The Einstein-Hilbert Action

The Einstein–Hilbert action is the lowest-derivative local scalar action built from a Lorentzian metric that produces nontrivial gravitational dynamics in four dimensions. Its Ricci scalar contains second derivatives of the metric, yet those derivatives combine into a boundary divergence; the bulk Euler–Lagrange equation is second order. This economy is powerful, but a finite region requires explicit boundary data and a boundary completion.

Required background. We use presymplectic potentials and variational boundary terms, the Riemann and Ricci tensors, and hypersurface normals and projectors.

Helpful background. Familiarity with special-relativistic field theory, tensor fields, and classical field actions helps separate bulk equations from boundary conditions.

The gravitational action and its normalization

For a four-dimensional region $\mathcal M$,

$$S_{\mathrm{grav}}[g] =\frac{1}{16\pi G} \int_{\mathcal M}d^4x\,\sqrt{-g}\,(R-2\Lambda).$$

The measure $d^4x\sqrt{-g}$ and scalar $R$ make the integral invariant under orientation-preserving coordinate changes. For a metric with fixed dimension and Lorentzian signature, a derivative-free natural scalar can only be a constant, so its action contribution is the cosmological term. The factor $1/(16\pi G)$ is fixed by matching the resulting field equation to Newtonian gravity.

In units with $c=1$ but $G$ explicit, $G$ has dimension $L^2$ in action units with $\hbar=1$, $R$ and $\Lambda$ have dimension $L^{-2}$, and $d^4x\sqrt{-g}$ has dimension $L^4$. The action is dimensionless. This check catches missing factors of $G$ or an incorrectly dimensioned cosmological parameter.

Reall introduces the action with this normalization and derives its metric variation Reall 2013, § 10.2, pp. 116–119, Eqs. (10.8)–(10.26), PDF.

Why the Ricci scalar is minimal

A local diffeomorphism-covariant metric Lagrangian must be a scalar built naturally from the metric and its derivatives. At a chosen event, any Lorentzian metric can be carried to the same Minkowski matrix by a linear change of frame. Covariance therefore prevents a derivative-free scalar from depending on particular metric components: after the dimension and signature are fixed, only a constant remains, giving the cosmological term.

Normal coordinates make the first metric derivatives vanish at that event. A purported natural scalar made only from $g_{ab}$ and $\partial_cg_{ab}$ would therefore vanish there in every coordinate system, so first derivatives alone supply no nontrivial term. At second derivative order, the tensorial information is curvature. The algebraic symmetries of $R_{abcd}$ make every complete contraction linear in curvature proportional to $g^{ac}g^{bd}R_{abcd}=R$, up to the curvature-sign convention. Adding a total divergence changes the boundary action rather than the interior Euler–Lagrange coefficient.

This is a local naturality argument under a fixed field content and derivative expansion, not the full Lovelock uniqueness theorem. It explains why $R$ is the first nontrivial scalar term; it does not classify every generally covariant action.

“Minimal” is not “uniquely compelled by covariance.” Scalars such as $R^2$, $R_{ab}R^{ab}$, and $R_{abcd}R^{abcd}$ are also covariant, but generally produce higher-derivative metric equations or, in special combinations and dimensions, boundary/topological or Lovelock contributions. Extra fields allow further terms. Selecting Einstein–Hilbert therefore uses a derivative-order and field-content restriction in addition to covariance.

Second derivatives and the boundary

Expanding $R$ in a coordinate chart gives the exact identity

$$\sqrt{-g}\,R =\sqrt{-g}\,g^{ab} \left( \Gamma^c{}_{ad}\Gamma^d{}_{bc} -\Gamma^c{}_{ab}\Gamma^d{}_{cd} \right) +\partial_c V^c,$$

with

$$V^c=\sqrt{-g}\left(g^{ab}\Gamma^c{}_{ab} -g^{ac}\Gamma^b{}_{ab}\right).$$

To obtain it, first collect the two $\partial\Gamma$ terms in the Ricci scalar into $\partial_cV^c$, then differentiate $\sqrt{-g}g^{ab}$ in the product rule. Metric compatibility turns the remaining terms into the displayed quadratic combination. The terms with second derivatives of $g_{ab}$ are therefore confined to the divergence. Neither the connection-quadratic density nor $V^c$ is separately a covariant tensor density; their sum has the required covariance. This distinction is why dropping the divergence changes the boundary problem even though it preserves the interior equations.

Consequently, varying the action gives

$$\delta(\sqrt{-g}R) =\sqrt{-g}\,G_{ab}\,\delta g^{ab} +\sqrt{-g}\,\nabla_a\Theta^a(g,\delta g).$$

The bulk equation contains at most second metric derivatives. The boundary term contains normal derivatives of $\delta g_{ab}$, so fixing only the induced metric does not make the bare action differentiable. One may use compactly supported variations, fix excessive boundary data, or add the Gibbons–Hawking–York term for a non-null Dirichlet boundary. These are different variational problems and must not be interchanged silently.

Matter and the bulk equation

Add a generally covariant matter action $S_{\mathrm m}[g,\psi]$ and define

$$T_{ab}=-\frac{2}{\sqrt{-g}} \frac{\delta S_{\mathrm m}}{\delta g^{ab}}.$$

For compactly supported metric variations, or after a suitable boundary completion,

$$\delta(S_{\mathrm{grav}}+S_{\mathrm m}) =\frac{1}{16\pi G}\int\sqrt{-g} (G_{ab}+\Lambda g_{ab})\delta g^{ab} -\frac12\int\sqrt{-g}\,T_{ab}\delta g^{ab}.$$

Stationarity for arbitrary interior $\delta g^{ab}$ yields

$$G_{ab}+\Lambda g_{ab}=8\pi G\,T_{ab}.$$

This derivation establishes the bulk dynamics. It does not select the allowed boundary geometry, establish a path integral, or prove that every solution is globally admissible.

Exercises

Explain why $\sqrt{-g}\,R$ may contain second metric derivatives even though the Einstein equation contains no derivatives higher than second order.

Solution

The curvature scalar contains $\partial\Gamma\sim\partial^2g$ and $\Gamma\Gamma\sim(\partial g)^2$. In the density $\sqrt{-g}R$, the second-derivative terms can be reorganized into a total divergence. Variation of that divergence contributes at the boundary, while the interior Euler–Lagrange coefficient is the Einstein tensor and contains at most second derivatives. Discarding the divergence is harmless only for compactly supported variations or after the boundary problem has been completed.

References

• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.