow that it equals

xc:ωkcωkyρ “2

peπω ´ e´πωq2(37)

One can see that this will be become a large number for ω ď 1. This means thatthe state will be quite singular: a Minkowski observer will observe very high energyquanta near the horizon.

8 Exercise class IV

8.1 Scalar field in de Sitter

Consider d “ 3` 1 de Sitter space in planar slicing

ds2 “ ´dt2 ` a2ptqd~x2, aptq “ H´1eHt. (38)

Here H is the Hubble parameter, a fixed constant.

a) Introduce a conformal time coordinate τ such that the metric becomes

ds2 “ a2ptpτqqp´dτ2 ` d~x2q. (39)

Find apτq.

b) Show that the massless scalar field equation lφ “ 0 leads to

φ2 ` 2a1

aφ1 ´∇2φ “ 0, (40)

where prime denotes differentiation with respect to conformal time. Make sure touse the definition of the d’Alembertian (‘box’) that is valid in curved spacetime.

c) Compare equation (40) to the equation of motion for a massless scalar field inMinkowski spacetime. In what do they differ, and how can this term be interpreted?

We are now going to solve the eom. Define a new field variable χ “ aptqφ. By definitionof the Fourier transform χ~k, we can write

χ “

ż

d3k

p2πq32

´

b~kχ~kpτqei~k¨~x ` b˚~kχ

˚~kpτqe´i

~k¨~x¯

. (41)

d) Show that this eliminates the term 2a1

a φ1 so that the eom now reads

χ2~k “

ˆ

2

τ2´ k2

˙

χ~k, k “ |~k|. (42)

The solutions of the previous equation are given by

χ~kpτq “ αe´ikτ?

2k

ˆ

1´i

kτ

˙

` βeikτ?

2k

ˆ

1`i

kτ

˙

. (43)

e) Typically one picks α “ 1 and β “ 0. Explain why this is natural choice of modefunction. The associated (de Sitter invariant) vacuum is often called the Bunch-Davies vacuum.

f) After quantizing the field, compute the spatial two-point function in the Bunch-Davies vacuum and bring it into the form

x0|φpτ, ~xqφpτ, ~yq|0y “

ż

d3k

8π3ei~k¨p~x´~yqσ2

~k. (44)

g) An observationally relevant quantity is the power spectrum P pkq ”k3σ2

k2π2 , which we

can measure by looking at the sky. Compute the power spectrum, assuming thatkτ ! 1.4

4Considering distance scales of observational interest today, let’s say ~x and ~y are two points on thesurface of last scattering which can be resolved by CMB observations. We know that the correspondingmomenta k were well inside this "frozen" regime at the end of inflation.