##2+1##-dimensional Einstein gravity has no(adsbygoogle = window.adsbygoogle || []).push({}); localdegrees of freedom. This can be proved in two different ways:

1. In ##D##-dimensional spacetime, a symmetric metric tensorappears to have##\frac{D(D+1)}{2}## degrees of freedom satisfying ##\frac{D(D+1)}{2}##apparently independentEinstein field equations. However, there is a set of ##D## constraints on the equations due to the invariance of the equations under diffeomorphisms, and a second set of ##D## constraints due to the conservation of the stress-energy tensor. Therefore, there are really only

$$\frac{D(D+1)}{2} - D - D = \frac{D(D-3)}{2}$$

degrees of freedom of the metric tensor satisfying ##\frac{D(D-3)}{2}## independent Einstein field equations.

2. In the ADM formulation in ##D##-dimensional spacetime, the metric induced on the spacelike hypersurfacesappears to have##\frac{D(D-1)}{2}## degrees of freedom. However, there is a set of ##D## constraints due to the ##D## Lagrangian multipliers in the Hamiltonian. Therefore, there are really only

$$\frac{D(D-1)}{2} - D = \frac{D(D-3)}{2}$$

degrees of freedom of the metric tensor.

The metric tensor of a manifold encodes information about the infinitesimal distance between nearby points on the manifold, so the ##\frac{D(D-3)}{2}## degrees of freedom are alllocaldegrees of freedom.

Therefore, it is said that ##2+1##-dimensional Einstein gravity is trivial locally.

But what does itmeanto say that ##2+1##-dimensional Einstein gravity isnon-trivial globally?

Why is the wordtopologicalused to describe ##2+1##-dimensional Einstein gravity?

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# A ##2+1##-D Einstein gravity is topological and only non-trivial locally

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