Vacuum spacetime ($T_{ab}=0$)

A vacuum spacetime has vanishing matter stress-energy tensor. In $D>2$ dimensions, tracing the Vacuum Einstein equations gives $R=2D\Lambda /(D-2)$ and hence $R_{ab}=2\Lambda g_{ab}/(D-2)$; in four dimensions this reduces to $R_{ab}=\Lambda g_{ab}$. With zero cosmological constant it is Ricci flat, although its full Riemann curvature tensor can be nonzero: a vacuum Kerr black hole and vacuum gravitational waves illustrate that distinction.