Einstein Field Equations - Vacuum Field Equations

Vacuum Field Equations

If the energy-momentum tensor is zero in the region under consideration, then the field equations are also referred to as the vacuum field equations. By setting in the trace-reversed field equations, the vacuum equations can be written as

In the case of nonzero cosmological constant, the equations are

The solutions to the vacuum field equations are called vacuum solutions. Flat Minkowski space is the simplest example of a vacuum solution. Nontrivial examples include the Schwarzschild solution and the Kerr solution.

Manifolds with a vanishing Ricci tensor, are referred to as Ricci-flat manifolds and manifolds with a Ricci tensor proportional to the metric as Einstein manifolds.