###
2.1
Why (2+1)-dimensional gravity is simple

In any spacetime, the curvature tensor may be decomposed into a
curvature scalar
, a Ricci tensor
, and a remaining trace-free, conformally invariant piece, the
Weyl tensor
. In 2+1 dimensions, however, the Weyl tensor vanishes
identically, and the full curvature tensor is determined
algebraically by the remaining pieces:
This means that any solution of the field equations with a
cosmological constant
,
has constant curvature: The spacetime is locally either flat (), de Sitter (), or anti-de Sitter (). Physically, a (2+1)-dimensional spacetime has no local degrees
of freedom: There are no gravitational waves in the classical
theory, and no propagating gravitons in the quantum theory.
This absence of local degrees of freedom can be
verified by a simple counting argument
[49, 94]
. In
dimensions, the phase space of general relativity is
parametrized by a spatial metric at constant time, which has
components, and its conjugate momentum, which adds another
components. But
of the Einstein field equations are constraints rather than
dynamical equations, and
more degrees of freedom can be eliminated by coordinate choices.
We are thus left with
physical degrees of freedom per spacetime point. In four
dimensions, this gives the usual four phase space degrees of
freedom, two gravitational wave polarizations and their conjugate
momenta. If
, there are no local degrees of freedom.

It is instructive to examine this issue in the
weak field approximation
[58]
. In any dimension, the vacuum field equations in harmonic gauge
for a nearly flat metric
take the form

where
and indices are raised and lowered with the flat metric
. The plane wave solutions of Equation (3) are, to first order,
Choosing a second null vector
with
and a spacelike unit vector
with
, we can construct a (2+1)-dimensional analog of the
Newman-Penrose formalism
[33]
; the polarization tensor
then becomes
apparently giving three propagating polarizations. There is,
however, a residual symmetry: A diffeomorphism generated by an
infinitesimal vector field
with
preserves the harmonic gauge condition of Equation (3) while giving a “gauge transformation”
. Writing
it is easy to check that
The excitations (5) are thus pure gauge, confirming the absence of propagating
degrees of freedom.
Fortunately, while this feature makes the
theory simple, it does not quite make it trivial. A flat
spacetime, for instance, can always be described as a collection
of patches, each isometric to Minkowski space, that are glued
together by isometries of the flat metric; but the gluing is not
unique, and may be dynamical. This picture leads to the
description of (2+1)-dimensional gravity in terms of “geometric
structures.”