Metric Tensor (general Relativity)

Curvature

The metric g completely determines the curvature of spacetime. According to the fundamental theorem of Riemannian geometry, there is a unique connection ∇ on any semi-Riemannian manifold that is compatible with the metric and torsion-free. This connection is called the Levi-Civita connection. The Christoffel symbols of this connection are given in terms of partial derivatives of the metric in local coordinates by the formula

The curvature of spacetime is then given by the Riemann curvature tensor which is defined in terms of the Levi-Civita connection ∇. In local coordinates this tensor is given by:

${R^\rho}_{\sigma\mu\nu} = \partial_\mu\Gamma^\rho {}_{\nu\sigma} - \partial_\nu\Gamma^\rho {}_{\mu\sigma} + \Gamma^\rho {}_{\mu\lambda}\Gamma^\lambda {}_{\nu\sigma} - \Gamma^\rho {}_{\nu\lambda}\Gamma^\lambda {}_{\mu\sigma}.$

The curvature is then expressible purely in terms of the metric and its derivatives.

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Metric Tensor (general Relativity) - Curvature