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A collapsed black hole in two­dimensional space­time
F. Vendrell 1
Blackett Laboratory, Imperial College, Prince Consort Road, London SW7 2BZ, U.K.
An imploding shell of radiation is shown to create a 2­D black hole within the framework
of the ``R = T '' theory. The radius of the horizon is given by r H = 1
2M , where M is the
mass of the black hole. The topology of the central singularity is that of a corner. The
radiation emitted very far from the black hole is thermal with temperature \Theta rad = ¯hM
2úk .
The back­reaction problem is solved to one­loop order.
This poster is a short version of Vendrell (1997b).
1 Introduction
Black holes in two dimensional space­times are toy models to understand the physics of their four
dimensional analogs. Since the general theory of relativity has no physical contents in two dimensions
(Collas (1977)), other 2­D gravity theories have been considered, in particular string theory for which
there is two dimensional black hole solutions (Callan et al. (1992)). It is possible, however, to extract
from the equations of general relativity a gravity theory by considering the formal limit D! 2 + , where
D is the dimension of space­time (Mann (1992), Mann & Ross (1993)). One obtains the equation
R(x) = 8úGT cl (x); (1)
where c = 1, G is Newton's constant and where T cl is the trace of the classical energy­momentum
tensor T cl
¯š . This equation is supplemented by the continuity equation
r ¯ T cl
¯š (x) = 0; (2)
which follows also from the equations of general relativity.
2 The collapsing shell
In a two­dimensional space­time we define:
ffl (x 0 ; x 1 ) = conformal global coordinates; x \Sigma = x 0 \Sigma x 1 .
ffl (t; r) = time and distance to an origin (r – 0).
ffl Localised massless collapsing shell (M ? 0, x +
0 ? 0):
T cl (x) = M
2úG ffi (x + \Gamma x +
0 ) (3)
\Lambda Boundary conditions for x + ! x +
0 :
\Pi ds 2 = dx + dx \Gamma
\Pi x \Sigma = t \Sigma r
\Lambda Problem calculate for x + ? x +
0 :
\Pi ds 2 =C(x)dx + dx \Gamma
\Pi t = t(x + ; x \Gamma ) and r = r(x + ; x \Gamma )
1 mailto:f.vendrell@ic.ac.uk
1

\Lambda Solution:
ds 2 =
8
? !
? :
dx + dx \Gamma ; if x + ! x +
0 ,
dx + dx \Gamma
M
\Gamma
x \Gamma
H \Gamma x \Gamma \Delta ; if x + ? x +
0 ,
(4)
where x \Gamma
H is a constant (Vendrell (1997a)) (The solution depends actually on the representation
of the delta function (eq.3), which is defined here by
R +1
\Gamma1 duf(u) ffi (u) = lim ffl!0 + f(ffl), where the
function f is continuous except possibly at u = 0.) The functions t = t(x + ; x \Gamma ) and r = r(x + ; x \Gamma )
are left undetermined for x + ? x +
0 , contrary to the 4­D case (Synge (1957)).
3 Systems of coordinates
ffl The coordinates y I and yE are defined for x \Gamma ? x \Gamma
H and x \Gamma ! x \Gamma
H respectively by
8
!
:
y +
I;E (x + ) = x +
y \Gamma
I;E (x \Gamma ) = \Sigma 1
M log
fi fi x \Gamma
H \Gamma x \Gamma fi fi
(5)
=) Line element for y + ? x +
0 :
ds 2 =
(
\Gammady +
I dy \Gamma
I ; if x \Gamma ? x \Gamma
H ,
dy +
E dy \Gamma
E ; if x \Gamma ! x \Gamma
H .
(6)
ffl The coordinates (t; r) are defined for x + ? x +
0 by
(
y 0
I;E = \Upsilont
y 1
I;E = r \Upsilon 1
2M log
fi
fi r \Gamma 1
2M
fi
fi
(7)
=) Line element for y + ? x +
0 :
ds 2 = sgn
`
r \Gamma
1
2M
' ''
dt 2 \Gamma dr 2
\Gamma
1 \Gamma 1
2Mr
\Delta 2
#
(8)
4 Black hole structure
See Figure 1.
ffl The accessible space­time (physical region) is defined by r(x 0 ; x 1 ) – 0.
ffl In the x coord., the metric (eq.4) is singular at x \Gamma = x \Gamma
H and x + ? x +
0 , where the apparent
horizon is located.
ffl In the y coord., the metric (eq.6) is Minkowskian for x + ? x +
0 . The y I /yE coordinates describe
the interior/exterior of the black hole. These two regions are causally disconnected.
ffl In the (t; r) coord., the metric (eq.8) is singular in the limits r ! 1
2M
and r ! 0 + , if x + ?
x +
0 . The horizon is located at r = 1
2M . It takes a finite amount of proper time for all time­
like geodesics with r ! 1
2M
to reach the central singularity at r = 0, where all the matter is
concentrated after the collapse.
2

x­
x­_H
Unphysical
region
x+_0
x+
Shock
wave
r = 0
r=0 (singularity)
(coordinate
singularity)
AH +
EH
r=1/2M
or
t=+infinity
EH
Unphysical region
region
Trapped
AH
Figure 1: Space­time diagram of the 2­D black hole. The light cones are looking upwards every­
where. The curves denoted by AH and EH are the apparent and event horizons respectively.
3

5 Thermal radiation and back­reaction
The energy­momentum tensor of the massless scalar field is calculated from the transformation
(x 0 ; x 1 ) 7! (r; t):
T rad
++ (t; r) ú 0; T rad
\Gamma\Gamma (t; r) ú M 2
48ú ; (9)
if r AE 1=(2M ) and x + ? x +
0 (Vendrell (1997a)). The black hole emits thus spontaneously a thermal
radiation (Hawking (1975)) of temperature
\Theta rad = M
2ú \Delta (10)
To the first loop approximation, the back­reaction of the massless radiation on space­time is
analysed by adding to eq.1 the trace anomaly T rad (x) = \Gamma ¯h
24ú R(x). One obtains
R(x) = ff 8úGT cl (x); (11)
where ff = 1
1+¯hG=3 ! 1. The mass and temperature of the black hole are renormalised and become
smaller
M br = ff M; \Theta rad
br = ff
M
2ú \Delta (12)
The radius r br
H = 1
2Mbr
of the black hole becomes larger.
Acknowledgments
I thank M.E. Ortiz and S. Schreckenberg for helpful discussions and pertinent comments.
References
Callan C.G. et al., 1992, PRD, 45, R1005.
Collas P., 1977, AJP, 45, 833.
Hawking S.W., 1975, CMP, 43, 199.
Mann R.B., 1992, GRG, 24, 433.
Mann R.B. & Ross S.F., 1993, CQG, 10, 1405.
Synge J.L., 1957, PRIA, 59 A, 1.
't Hooft G., 1996, IJMP, A11, 4623.
Vendrell F., 1997a, HPA, 70, 598, hep­th/9608007 2 .
Vendrell F., 1997b, hep­th/9705135 3 .
2 http://xxx.soton.ac.uk/abs/hep­th/9608007
3 http://xxx.soton.ac.uk/abs/hep­th/9705135
4