Feynman Path Integral Formulation

22 1 Continuum Formulation
Furthermore, in empty space T μν = 0, which then implies the vanishing of Riemann
there
R μνρσ = 0 (1.114)
As a result in three dimensions classical spacetime is locally flat everywhere outside
a source, gravitational fields do not propagate outside matter, and two bodies cannot
experience any gravitational force: they move uniformly on straight lines.
There cannot be any gravitational waves either: the Weyl tensor which carries information
about gravitational fields not determined locally by matter, vanishes identically
in three dimensions. One further surprising (and disappointing) conclusion
from the previous arguments is that black holes cannot exist in three dimensions,
as spacetime is flat outside matter, which always allows light emitted from a star
to escape to infinity. One interesting case though is three-dimensional anti-DeSitter
space, with a scaled cosmological constant λ = −1/ξ 2 . There one can show that
objects which could be described as black holes exist, with a black hole horizon
in the non-rotating case at r 0 = ξ √ MG, where M is the mass of the collapsed object
(Banados Teitelboim and Zanelli, 1992). Note that in three dimensions G has
dimensions of a length, so that the product MG ends up being dimensionless. The
scale of the horizon is therefore supplied by the scale ξ .
What seems rather puzzling at first is that Newtonian theory seems to make perfect
sense in d = 3. The Newtonian potential is non-vanishing and grows logarithmically
with distance,
∫
V (r) ∝ G d 2 ke ix·k /k 2 ∼ G logr . (1.115)
This can only mean that the Newtonian theory is not recovered in the weak field limit
of the relativistic theory (Deser, Jackiw and Templeton, 1982; Giddings, Abbott and
Kuchar, 1984).
To see this explicitly, it is sufficient to consider the trace-reversed form of the
field equations,
(
R μν = 8πG T μν − 1 )
d − 2 g μν T , (1.116)
with T = T λ λ , in the weak field limit. In the linearized theory, with h μν = g μν −η μν ,
and in the Hilbert-DeDonder gauge
one obtains the wave equation
✷h μν
∇ λ h λ μ − 1 2 ∇ μh λ λ = 0 , (1.117)
= −16πG
(
τ μν − 1 )
d − 2 η μν τ
, (1.118)
with τ μν the linearized stress tensor. After neglecting the spatial components of τ μν
in comparison to the mass density τ 00 , and assuming that the fields are quasi-static,
one obtains a Poisson equation for h 00 ,