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

In any spacetime, the curvature tensor may be decomposed into a curvature scalar R, a Ricci tensor Rmn, and a remaining trace-free, conformally invariant piece, the Weyl tensor s Cmnr . In 2+1 dimensions, however, the Weyl tensor vanishes identically, and the full curvature tensor is determined algebraically by the remaining pieces:
1- Rmnrs = gmrRns + gnsRmr - gnrRms - gmsRnr - 2(gmrgns - gmsgnr)R. (1)
This means that any solution of the field equations with a cosmological constant /\,
R = 2/\g , (2) mn mn
has constant curvature: The spacetime is locally either flat (/\ = 0), de Sitter (/\ > 0), or anti-de Sitter (/\ < 0). 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  [4994] . In n dimensions, the phase space of general relativity is parametrized by a spatial metric at constant time, which has n(n - 1)/2 components, and its conjugate momentum, which adds another n(n - 1)/2 components. But n of the Einstein field equations are constraints rather than dynamical equations, and n more degrees of freedom can be eliminated by coordinate choices. We are thus left with n(n - 1)- 2n = n(n - 3) 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 n = 3, 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 gmn = jmn + hmn take the form

[]h = O(h2), @ hmn = 0, (3) mn m
where hmn = hmn - 12jmnjrshrs and indices are raised and lowered with the flat metric j . The plane wave solutions of Equation (3View Equation) are, to first order,
ik.x 2 m hmn = emne with k = 0 and k emn = 0. (4)
Choosing a second null vector nm with n .k = - 1 and a spacelike unit vector mm with k .m = n .m = 0, we can construct a (2+1)-dimensional analog of the Newman-Penrose formalism  [33] ; the polarization tensor e mn then becomes
emn = Akmkn + B(kmmn + knmm) + Cmmmn, (5)
apparently giving three propagating polarizations. There is, however, a residual symmetry: A diffeomorphism generated by an infinitesimal vector field m q with m []q = 0 preserves the harmonic gauge condition of Equation (3View Equation) while giving a “gauge transformation” r dhmn = @mqn + @nqm - jmn@rq . Writing
ik.x qm = (akm + bnm + gmm)e , (6)
it is easy to check that
demn = 2iakmkn + ig(kmmn + knmm) + ibmmmn. (7)
The excitations (5View Equation) 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.”

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