This is a short note concerning the so called Einstein vacuum field equations where is nothing but the Einstein curvature tensor defined in this topic. I have not discussed either Einstein field equations or Einstein vacuum field equations yet, however, we can adop the following equation
to be the Einstein field equations (EFE) where is the Ricci curvature tensor, the scalar curvature, the metric tensor, is the cosmological constant, is the gravitational constant, the speed of light, and the stress-energy tensor.
One can write the EFE in a more compact form by defining the Einstein tensor
which is a symmetric second-rank tensor that is a function of the metric. The EFE can then be written as
where the cosmological term has been absorbed into the stress-energy tensor as dark energy.
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 full field equations, the vacuum equations can be written as
Taking the trace of this (contracting with ) and using the fact that , we get
Substituting back, we get an equivalent form of the vacuum field equations
The above equation is frequently used in the literature, sometimes, it is called the Einstein vacuum field equations, for example, we refer the reader to introduction part of the following paper due to James Isenberg. In the case of nonzero cosmological constant, the equations are
yielding the equivalent form
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.