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
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.”
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