Einstein–Hilbert action

Remarks

Let $g_t$ be a variation of the metric supported in the interior of $K$. Write $\delta g^{\mu\nu} = \frac{d}{dt}\big|_{t=0} g_t^{\mu\nu}$. Then $$\frac{d}{dt}\Big|_{t=0} S_{EH}[g_t] = \frac{1}{2\kappa} \int_K \Big(R_{\mu\nu} - \tfrac12 R\, g_{\mu\nu}\Big)\, \delta g^{\mu\nu} \sqrt{-\det g}\; d^4x,$$ where $R_{\mu\nu}$ is the Ricci tensor; the term with the variation of $R_{\mu\nu}$ is a divergence and drops out. So $g$ is a critical point of $S_{EH}$ for all such variations if and only if $R_{\mu\nu} - \tfrac12 R g_{\mu\nu} = 0$, the vacuum Einstein field equations. Example: Minkowski space $\R^4$ with $g = -dt^2 + dx^2 + dy^2 + dz^2$ is flat, so its Ricci tensor vanishes, and it is a critical point. Non-example: the Euclidean metric $dt^2 + dx^2 + dy^2 + dz^2$ is not Lorentzian, so the definition above does not apply to it; the analogous Riemannian functional $\int R \sqrt{\det g}\, d^4x$ is a different object. With a cosmological constant $\Lambda$ one replaces $R$ by $R - 2\Lambda$. Adding a matter action $S_M$ and setting $T_{\mu\nu} = -\frac{2}{\sqrt{-\det g}} \frac{\delta S_M}{\delta g^{\mu\nu}}$ gives $R_{\mu\nu} - \tfrac12 R g_{\mu\nu} = \kappa T_{\mu\nu}$. On a region with boundary, one adds the Gibbons–Hawking–York boundary term to obtain a well-posed variational problem with the metric fixed on the boundary.