- Email: [email protected]
Symmetries and solvable models for evaporating 2D black holes
ELSEVIER
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 57 (1997) 184-187
Symmetries and solvable models for evaporating 2D black holes* J. Cruz, J. Navarro-Salas a, M. Navarro b and C. F. Talavera c ~Departamento de Ffsica Tedrica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Facultad de Ffsica, Universidad de Valencia, Burjassot, 46100, Valencia, Spain. bInstituto de Matem&ticas y Ffsica Fundamental, CSIC, Serrano 113-123, 28006 Madrid, Spain. CDepartamento de Matem£tica Aplicada, E.T.S.I.I., Universidad Polit~cnica de Valencia, Camino de Vera, 46100, Valencia, Spain. We study the evaporation process of a 2D black hole in thermal equilibrium when the ingoing radiation is suddenly switched off. We also introduce global symmetries of generic 2D dilaton gravity models which generalize the extra symmetry of the CGHS model.
1. I N T R O D U C T I O N In recent years there has been a lot of activity in the study of two-dimensional dilaton gravity models. The string-inspired model of Callan, Giddings, Harvey and Strominger [1] is the simplest model that describes the formation of black holes by gravitational collapse. The existence of an extra s y m m e t r y for the CGHS model plays a fundamental role at the classical and quantum level. The one-loop q u a n t u m corrected model of Russo, Susskind and Thorlacius [2] was constructed to preserve the free field associated with the extra s y m m e t r y and the classical ground state. The model of Bose, Parker and Peleg [3] also preserves the extra s y m m e t r y but describes an evaporating black hole with a non-fiat remnant geometry. The content of this work is two- fold. In Sections 2 and 3 we study the one-parameter family of models interpolating between the B P P and R S T models [4] and the exponential model [5][6] respectively. However, instead of studying the evaporation of a black hole formed by gravitational collapse we analyze the semiclassical evaporation of a black hole initially in thermal equilibrium when the incoming thermal flux is suddenly switched off. In Section 4 we introduce extra global symmetries for generic 2D dilaton gravity *Work p a r t i a l l y s u p p o r t e d by t h e Comisidn Interministerim de Ciencia y Tecnologla a n d DGICYT. 0920-5632/97/$17.00 © Elsevier Science B.V All rights reserved. PII S0920-5632(97)00371 -X
models, in an a t t e m p t generalize the extra symmetry of the one-loop corrected theories of the CGHS model. 2. T H E R M A L BATH REMOVAL THE BPP-RST MODELS
IN
Let us consider the one-parameter family of models interpolating between the B P P and R S T models [4], S
=
(1)
So+SpW~/d2xv/-~
[(1 - 2 ¢ ) ( T ¢ ) 2 + (a - 1)¢R] , where So is the classical CGHS action So
--
1 Jd2xv/~[e-2¢(R
(2)
27r
N
1 + 4 ( V ¢ ) ~ + 4~ ~) - ~ ~ ( V f ' ) ~ ] i----1
' J
Sp is the Polyakov term and a is an arbitrary real parameter. In the conformal (ds 2 = -e2Pdx + d x - ) and Kruskal gauge (p = ¢) it is very easy to see that the above models admit the following solution e
•
:
+
M
(3)
which represents a black hole in thermal equilibrium at temperature T = 2A~. We can now remove
J. Cruz et al./Nuclear Physics B (Proc. Suppl.) 57 (1997) 184-187
185
N
the thermal bath by introducing a discontinuity in the boundary condition [7]
i=1
t~+
-
1 - 0 ( x + - ~0+), 4(x+) 2
t~-
=
0
(4) (5)
In this way we get the dynamical solution (for
N~
+9--~ f d2xvfL-g ( ¢ R + / 3 ( V ¢ ) 2 ) ' (12) where Sp is the Polyakov action. The classical limit
~+ > ~0+) e -2¢
+
N ] ~a ¢ = _N 48
--A2x+(x - -{- A) log--+l x+
(6)
4
M
O~
:
x+e~( "x- )
x+ int
-
4a~x~
where
1
°(
.
(8)
(9)
The evaporating solution can be continuously matched to the static solution e -2¢
+
Y]-~¢=a - 1 2 x + ( x - + A)
- ~ log ( - ~ x + ( x 4
log~-i
(13)
arises as a theory with extra conformal symmetry @,~
=
-c~g,~,
5¢
=
c.
(14)
The action (12) contains adequate counterterms to preserve this symmetry when we add the Polyakov action, in analogy to the BPP-RST models. In the absence of matter and taking the boundary conditions t± -- 0 we have the solution (in Kruskal gauge, 2p = ~¢)
ds2 =
dx+dx ~2cP7+ Cx+x- '
(15)
where 3 ' = 1 / ( 1 +N1--~2). If C < 0 (and A2 < 0) it represents a black hole in thermal equilibrium 2 with mass M = ~---~vrc and temperature TH =
+ A)) + c
+a,
,
(10)
where C= ~
V f/
2 i=1
(7) .
a = N~--~ 1 - logN
l fd2xx/-L--~(¢R+4A2e~¢ 2rr
-
+-A '
where A -N 48~2x~" It can be seen that the curvature singularity and the apparent horizon intersect at the point
x int +
S
(11)
with the emission of a thunderpop at the intersection point with negative energy Ethudo,~o~ = - A ~ , independent of the initial state. The remnant geometry (10) coincides with that obtained in the process originated by gravitational collapse.
4P-~M proportional to its mass. We now introduce the boundary conditions (4),(5) and (in order to simplify calculations) the incoming shock-wave 1 T++ -- 2x+/------~5 (x + - x +) .
(16)
With these ingredients and in the limit N/~ > > 1, we get the solution (for x + > x +) / x + \ 1/2
3. T H E R M A L BATH REMOVAL THE EXPONENTIAL MODEL
IN
The semiclassical theory of the exponential model [7] is given by the action
S=
1 / d2xv/..~(¢R + 4A2e~¢ 2---~
ds2=
~-~-~z) dx+dx - . ~2PTc + Cx+x- (l°g~-~o )
(17)
One can check that the curvature singularity and the apparent horizon do not intersect each other at any finite point. In contrast to the BPP-RST models the asymptotically fiat coordinate o"+ at
J. Cruzet al./Nuclear PhysicsB (Proc. Suppl.) 57 (1997) 184-187
186
past null infinity is not the same as for the initial solution implying that the incoming flux has not actually been removed. In fact, it can be seen that (with an adequate choice of the point x +) the incoming flux starts to decrease continuously at x + - x +, vanishes at a certain time x + and for x + > x + becomes small and negative, going to zero at infinity. The incoming flux cannot qualitatively affect the result because at later times it becomes negative and even so the black hole does not evaporate. We notice that a procedure of thermal bath removal [8] has recently been applied to the Schwarzschild black hole by applying an operator formalism [9] to quantize the Vaidya solution.
condition for the existence of a s y m m e t r y in these models [12] gives in this case d 2 log V(¢) -
d¢ 2
(23)
o,
which has the general solution V = 4)~2ez¢ with )L /3 constants. For/3 = 0 we recover the C G H S model while for /3 ~ 0 we have the exponential model. By direct computation one can check that the action (21) possesses the following extra symmetries for an arbitrary potential
51¢
=
0, gu~
The action of the exactly solvable models (1) can be rewritten as follows
S = So (~0~u,e-2¢-k -
e 2¢ , 2eg~,~e2¢.
d~xx/-J-g(OR+V(¢)).
(21)
In the conformal gauge this action can be identified with a 2D sigma-model action [12,6]
1/
S= ~
d2x
( - 4 0 + ¢ 0 _ p + ~ V1( ¢ ) e
c2v\(v¢)2
=
0,
)
2p
(3
_
63g~u
2
(24)
2 (re) 4
(25)
2v"v O)l (v¢) 4 ] J
(26)
[guu
+ J ( g"u (vCf
(20)
This simple reconstruction of the one-loop quant u m corrected CGHS models suggests that we investigate the existence of additional extra symmetries for generic 2D dilaton models to construct solvable semiclassical models. The general 2D dilaton gravity action can be written (through a conformal reparametrization of the metric [10,11]) as:
S= ~
63¢
=
19)
e
,
{
~-~ae-2¢) -k-Sp(guu), (18)
where ~uu = gu, e -2¢ is an auxiliary metric which is invariant under the s y m m e t r y transformation
6¢ = 5gu~, =
2:
(re)
4. EXTRA SYMMETRIES
where ~¢ --- V(¢), which form a closed nonabelian Lie algebra in which 52 is a central generator, but 51 and 53 generate the affine subalgebra
[51,5a] = 16 1.
(27)
These three symmetries imply the existence of three free field equations which are []
/
~
dr 2E + J (v)" = 0,
(28)
n + [] log ( r e ) 2 = 0,
(29)
DE = 0.
(30)
, (22) where E = ~
where (p, ¢) play the role of target-space coordinates and the target-space metric is flat. The
- J (¢
is not only a free
field but also a conserved quantity which can be interpreted as the local energy of the solutions.
J Cruz et at~Nuclear Physics B (Proc. Suppl.) 57 (1997) 184-187
The symmetry 5 = 52 - 4A251, when V = 4A2, is just the conformal symmetry (19),(20) and therefore generalizes this symmetry for an arbitrary model. Moreover 5Z = 52 + 2/353 when V = 4A2ez¢ is exactly the conformal symmetry of the exponential model (14). It is possible to construct an auxiliary metric gu~
-
,
2Ex ,(V¢)29u~
+ ( 12Ex
(31) 2 E ~ _}
where Ex = ~1 ( (xY¢)2 - J (¢)) + 2A2¢, which is invariant under the symmetry 5. Relation (31) can be easily inverted to give ,
(32)
+(
- 2(V¢)4
With the aid of this metric we can construct a semiclassical action invariant under 5 by coupling the matter conformally to this invariant metric s
= 2 i=l
+Sp (.~) ,
(33)
where SDG is the action (21). This action is invariant under the transformation 5¢ 5gu,~
= =
~, 0,
(34)
and is the natural generalization of the BPP model for an arbitrary model of 2D dilaton gravity. REFERENCES
1. C . G . Callan, S. B. GiddingS, J. A. Harvey and A. Strominger, Phys. Rev. D45 (1992) 1005. 2. J . G . Russo, L. Susskind and L. Thorlacius, Phys. Rev. D46 (1993) 3444; Phys. Rev. D47 (1993) 533.
187
3. S. Bose, L. Parker and Y. Peleg, Phys. Rev. D52 (1995) 3512; Phys. Rev. Lett. 76 (1996) 861; Validity of the Semiclassical Approximation and Back-Reaction, WIS-MILW-96-th10. 4. J. Cruz and J. Navarro-Salas, Phys. Lett. B 375 (1996) 47. 5. R.B. Mann, Nucl. Phys. B 418 (1994) 231 6. J. Cruz, J. Navarro-Salas, M. Navarro and C. F. Talavera, Symmetries and Black Holes in 2D Dilaton Gravity, hep-th/9606097. 7. J. Cruz and J. Navarro-Salas, Phys. Lett. B 387 (1996) 51 8. J. Cruz, A. Mikovic and J. Navarro-Salas, A quantum model for Schwarzschild black hole evaporation, hep-th/9611219 9. A. Mikovic and V. Radovanovic, Two-Loop Back Reaction in 21) Dilaton Gravity, hepth/9606098 (to appear in Nucl. Phys. B). 10. T. Banks and M. O'Loughlin, Nucl. Phys. B362 (1991) 649. 11. J. Gegenberg, G. Kunstatter and D. LouisMartinez, Phys. Lett. B321 (1994) 193; Phys. Rev. D51 (1995) 1781; Classical and Quantum Mechanics of Black Holes in Generic 2D Dilaton Gravity, gr-qc/9501017. 12. Y. Kazama, Y. Satoh and A. Tscuchiya, Phys. Rev. D51 (1995) 4265.
PROCEEDINGS SUPPLEMENTS
Nuclear Physics B (Proc. Suppl.) 57 (1997) 184-187
Symmetries and solvable models for evaporating 2D black holes* J. Cruz, J. Navarro-Salas a, M. Navarro b and C. F. Talavera c ~Departamento de Ffsica Tedrica and IFIC, Centro Mixto Universidad de Valencia-CSIC, Facultad de Ffsica, Universidad de Valencia, Burjassot, 46100, Valencia, Spain. bInstituto de Matem&ticas y Ffsica Fundamental, CSIC, Serrano 113-123, 28006 Madrid, Spain. CDepartamento de Matem£tica Aplicada, E.T.S.I.I., Universidad Polit~cnica de Valencia, Camino de Vera, 46100, Valencia, Spain. We study the evaporation process of a 2D black hole in thermal equilibrium when the ingoing radiation is suddenly switched off. We also introduce global symmetries of generic 2D dilaton gravity models which generalize the extra symmetry of the CGHS model.
1. I N T R O D U C T I O N In recent years there has been a lot of activity in the study of two-dimensional dilaton gravity models. The string-inspired model of Callan, Giddings, Harvey and Strominger [1] is the simplest model that describes the formation of black holes by gravitational collapse. The existence of an extra s y m m e t r y for the CGHS model plays a fundamental role at the classical and quantum level. The one-loop q u a n t u m corrected model of Russo, Susskind and Thorlacius [2] was constructed to preserve the free field associated with the extra s y m m e t r y and the classical ground state. The model of Bose, Parker and Peleg [3] also preserves the extra s y m m e t r y but describes an evaporating black hole with a non-fiat remnant geometry. The content of this work is two- fold. In Sections 2 and 3 we study the one-parameter family of models interpolating between the B P P and R S T models [4] and the exponential model [5][6] respectively. However, instead of studying the evaporation of a black hole formed by gravitational collapse we analyze the semiclassical evaporation of a black hole initially in thermal equilibrium when the incoming thermal flux is suddenly switched off. In Section 4 we introduce extra global symmetries for generic 2D dilaton gravity *Work p a r t i a l l y s u p p o r t e d by t h e Comisidn Interministerim de Ciencia y Tecnologla a n d DGICYT. 0920-5632/97/$17.00 © Elsevier Science B.V All rights reserved. PII S0920-5632(97)00371 -X
models, in an a t t e m p t generalize the extra symmetry of the one-loop corrected theories of the CGHS model. 2. T H E R M A L BATH REMOVAL THE BPP-RST MODELS
IN
Let us consider the one-parameter family of models interpolating between the B P P and R S T models [4], S
=
(1)
So+SpW~/d2xv/-~
[(1 - 2 ¢ ) ( T ¢ ) 2 + (a - 1)¢R] , where So is the classical CGHS action So
--
1 Jd2xv/~[e-2¢(R
(2)
27r
N
1 + 4 ( V ¢ ) ~ + 4~ ~) - ~ ~ ( V f ' ) ~ ] i----1
' J
Sp is the Polyakov term and a is an arbitrary real parameter. In the conformal (ds 2 = -e2Pdx + d x - ) and Kruskal gauge (p = ¢) it is very easy to see that the above models admit the following solution e
•
:
+
M
(3)
which represents a black hole in thermal equilibrium at temperature T = 2A~. We can now remove
J. Cruz et al./Nuclear Physics B (Proc. Suppl.) 57 (1997) 184-187
185
N
the thermal bath by introducing a discontinuity in the boundary condition [7]
i=1
t~+
-
1 - 0 ( x + - ~0+), 4(x+) 2
t~-
=
0
(4) (5)
In this way we get the dynamical solution (for
N~
+9--~ f d2xvfL-g ( ¢ R + / 3 ( V ¢ ) 2 ) ' (12) where Sp is the Polyakov action. The classical limit
~+ > ~0+) e -2¢
+
N ] ~a ¢ = _N 48
--A2x+(x - -{- A) log--+l x+
(6)
4
M
O~
:
x+e~( "x- )
x+ int
-
4a~x~
where
1
°(
.
(8)
(9)
The evaporating solution can be continuously matched to the static solution e -2¢
+
Y]-~¢=a - 1 2 x + ( x - + A)
- ~ log ( - ~ x + ( x 4
log~-i
(13)
arises as a theory with extra conformal symmetry @,~
=
-c~g,~,
5¢
=
c.
(14)
The action (12) contains adequate counterterms to preserve this symmetry when we add the Polyakov action, in analogy to the BPP-RST models. In the absence of matter and taking the boundary conditions t± -- 0 we have the solution (in Kruskal gauge, 2p = ~¢)
ds2 =
dx+dx ~2cP7+ Cx+x- '
(15)
where 3 ' = 1 / ( 1 +N1--~2). If C < 0 (and A2 < 0) it represents a black hole in thermal equilibrium 2 with mass M = ~---~vrc and temperature TH =
+ A)) + c
+a,
,
(10)
where C= ~
V f/
2 i=1
(7) .
a = N~--~ 1 - logN
l fd2xx/-L--~(¢R+4A2e~¢ 2rr
-
+-A '
where A -N 48~2x~" It can be seen that the curvature singularity and the apparent horizon intersect at the point
x int +
S
(11)
with the emission of a thunderpop at the intersection point with negative energy Ethudo,~o~ = - A ~ , independent of the initial state. The remnant geometry (10) coincides with that obtained in the process originated by gravitational collapse.
4P-~M proportional to its mass. We now introduce the boundary conditions (4),(5) and (in order to simplify calculations) the incoming shock-wave 1 T++ -- 2x+/------~5 (x + - x +) .
(16)
With these ingredients and in the limit N/~ > > 1, we get the solution (for x + > x +) / x + \ 1/2
3. T H E R M A L BATH REMOVAL THE EXPONENTIAL MODEL
IN
The semiclassical theory of the exponential model [7] is given by the action
S=
1 / d2xv/..~(¢R + 4A2e~¢ 2---~
ds2=
~-~-~z) dx+dx - . ~2PTc + Cx+x- (l°g~-~o )
(17)
One can check that the curvature singularity and the apparent horizon do not intersect each other at any finite point. In contrast to the BPP-RST models the asymptotically fiat coordinate o"+ at
J. Cruzet al./Nuclear PhysicsB (Proc. Suppl.) 57 (1997) 184-187
186
past null infinity is not the same as for the initial solution implying that the incoming flux has not actually been removed. In fact, it can be seen that (with an adequate choice of the point x +) the incoming flux starts to decrease continuously at x + - x +, vanishes at a certain time x + and for x + > x + becomes small and negative, going to zero at infinity. The incoming flux cannot qualitatively affect the result because at later times it becomes negative and even so the black hole does not evaporate. We notice that a procedure of thermal bath removal [8] has recently been applied to the Schwarzschild black hole by applying an operator formalism [9] to quantize the Vaidya solution.
condition for the existence of a s y m m e t r y in these models [12] gives in this case d 2 log V(¢) -
d¢ 2
(23)
o,
which has the general solution V = 4)~2ez¢ with )L /3 constants. For/3 = 0 we recover the C G H S model while for /3 ~ 0 we have the exponential model. By direct computation one can check that the action (21) possesses the following extra symmetries for an arbitrary potential
51¢
=
0, gu~
The action of the exactly solvable models (1) can be rewritten as follows
S = So (~0~u,e-2¢-k -
e 2¢ , 2eg~,~e2¢.
d~xx/-J-g(OR+V(¢)).
(21)
In the conformal gauge this action can be identified with a 2D sigma-model action [12,6]
1/
S= ~
d2x
( - 4 0 + ¢ 0 _ p + ~ V1( ¢ ) e
c2v\(v¢)2
=
0,
)
2p
(3
_
63g~u
2
(24)
2 (re) 4
(25)
2v"v O)l (v¢) 4 ] J
(26)
[guu
+ J ( g"u (vCf
(20)
This simple reconstruction of the one-loop quant u m corrected CGHS models suggests that we investigate the existence of additional extra symmetries for generic 2D dilaton models to construct solvable semiclassical models. The general 2D dilaton gravity action can be written (through a conformal reparametrization of the metric [10,11]) as:
S= ~
63¢
=
19)
e
,
{
~-~ae-2¢) -k-Sp(guu), (18)
where ~uu = gu, e -2¢ is an auxiliary metric which is invariant under the s y m m e t r y transformation
6¢ = 5gu~, =
2:
(re)
4. EXTRA SYMMETRIES
where ~¢ --- V(¢), which form a closed nonabelian Lie algebra in which 52 is a central generator, but 51 and 53 generate the affine subalgebra
[51,5a] = 16 1.
(27)
These three symmetries imply the existence of three free field equations which are []
/
~
dr 2E + J (v)" = 0,
(28)
n + [] log ( r e ) 2 = 0,
(29)
DE = 0.
(30)
, (22) where E = ~
where (p, ¢) play the role of target-space coordinates and the target-space metric is flat. The
- J (¢
is not only a free
field but also a conserved quantity which can be interpreted as the local energy of the solutions.
J Cruz et at~Nuclear Physics B (Proc. Suppl.) 57 (1997) 184-187
The symmetry 5 = 52 - 4A251, when V = 4A2, is just the conformal symmetry (19),(20) and therefore generalizes this symmetry for an arbitrary model. Moreover 5Z = 52 + 2/353 when V = 4A2ez¢ is exactly the conformal symmetry of the exponential model (14). It is possible to construct an auxiliary metric gu~
-
,
2Ex ,(V¢)29u~
+ ( 12Ex
(31) 2 E ~ _}
where Ex = ~1 ( (xY¢)2 - J (¢)) + 2A2¢, which is invariant under the symmetry 5. Relation (31) can be easily inverted to give ,
(32)
+(
- 2(V¢)4
With the aid of this metric we can construct a semiclassical action invariant under 5 by coupling the matter conformally to this invariant metric s
= 2 i=l
+Sp (.~) ,
(33)
where SDG is the action (21). This action is invariant under the transformation 5¢ 5gu,~
= =
~, 0,
(34)
and is the natural generalization of the BPP model for an arbitrary model of 2D dilaton gravity. REFERENCES
1. C . G . Callan, S. B. GiddingS, J. A. Harvey and A. Strominger, Phys. Rev. D45 (1992) 1005. 2. J . G . Russo, L. Susskind and L. Thorlacius, Phys. Rev. D46 (1993) 3444; Phys. Rev. D47 (1993) 533.
187
3. S. Bose, L. Parker and Y. Peleg, Phys. Rev. D52 (1995) 3512; Phys. Rev. Lett. 76 (1996) 861; Validity of the Semiclassical Approximation and Back-Reaction, WIS-MILW-96-th10. 4. J. Cruz and J. Navarro-Salas, Phys. Lett. B 375 (1996) 47. 5. R.B. Mann, Nucl. Phys. B 418 (1994) 231 6. J. Cruz, J. Navarro-Salas, M. Navarro and C. F. Talavera, Symmetries and Black Holes in 2D Dilaton Gravity, hep-th/9606097. 7. J. Cruz and J. Navarro-Salas, Phys. Lett. B 387 (1996) 51 8. J. Cruz, A. Mikovic and J. Navarro-Salas, A quantum model for Schwarzschild black hole evaporation, hep-th/9611219 9. A. Mikovic and V. Radovanovic, Two-Loop Back Reaction in 21) Dilaton Gravity, hepth/9606098 (to appear in Nucl. Phys. B). 10. T. Banks and M. O'Loughlin, Nucl. Phys. B362 (1991) 649. 11. J. Gegenberg, G. Kunstatter and D. LouisMartinez, Phys. Lett. B321 (1994) 193; Phys. Rev. D51 (1995) 1781; Classical and Quantum Mechanics of Black Holes in Generic 2D Dilaton Gravity, gr-qc/9501017. 12. Y. Kazama, Y. Satoh and A. Tscuchiya, Phys. Rev. D51 (1995) 4265.