This document summarizes the dimensional reduction of 4D general relativity to a (1+1) gravity model. It shows that the Einstein-Hilbert action in 4D can be reduced to an effective action for a (1+1) gravity coupled to a scalar field. The equations of motion for the reduced metric and scalar field are derived. The causal structure of the resulting (1+1)-dimensional spacetime is examined, with the regions above and below a critical radius value related by a change of coordinates. A Carter-Penrose diagram illustrates the causal structure.

1. Four Dimensional General Relativity Reduced As ( 1 + 1 ) Gravity
Roa, Ferdinand J. P.
I. Introduction
On its theoretical foundations, General
Relativity is a metric theory of gravity. In place
of Newtonian gravitational potential, the
components of the metric tensor are given
dynamical attributes that are governed by the
metric tensor’s own field equations- Einstein’s
Field Equations. These field equations are
derivable from Einstein-Hilbert action in which
the free Lagrangian (in the absence of interacting
matter terms and cosmological constant) simply
contains the curvature scalar.
By resorting to Kaluza-Klein type of
dimensional reduction, the originally 4D form (
or in originally higher dimensions) of Einstein-
Hilbert action, containing no other terms aside
from the curvature scalar, can be reduced as a 2D
gravity coupled to a scalar field or what is known
as the Einstein-Scalar system.
Two-D models of gravity have become
fashionable previously as toy models used to
study Hawking radiation along with its
associated problems of metric back-reaction and
information loss. The Callan-Giddings-Harvey-
Strominger (CGHS) model has been proposed to
deal with the cited problems. This model as a 2D
gravity is a renormalizable theory of quantum
gravity.
On the related problem of quantization
of gravity that is, reconciling General Relativity
with the postulates of Quantum Mechanics, there
was a consideration pointed out by ‘tHooft that
at a Planckian scale our world is not 3+1
dimensional. Rather, the observable degrees of
freedom can best be described as if they were
Boolean variables defined on a two-dimensional
lattice, evolving with time.
II. 4D General Relativity As A ( 1 + 1 )
Gravity
We start from Einstein-Hilbert action in
four dimensions
Rgxd
~~4
  , (1)
then obtain from this an effective action for a ( 1
+ 1 ) gravity upon consideration of the following
fundamental metric form
)sin(
~ 22222

ddedxdxgSd ji
ij 
(2)
where ),( 10
xxgij can be thought of as the
components of ( 1 + 1 ) dimensional metric and
together with field ),( 10
xx are functions of
the coordinates ),( 10
xx . We can think of
ji
ij dxdxg as the ( 1 + 1 ) dimensional space-
time, summation over i, j is from 0 to 1.
Proceeding from this set up, we obtain

sin~ 2
geg  with g being the
determinant of the lower dimensional metric.
The scalar curvature R
~
on 4d space-time can be
expressed as
  
 22)2(
2))((2
~
eRR L
L
)(2 22 
ee L
L 
. (3)
We can ignore the last major term in (3)
involving a covariant divergence so that we can
write the effective action as
  
)2(24
)( RegxdS effEH

  
sin2))((2 22 
 eL
L . (4)
This effective action, integration over a
differential four-volume, contains the ( 1 + 1 )
gravity as we take note that all the fields
],[ ijg involved are now solely functions of
the remaining (lower dimensional) coordinates
),( 10
xx . The equations of motion we get from
this action are classically solved for with these
coordinates on a ( 1 + 1 )-space-time.
Equation of Motion for the metric
The effective variation
0
)(






eff
ji
EH
g
S


,
which is equated to zero (by variational
principles) yields the field equations for the
metric
)2(2)2()2(
2
1
jijiji TegRR 
 . (5)
We choose to work in gauge coordinates

2. 10
xxx 
, where the components of the
metric are
0  gg ,
2
2
1
egg   , (6)
where in the ),( 10
xx coordinates these metric
components are
 ggg 21100 , 01001  gg .
In gauge coordinates, Einstein’s tensor
vanish. That is, 0
2
1 )2()2(
 jiji gRR . Thus,
the energy-momentum tensor
)2(
jiT becomes a
set of constraints, where in gauge coordinates
these constraints are
   ))((2)(20 22)2(

eT
22
)(   , (7a)
   ))((2)(20 22)2(

eT
22
)(   (7b)
and
  0)( 2)2(
  geT 
. (7c)
Equation of motion for 
While we have from the variation
0






 EHS
,
the equation of motion for the field 
 ))((2)( 2)2(22
 L
L
L
L Ree 
which, in the gauge coordinates, can be written
as
))(()()( 2
   .
(8)
Metric Ansatz
For our metric ansatz (2), we will
consider only the form in which

e is the
radius r on 2-sphere.
22
re 
. (9)
With this form, the non-vanishing components of
the metric in gauge coordinates can satisfy the
preceding constraints (7a, 7b, 7c) as well as the
equation of motion for the scalar field (8)
provided that
rc
rxc
rc
e
F
F
F
1
1)](exp[
1 12

. (10)
Note that in this solution rx 1
but
1
x is
related to r by
  )exp(1)exp( 1
rcrcxc FFF  (11)
for which the effective curvature in )(effEHS is
twice the curvature on 2-sphere
22
)( /44
~
reR eff   
. (12)
For later purposes we should also write the
relations of the gauge coordinates to r and
0
x ,
given (11):
  )](exp[1)exp( 0
rxcrcxc FFF 
,
(13a)
  )](exp[1)exp( 0
rxcrcxc FFF  
(13b)
III. Causal Structure of the ( 1 + 1 )
Space-time
As thought of,
ji
ij dxdxg is the ( 1 + 1
) dimensional space-time,







 dxdx
rc
dxdxg
F
ji
ij
1
1 . (14)
We can notice then that (11) can cover
only the region at and (or) above the critical
value of r, 1rcF . That is, relation (11) is for
the region where  
rcF
1
0 ,
 0
x ,  
x and
 
x .
To cover the region where r is below
1
Fc , we can switch sign in the originally time-
like ( 1 + 1 ) space-time (14) to become space-
like. Thus, in this region ( 10  rcF ), the (
 ) metric component is
2/)1/1(2/2
 rceg F

as a
consequence of the new relations
)exp()1()exp( 1
rcrcxc FFF  , (15a)
  )](exp[1)exp( 0
rxcrcxc FFF 
(15b)
and
  )](exp[1)exp( 0
rxcrcxc FFF  
(15c)
that can cover the region 10  rcF .
In both these regions, constraints (7a to
7c) and equation (8) still hold. That is, the
physics of both regions is still described by the
same set of equations.

3. Briefly, in ( 1 + 1 )-spacetime covering
the regions for which  
rcF
1
0 , on the
paths along tconsx tan
, all future
directed events )( 0
x happen with
1
 Fcr , while emerging from r-infinity in the
past )( 0
x . That is, these events describe
objects that are infalling towards
1
 Fcr . The
paths on tconsx tan
-surface all future
directed events are those with )( r and
such describe objects that are out-going.
Carter-Penrose diagram
The causal structure can be
conveniently constructed on   plane that
contains the triangular regions

 x2 (16a)
and

 x2 . (16b)
Then note that both in 10  rcF and
 
rcF
1
0 , the ( 1 + 1 ) coordinates can
span from  to  , so we resort to re-
defining these coordinates so as to bring the
coordinate values as finite and having to
compactify the ( 1 + 1 )-space-time in the form
given by
ji
ij dxdxg
x
xx
2
1
2
)sin(





  
where

 xx tan ,

 xx tan with
2/2/   
x and
2/2/   
x and the conformal factor
is parametrized by  and  :
2
coscos
2
)sin(
1
 





  
x
xx
. (17)
Figure 1 is the Carter-Penrose diagram
that illustrates the causal structure of the ( 1 + 1 )
space-time.
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Thorlacius, Black Hole Evaporation in 1+1
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Raphael Bousso, Stephen W. Hawking, Trace
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two dimensions, arXiv:hep-th/9705236v2
G. ’t Hooft, DIMENSIONAL REDUCTION in
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qc/9310026v1
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