Lorentz Anomaly and 1+1-Dimensional Radiating Black Holes

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(a) (b) (b)

G. Amelino-Camelia , L. Griguolo , and D. Seminara ,

(a) Theoretical Physics, University of Oxford, 1 Keble Rd., OxfordOX1 3NP, UK

(b) Center for Theoretical Physics, Laboratory for Nuclear Science, and Department of

Physics, Massachusetts Institute of Technology, Cambridge, Massachusetts 02139, USA

ABSTRACT

The radiation from the black holes of a 1+1-dimensional chiral quantum gravity mo del

is studied. Most notably, a non-trivial dep endence on a renormalization parameter that

characterizes the anomaly relations is uncovered in an improved semiclassical approximation

scheme; this dep endence is not present in the naive semiclassical approximation.

OUTP-95-42-P MIT-CTP-2484 Octob er 1995

Work supp orted in part by funds provided by the U.S. Department of Energy (D.O.E.) under co op erative

agreement #DE-FC02-94ER40818 and by the Istituto Nazionale di Fisica Nucleare (INFN, Frascati, Italy).

The realization that the study of some 1+1 dimensional quantum gravity theories, like

the CGHS mo del[1 ], might give the p ossibility to explore black hole quantum mechanics

in a rather simple (compared to the physical 3+1 dimensional context) setting, has led to

great excitement. (See Ref.[2] for a review of related results.) In particular, there have

b een several attempts[3] to mo dify the CGHS mo del in a way that would allow a consistent

analysis (from b eginning to end) of evap orating black holes. These attempts have b een

only partly successful[3], but there might still b e ro om for improvement since just a limited

class of mo di cations have b een considered. Other recent pap ers have b een devoted to the

investigation of the role of anomalies in the quantum mechanics of 1+1 dimensional black

holes. Most notably, again in studies of the CGHS mo del, it has b een found[4 , 5 ] that the

semiclassical (quantized matter elds in a background geometry) evaluation of the Hawking

radiation[6 ] is essentially insensitive to the addition of the lo cal counterterm that converts

the Weyl anomaly into an anomaly for di eomorphisms of nonconstant Jacobian[5 , 7, 8].

In this Letter, we consider a mo di cation of the CGHS mo del in whichchiral fermion

elds replace the usual b oson elds as the matter content, and (accordingly) the geometry is

a a b

describ ed by the zweibein e instead of the metric g .(g e e and diag (1; 1).)

We study the radiation of the black holes of this mo del in the semiclassical approximation,

with particular attention to the role played by the anomalies .

Chiral quantum gravity theories in 1+1-dimensions have a rich anomaly structure, and

no inconsistency has b een encountered in their previous studies (see, for example, Ref.[9,

10 ]); it is therefore quite natural to use one such theory for the investigation of the role of

anomalies in the quantum mechanics of 1+1-dimensional black holes. We also hop e that

the analysis here presented will motivate future studies in which the features of some chiral

quantum gravity theories b e exploited in obtaining a fully consistent description of black

hole evap oration.

The mo del we consider has action S + S + S , where

m m

i h

2 2 2 2

; (1) R +4(r) +4 S = d xE e

N Z

1 1 i 1i

5 5

() () a () 2 () a

S = : (2) @ @ d xE e

m n n n a n

2 2 2

n=1

e , , and are the zweibein, dilaton, and matter elds resp ectively. E denotes the de-

terminantofthezweibein (which equals det(g )). The curvature R can b e written in

a b

terms of the spin-connection ! = e @ e =E (wecho ose to work without torsion), as

R =2 @ ! =E . Note that S ,S are invariant under (tangent space) Lorentz transforma-

tions and di eomorphisms. S are also invariant under Weyl transformations.

With a straightforward generalization of the corresp onding analysis[2 ] of the CGHS

mo del, it is easy to verify that the classical equations of motion of the mo del have black

hole solutions. In particular, the gravitational collapse of the left-moving sho ckwave con-

sidered in Refs.[1, 2, 5] forms the same classical black hole that it forms in the CGHS mo del.

We shall limit our analysis to this example of black hole, so that our results can b e more

easily compared to those of [1, 2 , 5 ]; moreover, like in Refs.[1, 2 , 5 ], weintro duce light-cone

co ordinates and work in conformal gauge. [Note that, since ours is a zweibein theory, the

conformal gauge is characterized by e = e ;however, this implies that g = e =2,

g = g = 0.]

We refer the reader to Ref.[12] for a discussion of certain other asp ects (di erent from the ones we are

here concerned with) of the physics of black holes in presence of chiral matter; note, however, that in Ref.[12]

at the quantum level only the sp ecial case with equal numb er of left-moving elds and right-moving elds

was considered and therefore the structure of the anomalies was much simpler. 1

a a a

For an observere ^ ( ) that is zweibein minkowskian (^e = ) at past null in nity[5], the

solution of the classical equations of motion that corresp onds to the black hole considered

in Refs.[1, 2, 5] has conformal factor

1 a

+ + ( )

^ = ln 1+( ) e e 1 : (3)

In the semiclassical analysis of Hawking radiation one evaluates the quantum matter

energy-momentum tensor in the background geometry of the black hole. In our zweibein

mo del the matter energy-momentum tensor is

1 [S +S ]

m m

T < > ; (4)

E e

and for the example of black hole background that we are considering the most interesting

a 4

^ ^

physical information is enco ded[2] in the value of T T e^ g^ on I .

a R

From the results of Ref.[11] it follows that the di eomorphism invariant quantization of our

chiral elds in a background geometrye ^ leads to the anomaly relations (N.B.: X X (^e))

!^ e^ cq

^ ^ ^ ^

r T = R 2hr !^ ; (5)

24 2

i h

^ ^ ^

; (6) (c + h)R + cq r !^ e^ T =

1 cq

a b

^ ^ ^

e^ T = ; (7) R 2hr !^

b a

24 2

where c (N + N )=2, q (N N )=(N + N ), and h is a free renormalization

+ + +

parameter (the co ecient of a lo cal counterterm). The emergence of the free parameter

h in chiral quantum gravity theories should b e understo o d in complete analogy with the

emergence of the Jackiw-Ra jaraman parameter in the quantization of the matter elds of

the Chiral Schwinger mo del[14 ].

For arbitrary c,q ,h the right-hand sides of Eqs.(5), (6), (7) do not transform covariantly

under Lorentz and Weyl transformations, also T do es not transform covariantly under these

transformations. Instead, in spite of the unusual asp ect of Eq.(5) (that is due to the Lorentz

anomaly), T transforms covariantly under di eomorphisms. For h = q = 0 (di eomorphism

and Lorentz invariant Dirac fermions) the anomaly relations (5), (6), (7) are equivalent

to the ones encountered[1, 2] in the CGHS mo del, and T transforms covariantly under

b oth Lorentz transformations and di eomorphisms. Also sp ecial is the case q = 0 (Dirac

fermions), h = c which leads to a T that is traceless and covariant under di eomorphism

transformations, but Lorentz anomalous.

The Eqs.(5), (6), (7) can b e used to evaluate T on I . Indeed, these anomaly relations

completely determine, mo dulo the imp osition of physical b oundary conditions at past null

3 +

Like in Refs.[2, 5], a denotes the magnitude of the sho ckwave, and the sho ckwavemoves along = .

Like in Ref.[2,5], I (I ) is the future (past) null in nity for right-moving lightrays, and analogously

R R

I (I ) is the future (past) null in nity for left-moving lightrays.

L L

Obviously, here \free" should b e understo o d in the usual eld theoretical sense, i.e. one can cho ose

freely among the values of h consistent with unitarity (for example in ordinary theories one cannot cho ose a

negative mass parameter). In Ref.[9]itwas argued that only h>0would lead to a unitary chiral quantum

gravity theory; we shall come back to this later. 2

in nity, the four comp onents of T . This is analogous to the ordinary CGHS case, the only

di erence b eing that here wehave one more comp onent of the energy-momentum tensor

(the antisymmetric comp onent coming from the Lorentz anomaly) and one more anomaly

relation (the one expressing the Lorentz anomaly).

Imp osing as physical b oundary condition[2 , 5] that T vanish on I (i.e. = 1),

a L

and that there b e no incoming radiation along I (i.e. = 1) except for the classical

sho ck-waveat = , with a straightforward (but lengthy) derivation one nds that

1 cq cq

2 2 2

T = c + h @ ^ (@ ^) (@ ^) ; (8)

12 2 2

The physical interpretation of the energy ux seen on I is clearest[2] to observers that

are asymptotically minkowskian on I , but from Eq.(3) one sees that thee ^ ( ) observer

is not minkowskian on I (whereas it is minkowskian on I ). An observere (y ) that

R R;L

is zweibein minkowskian on I can b e obtained frome ^ ( )by combining the (conformal)

di eomorphism ! y , where

+ +

y = ; y = ln (e a=)= ; (9)

y 1=2

and the Lorentz transformatione (y ) ! (1 + ae =) e (y ).

Using the fact that the energy-momentum tensor transforms covariantly under di eomor-

phisms, but under Lorentz transformations it follows the transformation rules implied by the

anomaly relations (5), (6), (7), one nds that the observere (y ) sees the following T

e^!e 2

T (y )= [T ( (y )) + ( (y ))](d =dy ) (10)

where

8 9

" !# ! !

< =

dy dy dy 1

2 2 e^!e

ln ln h @ (2h + cq ) @ +2cq @ @^ ln : (11) =

: ;

48 d d d

Since on I (i.e. y = 1) one nds that ^ =(1=2) ln (dy =d ), from Eqs.(8)-(11) it follows

that T is h-indep endenton I

h i i h

y 2

T = c(1 q ) : (12) 1 (1 + ae =)

Notice that this result repro duces the corresp onding result[1, 2 ] for the CGHS mo del

up on replacing c(1 q )= N with N , the numb er of b oson elds of the CGHS mo del.

This might indicate that, in spite of the Lorentz anomaly, in the present approximation

left-movers and right-movers are still decoupled[2]; in fact, such a decoupling implies that

the Hawking radiation on I due to our black hole (formed by gravitational collapse of a

left-moving sho ckwave) should only b e sensitiveto N .

The fact that the complicated structure of the anomalous Lorentz transformations of

the energy momentum tensor and the anomaly relations (5), (6), (7), conspire to give the

h-indep endent result (12) is consistent with the ndings of Ref.[5], where the semiclassical

analysis of the black hole radiation in the ordinary CGHS mo del was shown to b e insensitive

to the value of the co ecient of a lo cal regularization counterterm. However, the similarities

between our mo del and the Chiral Schwinger mo del (ours is essentially a gravitational version 3

of the Chiral Schwinger mo del) suggest that the parameter h should have some non-trivial

physical role; for example, the Jackiw-Ra jaraman parameter a ects the value of the mass

emergent[14] in the Chiral Schwinger mo del. Moreover, investigations[10 ] of related chiral

quantum gravity theories have found that h do es have a non-trivial role in the fully quantized

theory; most notably, the central charge has b een found to dep end on it. We therefore exp ect

that in the fully quantized theory the black hole radiation dep ends on h even though this is

not seen in the naive semiclassical analysis. To test this exp ectation, in the following we shall

go b eyond the naive semiclassical approximation; sp eci cally,we shall exploit the fact that it

is rather simple to p erform the functional integration over one of the four elds characterizing

the zweibein, and include the corresp onding quantum correction in our analysis.

A general zweibein e can b e written in terms of a eld , whichwe shall call Lorentzon,

and a Lorentz gauge- xed zweibein [^e ] (i.e. a zweibein whose orientation in the tangent

space is prescrib ed, and therefore is characterized by only three elds), as follows

e = e [e ] : (13)

In our theory it is rather simple to integrate out the eld ; in fact, the corresp onding

functional integration can b e cast in Gaussian form[10], and cho osing a di eomorphism and

Lorentz invariant measure of the typ e prop osed in Ref.[15], one nds that, as we shall showin

detail in Ref.[13], the Lorentzon-integrated energy-momentum tensor satis es the following

7 a

anomaly relations in the black hole backgrounde ^

^ ^

r T = 0 ; (14)

a 2 2

^ ^

e^ T = [c + h + c q =(4h)]R=(24 ) ; (15)

a b

e^ T = 0 : (16)

b a

One can recognize that these anomaly relations b ecome equivalent to the ones encountered[1,

2 2

2] in the ordinary CGHS mo del, up on replacing c c + h + c q =(4h) with N . Therefore,

ef f

once the Lorentzon is integrated out, the semiclassical analysis of black hole radiation in

our mo del can b e p erformed exactly as in the ordinary CGHS mo del (in particular the

Lorentzon-integrated T transforms covariantly under b oth Lorentz transformations and

di eomorphisms); this leads to the result that for the \go o d observer"e (y )

" #

2 2 2

i h i h

c q

y 2

: (17) T = c + h + 1 (1 + ae =)

4h 48

Hence, we nd again that the black hole radiation on I has the same functional dep endence

on y as in the semiclassical analysis[1 , 2] of the CGHS mo del, but in the present case the

overall co ecient has an h-dep endence.

The sp eci c dep endence on h of the overall co ecient c is quite interesting. In partic-

ef f

ular, c diverges at h = 0, and c N whenever h<0; this app ears to b e in agreement

ef f ef f

The Lorentzon corresp onds to the degree of freedom \turned on" by the anomaly. The consequences of

integrating it out are therefore particularly interesting.

Note that these anomaly relations receive no contribution from the Lorentzon measure and the Lorentz

ghosts. If, instead of following the prop osal of Ref.[15], one de nes the measures in the ordinary way (with

the real metric), the Lorentz ghosts still do not contribute to the anomaly relations, but a contribution from

the Lorentzon measure app ears. Since this contribution can b e reabsorb ed by increasing the value of c by1,

our analysis would not b e qualitatively mo di ed. 4

8 9

with the exp ectation that the theory b e not unitary (the Lorentzon is ghost-like[9 ]) when

h 0. Instead, c N (consistently with the absence[9] of ghost-like elds) whenever

ef f

h>0.

Our result that after integrating out the Lorentzon the radiation has an h-dep endence,

which is not present in the naive semiclassical approximation, also allows to reconcile the

ndings of Refs.[4, 5] with the ones of Refs.[9, 10 , 14]. Notably, the T in Eq.(17) takes the

same value in corresp ondence of pairs of values of h, just like paired values of the Jackiw-

Ra jaraman parameter lead to the same value of the mass emergent in the Chiral Schwinger

mo del[14 ].

In conclusion, wewant to emphasize that in this pap er wehave only explored some asp ects

of a torsionless chiral quantum gravity theory which are relevant to the issue of the p ossible

role of renormalization parameters (co ecients of lo cal counterterms) in 1+1-dimensional

black hole radiation; however, it is our exp ectation that other problems of 1+1-dimensional

black hole quantum mechanics mightbeinvestigated by exploiting the rich structure of chiral

quantum gravity theories. For example, it would b e interesting to check whether a fully

consistent analysis of the back reaction[3] (a necessary step toward a rigorous description of

black hole evap oration, which has b een attempted without success in the CGHS mo del) is

p ossible in some (not necessarily torsionless) chiral quantum gravity theory.

We happily acknowledge conversations with R. Jackiw, E. Keski-Vakkuri, S. Mathur, M.

Ortiz, and B. Zwiebach.

Work is in progress[13] attempting to test more rigorously whether the theory with negative h is physical

2 1=2 2 1=2

at least when c 0, i.e. c[1 + (1 q ) ]=2 h c[1 (1 q ) ]=2.

ef f

Note that, in the spirit of Ref.[15], wehave included no contribution from the measures of the Lorentzon,

dilaton, conformal factor, and ghosts. However, the (di eomorphism invariant) quantization of the chiral

matter elds (whose measure has to involve the physical metric[15]) still induces a kinetic term for the

Lorentzon. 5

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