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Four-dimensional General Relativity reduced as (1+1) gravity
Ferdinand Joseph P. Roa
Independent physics researcher
rogueliknayan@yahoo.com
Abstract
This is an exploratory paper that tackles in brief dimensional reduction of gravity starting with Einstein-Hilbert
action originally in its 3+1 form though without matter and cosmological terms. The reduction is of Kaluza-Klein
type, wherein we are to define a starting metric for the fundamental line element given for the 3+1 space-time and
such space-time encodes the metric for the 1+1 space-time of interest.
Keywords: action, metric
1 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 4-dimensional (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 2-dimensional (2D) gravity coupled to a scalar field or what is
known as the Einstein-Scalar system.
2D 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.
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.
2 Four-dimensional General Relativity as a (1+1) gravity
We start from Einstein-Hilbert action in four dimensions
∫ 𝑑4
𝑥 √−𝑔̃ 𝑅̃ (1)
then obtain from this an effective action for a (1+1) gravity upon consideration of the following
fundamental metric form
𝑑𝑆̃2
= 𝑔𝑖𝑗 𝑑𝑥 𝑖
𝑑𝑥 𝑗
+ 𝑒2𝛽𝜙( 𝑑𝜃2
+ 𝑠𝑖𝑛2
𝜃 𝑑𝜓2) (2)
where 𝑔𝑖𝑗(𝑥0
, 𝑥1
) can be thought of as the components of (1+1) dimensional metric and together with the
scalar field 𝜙(𝑥0
, 𝑥1
) are functions of the coordinates (𝑥0
, 𝑥1
). We can think of 𝑔𝑖𝑗(𝑥0
,𝑥1
) as the metric
of the (1+1)-dimensional space-time, summation over i, j is from 0 to 1.
Proceeding from this set up, we obtain √−𝑔̃ = 𝑒2𝛽𝜙
√−𝑔 𝑠𝑖𝑛 𝜃 with 𝑔 being the determinant of
the lower dimensional metric. The scalar curvature 𝑅̃ on 4d space-time can be expressed as
𝑅̃ = 𝑅(2)
+ 2𝛽2( 𝜕𝐿 𝜙)( 𝜕 𝐿
𝜙) + 2𝑒−2𝛽𝜙
− 2𝑒−2𝛽𝜙
∇ 𝐿( 𝜕 𝐿
𝑒2𝛽𝜙) (3)
We can ignore the last major term in (3) involving a covariant divergence so that we can write the
effective action as
𝑆 𝐸𝐻 = ∫ 𝑑4
𝑥 √−𝑔 𝑒2𝛽𝜙( 𝑅(2)
+ 2𝛽2( 𝜕𝐿 𝜙)( 𝜕 𝐿
𝜙) + 2𝑒−2𝛽𝜙) 𝑠𝑖𝑛 𝜃 (4)
This effective action, integration over a differential four-volume, contains the (1+1) gravity as we take note
that all the fields ( 𝑔𝑖𝑗, 𝜙 ) involved are now solely functions of the remaining (lower dimensional)
coordinates (𝑥0
, 𝑥1
). The equations of motion we get from this action are classically solved for with these
coordinates on a (1+1) space-time.

3 Equation ofMotion for the metric
The effective variation
(
𝛿𝑆 𝐸𝐻
𝛿𝑔 𝑗𝑖
)
(𝑒𝑓𝑓)
= 0 (5)
which is equated to zero (by variational principles) yields the field equations for the metric
𝑅𝑗𝑖
(2)
−
1
2
𝑅(2)
𝑔𝑗𝑖 = 𝑒−2𝛽𝜙
𝑇𝑗𝑖
(2)
(6)
We choose to work in gauge coordinates, 𝑥±
= 𝑥0
± 𝑥1
, where the components of the metric
are
𝑔++ = 𝑔−− = 0, 𝑔+− = 𝑔−+ = −
1
2
𝑒2𝜌 (7)
where in the (𝑥0
,𝑥1
) coordinates these metric components are
𝑔00 = −𝑔11 = 2𝑔+−, 𝑔01 = 𝑔10 = 0 (8)
In gauge coordinates, Einstein’s tensor vanishes. That is, 𝑅𝑗𝑖
(2)
−
1
2
𝑅(2)
𝑔𝑗𝑖 = 0. Thus, the
energy-momentum tensor 𝑇𝑗𝑖
(2)
becomes a set of constraints, where in gauge coordinates these constraints
are
𝑇++
(2)
= 0 = 2𝑒2𝛽𝜙( 𝛽( 𝜕+
2 𝜙) − 2𝛽( 𝜕+ 𝜌)( 𝜕+ 𝜙) + 𝛽2(𝜕+ 𝜙)2) (9)
𝑇−−
(2)
= 0 = 2𝑒2𝛽𝜙( 𝛽( 𝜕−
2
𝜙) − 2𝛽( 𝜕− 𝜌)( 𝜕− 𝜙) + 𝛽2
(𝜕− 𝜙)2) (10)
𝑇+−
(2)
= 0 = −( 𝜕+ 𝜕− 𝑒2𝛽𝜙
− 𝑔+−) (11)
4 Equation of motion for 𝜙
While we have from the variation
(
𝛿𝑆 𝐸𝐻
𝛿𝜙
) = 0 (12)
the equation of motion for the field 
    ∇ 𝐿 ( 𝜕 𝐿
𝑒2𝛽𝜙) = 𝑒2𝛽𝜙 ( 𝑅(2)
+ 2𝛽2( 𝜕𝐿 𝜙)( 𝜕 𝐿
𝜙)) (13)
which, in the gauge coordinates, can be written as
− 𝜕+ 𝜕− 𝜌 = 𝛽( 𝜕+ 𝜕− 𝜙) + 𝛽2( 𝜕+ 𝜙)( 𝜕− 𝜙) (14)
5 Metric solution
For our metric ansatz (2), we will consider only the form in which 𝑒 𝛽𝜙 is the radius r on 2-sphere.
𝑒2𝛽𝜙
= 𝑟2
(15)
With this form, the non-vanishing components of the metric in gauge coordinates can satisfy the preceding
constraints (9, 10, 11) as well as the equation of motion for the scalar field (14) provided that
𝑒2𝜌 =
1
𝑐 𝐹 𝑟
𝑒𝑥𝑝( 𝑐 𝐹 ( 𝑥1 − 𝑟)) = 1 −
1
𝑐 𝐹 𝑟
(16)
Note that in this solution 𝑥1
≠ 𝑟 but 𝑥1
is related to r by
𝑒𝑥𝑝( 𝑐 𝐹 𝑥1) = ( 𝑐 𝐹 𝑟 − 1) 𝑒𝑥𝑝( 𝑐 𝐹 𝑟) (17)
for which the effective curvature in 𝑆 𝐸𝐻 is twice the curvature on 2-sphere
𝑅̃(𝑒𝑓𝑓) = 4𝑒−2𝛽𝜙
= 4/𝑟2
(18)
For later purposes we should also write the relations of the gauge coordinates to r and 𝑥0
, given (11)
𝑒𝑥𝑝( 𝑐 𝐹 𝑥+) = ( 𝑐 𝐹 𝑟 − 1) 𝑒𝑥𝑝( 𝑐 𝐹( 𝑥0
+ 𝑟)) (19)
𝑒𝑥𝑝(−𝑐 𝐹 𝑥−) = ( 𝑐 𝐹 𝑟 − 1) 𝑒𝑥𝑝(−𝑐 𝐹( 𝑥0
− 𝑟)) (20)
References
[1] Curtis G. Callan, Jr., Steven B. Giddings, Jeffrey A. Harvey, Andrew Strominger, Evanescent Black
Holes, arXiv:hep-th/9111056 v1
[2] Jorge G. Russo, Leonard Susskind and L´arus Thorlacius, Black Hole Evaporation in 1+1 Dimensions,
arXiv:hep-th/9201074 v1
[3] Raphael Bousso, Stephen W. Hawking, Trace anomaly of dilaton coupled scalars in
two dimensions, arXiv:hep-th/9705236v2
[4] G.’t Hooft, DIMENSIONAL REDUCTION in QUANTUM GRAVITY, arXiv:gr-qc/9310026v1
[5] Kaluza-Klein Theory, http://faculty.physics.tamu.edu/pope/ihplec.pdf

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