Gauge invariance/More on General relativity
Note: in this appendix, the convention of summation over lower and upper repeated indices is used.
Given an affine connection \(\mathbf{\Gamma}_\mu\) on a differential manifold \(\mathcal{M}\), 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 \(V^\lambda\) is \[(\mathrm{D}_\nu V)^\lambda:=\partial_\nu V^\lambda +\Gamma^\lambda_{\mu\nu}V^\mu.\] The curvature Riemann tensor is then defined by \[ \mathbf{R}_{\mu\nu}:=[\mathrm{D}_\mu,\mathrm{D}_\nu]=\partial_\mu \mathbf{\Gamma}_\nu- \partial_\nu \mathbf{\Gamma}_\mu+[\mathbf{\Gamma}_\mu,\mathbf{\Gamma}_\nu],\] or in component form \[ R_{\sigma\mu\nu}^\rho = \partial_\mu\Gamma^\rho_{\sigma\nu}- \partial_\nu\Gamma^\rho_{\sigma\mu}+\Gamma^\rho_{\tau\mu}\Gamma^\tau_{\sigma\nu}-\Gamma^\rho_{\tau\nu}\Gamma^\tau_{\sigma\mu}\,.\] The covariant trace of the curvature yields the Ricci tensor \[R_{\mu\nu}:= R_{\mu\nu\rho}^\rho\] In a (pseudo-) Riemaniann manifold one can use the metric tensor \(g_{\mu\nu}(x)\) (its inverse \(g^{\mu\nu}(x)\)) to take the covariant trace of the Ricci tensor \(R_{\mu\nu} \) and obtains the scalar curvature \[ R:=g^{\mu\nu}R_{\mu\nu}\]
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 \(g_{\mu\nu}\); for example, \[\Gamma^\lambda_{\mu\nu}=\frac{1}{2}\,g^{\lambda\rho}\left(\partial_{\mu}\,g_{\rho\nu}+\partial_{\nu}\,g_{\mu\rho}-\partial_{\rho}\,g_{\mu\nu}\right) \,.\] 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 \(\mathbf{g}\equiv\{g_{\mu\nu}(x)\}\) read \[ R_{\mu\nu}({\mathbf g} (x) )-{1\over2}R({\mathbf g} (x))g_{\mu\nu}=0\,. \] These equations can be derived from Einstein-Hilbert's action, \[\mathcal{S}({\mathbf g})=\int {\mathrm d}^4x\, (-g(x) )^{1/2} R ({\mathbf g} (x) ),\] where \(g(x)\) is the determinant of the metric tensor and we assumed a four dimensional pseudo-Riemann manifold with metric signature \((+,-,-,-)\). These equations can be generalized to include a cosmological constant and matter fields.
