Cosmological constant

Remarks

Writing the term on the right gives $G_{\mu\nu} = \kappa (T_{\mu\nu} + T^{\Lambda}_{\mu\nu})$ with $T^\Lambda = -\frac{\Lambda}{\kappa} g$. This is the stress–energy tensor of a perfect fluid with energy density $\rho_\Lambda = \Lambda/\kappa$ and pressure $p_\Lambda = -\rho_\Lambda$, the vacuum energy. Example: de Sitter space with metric $-dt^2 + e^{2Ht}(dx^2 + dy^2 + dz^2)$, in units with $c = 1$ and $H > 0$, satisfies $\mathrm{Ric} = 3H^2 g$ and $R = 12H^2$, so $G = -3H^2 g$; it solves the equations with $T = 0$ and $\Lambda = 3H^2 > 0$. Non-example: a nonconstant function $\Lambda(x)$ in place of $\Lambda$ is not allowed when $\nabla^\mu T_{\mu\nu} = 0$: taking the divergence and using $\nabla^\mu G_{\mu\nu} = 0$ gives $\partial_\nu \Lambda = 0$.