Upload
y
View
218
Download
2
Embed Size (px)
Citation preview
PHYSICAL REVIEW D 15 JANUARY 1998VOLUME 57, NUMBER 2
Black holes and two-dimensional dilaton gravity
T. FutamaseAstronomical Institute, Faculty of Science, Tohoku University, Sendai 980-77, Japan
M. HottaDepartment of Physics, Faculty of Science, Tohoku University, Sendai 980-77, Japan
Y. ItohAstronomical Institute, Faculty of Science, Tohoku University, Sendai 980-77, Japan
~Received 8 September 1997; published 22 December 1997!
We study the conditions for two-dimensional dilaton gravity models to have a dynamical formation of blackholes and construct all such models. Furthermore we present a parametric representation of the general solu-tions of the black holes.@S0556-2821~98!02002-5#
PACS number~s!: 04.70.Dy, 04.60.Kz
atofnsmis
-va
uiee
oeveio
tytseufotithle
psg
oecthpr
reIIthth
imo
la-
is
itheorbearen-
tyos-
ib-
ita-
ic:
I. INTRODUCTION
Two-dimensional dilaton gravity has attracted muchtention in the context not only of string theory but alsoHawking radiation. It turns out that even in two dimensiothe event horizon can be dynamically formed and the therradiation is emitted from the hole if the quantum effecttaken into account@1–3#. It is widely believed that the physical essence of two-dimensional Hawking radiation is equilent to four-dimensional physics.
Dilaton gravity in two dimensions does not have a uniqaction and an infinite number of variations of the theorexist. Thus it is quite crucial to extract aspects independof the details of the models.
Much effort has already been made to solve the mgeneral dilaton gravity model and much progress has bachieved@4–8#. Especially the general static solutions habeen formally obtained using the conformal transformattechnique@8#.
However, the question of what kinds of dilaton gravimodels survive after imposing the following requiremenwhich are quite reasonably expected to hold has not bfully addressed. First, the model should have a flat vacusolution in order to describe the state before black-holemation. Secondly, the model should admit a family of stablack-hole solutions with regular horizons. The mass ofhole can get heavier without bounds. Finally, the black hoshould be created in a process of the gravitational collaFor example, the infalling matter shock wave will brinabout the formation of a black hole.
In this paper we address and solve this question. We cstruct all the dilaton gravity theories containing at most sond derivatives of the fields in an action which satisfiesabove requirements. Moreover, we give a parametric resentation of the general black-hole solutions.
This paper is organized as follows. We give a shortview on the most general model of dilaton gravity in Sec.In Sec. III we analyze the conditions for the existence offlat vacuum as well as a black hole, and then construct alltheories which admit at least one black-hole solution byposing an appropriate boundary condition at the black h
570556-2821/97/57~2!/1129~7!/$15.00
-
al
-
esnt
sten
n
enmr-cese.
n--ee-
-.ee-le
horizon. It is shown that the theories can be completelybeled using the two parametersg and m. In Sec. IV wediscuss the dynamical formation of the black hole. In thargument the parameterm is identified as the black-holemass. In Sec. V we obtain general black-hole solutions wlarger masses thanm. Furthermore, it will be shown that thparameterg is fixed if the event horizon does not vanish fan arbitrarily large mass of the hole. Consequently, it willargued that all the theories satisfying the natural criteriacompletely fixed by giving an arbitrary input black-hole cofiguration and its massm. In Secs. VI and VII we discuss twoexamples in detail.
II. TWO-DIMENSIONAL DILATON GRAVITY
Let us first write down the action of the dilaton graviincluding only second derivative terms and the dilatonic cmological term:
S51
2p E d2xA2g„F~f!R
14G~f!~¹f!214l2U~f!2 12 ~¹ f !2
…. ~1!
Heregab is the two-dimensional metric andf is the dilatonfield. The forms of the functionsF, G, andU are arbitraryso far, but will be specified later to admit solutions descring the black-hole formation process. The constantl has onemass dimension. A massless matter fieldf is included in theabove action since we shall consider the process of gravtional collapse in a later section.
Now we adopt the conformal gauge fixing for the metr
x65t6r ,
ds252e2rdx1dx2. ~2!
Note that there remains a residual gauge freedom in Eq.~2!,that is, the conformal gauge transformation
x865x86~x6!, ~3!
1129 © 1997 The American Physical Society
at
p
ioas
onden:
-m
thra
aen-ion,
the
at
by
the
k-of
ion
-
1130 57T. FUTAMASE, M. HOTTA, AND Y. ITOH
r85r21
2lnFdx81
dx1
dx82
dx2 G . ~4!
We do not fix this gauge freedom and change the coordinat our convenience.
The connections and the scalar curvature can be simcalculated from the above metric:
G111 52]1r,
G222 52]2r,
R58e22r]1]2r.
The equations of motion are derived by taking a variatof the action~1!. In the conformal gauge they are written
2F8]1]2r18G]1]2f14G8]1f]2f1l2U8e2r50,~5!
]1]2F1l2Ue2r50, ~6!
]62 F22]6r]6F24G~]6f!252
1
2~]6 f !2. ~7!
Equation~7! is the first constraint of the system and Eq.~5!is automatically statisfied if Eqs.~6! and ~7! hold.
For static solutions without matter, Eqs.~5!–~7! can berewritten in the following forms:
F8r14Gf52l2U8e2r22G8f2, ~8!
F8f54l2Ue2r2F9f2, ~9!
l2Ue2r5Gf211
2F8rf. ~10!
Here the prime means the derivative with respect tof andthe overdot signifies the spacelike variabler .
III. EXISTENCE OF THE FLAT VACUUM STATEAND A BLACK HOLE SOLUTION
In order to restrict the general two-dimensional dilatgravity described in the previous section, we first consithe condition for the existence of the flat vacuum solutio
r5C5const, ~11!
f5fvac~r !. ~12!
Substituting Eqs.~11! and ~12! into the equations of motions ~8!–~10!, we get the necessary condition of the vacuuexistence:
U8
U5
G8
G22
F9
F818
G
F8. ~13!
Furthermore, it is easily seen that the condition~13! is usedto show that the theory has the flat vacuum solution anddilaton field in the vacuum state is then obtained by integing the following differential equation:
es
ly
n
r
et-
df
dr5leCS U~f!
G~f! D1/2
. ~14!
Next we consider the condition for the existence ofblack hole solution. The black-hole configuration can be gerally expressed, using the conformal gauge transformatas
f5f~r !, ~15!
r~r;2`!5lr 1ro1r1e2lr1r2e4lr1••• , ~16!
limr→`
r5r~`!5finite. ~17!
Otherwise the functionr(r ) is chosen arbitrary except thar(`).r(r ) which is needed to ensure the regularity of tdilaton field. In this expression the event horizon staysr 52` and the scalar curvature on the horizon is28l2r1 .It is of course possible to investigate inside the horizonchanging the coordinates:
x6561
le6lX6
.
Then the horizon is mapped into null surfacesX150 andX250.
Let us assume the configuration to be a solution ofequations of motion~8!–~10!. If we introduce three functionsof r ,
F5F„f~r !…, ~18!
G5f2G„f~r !…, ~19!
U5U„f~r !…, ~20!
then the equations of motion are rewritten as
F r52l2Ue2r22G, ~21!
F54l2Ue2r, ~22!
l2Ue2r5G11
2F r. ~23!
It is quite essential to realize the fact that giving a blachole solution imposes restrictions on the theory. The formr(r ) which satisfies the boundary condition~16!–~17! leadsus to explicit forms ofF, G, andU.
Let us prove this statement. From the equations of mot~22! and~23! we can expressG(r ) andU„f(r )… by F„f(r )…andr(r ):
G51
4@ F22rF#, ~24!
U5e22r
4l2 F. ~25!
The functionsF, G, andU are also subject to the vacuumexistence condition~12!. Equation~12! is reexpressed as
he
h
y
on
.en
fa
o
hen
:
the
to
lu-
t ack
di-as
te
57 1131BLACK HOLES AND TWO-DIMENSIONAL DILATON GRAVITY
U
U2
G
G12
F
F58
G
F. ~26!
By substituting Eqs.~24! and~25! into Eq.~26! we obtain theequation to determineF„f(r )… from r(r ):
F ~3!
F22
F
F5
r
r22r. ~27!
The differential equation~27! can be straightforwardly in-tegrated and we get the general solution as follows:
F52gm
e22r2g2 , ~28!
F5F01E 2gm
e22r2g2 dr, ~29!
wherem, g, andF0 are integration constants. Note that in tcase thatg→0 with gm fixed the solution~28! takes theform F5Ae2r, but it leads toG50 which is not interestingin the present discussion. Thus we will not consider succase henceforce.
The integration constantF0 above does not generate anphysical effect because the variation of
E A2gF0Rd2x
in the action yields just a boundary term and does not ctribute to the equations of motion.
Substituting Eq.~28! into Eqs. ~24! and ~25! yields thefollowing expressions:
G5gr
4m F 2gm
e22r2g2G2
, ~30!
U5gm
l2
re24r
@e22r2g2#2 . ~31!
Equations~29!, ~30!, and~31! are the required expressionsIt is notable thatf(r ) is not yet specified though we hav
consumed all the information of the equations of motion athe vacuum-existence condition. This is reflected by thethat dilaton-field redefinition
f5f~f!
does not change the theory itself. The functional formf(r ) depends on the definition adopted by the dilaton fieldfand thus has no physical meaning. On the other hand, trdependences ofF, G, andU are independent of the dilatoredefinition and real physical quantities.
We can take the form off(r ) as we wish, and impose
f~r !.0. ~32!
Then r is solved byf keeping one-to-one correspondence
r 5r ~f!. ~33!
a
-
dct
f
Using this and Eqs.~29!–~31!, we reconstructF(f), G(f),andU(f) as follows:
F5F@f„r ~f!…#, ~34!
G5G„r ~f!…
f„r ~f!…2, ~35!
U5U@f„r ~f!…#. ~36!
As is easily understood from the action~1!, the inverse ofF can be identified as the gravitational constant. Thus innormal situation it is natural to take
F>0, ~37!
F.0. ~38!
This property ofF enables us to define the dilaton fieldsatisfy
F~f!5f,
holding Eq.~32!. We adopt this definition forF in the fol-lowing sections.
IV. GRAVITATIONAL COLLAPSE SOLUTION
The theory we discussed in Sec. III has the vacuum sotion @rvac(r )5const,fvac(r )# and a black-hole solution@r(r ),f(r )#. From these two solutions we can construcsolution which describes the dynamical formation of a blahole caused by a shock-wave-type source:
1
2~]1 f !25«d~x12z1!.
The spacetime is in the vacuum state forx1,z1 and ablack hole appears forx1.z1. Thus the metric of this pro-cess may be written as
ds252dx1dx2@Q~x12z1!e2r@r 5~1/2!~x12x2!#
1Q~z12x1!e2r@r 5~1/2!~z12x2!##,
whereQ is the step function.For the vacuum portion there exist the canonical coor
natess6 where the metric and dilaton can be expressedfollows:
ds252e2Cds1ds25e2C~2dT21dX2!, ~39!
dfvac
dX5leCS U~fvac!
G~fvac!D 1/2
. ~40!
Integration of Eq.~40! together with Eqs.~30! and ~31!yields the expression forX in terms ofr :
X5ge2CE r ~fvac!
e2r~r !dr. ~41!
It is also easy to write down explicitly the coordinatransformation betweenx6 ands6:
n
g
ktt-
e.
q.
t
1132 57T. FUTAMASE, M. HOTTA, AND Y. ITOH
s15x1, ~42!
s25E expF2rS r 51
2~z12x2! D22CGdx2. ~43!
Using the relation
X51
2@x12s2~x2!#, ~44!
we get the nontrivial relation
1
2@x12s2~x2!#5ge2CE r @fvac~x1,x2!#
e2r~r !dr. ~45!
Differentiation of Eq.~45! with respect tox6 yields
]fvac
]x1 5eCm
12g2e2 r~x1,x2!, ~46!
]fvac
]x2 52e2C12r@r 5~1/2!~z12x2!#m
12g2e2 r ~x1,x2!,
~47!
r ~x1,x2![r$r 5r @fvac~x1,x2!#%, ~48!
where we used Eq.~28! for F5f.The shock-wave contribution appears in11 part of the
equations of motion~7! as follows:
]12 f22]1r]1f24G~]1f!252«d~x12z1!. ~49!
From this a junction constraint must be imposed:
]f
]x1 Ux15z1
2]fvac
]x1 Ux15z1
52«. ~50!
On the other hand, the remaining part of Eq.~7! leads tothe continuity condition
]f
]x2 Ux15z1
2]fvac
]x2 Ux15z1
50. ~51!
Noticing
]6f561
2
df
dr56
gm
e22r~r !2g2
and substituting Eqs.~46! and ~47! into Eqs.~50! and ~51!,the following relations are obtained:
«5m
g, ~52!
e2C51
g2 , ~53!
r ~z1,x2!5rS r 51
2~z12x2! D . ~54!
Equation ~54! is automatically satisfied by the definitio~48!.
From Eqs.~52! and ~53! the total mass of the collapsinbody is evaluated as
M5e2C«5m.
Thus the parameterm is identified as the mass of the blachole. This is also confirmed by evaluating the ArnowiDeser-Misner~ADM ! mass.
We shall prove thatm is in fact the ADM mass. Accord-ing to Mann@7#, the conserved ADM massM is constructedfrom the conserved current
Jm5Tmnena¹aK~f!, ~55!
whereTmn is the stress-energy tensor andK takes the fol-lowing form in our case:
K5K0Ef
dfe24*fG, ~56!
where the constantK0 is chosen by the condition that thvector jm5ena¹aK is normalized unity at spatial infinityThen the mass functionm is introduced asJm5em
n ¹nmwhich is written in our present case withF5f as follows:
m5K0
2 F4l2Ef
dfUe24*fG2~¹f!2e24*fGG . ~57!
The ADM mass is evaluated as the limit ofm at the spatialinfinity. In our static situation forx1.z1, the mass functionis easily evaluated as
m5K0
2 Fl2S U
GD 1/2
2e22rf2S G
U D 1/2G5K0
2
l
gf@e22r2g2#
5K0lm, ~58!
where we have used the relation
U5GexpF8Ef
Gdf G ~59!
which is obtained from the vacuum existence condition~13!,and the relations~28!, ~30!, ~31! and~53!. The normalizationconstantK0 is fixed as follows. Using the above relation, E~56! takes the form
K5K0Ef
dfS G
U D 1/2
. ~60!
This gives us the timelike Killing vectorj whose componenat the spatial infinity is
K~`!5K0S G
UD 1/2U
r 5`
5K0
l
g. ~61!
Since the metric takes the forme2C5g22 at the spatial in-finity, the normalization constant is chosen asK051/lwhich gives us the required result for the mass.
nd
kono
us
on
the
rerary
he
57 1133BLACK HOLES AND TWO-DIMENSIONAL DILATON GRAVITY
V. BLACK HOLES WITH LARGER MASS
As seen in Sec. IV, giving a black-hole configuration aspecifyingg and the massm is equivalent to fixing the actionof the dilaton gravity possessing the Minkowski and blachole configurations as solutions of the equations of motiIn this section we prove that the theory has static black-hsolutions with larger masses thanm.
Taking the dilaton definition asF5f, the following re-lations come from Eqs.~28!, ~29!, and~30!:
f5f01E2`
r 2gm
e22r2g2 dr, ~62!
r 5r ~f2f0 ,m,g!, ~63!
G5G@f2f0 ,m,g,r~r !#, ~64!
U5U@f2f0 ,m,g,r~r !#. ~65!
Let us consider another static configuraton@r8(r 8),f8(r 8)#as a solution of the same theory;@F5f,G(f),U(f)#. Fromthe same argument in Sec. IV the following relations mhold:
df8
dr85
2gbM
g2e22r82b2, ~66!
r 85r 8~f82f80 ,M ,b!, ~67!
G@f82f80 ,M ,b,r8~r 8!#5b
4gM
dr8
dr8, ~68!
U@f82f80 ,M ,b,r8~r 8!#5g3bM
l2 F e22r8
g2e22r82b2G 2dr8
dr8,
~69!
whereb andM are constants.In order to ensure that the configurationsr(r ) andr8(r 8)
are the solutions of the same system, the following relatiare needed:
G@f82f80 ,M ,b,r8~r 8!#5G@f2f0 ,m,g,r~r !#,~70!
U@f82f80 ,M ,b,r8~r 8!#5U@f2f0 ,m,g,r~r !#.~71!
These yield the equations
dr8
dr5
bm
g2M
dr8
dr, ~72!
e2r85g2
b2 F12M
m~12g2e2r!G . ~73!
By substituting Eq.~73! into Eq. ~72! and integrating itwe acquire the transformation relation betweenr and r 8
r 82r 805bE e2r
12 ~M /m! ~12g2e2r!dr. ~74!
-.
le
t
s
Now let us concentrate on the parameter space with
M.m.
Then there exists a pointr 5r H , where
e2r~r H!51
g2 F12m
M G . ~75!
Whenr approachesr H , the value ofr diverges to2`. Ifwe take
b5g2M
ml
dr
dr~r H!,
the asymptotic behavior is given as
r8~r 8;2`!5lr 81r801r81e2lr 81••• . ~76!
This is precisely a black-hole configuration. Repeatingjunction argument in Sec. V for this black hole,M can beexactly proven as the mass of the hole.
Note that the regularity of the dilaton field everywheoutside the horizon and the permission to have an arbitheavy mass of the black hole are guaranteed by setting
g5e2r~`!,
which is derived from Eqs.~62! and ~73!.Then the functions specifying the action,F, G, and U,
are given by a parametric representation:
F~f!5f, ~77!
G~f!5e2r~`!
4mr~r !, ~78!
U~f!5m
l2
e2r~`!r~r !
~12e2r~r !22r~`!!2 , ~79!
f5f012mE2`
r e2r~`!
e22r~r !2e22r~`! dr. ~80!
Finally we obtain a parametric representation of tblack-hole solution:
f8~r 8!5f~r !5f01E2`
r 2me2r~`!
e22r~r !2e22r~`! dr, ~81!
e2r8~r 8!5l2m2e2r~`!
M2r~r H!2 F12M
m~12e2r~r !22r~`!!G , ~82!
r 82r 805M
m
r~r H!
l E e2r~r !22r~`!
12 ~M /m! ~12e2r~r !22r~`!!dr,
~83!
where
e2r~r H!22r~`!512m
M.
u
e
et
1134 57T. FUTAMASE, M. HOTTA, AND Y. ITOH
This is our main result. Once the input black-hole configration r(r ) and its massm are given arbitrarily and thetheory is fixed by Eqs.~77!, ~78!, and~79!. Then Eqs.~81!–~83! generate black holes withM.m.
VI. CGHS BLACK HOLES
In this section we check formulas~78!–~83! for the well-known Callan-Giddings-Harvey-Strominger~CGHS! blackhole @2#.
Let first give the CGHS black hole with massm5l.
e2r51
11e22lr . ~84!
Then the horizon for the black hole with massM is evalu-ated as
r H521
2llnS l
M2l D ,
r~r H!5l2
M.
Equation~80! with
f051
yields
f511e2lr , ~85!
r 51
2lln@f21#. ~86!
Equations~78! and ~79! give
G51
4
1
11e2lr 51
4f,
U511e2lr5f.
Thus, Eqs.~78!–~80! reproduce the precise form of thCGHS action.
From Eqs.~80!–~82! with
- r 8050,
the following relations come out:
r 851
2llnFe2lr112
M
l G ,e2r85
1
11~M /l!e22lr 8,
f85e2lr 81M
l.
Thus CGHS black-hole solutions with massM are exactlyreproduced.
VII. MODIFIED CGHS BLACK HOLES
In this section we discuss a more nontrivial example. Lus set up the metric as
e2r511ee2lr
11e22lr1ee2lr .
Also let its mass
m5l
and take the pamameters as
f051,
r 8050.
From Eq.~80! the dilaton field is solved as
f~r !511e2lr1e
2e4lr , ~87!
r 51
2llnF1
e~A112e~f21!21!G . ~88!
Substituting Eq.~88! into Eqs.~78! and ~79! yields
F~f!5f, ~89!
G~f!51
4
@A112e~f21!2 1/2#@11e~2f21!1A112e~f21!#
@f1~e/2!~2f21!2#A112e~f21!, ~90!
U~f!54
e2 G~f!@11e~2f21!2A112e~f21!#2. ~91!
These determine the action of the theory.Next consider the black holes with massM.m. The horizon is given as
r H51
2llnH 1
2e FA114eS M
l21D21G J , ~92!
r~r H!5l3
2eM ~M2l!A114eS M
l21D FA114eS M
l21D21G .
57 1135BLACK HOLES AND TWO-DIMENSIONAL DILATON GRAVITY
Then the final result is expressed as
f511s1e
2s2,
e2r5112e~M /l21!1A114e~M /l21!
2@114e~M /l21!#
s1es2112M /l
11s1es2 ,
e2lr 85H s11
2e F12A114eS M
l21D G J 12bH s1
1
2e F11A114eS M
l21D G J b
,
b51
2 F121
A114e~M /l 21!G ,
lu
nacluo-de
k-a
ackdedrib-r-is
-ant-n,
wheres5e2lr .This is a parametric representation of the black-hole so
tion with massM in the theory with Eqs.~89!–~91!. Theparameters runs in the region
1
2e FA114eS M
l21D21G<s<` ~2`<r<`!.
VIII. SUMMARY
We have investigated the conditions for two-dimensiodilaton gravity to have a vacuum solution as well as a blahole solution. The general expression for black-hole sotions is obtained by giving an arbitrary function with apprpriate boundary conditions. Then we derive the solution
er
-
lk-
-
scribing the dynamical formation of a black hole by a shocwave-type material source. Furthermore we obtainedparametric representation of the general solution of blholes with arbitrary mass. Thus our work should be regaras the general formalism to derive general solutions descing not only a static black hole but also the dynamical fomation of a black hole. The quantum effect in our modelquite interesting; this remains a future problem.
ACKNOWLEDGMENTS
We would like to thank M. Morokawa for fruitful discussions. This work was supported in part by a Japanese Grin-Aid for Science Research fund of Ministry of EducatioScience and Culture Grant No. 09640332~T.F.!.
@1# C. G. Callan, S. B. Giddings, J. A. Harvey, and A. StromingPhys. Rev. D45, R1005~1992!.
@2# W. T. Kim and J. Lee, Int. J. Mod. Phys. A11, 553 ~1996!.@3# J. G. Russo, L. Susskind, and L. Thorlacius, Phys. Lett. B292,
13 ~1992!; Phys. Rev. D46, 3444~1992!; A. Bilal and C. G.Callan, Nucl. Phys.B394, 73 ~1993!; M. Hotta and M.Yoshimura, Prog. Theor. Phys.91, 181 ~1993!.
@4# T. Banks and M. O’Loughlin, Nucl. Phys.B362, 649 ~1991!.
, @5# V. Frolov, Phys. Rev. D46, 5383~1992!.@6# O. Lechtenfeld and C. Nappi, Phys. Lett. B288, 72 ~1992!; S.
D. Odintsov and I. L. Shapiro, Mod. Phys. Lett. A7, 437~1992!.
@7# R. B. Mann, Phys. Rev. D47, 4438~1993!.@8# D. Louis-Martinez and G. Kunstatter, Phys. Rev. D49, 5227
~1994!.