Black Holes and Moving Mirrors

Niels Tuning

Abstract

In this rep ort I treat the sub ject of black hole radiation Until the early seventies

black holes were considered to b e eternal ob jects in our universe but the striking

analogy with the laws of thermo dynamics and the explicit calculation of the black hole

radiation due to quantum eects by Hawking showed a dierent scenario Hawking

recognized already the problems concerning quantum purity due to thermal radiation

o the black hole and the o ccurence of divergent energies due to the gravitational blue

shift

Backreaction eects were neglected by Hawking but could lead to corrections of

Hawkings result and resolve the information problem Furthermore including backre

action ensures energy conservation I included backreaction eects in the simple moving

mirror mo del and found that the temp erature of the mirror is reduced by a factor

relative to the standard result However the WKB approximation I used breaks down

in the case of dilaton gravity and therefore I could not extend the result to real black holes

Contents

Introduction

Black Hole Thermo dynamics

Black Hole

Thermo dynamics

Information Problem

Remnants

Nice Slices

Complementarity SMatrix Ansatz

The Unruh Eect

Particle Detector

The Hawking Eect

Stress Tensor

Moving Mirrors

Fixed Trajectory

Saddlep oint Approximation

Uniform Acceleration

Backreaction on the Mirror

Dilaton Gravity

Discussion

Conclusion

A WKB Approximation

B Thermal Sp ectrum

Preface

This thesis is the conclusion of a year work under sup ervision of profdr Herman Verlinde

on the Institute of Theoretical Physics of the University of Amsterdam The thesis is

the nal pro ject of my study of theoretical physics

I tried to write this thesis on the level of graduate students At the b eginning of

my research I encountered some problems and questions some of them trivial without

nding explicit answers I tried to answer these questions and hop e that it might help

other students who are trying to understand the dicult sub ject of evaporating black

holes

Amsterdam August

Chapter

Introduction

A great deal of eort has b een made the last few decades to construct a unied theory

of the forces of nature In Glashow prop osed the gauge symmetry SUU

as the unication of the weak interactions and the theory of quantum electro dynamics

also known as QED which has gauge symmetry U The gauge theory with sym

metry SU quantum chromodynamics QCD b ecame in a strong candidate for

the strong interactions The standard electroweak theory and QCD together form the

standard mo del A p ossible candidate for a grand unied theory GUT ie a gauge

theory with a single coupling constant is a theory with gauge symmetry SU The

last few years there has also b een a revival of string theory which incorp orates gravity

and is considered as another serious candidate for the Theory of Everything

Until now it has not b een p ossible to include the fourth force namely gravity Prob

lems arise while constructing a quantum theory of gravity b ecause it app ears that

the coupling constant is not dimensionless making p erturbation expansion imp ossible

Another problem is that gravity couples to everything b ecause it inuences the metric

A graviton is thus b oth a source of gravitation and sub ject to gravitation

Therefore one considers the gravitational eld as a classical background obtaining a

semiclassical theory Hawking considered quantum pro cesses in a classical background

geometry and found that vacuum uctuations near the black hole horizon results in

thermal radiation o the hole Because of those quantum pro cesses the black hole

lo oses mass and can eventually disapp ear while black holes were considered classically

as eternal ob jects

Hawkings result op ened a new area in physics and is derived in dierent ways by

several authors One of the strongest arguments to supp ort the Hawking eect is the

deep connection b etween the laws of black hole physics and the laws of thermo dynamics

This is also the ma jor reason why Hawkings discovery is now widely b elieved to b e real

The pap er is organized as follows I will start more or less historically by discussing

the sub ject of black hole thermo dynamics developed in the late sixties anticipating

Hawking by suggesting that black holes b ehave as black b o dies with a temp erature

CHAPTER INTRODUCTION

In chapter I discuss briey the problems which arise assuming black holes having

a temp erature and emitting thermal radiation The dierent p oints of view in litera

ture are given Instead of treating subsequently the Hawking eect I will discuss the

Unruh eect in chapter The Unruh eect is in some sense more general b ecause

it states that particle pro duction o ccurs wherever a bifurcate killing horizon app ears

Even an accelerating observer in Minkowski spacetime detects thermal radiation The

breakdown of Poincare invariance as we have in curved spacetime is already manifest

in noninertial systems in at spacetime In chapter the Hawking eect is explic

itly calculated giving a strong supp ort to the theory of black hole thermo dynamics of

chapter In Hawkings calculation pure states can evolve into mixed states hereby

violating the basic principle of unitary evolution in quantum mechanics In chapter

it b ecame clear that backreaction eects are imp ortant and may lead to corrections

of Hawkings result Furthermore due to the gravitational redshift the energy near

the horizon diverges In order to investigate back reaction eects and to implement

energy conservation I treat in chapter a mo del which mimics many of the features of

gravitational collapse With this moving mirror mo del I showed that including back

reaction eects and thus energy conservation leeds to a correction of the temp erature by a factor half relative to the standard result

Chapter

Black Hole Thermo dynamics

Black Hole

Before we start talking ab out black holes I will give a short review of what a black hole

is I will use units where h c

In Einstein found his classical eld equations of general relativity

G T

If one imp oses the metric to b e static and spherically symmetric one nds an exact

solution for the vacuum eld equations G namely the Schwarzschild solution

which has line element

m

dt ds dr r d sin d

m

r

r

This solution is physically interesting b ecause the ob jects in our universe are mostly

spherically symmetric and the surrounding space can b e considered empty We see

that the line element is singular at the co ordinates r and r m The coordinate

singularity at r m is however removable by p erforming a co ordinate transforma

tion The other singularity at r is an intrinsic singularity and is not removable

by co ordinate transformations Although r m is no real singularity it remains an

future singularity singularity r=const.

t=const.

I- horizon

r=0 r=2m past singularity

Figure Two dierent gures of a black hole a Schwarzschild solution b Compactied Kruskal

solution

CHAPTER BLACK HOLE THERMODYNAMICS

interesting p oint or sphere called the Schwarzschild radius If one writes the solu

tion in EddingtonFinkelstein co ordinates one sees the lightcones p ointing towards the

center of the hole and it b ecomes clear that nothing can come back after passing the

Schwarzschild radius r m is therefore called an event horizon It is imp ossible to

b ecome aware of events that take place b ehind the horizon Therefore this ob ject is

referred to as a black hole

By p erforming a suitable co ordinate transformation one removes the co ordinate

singularity at r m obtaining the EddingtonFinkelstein co ordinates The Kruskal

solution is the maximal analytic extension of the geometry ie every geo desic ends in

innity or in a singularity and can b e compactied by mapping the innite interval to

a nite interval In this way one obtains the p enrose diagram of gure b

We can imagine dierent types of black holes Beside the collapsing star we can

imagine huge black holes formed after a collapsing star cluster On the other hand we

can imagine so called primordial black holes with masses of kg One often hears

the question if elementary particles could b e black holes The answer is no b ecause the

Compton wavelength of a particle is much bigger then its Schwarzschild radius Those

p

two length scales are equal up to a factor at the Planck mass

Gm h

m k g

P l

c mc

By combining the fundamental constants c h and G one nds units of length time and

mass the so called Planck units

s

ch

m k g

P l

G

s

Gh

l m

P l

c

s

Gh

t s

P l

c

E m c GeV

P l P l

The Planck values mark the frontier where a full theory of quantum gravity is indis

p ensable

As we will see in the following chapters quantum eects alter the course of gravita

tional collapse As seen by an asymptotic observer collapsing matter evaporates b efore

forming a real curvature singularity see gure However after stellar collapse the

evaporation of a large heavy ob ject pro ceeds very slowly so we refer to this ob ject as a black hole keeping in mind that it is a rather classical notion

THERMODYNAMICS

Thermo dynamics

In the late sixties and b egin seventies a lot of work has b een done on the theory of

black holes For example one has calculated stimulated emission in the ergo region

of a rotating black hole or even a neutron star the so called superradiance One

of the most remarkable results obtained was the striking analogy b etween the laws of

thermo dynamics and the black hole laws see for a recent review

It was found that classically energy extraction from black holes is p ossible as long

as the black hole area do es not decrease This can b e done for example by dropping an

opp osite charged particle in a charged black hole by diminishing angular momentum

of a rotating black hole or by letting two black holes collide which form one bigger

black hole The analogy b etween entropy and the horizon area of the black hole was

made

A S

The Second Law of thermo dynamics stating that the total entropy never decreases

has now an analog in black hole physics with Hawkings area theorem

It turns out that the analogy of the First Law dE T dS in black hole theory

b ecomes

dM dA dJ dQ

G

where A J and Q are resp ectively the area angular momentum and charge and and

are the angular velocity and electric p otential of the horizon The surface gravity is

denoted by M for a nonrotating neutral black hole This formula suggests

that plays the role of temp erature By introducing quantummechanics Hawking

found in that a black hole has indeed a temp erature k T This equivalent of

H

the rst law ensures energy conservation during black hole pro cesses like sup erradiance

or radiation from rotating and charged black holes whereby the hole lo oses angular

momentum or charge

The constancy of on the horizon is the analogy of the Zeroth Law which states

that the temp erature is uniform everywhere in a system in thermal equilibrium

The Third Law the temp erature can not b e reduced to zero in a nite number of

steps has its analogy in the fact that zero surface gravity requires a black hole with

innite mass There are however some diculties with charged black holes For an

extremal black hole a hole which is maximally charged e M the temp erature

would b e reduced to zero see The question is if the extremal limit can in

principle b e reached

Concluding we can write down an explicit expression for the entropy by comparing

mass and thermo dynamic energy which is not so strange knowing that the mass of a

black hole is the same quantity as its energy so we nd for the entropy

dE

dS M k dM

B

T

k A S k M

B bh B

CHAPTER BLACK HOLE THERMODYNAMICS

This is the famous BekensteinHawking entropy When a black hole radiates the mass

of the black hole and thus the entropy decreases To ensure the validity of the second

law which states that entropy always increases we get the Generalized Second Law

S S S

th bh

with S the thermo dynamic entropy The black hole entropy radiates away in the form

th

of thermo dynamic entropy The laws of thermo dynamics and the laws of black holes are

so strongly related that the laws of black holes actually are the laws of thermo dynamics

In chapter I will give the derivation of the Hawking eect and will show that in

3

hc

SI units the Hawking temp erature is given by T and thus the expression

M k G

B

for the black hole entropy S b ecomes

bh

A

S k

bh B

Ghc

In this form it b ecomes visible that the area is expressed in units of the Planck length

squared Entropy is a measure of the containing information and therefore it is sug

gested that a Planck length squared contains a limited amount of information This is

in contradiction with the idea that a black hole can b e made in unnumerable dierent

ways and that therefore the entropy should b e innite The problem is our lack of

knowledge ab out Planck scale physics but a p ossible candidate to resolve these prob

lems is string theory More details on the information problem are given in chapter

It is also visible that is clearly a quantum mechanical result in the classical limit

h the entropy go es to innity

1

The subscript bh conveniently stands for b oth BekensteinHawking and black hole

Chapter

Information Problem

In the last chapter I mentioned the fact that black holes have a temp erature T

H

and radiate thermal radiation just as a black b o dy of temp erature T Imagine

H

M

now forming a black hole out of pure quantum states The black hole subsequently

starts radiating thermal radiation which are mixed states This is a serious violation

of quantum mechanics where pure states evolve unitarily to pure states U j i

i t

e j i j i Furthermore we see that probability is conserved when the evolution

t

y

op erator is unitary h j i h jU U j i h j i One could also say that

t t

information is lost in the Hawking pro cess

You might wonder what the p oint is if you consider the following pro cess where

information is absorb ed thermalized and radiated away Consider a cold piece of black

cole which absorbs the signal of a coherent laser starts burning and subsequently emits

the laser signal in the form of thermal radiation The dierence b etween this pro cess

and the Hawking pro cess is that the former is a macroscopic description without sp ec

ifying the microscopic pro cess of the coal In the black hole case the outside observer

sees the information b eing stored somehow microscopicly on the stretched horizon b e

fore b eing radiated away The stretched horizon is a physical membrane just outside

ab out a few Planck lengths the event horizon

To give some insight in how a pure state could apparently evolve in a mixed state

I will give a general mixing mechanism Consider a quantum theory with Hamilto

nian H A pure state evolves for each in a pure state t at time t

iH t

t e

But supp ose that is p o orly known and that it has a probability distribution P

The exp ectation value of an op erator O b ecomes a mixed state

X

hO i P h tjO j ti

A

A black hole can b e made in approximately e dierent ways assuming that a black

hole has entropy A Recall that the entropy of a macrostate is equal to the loga

A

rithm of p ossible microstates S l nms The number of microstates can b e

CHAPTER INFORMATION PROBLEM

identied with the dierent p ossibilities to make a black hole The uncertainty in the

Hamiltonian H which forms the black hole p ossibly gives rise to a mixed state

Discussions ab out the information problem resulted in three dierent directions which

may lead to the solution of the problem

Remnants

The rst p ossibility is that the information is stored in the last g the Planck

mass Describing such a system would require knowledge of Planck scale physics

which we do not have

What we can say is that if all the information is stored in the last g then

the information could only b e radiated away in the form of a very large number of

so called soft particles for example photons with low energy The time to radiate all

the information in this way turns out to b e very large

M

t

5

pl

g

m

pl

with M the original mass of the hole which gives rise to the idea of a longlived remnant

A black hole may even leave a stable remnant at the end of his live

This solution might seem to b e an easy way out but at least the basic principles of

quantum mechanics are not violated

However one other principle is now violated namely crossing symmetry A particle

that couples to the electromagnetic eld implies by crossing symmetry Schwinger pair

pro duction The gravitational analog is Hawking radiation The decay rate of a black

hole is prop ortional to the number of sp ecies which can b e pro duced A small black

hole should therefore decay with a high rate Leaving a remnant would mean violation

of crossing symmetry

Nice Slices

Another p ossibility is that the black hole disapp ears and the information is simply lost

An argument to supp ort this idea is the so called nice slice argument

The argument is based on the fact that almost the entire evolution of a black hole

involves lowenergy particles ie energies far b elow the Planck scale Only in the

nal stage of the evaporation pro cess one has to deal with high energies but there

would b e to o little particles left to carry all the information and is thus considered

to b e irrelevant The history of a black hole may therefore b e describ ed by ordinary

lowenergy eective eld theory ie a eld theory with a cuto in order to avoid large

energies

To show that only low energy physics is involved we foliate the geometry with a

family of Cauchy surfaces We call a family of surfaces nice slices if the surfaces avoid

regions of strong curvature and if b oth the infalling matter and the Hawking particles

COMPLEMENTARITY SMATRIX ANSATZ

have low energies in the frame of the surface Hence a nice slice do es not contain the

region inside the black hole If one chooses the nice slices to agree with surfaces of

constant time far from the hole then one could parametrize the surfaces by t and an

example of a full set of nice slices could b e the following

tGM

uv R v e u

tGM tGM tGM

e u e v R v e u

At this p oint one may introduce the nice slice Hamiltonian which generates the time

evolution The Hamiltonian maps the state on one slice to a state on the neighboring

slice

i ji H ji

t ns

One has to realize that the nice slice argument involves information loss which implies

a violation of unitarity a breakdown of quantummechanics

The niceslice theory is a lo cal quantum eld theory and therefore spacelike sepa

rated op erators will commute Consider a late time slice and separate it in two parts

one part b ehind the horizon and one outside the hole According to the

in out

states evolve linearly to another time slice but according to the no quantum Xerox

principle the initial information can not evolve completely to two separated sets of

commuting degrees of freedom The problem arises if one considers the infalling infor

mation b eing recorded inside the hole Since an infalling observer sees nothing unusual

happ en while passing the horizon little or no information can b e found outside the

horizon in the form of Hawking radiation or stored in the stretched horizon a surface

which is a few Planck lengths away from the event horizon The information do es not

come out and is thus lost after the evaporation of the hole

Furthermore information transmission requires energy and therefore also energy

conservation is violated If evolution is not describ ed by a unitary op erator then the

Hamiltonian is time dep endent and energy is not conserved But as we will see none

of the p ossible solutions are ideal

Complementarity SMatrix Ansatz

The third p ossibility is that the information is stored in some way in the outgoing

Hawking radiation The keypoint is to take interactions into account which were

neglected by Hawking see Massless scalarelds can b e regarded as a

sup erp osition of left and rightmoving parts Leftmoving waves after a certain time

after gravitational collapse will disapp ear b ehind the horizon Those waves however

could inuence the outgoing waves and hence the thermal sp ectrum Another way of

seeing it is that incoming particles interact with the outgoing particles b ecause the

infalling matter enlarges the Schwarzschild radius or v and changes the total black

hole geometry The variation in v is very small but the eect on the wavefunction of

an outgoing particle is enormous Taking this small quantum contribution into account

one nds that the in and outgoing elds no longer commute

CHAPTER INFORMATION PROBLEM

The idea is to assume that the evolution of the black hole Hilb ert space is describ ed

by a unitary op erator U t One refers to this approach as the Smatrix Ansatz where

the Smatrix working on the instate gives the outstate

jouti S jini

The Smatrix Ansatz only makes sense if one takes interactions into account the in

and outstates are indep endent without interaction b etween them and if so there could

b e no matrix connecting them

Because of the unitary evolution of black hole states the Hilb ertspace is only

allowed to contain either all instates or all outstates but not b oth sets simultaneously

since we exp ect the Smatrix to connect these Hilb ertspaces The instates evolve either

into the black hole or out to null innity J dep endent on the p osition of the observer

In fact those two worlds are complementary the information is b oth evolved into

the hole as well as out to innity An infalling observer will see the infalling matter

inclusive information propagate without interaction or p erturbation into the hole

The observer at J the asymptotic observer sees Hawking radiation and the infalling

matter app ears to evaporate inclusive information completely b efore it falls in the

hole Actually one might say that there is no singularity formed at all It lo oks like a

quantum copying machine where the infalling matter is the complementary description

or the duplicate of the outgoing matter A justication for this view can b e the fact

that so called sup erobservers can not exist ie it is not p ossible for ordinary p eople to

do exp eriments simultaneously on b oth sides of the horizon A similar paradox o ccurs

in quantum mechanics where we can consider light also in two dierent ways as a particle or as a wave

evaporation

unitary evolution t

infalling

matter

Figure Infalling matter evolves unitarily to Hawking radiation without having formed a singu

larity

General relativity seems to b e violated by having created two dierent complemen

tary worlds There is not necessarily a relation b etween the two dierent Hamiltonians

of the two worlds ie there is not necessarily a transformation from one world to its

complementary partner and reality is describ ed by only one Hamiltonian Furthermore

the fact that the in and outgoing elds no longer commute seems to violate lo cality

and causality b ecause events b ehind the black hole horizon are spacelike separated

1

Here we mean with instates those instates which are evolved into the hole

2 The causality condition means that no signal can travel faster than the sp eed of light

COMPLEMENTARITY SMATRIX ANSATZ

from events outside the horizon and therefore they can not inuence each other How

ever commutators in light front string theory remain large even when observers are

separated by a macroscopic distance String theory can p ossibly solve the paradox and is a strong candidate for a quantum theory of gravity

CHAPTER INFORMATION PROBLEM

Chapter

The Unruh Eect

As a result of eorts to understand the Hawking eect Unruh discovered the eect that

a

a uniformly accelerated observer detects thermal radiation of temp erature T

U

Although the Unruh eect was discovered after the Hawking eect I will treat

rst the Unruh eect b ecause it is in some way underlying the Hawking eect The

p oint is that even our well known Minkowski vacuum has a thermal character

Consider Minkowski spacetime in dimensions for convenience with line element

ds dt dx We transform it conformally to the socalled Rindler co ordinates

a

ds e d d

where the co ordinate transformation is given by

a

t e sinh a

a

a

x e cosh a

a

In this co ordinate system a uniform accelerating observer travels along a line with

a

constant see gure with prop er acceleration ae Furthermore we see that

in this form the b o ost symmetry generated by the killing vector is manifest A

noninnitesimal Lorentz b o ost along the xaxis is a transformation from system S to

system S with relative velocity v in the xdirection b etween the systems Note that

the translation symmetries generated by and are not manifest anymore

t x

We now have constructed the socalled Rind ler space This spacetime is said to

b e globally hyperbolic b ecause it admits a Cauchy surface A Cauchy surface is a

sort of time slice More precisely if every past or future inextendible causal curve

through an arbitrary p oint of spacetime intersects a surface then is said to b e

1 2

Conformal transformation g g The lightcones remain unchanged in a conformal

ab ab

transformation

2

Recall that a b o ost along the xaxis is equivalent to an imaginary rotation in x itspace sinh

i sin i

CHAPTER THE UNRUH EFFECT

a Cauchy surface For examples the lines const are Cauchy surfaces in Rindler

space Because of the admittance of Cauchy surfaces one has a well dened classical

evolution on globally hyperb olic spacetimes and Rindler space is thus a spacetime in

its own right We can therefore apply the standard pro cedure to construct quantum

elds If b oth sets are complete one can expand one set in terms of the other set and

conversely For example if we have two expansions and with sets of mo de solutions

u and u one can write the relation b etween the two sets as follows

k k

X

u u u

j j i i j i

i

i

X

u u u

j j i i

j i j

j

with the co ecients and

u u u u

ij i j ij i

j

These relations are known as Bogolyubov transformations with and the Bo

ij ij

golyubov coecients With these equations in hand and knowing that and describ e

the same quantumeld one obtains relations b etween the creation and annihilation op

erators from the dierent expansions

X

y

a a a

j i j i

j i

j

j

X

y

a a a

j i

j i j i

i

i

The co ecient is imp ortant b ecause it gives you the number of umo de particles

present in the vacuum ji

X X

y y

j j j j hja a ji hja a ji

j i j i j i

j i

j j

y

Recall that the number op erator N a a counts the number of iparticles

i i

i

Now I will construct the quantum elds explicitly A massless quantum eld in

dimensional Minkowski spacetime satises the KleinGordon equation

2

t x

with mo de solutions

ik xi t

p

u t x e jk j

k

The mo des are said to b e p ositive frequency mo des with resp ect to the Minkowski

Note that if time t b ecause they are eigenfunctions of the killing vector t

t F | ξ=const

η=const L

----> x R

P

Figure a The tra jectory of a uniformly accelerating observer b Rindler space with lines of

constant and

then u is a mixture of p ositive frequency mo des u and negative frequency mo des u

see So if a pure p ositive frequency mo de in one system corresp onds to a mixture

of p ositive and negative frequency mo des in another system then the vacua in b oth

systems will dier

The system is quantized by treating the eld as an op erator may b e expanded as

follows

X

y

t x u t x a u t x a

k k

k

k

k

y

with u the mo de solutions and a a the annihilation resp ectively the creation

op erators We dene as the usual scalar pro duct and and a satisfy the

canonical commutation relations

t x t x

y

0

a a

0

k

k k

k

This pro cedure is also known as second quantization in order to distinguish the quan

tization in quantum eld theory with an op erator from the rst quantization

in ordinary quantum mechanics where one imp oses commutation relations up on the

observables and where is a wavefunction

In exactly the same way we can construct quantum elds in Rindler co ordinates

The equation satised by the massless KleinGordon eld in Rindler co ordinates is

a

e 2

with mo de solutions

ik i

p

e u

k

Note that we obtained here two classes of mo de solutions The upp er sign applies in region L the lower sign in region R b ecause the killing vector is pastdirected in region

CHAPTER THE UNRUH EFFECT

L with increasing the slop e of the line const is increasing and one can see from

gure that is then pastdirected in region L Note also that Rindler observers in

region R are completely disconnected from region L and vice versa

We can expand the eld in this second set of mo des

X

Ry Ly

R L R R L L

x t u u b u b b u b

k k k k k k

k k

k

As b oth sets and are complete one can expand the Rindler mo des in terms

of the Minkowski mo des and conversely Constructing the Bogolyub ov transformations

one gets the relations b etween the dierent sets of mo de solutions Equating

and and using the Bogolyub ov transformations it is p ossible to deduce relations

b etween the ladderop erators and thus one can express the Minkowski vacuum in terms

of Rindler mo des as in

Here I will use a slightly dierent metho d which was used by Unruh by rst

making use of analytic continuation across the horizons of the mo de solutions to

the regions F and P obtaining an imaginary a Then we construct the Bogolyub ov

transformations b etween the new mo des and the Rindler mo des The tra jectory of a

uniform accelerating observer approaches the lines u and v for which

The Rindler mo des are thus nonanalytic there but sp ecial linear combinations

are The new mo des are analytically continued to the other side of the horizon The

app earance of horizons is crucial to get thermal radiation The new mo des have the

same analycity prop erties as the Minkowski mo des and therefore they share the same

vacuum provided that the analytic continuation is p erformed in the lower half complex

plane ie ln i not i xing the sign of the exp onent which gives rise to

the thermal sp ectrum

The two combinations are

R a L

u e u

k k

and

R a L

u e u

k k

One can see why these combinations are analytic and b ounded in the lower half complex

Minkowski plane just as the Minkowski mo des by writing

i a

v k

a a R a L

e v u e u

k k

i a

u k

with u v the Minkowski null co ordinates u t r and v t r

One can expand the eld in the new mo des as follows

X

p

sinh a

k

i h

a R a L a R a R

e u e u hc e u e u d d

k k k k

k k

The Bogolyub ov transformations b etween the new mo des and the Rindler mo des

R L

are easily obtained By taking the inner pro ducts u u with from the

k k

new expansion and from the Rindler expansion one nds the Bogolyub ov transforma

tion b etween the Rindler op erators b and the op erators d which act on the Minkowski

vacuum

y

a a

p

b e d e d

k k k

sinh a

To know what Rindler particles are present in the Minkowski and simular for b

k

vacuum we calculate

a

e

LR LRy

p

j i b h jb

M M

k k

a

e

sinh a

a

see which is precisely the Planck sp ectrum of a black b o dy with temp erature T

U

app endix B

A uniformly accelerated observer or Rindler observer sees the redshifted temp er

ature

a

ae

T

U

a

The prop er acceleration is ae since the lines with constant are the hyperb olae

and the temp erature seen by the accelerating observer is prop ortional x t

2 2a

a e

p

to the proper acceleration Furthermore the redshifted temp erature is T T

g

00

a

and from we see that g e In the same way one can show that an inertial

observer sees himself immersed in a thermal bath of Rind ler particles

A way to understand the Unruh pro cess is to imagine vacuum pro cesses where pairs

of virtual particles and antiparticles are created One of the particles is detected by the

accelerating observer while its counterpart passes the horizon u and gets eternally

disconnected from the world of the ever accelerating observer wedge R

In SI units the temp erature is given by

K ha

T a

U

k c ms

B

which shows that it is very dicult if not imp ossible to see the Unruh eect in the

lab oratory However recently the Unruh eect was mentioned in the context of sonolu

minesence a phenomenon which is still unexplained A related geometrical eect

which is detected exp erimentally is the Casimir eect By imp osing certain b ound

ary conditions one alters the top ology and can give nonzero vacuum stress energy

Casimir calculated that two conducting surfaces at close distance m feel at

tractive forces in the vacuum due to vacuum uctuations Between the plates we have

the situation that not al l the uctuations are aloud b ecause the waves have to t

b etween the plates Outside the plates there are more uctuations which results in a pressure and a measurable force

CHAPTER THE UNRUH EFFECT

But there are more metho ds to derive the Unruh eect In statistical mechanics

we have the KMScondition named after Kub o Martin and Schwinger which says

that the prop erty of p erio dicity in imaginary time is asso ciated with states of nite

temp erature in eld theory Formally one can write the KMScondition as follows

For each pair of local observables A B there is a function Fz of the complex variable

z that is analytic in the strip Im z and continuous on its boundaries such that

F t AB and F t i A B for real t

t t

Simply one writes

hA B i hBAi

i

dening hAi to b e the exp ectation value TrA the inverse temp erature and A

t

to b e the time translation by t of the op erator A I will prove that the KMScondition

is equivalent to that of lo cal thermo dynamical stability

We have

itH itH

A e Ae

t

so with t i

H H

hA B i Z Tre Ae B

i

with the density matrix In order to satisfy the KMS condition we get the condition

H H

Tre Ae B TrB A

or by using the cyclical prop erty of the trace

H H H H

e B e B e B B e

H H

By putting B I one sees that commutes with e This means that e com

mutes with arbitrary B and must b e of the form cI with c a scalar As Tr must

H

b e the canonical density matrix Z e In other words is the only density

c c

matrix that satises the KMScondition

Because of G f sinh and sinh x i sinh x it can b e shown that

the twopoint function G h i along a uniformly accelerated worldline

in Minkowski vacuum satises the KMS condition

i

G G

a

The twopoint correlation function is a Green function and is an imp ortant entity

The twopoint function is dened as the vacuum exp ectation value of the time ordered

pro duct of the elds x and y Gx y hjT xy ji and give information

ab out the system With the twopoint correlation function in hand one knows the

PARTICLE DETECTOR

evolution of the system and one can calculate the observables Along a uniformly

accelerated worldline the Minkowski vacuum is a thermal state

i h

Y X

n a

j j

j i e jn Ri jn Li N

M j j

n

j

j

Another way to analyse the thermal prop erties of the Minkowski vacuum is to lo ok at

the intimate relation b etween the top ological structure and the particle concept We

already noticed the imp ortance of horizons In fact the key geometrical structure in

the Minkowski spacetime analysis which gives rise to the thermal state prop erty of the

vacuum is the bifurcate kil ling horizon generated by the Lorentz b o ost isometries In

Rindler Kruskal and De Sitter spacetime two killing horizons app ear which divide the

spacetime into dierent sections or wedges A pair of intersecting killing horizons is

called the bifurcation horizon The killing eld is normal to null plane horizons

and therefore we get nonanalytic b ehaviour across the horizon Analytic continuation

k

across the horizon gives rise to the factor e which is resp onsible for the thermal

sp ectrum

The notion of vacuum has b ecome sub jective We can not sp eak anymore ab out the

vacuum So when one sp eaks ab out the vacuum what do they mean The conven

tional vacuum the vacuum is the agreed vacuum for all inertial observers Both

the conventional vacuum and the set of inertial observers in Minkowski space are

invariant under the p oincaregroup From we see that the temp erature indeed

vanishes if the acceleration is equal to zero Furthermore the vacuum b ecomes only

ill dened when huge accelerations or very strong gravitational elds are involved

Particle Detector

In order to understand the Unruh eect b etter and to clarify the obscure notion of

particles one has developed mo dels of particle detectors The essential feature of

a particle detector is the interaction with the eld After interaction the state of the

detector has changed One can show that a noninertial detector will not detect the

same particle density as an inertial one

In literature one nds two mo dels One mo del is a particle detector which consists

of an idealized p oint particle with internal energy levels The interaction Lagrangian is

describ ed by m x with m the detectors monop ole moment and the detector

couples via monop ole interaction to the eld The other mo del Unruh and Wald

is essentially a particle in a b ox detector which is simplied by allowing the

detector only two energy levels I will treat the second mo del in more detail We nd

3

A p oincare transformation is a prop er Lorentz transformation followed by a translation The

generators of the p oincare group are the six generators of the lorentz group from rotations and

from b o osts plus the four generators of the translation group The p oincaregroup is thus the group

of rotations spacetime translations and Lorentz b o osts

4 Dont confuse here particle detectors with bubble chambers

CHAPTER THE UNRUH EFFECT

a remarkable paradox involving the change of the eld energy during detection In the

following it will b e shown that the detector detects ie the detector will b e found

in its excited state j i when it absorbs a quantum of the eld

The total hamiltonian of the elddetector system is

H H H H

F D I

R

where H is the free KleinGordon hamiltonian of the massless eld H dx

F

r For the detector hamiltonian H we take

D

y

H A A

D

The raising and lowering op erators are dened by

y

A j i j i

Aj i j i

where the kets represent the state of the detector The coupling of the detector to the

eld is assumed to b e given by the interaction Hamiltonian

Z

p

y

g d x H t xxA xA

I

where x is a smo oth function which vanishes outside the detector and t is the

coupling constant with explicit time dep endence in order to enable us to turn the

detector on and o

The system ie a state jsi at time t evolves to time t via

R

t

2

H dt i

t

1

js t i js t i e

Let us take the initial state at early times to b e a state where the detector is in its lowest

state while the eld is containing n particles thus jn i At late times the system will

b e found in rst order in in the state

Z

p

0

it

e t x x js i jn i i g d x dt jn i

It can b e shown with some tedious algebra that

Z

p

g d x a f

it

for any function f for example f e Here a is the annihilation op erator for

the free eld

We can conclude that our simple mo del do es function prop erly as a detector b ecause

the detector will b e excited if and only if one quantum of the eld is absorb ed due to

the app earance of the annihilation op erator a

PARTICLE DETECTOR

We now let the detector uniformly accelerate by taking the b o ost killing eld as

its killing time Cho osing the Minkowski vacuum j i as initial state and letting the

M

detector uniformly accelerate which results in the fact that the annihilation op erator

in b ecomes the Rind ler annihilation op erator in the right wedge R a we get

R

js i j i ia j i

M R M

The Minkowski vacuum is equivalent to a thermal bath of Rindler particles see

and hence we see that the detector detects absorbs indeed particles b ecause of the

annihilation op erator a

R

On the other hand we can by using the Bogolyub ov transformations express the

Rindler operator in terms of Minkowski operators instead of expressing the Minkowski

vacuum state in terms of Rindler states One nds that the detection of a particle ie

the absorption of a Rindler quantum corresp onds to the emission of a Minkowski

particle

An inertial observer sees the emission of a particle while an accelerating observer

in the detector frame sees absorption The apparent paradox whether the eld wins

or lo oses energyh can b e resolved as follows According to the inertial observer the

eld energy increases by one quantum The Rindler observer has a dierent interpre

tation but agrees that the eld energy increases The initial state of the eld is not an

eigenstate of energy and thus the act of detection the measurement indicates that a

larger number of particles then originally exp ected were present initially

CHAPTER THE UNRUH EFFECT

Chapter

The Hawking Eect

Closely related to the Unruh eect is the Hawking eect which states that at suciently

late times after gravitational collapse a black hole will emit thermal radiation at a

due to quantum eects This eect strongly supp orts the temp erature T

H

relationship b etween thermo dynamics and black hole physics as describ ed in chapter

Secondly one would exp ect according to the principle of equivalence that also in

strong gravitational elds some thermal eects could o ccur

In his calculation of the outgoing ux Hawking worked with a xed background

geometry He neglected backreaction eects of the quantum elds on the black hole

The reason for this approximation is the absence of a quantum theory of gravity Before

the theory of quantum electro dynamics was developed one used successfully the same

pro cedure by considering the electromagnetic eld as a classical background eld

Without taking backreaction eects into account however it seems unavoidable

that pure states evolve into mixed states and thereby violating quantum mechanics I

discussed this problem in chapter

Now I will give you the derivation of Hawkings result following I will work

in dimensions with the advantage that it enables us to say something ab out the

energystresstensor The result is the same as in four dimensions Secondly we restrict

our attention to massless scalarelds Note that the solutions of the KleinGordon equa

tion for massless scalarelds are sup erp ositions of left and rightmoving

u v

waves

Consider a collapsing spherically symmetric ball of matter forming a black hole

i v

One can imagine nullrays e coming in from J the rightmoving part is

in

irrelevant b eing reected in the center of the ball and b ecoming outgoing waves

out

i pu

e However nullrays which are emitted after v v will pass the horizon and fall

into the singularity Hawking lo oked at the time reversal pro cess where an outgoing

0 0

i u i f v

wave e reects at the origin and b ecomes the ingoing wave e

out in

Here I have used a WKB approximation see app endix A by taking the classical

1

A frame linearly accelerated relative to an inertial frame in sp ecial relativity is lo cally identical

to a frame at rest in a gravitational eld

CHAPTER THE HAWKING EFFECT

singularity

Hawking radiation

I+ horizon e-iwp(u) e-iw'u

r=0,

center of I- matter ball e-iwv e-iw'f(v)

v=vo

surface of

collapsing star

Figure a A collapsing star forming a black hole b Incoming test particles photons for example

which reect at r c The time reversed pro cess

path v pu of the reecting wave resulting in the waves describ ed ab ove Hawking

referred to this approach as the geometrical optics approximation which means that

one takes the waves at one frequency without taking disp ersion into account

Before we can calculate the Bogolyub ov co ecients we will have to nd the phase

pu of the outgoing wave The recip e is as follows

We b egin by writing down the expressions of the line element for resp ectively the

regions in and outside the matter ball

ds AU V dU dV

ds C r dudv

M

for a Schwarzschild black hole and with C

r

u t r R v t r R

U r R V r R

dr

where dr since C dudv C dt dr R is the initial p osition of the

C C

surface of the collapsing ball and the relation b etween R and R is the same as

the relation b etween r and r

The co ordinate transformation is given by

U u v V

At the center of the matter ball at r one has v V U R

u R pu b ecause the quantumeld has to ob ey the reection con

dition ie to vanish at the reection p oint r In fact the expression for

the total massless scalareld is a sup erp osition of left and rightmoving waves

i v i pu

e e This shows that the quantumeld indeed vanishes at the

reection p oint We neglect one term of the eld b ecause it do es not contribute

to the reection pro cess

Secondly the line elements have to match on the collapsing surface r R

C Rdt dR Ad dR

With this relation in hand we nd

dU d dr d RC RC

1

du dt dr

C dt Rd 2

AC R R R

where R R

Finally note that we only consider the particle ux in the asymptotic region ie

near the horizon C or equivalent at late times u hence we can

approximate by

R C R dU

du

R

We nd

R

C du dU

R

In order to nd an expression for R one can expand R near the horizon

R R R O

h h h

and thus

C

u l njU R j const

h h

r

u

u U e const

The phase pu of our outgoing wave now nally b ecomes

u

pu C e D

We used v V V b ecause at late times an asymptotic observer sees the matter

ball hardly change due to time dilation

The constant is the so called surface gravity of the horizon and is dened by

C

j

r R

h

r M

The surface gravity is a measure of the strength of the gravitational eld at the horizon

More precisely is the magnitude of the acceleration with resp ect to killing time of

CHAPTER THE HAWKING EFFECT

a stationary particle just outside the horizon This is the same as the force p er unit

mass that must b e applied at innity in order to hold the particle on its path

Formally the surface gravity is dened as the magnitude of the gradient of the norm

a

of the horizon generating killing eld evaluated at the horizon

a

r jjr jj

a

and the constant D is the horizon v The constant C in turns out to b e

As u v approaches indeed the horizon v as we can see in gure Further

more on physical grounds we expected already the phase factor of the outgoing wave

pu to b ecome exp onentially small as the surface of the matter ball approaches the

event horizon due to the gravitational redshift

0

i u

If we consider an outgoing wave e then the corresp onding ingoing wave

out

0 1

i p v

had to b e e We write f v for the inverse function of pu

in

f v ln v v v

We wish to calculate the Bogolyub ov co ecients and which relate the dierent

expansions of the quantum eld One of the expansions is in terms of outgoing mo des

and the other in terms of ingoing mo des But instead of the outgoing mo de in the form

0 0

i u i f v

e we take its incoming form e which is of course the same

out in

reected wave The Bogolyub ov co ecients are scalar pro ducts of wave mo des and

give therefore the overlap b etween the mo des We now wish to calculate the overlap

b etween the incoming waves from the dierent expansions

0

i v i f v

0

e e

out

in

0

i v i f v

0

e e

out in

After an integration by parts and including the normalization factor one obtains

r

Z

v

0

i

0 0

i f v i v

e dv

0

r

Z

v

0

0 0 0 i

i iv i v

0

v e dv

0

Z

0

i

i v

0

i

0 0

i x iv

0

p

x e dx e

R

n x

With the denition of the Gamma function n x e dx and with the knowl

edge that lightrays after v v do not contribute to the integral we nd

0 0 0 i i

0

iv i

0

p

e e

0

As we saw in chapter the outgoing radiation sp ectrum is given by the Bogolyub ov

co ecient

X

0 0

j i F j hN

After a straightforward calculation we nd for the ux coming out of a neutral non

rotating black hole

F

0

e

The same result is found if we use the completeness relation

X

0 0

j j j j

0 0

j are indep endent of j and j and the fact that j

X

0

j F j

0

2

j j

0

e

2

j j

0

This is precisely the sp ectrum you would exp ect for b osonic thermal radiation at a

temp erature

K k g T

H

k k M M

B B

Of course we assumed that the black hole is surrounded by empty space otherwise

a black hole heavier then M ab out half of the mo on would absorb the

K background radiation The notion of thermo dynamic equilibriu m b ecomes curious

however b ecause a big cold black hole would absorb the Kradiation and instead

of b ecoming hotter as one would exp ect according to the Laws of thermo dynamics it

b ecomes bigger and thus colder This might seem as a violation of the Second Law

but it is not according to the Generalized Second Law A black hole acts as a vacuum

cleaner of entropy storing it on its area

Although one encounters some technical diculties in the four dimensional case

the derivation is roughly the same and the result is identical It is also p ossible to

generalize the result to charged and rotating black holes A rotating hole emits a ux

F

0

m

e

where m is the azimuthal quantum number m l l and the angular sp eed

at the event horizon The temp erature of a charged black hole turns out to b e

e

T

k M A

B

with A the area of the event horizon and e the charge of the hole

We restricted our attention to massless scalarelds which is actually quite unphys

ical the only scalar particles we know of are Higgs b osons which are still not found

CHAPTER THE HAWKING EFFECT

exp erimentally and have big masses However having the fundamental thermo dy

namic nature of quantum particle pro duction in mind the calculations should also b e

valid for higher spin elds Because of the anticommuting nature of fermions we would

get a sign in resulting in the appropriate Planck factor for Fermi statistics see

app endix B

Due to the radiation the black hole lo oses mass and gets hotter For a hole with

mass M kg the creation of electronp ositron pairs b ecomes p ossible at ab out

M kg the Schwarzschild radius approaches the range of strong interactions and

when the hole has reached the Planck mass M kg nob o dy knows what the

outcome is a nal burst of huge energy or maybe a stable remnant as describ ed in

section

Stress Tensor

How real is this black hole radiation actually A heuristic way to understand the

Hawking pro cess was given by Hawking p ersonally All the time and in the whole

of space we have quantum pro cesses in the vacuum where pairs of virtual particles

and antiparticles are created These vacuum uctuations are p ossible b ecause the

uncertainty relation of Heisenberg states that one can not measure the total energy

with innite accuracy at one moment E t h and therefore energy conservation

can b e violated in short time intervals Hence these pairs annihilate immediately

typically in s

Now imagine that this happ ens just outside the event horizon Due to the strong

gravitational eld these particles can b e separated causing one particle to fall in the

hole while the other particle manages to escap e to innity constituting to the thermal

radiation Inside the black hole the virtual antiparticle b ecomes real without loosing

Figure Pair creation of virtual particles and anti particles near the horizon

its negative energy This is p ossible b ecause the time translation generating killing

eld b ecomes spacelike inside the horizon allowing negative values This is a heuristic

description of the pro cess and should not b e taken to o literally b ecause one could also

imagine the particle falling into the hole and increasing the mass of the hole while the

antiparticle escap es

Investigating the energy stress tensor T is a more rigorous and relevant metho d

to get more insight in the physical existence of the Hawking radiation We will

STRESS TENSOR

nd an expression for the energymomentum tensor regularized by the p ointsplitting

pro cedure As b efore we will restrict our attention to the quantum theory of a massless

scalar eld in two dimensions

The classical expression for the stress tensor is

T x g

where are normal mo des The op erator can b e expanded in those mo des For the

operator T we get

g T x

Now consider a p oint x at a prop er distance jj from x We denote the normalized

tangent vector at x by

dx

t

d

as the matrix which maps the vectors at x to those at Next we dene the matrix e

x Evaluating one eld at x and the other at x we nd for the exp ectation value

of T

e hT x i hj x x ji e e g e

After an expansion in p owers of and using the fact that every two dimensional metric

is conformally at ie for any co ordinate system we can write ds C u v dudv we

get

R t t

hT x i O g

t t t t

where R is the curvature scalar and the tensor has the comp onents in the sp ecial

i v i u

null co ordinates u v such that the normal mo des are simple exp onentials e e

on J and J resp ectively

p

1

2

C C

u u

u

p

1

2

C C

vv

v

u v vu

One sees immediately that the result diverges as Moreover the result contains

terms which dep end on the arbitrary direction of p oint separation Therefore we

will simply discard the terms containing or the tangent vector t

R

g hT xi

2

for xed in a given co ordinate system is the contravariant vector x This means that e

obtained by parallel transp ort along the geo desic from x of the basic vector in the direction

CHAPTER THE HAWKING EFFECT

As in we consider the metric

ds C r dudv

Because of the fact that any two dimensional co ordinate system is conformally related

to Minkowski spacetime which results in R and b ecause of

C CC

r

r u

we nd for the stress tensor

M M

hT i CC C

uu

r r

So far there has b een no collapse and still no black hole is formed If there is a

gravitational collapse then the co ordinates u v are no longer appropriate b ecause the

i u i pu

outgoing mo des e b ecome complicated exp onentials e In other co ordinates

u pu and v v the metric b ecomes

ds C u vdud v

uM

With pu e v as in

M M du

C u v C r

du r v u

we nd for the energy tensor

M M

hT i

uu

r r M

For large r we see an outgoing ux

hT i

uu

M M

This is exactly the energy ux for a b o dy with temp erature T Near the horizon

M

however r M we see a negative energy ux going into the horizon

M M

hT i

v v

r r M

Indeed one could in some way describ e this pro cess as the creation of particles and

antiparticles near the horizon Particles carrying the p ositive energy escap e to innity

while their counterparts with negative energy fall through the horizon into the black hole

Chapter

Moving Mirrors

In Davies and Fulling developed a simple moving mirror mo del which

provides a very useful analogy to the black hole situation see also By

choosing suitable tra jectories the moving mirror mimics many of the features of black

hole radiance with the dierence that the quantum elds propagate in at Minkowski

space instead of complicated geometries with high curvatures First we will treat the

case with a xed tra jectory which represents a xed background geometry Then we

take the rest mass of the mirror nite and calculate classically the back reaction eect

of the reection of massless scalar elds o the b oundary

Fixed Trajectory

Consider a reecting b oundary in dimensions following a tra jectory at late times

for example z t ln cosh t that mimics the eect of the gravitational collapse

geometry

t

z t t e v

In the black hole case the incoming waves travel through the collapsing matter and

reect at r The mirror tra jectory represents thus the origin of the black hole

geometry The last term v is the horizon after which the lightrays travel undisturb ed

to the left without b eing reected This represents the black hole event horizon

Furthermore we see that the mirror velocity approaches the sp eed of light exp onentially

fast

dz t

t

e vt

dt

Secondly we have a massless scalareld which reects o the mirror We imp ose

the reection b oundary condition that the eld has to vanish on the b oundary

t z t

One could see that this is a reection condition by imagining a running wave on a rop e

with a xed end The wave vanishes at the xed end and reects

CHAPTER MOVING MIRRORS

A late outgoing wave along u u hits the mirror approximately at time t

u v From we deduce the following relation at the reection p oint u v

u v

0

e u v lnv v f v v v

This classical calculation gives us the classical tra jectory f v u of an mono chro

matic outgoing wavefunction mono chromatic means that the wave function is sharply

p eaked at one frequency which reects o the mirror Taking the classical tra jectory

as the phase of the wavefunction corresp onds to making use of the WKB approxima

tion as describ ed in app endix A Due to this classical approximation we have the

mono chromatic outwavefunction

0

out i u

e

and the corresp onding complicated inwave

0

in i f v

e

Note that we have constructed the mirror tra jectory in such way that the function f v

is identical to the phase of the wavefunction in The Bogolyub ov co ecients are

therefore the same as in

0 i 0 0

i iv

0

p

0

e e

and we nd for the outgoing ux

X

0

j F j

0

e

However which is the ux of the radiation of a black b o dy with temp erature T

is not the surface gravity but a measure of the acceleration

out - region ----> t

V=Vo

in-region

----> x

Figure The mirror tra jectory z t with reecting particle

FIXED TRAJECTORY

Saddlep oint Approximation

For later comparison I will calculate the Bogolyub ov co ecients again but now by using

the metho d of saddlep oints The idea is to approximate the integral by taking

only the biggest contribution of the integral into account The biggest contribution of

the integral is at the socalled saddlep oint and is found in our case where the derivative

of the exp onent vanishes Note that it makes no dierence if we take the outgoing

0 0

i u i pu i f v i v

forms of the waves e e or the ingoing forms e e

0 0

lie resp ectively at u u with and The saddlep oints for

puj u v log

u u



0

corresp onds to the physical reection The saddlep oint u u that contributes to

time which is uniquely determined for given in and outfrequency via the Doppler

relation

v

v

with v the velocity of the mirror given in This means that one could also deduce

by using The reection p oint satises the equation pu v and we see

t r v

pu v

u u

v

t r

0

on the other hand has an imag The saddlep oint u u that contributes to

log i The saddlep oint inary part equal to b ecause of the term

u corresp onds therefore to the virtual reection pro cess that gives rise to the particle

creation phenomenon These outgoing reection times corresp ond to ingoing timesv

with

v v

Hence for the Bogolyub ov co ecients we nd

0 0 0

i i

0 0

i u i v ln iv

+ +

0

0

e e

0 0 0

i i

0 0

ln iv i u i v

0

0

e e

The outgoing ux is again our well known Planck sp ectrum

F

0

0

j j e

0

1

Instead of we take here its conjugate This means nothing more then lo oking at the outgoing

0

wave where the ingoing wave had to have negative energy In we saw that the outgoing energy

changed sign One could say that we lo ok at the time reversed pro cess Lo oking at the time reversed

pro cess has no inuence on the mirror tra jectory b ecause we take it xed In the next section however

we will have to b e consistent and take the mirror tra jectory at late times determined in order to b e able to lo ok at the time reversed pro cess

CHAPTER MOVING MIRRORS

Uniform Acceleration

If one takes the acceleration of the mirror uniform one nds that the Bogolyub ov

co ecient is nonzero and that the mirror therefore emits particles However the

energy stress tensor vanishes Recall the relation for the exp ectation value of the stress

tensor

hT i

which b ecomes after a co ordinate transformation

u f u v v u pu v v

to a co ordinate system u v in which the mirror remains at rest x

1 1

2 2

hT i f u f u

uu

u

Transforming to the u v system gives

1 3

2 2

f u f u hT i

uu

u

and with z t the mirror tra jectory we nd in the x t system

1

2

z d z

hT i hT i

tt xx

3

z d

2

u

z

d

For nonrelativistic motion this is simply z We see that a uniform accelerating

dt

mirror will not radiate energy although we are dealing with noninertial motion of

the mirror and particle creation o the mirror o ccurs The problem is that in curved

spacetime or involving noninertial motions the particle concept b ecomes bad dened

and dierent from our conventional view The statement energy h p er quantum

do es not apply in this case

Backreaction on the Mirror

In this section we will rep eat the calculation from section but now by taking

the backreaction of the radiation o the mirror into account This section is largely

based on Backreaction eects are imp ortant for two reasons First we have the

information problem as describ ed in chapter Including backreaction eects could

lead to a solution of the problem see section Secondly there is the problem of

divergent energies as v v The radiation coming o a mirror or out of a black hole

at late times had to b e innitely blueshifted incoming waves due to the Dopplershift

or gravitational shift

A mirror with innite mass corresp onds to a xed tra jectory b ecause an ob ject

with innite mass do es not feel the b ounce from particles I will therefore take the

mirror mass to b e nite and imp ose energy conservation up on reection However the

BACKREACTION ON THE MIRROR

γmv ω' reflected photon ----> t

mirror particle in-coming photon ω

----> x

Figure The reection of a photon o a mirror particle with nite mass

Bogolyub ov co ecients are not exactly soluble anymore which forces us to use approx

imations We will use here the saddlep oint approximation

Consider an incoming photon with energy h which collides elastically with a mir

ror particle with mass m After the collision the mirror particle has energy vmc

and the photon reemerges with a energyh From now we put againh c We

want to express the energy in terms of and the mirror velocity v just after the

collision With the condition for energy conservation in two dimensions p k p k

classically one nds

k k p k k

and with k k and p m m jvj

m

with the Doppler factor

s

v

v

Note that for v we get and for m we nd the quadratic Doppler

shift as exp ected see We now wish to consider the same classical mirror tra jec

tory as in the previous section We imagine therefore that there is some external force

that acts on the mirror particle which in the absence of other forces such as those

due to p ossible virtual collisions with the photons will keep it in the given tra jectory

More sp ecically in determining the relevant classical WKBtra jectory we keep

the mirror tra jectory after the collision with the photon xed so that it always remains

asymptotic to v v From we deduce

tv

0

v e v

2

Note that we take now a shifted tra jectory t t v 0

CHAPTER MOVING MIRRORS

tv

0

v v e

and by using and we nd for late times

uv

0

2

p

e

v v

Thus applying the general formula we get the following relation b etween the in

and outgoing energies in terms of the outgoing time u

uv

uv 0

0

2

e e

m

and in terms of the incoming time we have

p

0

v v v v

m

We see now that the energy of the incoming photon can not b ecome innitely

large neither larger then the mirror energy after collision Energy is indeed conserved

If we keep xed then for late times approaches this maximal value

m

m

since v for late times

We would again like to use a combination of a WKB approximation for determining

the relation b etween the in and outgoing wave packets and a saddlep oint approxima

0

tion for the integrals in the expressions of the corrected Bogolyub ov co ecients

0

In fact one could say that the two approximations are related b ecause the and

saddlep oint approximation gives us the classical path as do es the WKB approximation

Following the same steps as in section we can again express these co ecients in

terms of the reection p oints u v as

0

i u i v

+ +

0

e

0

i u i v

0

e

That is the co ecients are just equal to the phase jump b etween the incoming and

outgoing wave at the reection p oint Note that the incoming frequency changes sign

when we calculate the saddlep oints u v for teh Bogolyub ov co ecient b ecause

we lo ok at the time reversed pro cess Furthermore I take solution b ecause in the

limit m one would like to get the old result back and avoid negative values of the

exp onent

Using we nd the new saddlep oints u at

p

u v log log x x

BACKREACTION ON THE MIRROR

and with we obtain

p

v v x x x

with

x

m

As b efore the physical reection p oint u v that contributes to the Bogolyub ov

0 0

is always real while the saddlep oint u v that contributes to co ecient

again has an imaginary part This imaginary part however is no longer constant but

dep ends on the variable x Note further that the virtual reection p oint u v is

related to the real one by changing the sign of the ingoing frequency Indeed the out

going radiation is pro duced b ecause negative energy incoming mo des can propagate

via virtual tra jectories into p ositive energy outgoing mo des

It is easy to see that if we keep the outgoing frequency constant which is still

physically reasonable the incoming frequency will at late times still grow exp o

nentially in the outgoing time but now with half the efolding time This implies in

particular that at late times that is for u v we have the relation

u log x const

b etween the parameter x and the reection time u In the following we will make use

of this relation to read o the time dep endence of the radiation sp ectrum given its

expression in terms of the in and outgoing frequencies

A straightforward calculation now gives the following result for the new Bogolyub ov

co ecients

i h

p p

i

0 0

x x log x x exp x

and

h i

p p

i i

0 0

exp x x x log x x

0 0

denote the uncorrected co ecients given in section Note that and where

we indeed get the old result back in the limit x which corresp onds to the innite

mass limit m For the ratio of the absolute values of the new co ecients we nd

h i

0

j j

exp Rx

0

j j

with

p

x for x Rx arccos x x

for x

3

Note that the dep endency is transformed into the time dep endency x which justies this pro cedure

CHAPTER MOVING MIRRORS

κ/2π T B

κ π ----> k /4

----> time

2 2

log x Figure The temp erature of the radiation as a function of the outgoing time u

p p

x arctanx x arccos x We now where we used Im log x i

dene the outgoing sp ectrum F x as a function of the time parameter x as

F x

0

2

j j k T x

0

B

e

2

j j

0

where we identied the timedep endent temp erature as

k T x

B

Rx

The temp erature as a function of the outgoing time u is plotted in gure

We read o that after a relatively short amount of time of the order of

m

u v log

the mirror radiates a constant ux of thermal radiation at a temp erature k T

B

where I used with x Recall that the typical frequency of the outgoing

radiation is of the order of see app endix B

In conclusion we see that introducing a reecting b oundary with nite mass and

imp osing energy conservation up on reection leads to back reaction eects with dra

matic consequences the temp erature of the mirror is half of the original result This

result could have b een anticipated from equation For late times ie in the limit

v v the second term in the denominator determines the corrected Doppler rela

tion b etween the in and outfrequencies In other words the magnitude of the ingoing

frequency for given grows linearly with the Doppler factor instead of quadratically

Dilaton Gravity

A close related mo del to the one of moving mirrors is dilaton gravity see

This mo del is just complicated enough to contain black hole solutions as well as Hawking

radiation Again we restrict our attention to two dimensions

DILATON GRAVITY

The action which gives us the Einstein equations is

Z

p

S d x g R f

with g the metric R the Ricci or curvature scalar and f the matter elds The action

for dilaton gravity is given by

Z

p

g e R r S rf d x

where is the dilaton and a cosmological constant which sets the scale of the surface

gravity and e plays the role of gravitational coupling A factor in front of R in the

action scales the metric giving rise to dilatation of spacetime is therefore called a

dilaton A crucial prop erty is the fact that we can rescale the metric obtaining a at

spacetime

ds e dx dx

x x x x

with x x x Now one can solve the classical equations of motion for e

Z Z

+

x

dy dz T e M x x dy dz T

x

Putting the energymomentum tensor equal to zero we get the static black hole solution

see Without any matter present ie T M one gets the the two

dimensional Minkowski vacuum or linear dilaton vacuum

dx dx

ds

x x

which b ecomes with

r t

x e

our familiar expression for the Minkowski line element

ds dt dx

One obtains the Hawking radiation by considering infalling matter Before the matters

comes in we have simple Minkowski spacetime The infalling matter forms a black hole

in the mo del of dilaton gravity as can b e seen from the metric after some co ordinate

transformations If one now calculates the matter stress tensor of the right movers one

nds the thermal Hawking ux In the left asymptotic region of the Minkowski plane

x r the dilaton takes a large value as can b e seen from

4

We take this action simply b ecause it gives us a black hole solution Furthermore this action is

closely related to an action for some sort of strings The lowest order string eective action lo oks like

R

p

d+1 2 2

S d x g e R r V l oops

5

In YangMills theory the general gauge theories for example one writes the Lagrangian as L

1

with g the coupling constant

2 g

CHAPTER MOVING MIRRORS

Including the reection condition ie energy conservation and imp osing a cuto on

+

x x

we nd the b oundary tra jectory x where e takes its critical value

cr

cr

e This is closely related to the moving mirror tra jectory or the r p oint in the

black hole geometry

Ab ove we considered T If we now consider an incoming sho ck wave T

p x q as the photon in the previous section one sees that the b oundary

tra jectory acts as a mirror with negative mass

In particular one nds that all incoming particle wave of energy b elow some critical

value will reect back to innity provided it do es not disapp ear b ehind the horizon

cr it

of an already existing black hole see The tra jectory is given by

m

x p x

m

x p x

with

p q

p

m q p

2

m

we see that the tra jectory For incoming energies larger than p q

2 cr it

after the collision b ecomes spacelike For larger energies than the critical value

cr it

the particle wave will never reect back but always lead to black hole formation In

the subcritical regime one can calculate the relation b etween the in and outgoing

frequencies by solving for the corresp onding classical geometry After shifting x by

v

0

x y P dening P e and by using we can rewrite the tra jectory

in the notation of the forgoing sections

u v u v

0

e e

cr it

The relation x p y p b ecomes

v u u v

0 0

e e

In the background geometry of a black hole of mass M we send a signal with frequency

backwards in time from an outgoing time u It will b ounce o the b oundary and

pro duce an incoming signal at past innity The relation b etween the initial frequency

and nal frequency can thus b e written for late times u as

log u v

with the dilaton gravity cosmological constant which sets the scale of the surface

gravity at the black hole horizon The time v is roughly the black hole formation time

v given by The reection o the b oundary takes place at an ingoing time

cr it

log v v

DISCUSSION

Somewhat surprisingly we see in that the backreaction of the geometry do es not

mo dify the old linearized relation b etween the in and outgoing frequencies Hence if

we would use this relation to compute the Bogolyub ov co ecients and the outgoing

sp ectrum we nd no interesting corrections to the outgoing sp ectrum We have for

cr it

0 0

i i

0 0

iv u i v i log log

+ +

0 cr it

0

e e

0 0 0

i i

0 0

i u i v iv log log

0 cr it

0

e e

So that

0

2 0

j j

e

0

j j

which seems to indicate that the outgoing thermal sp ectrum receives no corrections

whatso ever

However the result for the classical reection time is only valid in the sub

critical regime As so on as

cr it

we can no longer use the ab ove formulas b ecause there no longer exists a classical

tra jectory that relates an outgoing wave of this frequency to a regular incoming wave

Indeed as seen from there is no longer a physical reection time Instead the

only classical solutions that contain outgoing particles in the sup ercritical regime

are solutions that contain either white holes or naked singularities In neither case

however we know of go o d physical principles that provide concrete initial conditions

for the outgoing state This sup ercritical regime is reached very quickly since

in terms of the outgoing frequency the inequality reads

uv

0

e

cr it

Thus we are forced to conclude that the WKB metho d breaks down when applied to

twodimensional dilaton gravity

Discussion

In app endix A I discuss the WKB approximation It is shown that the approximation

is justied provided that the frequency varies slowly relative to its magnitude In our

case the frequency increases exp onentially and thus the variation is big At the same

time however the frequency itself is very large satisfying the condition

Another justication for using this approximation is the fact that this same ap

proximation was used by the physicists who invented the moving mirror mo del I just

included backreaction eects in the same calculation However I do not deny that

other approximation schemes could give other interesting results

The combination of the WKB approximation and the saddlep oint approximation should not b e viewed as succesive approximations but rather as related approximations

CHAPTER MOVING MIRRORS

As describ ed in the app endix the WKB approximation is a classical approximation by

taking the classical path of the quantumelds On the other hand the saddlep oints of

the integral are the classical reection p oints on the mirror

Chapter

Conclusion

In this rep ort I gave a short review of the facts which led to Hawkings discovery and

the explicit calculation of the Hawking eect is presented Through the years their has

b een done a lot of work on the sub ject and the eect is supp orted by many authors At

the same time one recognized the problems coming along with the acceptance of black

hole evaporation The violation of unitarity and the inevitable o ccurrence of divergent

energies are problems which are still unresolved due to the absence of a theory of

quantum gravity and the lack of knowledge of Planck scale physics

I tried to include backreaction eects in the moving mirror mo del in order to imple

ment energy conservation in the particle creation pro cess and found that backreaction

eects indeed aect the outcome

In the calculations of the corrections of the original result I was forced to use

approximations In itself the result is promising and could give an indication to other

mo dels However it remains a challenge to improve the approximations in order to

strengthen the result and gain more insight in the whole pro cess This area in physics

where general relativity and quantum mechanics meet each other could give more

insight in what prop erties a unied theory or a quantum theory of gravity should

p ossess

CHAPTER CONCLUSION

App endix A

WKB Approximation

Consider the Schrodinger equation in dimension where the p otential energy do es

not have a simple form

d

E V x A

dx h

i

k x

h

If V const the equation A has simple solutions e This suggests to try a

solution of the form

i

ux

h

A x e

Substitution of A in A gives us an equation for the xdep endent phase ux

du d u

h k x A ih

dx dx

where

q

E V x E V x

2

h

q

k x

i V x E E V x

2

h

At this p oint we make the approximation that the p otential varies slowly A rst crude

approximation is to omit the rst term in A In rst order u we nd thus

Z

x

u h k x dx C A

To improve the approximation in second order we write rst

s

i du

k x h u x

dx h

and make the approximation

s

Z Z

q

x x

i

x u x dx C h u x h k k x ihk x dx C A

h

APPENDIX A WKB APPROXIMATION

The correction is baseless unless u is close to u x ie unless

jk xj jk xj A

This means that k x may only vary slowly With condition A we can approximate

A

Z Z

x x

i k x ih

u x h k x dx C h log k x C A k x dx

k x

which is known as the WKB approximation named after Wentzel Kramers and Bril

louin As we see in A ux can b e regarded as the action S One expands S in

p owers of h

h

S S hS S

Note that S h S u x

The approximated wave function b ecomes thus

R

x

i k xdx

e in zeroth order in h

R

x

A x

i k xdx

p

e in rst order in h

k x

The WKB approximation is also refered to as classical approximation stationary state

approximation or geometrical optics approximation In the classical limit h we

obtain the classical solution and the rays asso ciated with orthogonal tra jectories to

the surfaces with constant phase are the p ossible paths of the classical particle If we

compare those wavefunctions with the wavefunctions in chapters and we nd

R

1

0

0

i dv

i f v

(v v )

0

e e A

R

u

i e du

i pu

e e

The condition A b ecomes in our case

A

u

Note that e

In path integral language the transp ort of a wave function is describ ed as

Z Z

i h

i

S r

h

A D r e t dr r

ihS r

0

with r e and S the action of the path r For small h thus in the

classical limit the extrema of the action dominate the path integral and the integral

can b e approximated by the saddlep oint phase

R

r

S

i

0

dr E t S r

0 0

h r

r

0

0

A r t e

This formula means that the energy at r is transp orted along the classical tra jectory

to r t However the frequency of the wave function must vary suciently slowly

Imp osing that E is constant giving an energy eigenstate yields the WKB expression

for eigenstates in one dimensional quantum mechanics

App endix B

Thermal Sp ectrum

In this app endix I will calculate the exp ectation value of the number op erator in an

ideal gas and give the expression for the Planck sp ectrum of a black b o dy

The partition function Z is given by

H N

Z Tre B

where the number op erator and the hamilton op erator are given by

X X

y y

N a a H a a

p p p

p p

p p

Hence we nd for the exp ectation value

P

N

y y

i i

i

a B a a i ha Tre

q q

p p

Z

An imp ortant relation is

A A

A B B e B e e B

applied to

y y

N a a B

p

p p

Using this in the rst step

P P

N y N y

p

i i i i

i i

a a B a a e Tre Tre

q q

p p

P P

N N

y

i i p i i

i i

Tre a a e Tre

q pq

p

Z

gives us the nal result

pq

y

ha a i B

q

p

p

e

where the sign is for fermions and the sign for b osons and with the inverse tem

p erature kT

APPENDIX B THERMAL SPECTRUM

The precise Planck expression for the density of radiation

h

E

0

h k T

c

e

With h the maximum of this function lies at k T so the statement that the

typical frequency of the outgoing radiation is of the order in the case of the moving mirrors is justied

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at the URL httpxxxlanlgov