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  . In any dimension, the vacuum field equations in harmonic gauge for a nearly flat metric take the form ; the polarization tensor then becomes
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.”
|http://www.livingreviews.org/lrr-2005-1||© Max Planck Society and