Einstein field equations

Remarks

Taking the trace with respect to $g$ in dimension four shows that the vacuum equations are equivalent to $\mathrm{Ric}_g = 0$. In local coordinates the equations read $R_{\mu\nu} - \tfrac{1}{2} R g_{\mu\nu} + \Lambda g_{\mu\nu} = \kappa T_{\mu\nu}$, ten equations for the ten components of $g$.

Example: Minkowski space, $\R^4$ with $g = -dt^2 + dx^2 + dy^2 + dz^2$, is flat, so $\mathrm{Ric}_g = 0$ and it solves the vacuum equations.

Non-example: de Sitter space has $\mathrm{Ric}_g = \Lambda_0 g$ with a constant $\Lambda_0 > 0$, so it does not solve the vacuum equations. It solves the equations with $T = 0$ and cosmological constant $\Lambda = \Lambda_0$.

The left side has zero divergence by the contracted Bianchi identity, which forces the conservation law $\nabla^\mu T_{\mu\nu} = 0$. Yvonne Choquet-Bruhat proved that the vacuum equations have a well-posed initial value problem for suitable initial data. The Riemannian analogue of $\mathrm{Ric}_g = \lambda g$ defines an Einstein manifold.