More on General relativity - davidar/scholarpedia GitHub Wiki

Note: in this appendix, the convention of summation over lower and upper repeated indices is used.

Given an affine connection <math>\mathbf{\Gamma}_\mu</math> on a differential manifold <math>\mathcal{M}\ ,</math> the parallel transport of differentiable tensor fields can be locally defined with the use of covariant derivatives. For example, the form of the covariant derivative acting on a vector field <math>V^\lambda</math> is

<math>(\mathrm{D}_\nu V)^\lambda:=\partial_\nu V^\lambda +\Gamma^\lambda_{\mu\nu}V^\mu.</math>

The curvature Riemann tensor is then defined by

<math> \mathbf{R}_{\mu\nu}:=[\mathrm{D}_\mu,\mathrm{D}_\nu]=\partial_\mu \mathbf{\Gamma}_\nu- \partial_\nu \mathbf{\Gamma}_\mu+[\mathbf{\Gamma}_\mu,\mathbf{\Gamma}_\nu],</math>

or in component form

<math> R_{\sigma\mu\nu}^\rho = \partial_\mu\Gamma^\rho_{\sigma\nu}- \partial_\nu\Gamma^\rho_{\sigma\mu}+\Gamma^\rho_{\tau\mu}\Gamma^\tau_{\sigma\nu}-\Gamma^\rho_{\tau\nu}\Gamma^\tau_{\sigma\mu}\,.</math>

The covariant trace of the curvature yields the Ricci tensor

<math>R_{\mu\nu}:= R_{\mu\nu\rho}^\rho</math>

In a (pseudo-) Riemaniann manifold one can use the metric tensor <math>g_{\mu\nu}(x)</math> (its inverse <math>g^{\mu\nu}(x)</math>) to take the covariant trace of the Ricci tensor <math>R_{\mu\nu} </math> and obtains the scalar curvature

<math> R:=g^{\mu\nu}R_{\mu\nu}</math>

In the special case of the Levi-Civita (metric, torsion-less) connection on a (pseudo-) Riemaniann manifold, the connection and, thus, the Riemann, Ricci tensors and the scalar curvature can be entirely expressed in terms of the metric tensor <math>g_{\mu\nu}\ ;</math> for example,

<math>\Gamma^\lambda_{\mu\nu}=\frac{1}{2}\,g^{\lambda\rho}\left(\partial_{\mu}\,g_{\rho\nu}+\partial_{\nu}\,g_{\mu\rho}-\partial_{\rho}\,g_{\mu\nu}\right) \,.</math>

The Ricci tensor then becomes a symmetric tensor.

In the absence of matter, the equations of the classical motion of General relativity for the metric tensor <math>\mathbf{g}\equiv\{g_{\mu\nu}(x)\}</math> read

<math> R_{\mu\nu}({\mathbf g} (x) )-{1\over2}R({\mathbf g} (x))g_{\mu\nu}=0\,. </math>

These equations can be derived from Einstein-Hilbert's action,

<math>\mathcal{S}({\mathbf g})=\int {\mathrm d}^4x\, (-g(x) )^{1/2} R

({\mathbf g} (x) ),</math> where <math>g(x)</math> is the determinant of the metric tensor and we assumed a four dimensional pseudo-Riemann manifold with metric signature <math>(+,-,-,-)\ .</math> These equations can be generalized to include a cosmological constant and matter fields.