fermion quantum field theory in black hole spacetimes

8/13/2019 Fermion Quantum Field Theory in Black Hole Spacetimes

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Fermion Quantum Field Theory In Black Hole Spacetimes And

The Information Loss Paradox

Syed Alwi B Ahmad*

Singapore

Abstract

We study the Dirac equation in black hole spacetimes and vacuum, stationary axisymmetric

spacetimes in general. We solve the Dirac equation in such spacetimes via a factorization ansatz.

The thermal Hawking-Unruh flux is confirmed for the case of Schwarzschild spacetime arrived

at via gravitational collapse. Then we apply the factorization ansatz to Kerr spacetime and show

how it works there. The Dirac equation is then studied in the Eddington-Finkelstein spacetimewhere it is suggested that a semi-classical gravitational back-reaction may be computed

numerically in the one particle case only and may represent the emission of gravitational waves.

In particular the mode solutions to the Dirac equation in Eddington-Finkelstein spacetime, are

completely regular at the horizon and the infalling particle encounters nothing unusual there. The

semi-classical calculation breaks down however, for the case of the infall of one half of an

entangled pair. Thus on one hand we see nothing unusual at the horizon in the infall of one

particle and yet the break down of the semi-classical calculation in the entangled case, requires a

new ingredient if we are to retain unitary evolution in the presence of a black hole.

(*[email protected])

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1. IntroductionThe gravitational collapse of compact objects (white dwarfs, neutron stars) to form black

holes still remains much of a problem in modern Physics [1]. A detailed description of such a

collapse is still missing. In part, this is due to the extreme conditions found on these compact

objects. Typically a neutron star is between 1 to 3 solar masses, with a radius that is 510 of the

solar radius, is of nuclear densities ( 1510 gcm 3 ) and has a surface gravity510 times solar.

With surface gravities like this General Relativity (GR) is an integral part of the description of

neutron stars. At the same time, the nuclear density of neutron stars necessitates a quantum

mechanical description of neutron star matter. Indeed the star is supported against collapse

primarily by the quantum mechanical, neutron degeneracy pressure.

It is clear then, that detail of the collapse is sensitive to the elementary particle physics relevant

at each stage of the process. So here is a natural laboratory to study elementary particle physics

in a strong gravitational field. There are many interesting and deep theoretical questions that one

can pose in this situation. For example, one may ask about the role which current algebra plays

during the gravitational collapse since after all, gravity couples to the energy-momentum tensor.

Or one may ask regarding the implications of CP violation and CPT invariance on the collapsing

matter. And if supersymmetry (SUSY) or supergravity (SUGRA) is present, then one would like

to know how it will affect the process of collapse. Thus many interesting questions arise

naturally in this context of gravitational collapse.

Besides elementary particle physics, the interplay between a quantum field theory of the

collapsing matter and the general relativistic spacetime, can lead to novel effects. As is well

known, the black hole formed will emit the Hawking-Unruh radiation and the thermal nature of

this radiation can lead to the Information Loss problem. In particular one would like to be able to

compute the gravitational back-reaction of the black hole, due to the infalling quantum matter.

So whether we are studying elementary particle physics in the gravitational collapse of compact

objects, or the Information Loss problem, we need to be able to do quantum field theory in a

black hole background. In particular, we need to know how to construct afermionquantum field

theory in a black hole background.

In the semi-classical framework of quantum field theory in classical curved spacetimes [2][3],

there have been much previous work on scalar fields in black hole backgrounds (see [4] for

example). But the quantization of fermionic fields in black hole spacetimes however, is lessstudied and two early work includes [5] and [6].

In this paper we show how to solve the Dirac equation and quantize the resulting fermionic

theory on a stationary axisymmetric spacetime satisfying the vacuum Einstein equation. We do

notmake use of the Newman-Penrose formalism (see [7] and [8]) as it is not well adapted for

computations in elementary particle physics. Instead we employ a factorization ansatz which

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capitalizes on the stationary axisymmetric nature of the spacetime, to give us modes that we use

in a canonical quantization of the fermionic theory. We do this for Schwarzschild and Kerr

spacetime. Modulo the factorization, these modes are the ones found in [6]. And herein lies the

problem; these modes are divergent at the event horizon and this divergence is made explicit by

the factorization terms (which was not explicit in [5] and [6]). In Schwarzschild spacetime, this

is due to the use of the Schwarzschild coordinate system attached to an observer in the

asymptotically Minkowskian region far away from the black hole. As is well known, an observer

far away only sees the piling up of the infalling matter red-shifted as it nears the horizon and

hence the divergence of these modes there. In particular the energy-momentum tensor based on

these modes, cannot be used to compute any kind of back reaction meaningfully.

However this does not mean that we cannot extract any useful information from the energy-

momentum tensor, T , constructed from these modes. It was shown in [6],[9] and [10] that the

renormalized energy-momentum tensor,ren

T , does not vanish as r and in fact,

reproduces the famous Hawking-Unruh flux that escapes from the black hole to infinity. This is

an alternative derivation of the Hawking-Unruh radiation. But since we are interested in the

process of gravitational collapse, the method presented in [2] and [4] which studies the case of

spherical collapse, was used to derive the Hawking-Unruh radiation.

To address the problem of the gravitational back reaction of the black hole in response to the

infalling fermion, we consider the radial infall of a spin-half Dirac fermion into a static, non-

rotating Schwarzschild black hole. In the far future we expect the black hole to be of Kerr

rotating type because we expect the spin angular momentum of the fermion to be converted into

the rotational angular momentum of the black hole due to angular momentum conservation. This

suggests that the gravitational back reaction due to the infall of a single Dirac fermion is a

transient effect, which we expect to be the emission of gravitational waves from the perturbed

horizon, that connects the two asymptotic black hole states ; Schwarzschild in the past and Kerr

in the future. In this manner, we speculate that the information carried by the single infalling

particle leaks away or is radiated away from the black hole via the emitted gravitational wave

transient. Unfortunately, as we will discuss later, the semi-classical approximation breaks down

when we have to deal with entangled pairs of particles.

To compute the transient back reaction in the one particle case however, requires modes that do

not diverge at the event horizon. This is turn, requires a coordinate system well-behaved there

and the Eddington-Finkelstein system is well adapted for this. It turns out that the solutions to the

Dirac equation in the Eddington-Finkelstein coordinate system are indeed regular at the horizon

and can thus be used to compute, semi-classically, the back reaction due to the radial infall of a

single Dirac fermion. However, earlier straight-forward application of the semi-classical

approximation does not reveal this transient back-reaction for various reasons. First we have to

renormalize the energy-momentum tensor,ren

T , before we compute the semi-classical back-

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reaction [10][11][12][13]. In some of these early works,ren

T was computed in the vacuum

Schwarzschild case with the assumption of spherical symmetry and time independence [10][13].

But this cannot hold in the presence of an infalling particle. One would have to sandwich T

between infalling one-particle states and then perform a renormalization procedure thus making

the problem time-dependent as well as lacking in spherical symmetry. Indeed if the information

is radiated away in the transient back-reaction, then both assumptions of spherical symmetry and

time-independence cannot hold. This makes the problem difficult to study analytically and

perhaps only a numerical study can reveal the nature of the transient back-reaction.

In the concluding section, we speculate on the Information Loss paradox beyond the semi-

classical approximation and why it breaks down when we consider an entangled pair of particles.

2. Vacuum, Stationary Axisymmetric SpacetimesStationary axisymmetric spacetimes form an important class of solutions to the vacuum

Einstein equations. It has been studied and discussed extensively in the literature [14] [15]. We

follow closely the notation in [14] except for the appropriate changes to the signature of the

metric which we take to be (+,-,-,-) as is usually done in particle physics. A spacetime is

stationary and axisymmetric if it has a pair of commuting Killing vectors corresponding to

azimuthal rotations and time translations. If these Killing vectors satisfy the Frobenius condition,

then there exists a coordinate system, 0 1 2 3, , ,x t x x x = = such that the Killing vectors are :

3 0m i.e. m and k i.e. k t

= = = =

.

In addition, the metric tensor is a function of 1 2andx x alone with01 02 31 32g g g g 0= = = = and

the line element is

2 2 2ds Vd 2Wd d Xd g d d A BABt t x x = + + + (1)

where , 1,2A B= and

00

03

33

V g k k 0

W g k m g

X g m m g 0

g

= = >

= =

= =

= =

= =

+ <


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5. Schwarzschild SpacetimeIn Schwarzschild coordinates the vierbeins are diagonal with,

1/2

00 21 Me

r

=

,

1/2

11 21 Me

r

= , 22e r= and 33 sine r = .

From (2),

2V 1

M

r= , W 0= , 2 2X sinr = and

21

M

r= .

Thus we find that

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1 M

hr

=

and 1/21/2

1

sine

r

= . As in the Minkowskian case,

( ) 51

0

4

abcd

ab dc = and so we are left with the equation, 0 0

c

ci E m

= . The spherical

symmetry of the Schwarzschild spacetime can be exploited and one can follow identicalsteps as

in the Minkowskian case [18], to obtain a singular Sturm-Liouville boundary value problem for

the radial piece, valid for 2r M> . Solutions to the radial equation that are regular for 2r M>

exist and are asymptotic to the spherical Bessel functions as r . We shall denote them by

( )j

aZ prand ( )

jbZ pr

. They oscillate with increasing frequency as 2r M . Along with the bi-

spinor spherical harmonics, we can construct fermion field operators as before, using the

normalized mode solution

( ) ( )1/2

0

2( ) ( )j j m mj j j j j j

iEt

Ejm a b

Ee h e Z pr Z pr m

+

= +

where the radial functions must obey the normalization,

1/2 1/2

2

' '20

2 2d 1 ( ) ( ') ( ') 1

2l l ll

M Mp p Z pr Z pr r r

r r r

=

and perform a canonical quantization of the theory for 2r M> . The factor, h , diverges at the

event horizon and therefore the quantized theory is only valid for 2r M> . This is good as it is

consistent with the Euclidean Path Integral approach (see [20]), where the entire region of

2r M is mapped onto a point at the origin in the Euclidean section of the Schwarzschild

spacetime. As discussed in the introduction, these modes cannot be used to compute any kind of

semi-classical back reaction due to its divergence at the event horizon.

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6. Kerr SpacetimeIn Boyer-Lindquist coordinates, the line element in Kerr spacetime is

( )2 2 2 2 22

2 2 2 2 2 2 2 2 2 2 2

2 2 2

sin sin 2 sinds d sin d d d d d

a at r a a r a t

= + + +

where 2 2 2r a Mr = + and 2 2 2 2cosr a = + . Thus,

( )

2 2

2

22 2

2

22

2 2 2 2

2

sinV

sinW

sinX sin

a

ar a

r a a

=

= +

= +

If we apply the factorization ansatz to the the Dirac equation in Kerr spacetime, we find that the

term ( ) 51

4

abcd

ab dc does not vanish. It turns out that another factorization to remove the

offending term is possible, in the masslesscase. In that situation, (14) becomes

( ) 51

0

4

c abcd

c ab d ci E = (20)

We consider the left and right handed5 eigenstates with eigenvalue 1 respectively,

andL R

= =

where is a 2-component spinor. Then in the left handed case, equation (20) becomes

( )1

04

c abcd

c L ab d Lci E + = (21)

In the absence of torsion, we expect to be able to remove the second term in (21) by another

factorization. To this end, set

( , )L L

g r

=

where ( , )L

g r = and is also a 2-component spinor and ( , )L

g r is a

scalar function. Substitute this into (21) and set

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0cc

i E

=

(22)

to obtain the equation

( )1

04

c abcd

c L ab d Lci E g g + = (23)

(23) can be further simplified by multiplying by another gamma matrix and taking the trace to

get

( )1

log4

df abcd

f L ab ci E g = (24)

Following [6], we choose our vierbeins to be

0 0 2 3

0 3 0

2 23 1 2

3 1 2

sin, sin ,

sin , ,

ae e a e

r ae e e

= = =

+= = =

so that we get,

( )11/2 costan1/2

1/2 2 2, sin ,

i a

rLh e g e

= = =

leaving us with (22) to solve. If we choose the ansatz,

( , )

( , )

i t im u r

e ev r

=

with m an integer, then we obtain precisely the solutions found in [6] modulo the factorization

terms. Notice that h diverges at the event horizon and reduces to the Schwarzschild case when

there is no rotation. We also note in passing, that this method can also be applied to the Taub-NUT spacetime [20].

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7. Hawking-Unruh RadiationWe can derive the thermal Hawking-Unruh radiation and its properties by studying the

spherical collapse of a star by following the method in [2] and [4]. Of course we have to adapt

the method to fermions and make the appropriate changes such as using equal-time anti-

commutation relations and so on. Indeed, by using the modes that we found for Schwarzschild,

the calculations in [2] and [4] can be carried out in detail and we find the thermal Fermi-Dirac

spectrum and the corresponding thermal density matrix [4]. We shall not repeat it here and we

only comment that (a) the calculation depends on the fermion fields being massless (in order to

make conformal maps to obtain the necessary Penrose diagrams) and (b) the fermionic theory

has to be time-reflection invariant to match the time reflection invariance of General Relativity

(GR).

The free, massless Dirac theory obviously satisfies these two conditions. But what happens in the

case of a CP-violating theory during gravitational collapse (assuming that CPT holds) is an openquestion. How CP-violating theories would behave during gravitational collapse is an interesting

question that requires further study.

8. Eddington-Finkelstein SpacetimeAs mentioned earlier, the possibility of calculating the back-reaction due to the radial infall

of a single Dirac fermion, requires modes that are regular at the event horizon. The ingoing

Eddington-Finkelstein spacetime is especially well-suited for this and it is physically meaningful

as it represents a coordinate frame attached to an infalling photon. One does not have to dealwith the Einstein-Rosen bridge nor the white hole extension. The line element is given by

2 2 2 22d 1 d 2d d d M

s v v r r r

=

where *v t r= + and * 2 ln 12

rr r M

M= + . This metric is regular at the horizon and it is

spherically symmetric. Though00

2g 1

M

r

=

vanishes at the horizon, the metric remains non-

degenerate due to the off-diagonal term in d dr v . At the same time, this off-diagonal term makes

the metric non-stationary. It is therefore not a stationary axisymmetric spacetime and the

factorization ansatz does not apply. However it is spherically symmetric and with a suitable

choice of vierbeins, we hope to be able to apply a separation of variables in a partial wave

expansion using bi-spinor harmonics and the appropriate radial function. From the metric, we see

that the vierbeins must take the form,

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0 0

0 1

1 1

0 1

2

2

3

3

0 0

0 0

0 0 0

0 0 0

a

e e

e ee

e

e

=

We shall impose the condition that our time-like vector field, 0E

, be regular everywhere

except at the singularity present at 0r= . This leaves us with very few choices and we take

0 1 1 1 0 0

1 1 1 0 1 0, 1, , 1, 1M M

e e e e e er r

= = = = =

With this choice of vierbeins, the vector field 0E

is a unit time-like vector field that is regular

everywhere except at the singularity (i.e. at 0r= ). So this choice of vierbeins, although notsymmetric, leads to nice behavior at the event horizon. Now observe that

( ) ( )2ab c ac b ab cbc bca a =

so that with the help of (13), the Dirac operator becomes,

( ) ( ) ( )0 5 01 1

2 4

a ab c abcd

a bc bc d a aiD m i E i m / = + (25)

But ( ) ( ) ( )lnab abc c c ca a E E e

= = + and we can explicitly verify that the term,

( ) 51

4

abcd

bc da , vanishes. This is not surprising as after all, the Eddington-Finkelstein is an

extension of the Schwarzschild spacetime. The Dirac equation can then be written as

( )0 01

ln 02

a a

a a aiD m i E i E E e m

/ = + + = (26)

Put1/2

1

sinr

= and substitute this into (26) and simplify to get,

1 2 3

2 2

02

1 11

2 sin

2

M M M Mi i i i i i

v r r r v r r r r r

Mi m

r

+ = + +

+ +

where 0k k = and 0 = are the Dirac matrices. The previous equation can be rewritten in

an explicitly spherically symmetric form as (see [18] and [19]),

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( ) 02 21 1

12 2

r

M M M Mi i i i e K m

v r r r v r r r r r

+ = + + + +

(27)

where ( )2 1K S L= +

. We consider the partial wave,

( ) ( )( , ) ( , )j j m j j j m j j jj j

jm m mf r v f r v

+ + = + (28)

which upon substitution into (27) yields the pair of equations,

02 2

11

2 2

j j j j j j j j

j j j j j j j j

m m m m j

m m m m

f f f fM M M Mi i f i m f f f

v r r r v r r r r

+ +

+ +

+ = (29)

02 2

11

2 2

j j j j j j j j

j j j j j j j j

m m m m j

m m m m

f f f fM M M Mi i f i m f f f

v r r r v r r r r

+ +

+ + +

+ + = + + + (30)

Choose the ansatz,

( , ) ( )m j jj

iEvf r v e a r

+ = and ( , ) ( )m j jj

iEvf r v e b r

= (31)

to obtain the pair of equations from (29) and (30),

( ) ( ) ( )0 2d 1 d

d d 2

j

j j j j j j j

ja M M

a E m b iEa ib a ib ar r r r r

++ + = + (32)

( )

( ) ( )0 2

db 1 d

d d 2

j

j j j j j j j

j M Mb E m a iEb b ia b ia

r r r r r

+ + = + + (33)

The homogeneous version of (32) and (33),

( )0d 1

0d

j

j j j

ja

a E m b iEar r

++ + =

( )0db 1

0d

j

j j j

jb E m a iEb

r r

+ + =

possess exact solutions. Indeed one can verify that the solutions to the preceding homogeneous

equations are,

( )j j

iEr

aa e j pr

= and ( )

j j

iEr

bb e j pr =

where the spherical Bessel functions retain their same labeling and argument as in the

Minkowskian case. This suggests that we may write,

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( ) ( ) ( )j j jk k

a r p r a r = and ( ) ( ) ( )j j jk kb r q r b r = (34)

Substitution of (34) into (32) and (33) yields the pair,

2 2

d d d21

d d 2 2 d j j j

j j j j j

p a b

M M M M Ma a p i b i qr r r r r r r r

= +

(35)

2 2

d d d21

d d 2 d 2

j j j

j j j j j

q a bM M M M Mb i i a p b q

r r r r r r r r

= +

(36)

This first-order system for ( )j

p r and ( )jq r , convertible to second-order equations

individually, contain a regular singular point at 2r M= and may be solved by the series method

of Frobenius [21]. In particular, if we demand that the first derivative of both ( )j

p r and ( )jq r

be continuous at 2r M= , then (35) and (36) imply that both ( )jp r and ( )jq r must vanishthere

as some power of2

1 M

r

. This means that everypartial wave (28) is completely regular and

in fact, vanishes, at the event horizon. This gives hope that the renormalized energy-momentum

tensor sandwiched between infalling, one-particle states may be sensibly contructed and a back-

reaction transient, numerically computed, as was suggested in the Inroduction.

9. Conclusions and SpeculationsWe have seen that it may be possible to compute numerically the gravitational back

reaction due to the radial infall of a single Dirac fermion in the Eddington-Finkelstein spacetime.

However if we have an entangledpair of fermions in which one falls into the black hole, while

the other escapes, then the semi-classical framework breaks down because it cannot capture and

deal with the non-local correlations between the entangled pair [22][23]. This is not surprising

since entanglement has no semi-classical description, being a purely quantum phenomenon.

But the infall of a single particle in which the semi-classical approximation may be valid, does

not suggest anything unusual at the event horizon such as a Firewall [24]. Indeed in [25] a twodimensional model was considered and no Firewalls were seen. It is difficult to imagine that

something radically different occurs in the case of the entangled pair. Since we need a fully

quantum treatment to deal consistently with entanglement, this must mean that a fully quantum

treatment of the back-reaction is required.

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In the one particle case, the back-reaction is expected to be a transient gravitational wave. In a

fully quantum treatment, one would expect the emission of gravitons or other particles as

determined by the relevant theory. Of course this leaves open the difficult question of how the

correlations between the infalling particle and the escaping particle, is transferred to between the

emitted gravitons etc on one hand, and the previous escaping particle on the other. Indeed such a

mechanism may be non-local [26][27]. Clearly more work needs to be done to understand the

quantum dynamics of the horizon interactions with the infalling matter.

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fermion quantum field theory in black hole spacetimes

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