Einstein-Hilbert Action and Boundary Terms

The Einstein–Hilbert action packages Einstein dynamics into a covariant variational principle, but its apparent simplicity hides the most important technical issue: curvature contains second derivatives of the metric. In a finite region those derivatives become boundary variations. This chapter derives the bulk equation without losing them, completes non-null Dirichlet boundaries, treats null segments and joints, and states exactly when an independent-connection variation is equivalent.

Helpful background. The chapter uses special-relativistic field theory, tensor fields, presymplectic potentials and variational boundary terms, curvature, hypersurface geometry, and classical field actions.

Enter the variational problem

For a metric and matter fields,

$$S=\frac{1}{16\pi G}\int_{\mathcal M}\sqrt{-g}(R-2\Lambda)d^4x +S_{\mathrm m}+S_{\partial\mathcal M}.$$

The bulk normalization, stress-tensor sign, and boundary completion are one coupled convention set. Varying the inverse metric with

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

yields $G_{ab}+\Lambda g_{ab}=8\pi GT_{ab}$ only when the surface divergence has been made to vanish by compact support or by boundary terms consistent with the chosen data.

Reall derives the metric identities, matter variation, and normalized bulk equation in a compact sequence Reall 2013, §§ 10.2–10.3, pp. 116–120, Eqs. (10.14)–(10.34), PDF. Lehner and collaborators give the non-null and null segment terms, the null-joint gluing rules, and the reparameterization counterterm in Lehner et al. 2016, §§ II B–C and II H, pp. 6–27, and Appendices B–C, pp. 42–44, PDF.

Six pages in derivational order

1. The Einstein-Hilbert Action explains why the Ricci scalar is the minimal nontrivial local metric scalar at second derivative order and why its highest derivatives live at the boundary.
2. Metric Variation Identities derives inverse-metric, determinant, connection, Ricci, and Palatini identities without discarding the divergence.
3. Deriving the Einstein Equation from the Action combines gravitational, cosmological, and matter variations and checks the $8\pi G$ factor against Poisson gravity.
4. The Gibbons-Hawking-York Boundary Term declares $K_{ab}=h_a{}^ch_b{}^d\nabla_cn_d$ and cancels the normal derivative of $\delta h_{ab}$ under non-null Dirichlet data.
5. Null Boundaries, Corners, and Action Additivity replaces unit-normal formulas with generator nonaffinity, screen expansion, logarithmic joints, an optional reparameterization counterterm, and the limits of ordinary additivity.
6. Metric and Palatini Variations Compared varies a torsion-free connection independently and isolates the dimension, matter, torsion, and boundary assumptions needed for equivalence.

The order is cumulative. Null joint signs make sense only after the non-null boundary problem has been fixed, and the Palatini comparison is meaningful only after the metric variation’s connection dependence is visible.

Bulk identities and boundary data

The decisive identity is

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

The divergence contains a normal derivative of the metric variation. Fixing an induced metric at a boundary makes $\delta h_{ij}$ zero there but does not fix its normal slope. Adding

$$S_{\mathrm{GHY}} =\frac{1}{8\pi G}\int_{\Sigma}\varepsilon\sqrt{|h|}\,K\,d^3y$$

with outward unit normal, $\varepsilon=n^2$, and the declared $K_{ab}$ convention cancels that slope term. Joints are then required when the normal jumps between segments.

Null hypersurfaces are structurally different. Their normal is a tangent generator $k^a$, the induced metric is degenerate, and $k^b\nabla_bk^a=\kappa k^a$ depends on parameterization. Segment and joint terms must be varied and glued as a set. A logarithmic expansion counterterm can remove generator-reparameterization dependence, but introduces a reference scale that must be stated. Lehner et al.’s null-joint prescriptions restore composition for the spacelike-joint gluing class they analyze when common null tangents are normalized consistently; they do not remove the ordinary-additivity obstruction at timelike joints. Brill and Hayward instead retain the finite residue as an action assigned to the identification surface, which they call generalized additivity Brill and Hayward 1994, §§ 1 and 5, pp. 1–14, Eqs. (14)–(16), Open PDF.

Three boundaries of the result

First, a stationary Lorentzian action is not generally minimized. “Least action” should not replace the Euler–Lagrange statement. Second, a differentiable action does not prove that the resulting initial-boundary value problem is well posed. Third, a classical composition rule for on-shell actions does not define a gravitational path integral.

Likewise, metric–Palatini equivalence is conditional. With nondegenerate metrics in $D>2$, a torsion-free independent connection, Einstein–Hilbert dependence, and matter with no connection hypermomentum, the connection becomes Levi-Civita. Allowing an unrestricted connection in pure Einstein–Hilbert gravity instead leaves a projective gauge field without new physical content Dadhich and Pons 2012, §§ 1–2 and 4.1, pp. 2–7, Open PDF. Connection-dependent matter must respect the resulting projective consistency condition or the theory must be modified Sotiriou and Liberati 2007, § 4.2, pp. 18–20, Eqs. (40)–(47), Open PDF; $f(R)$ dynamics and two-dimensional degeneracy also change the conclusion.

Review the chapter

Why is imposing only $\delta h_{ij}=0$ on a smooth non-null boundary sufficient after adding the GHY term but insufficient for the bare Einstein–Hilbert action?

Solution

The bare curvature variation contains $n^a\nabla_a\delta h_{ij}$. Dirichlet data fix the value of $h_{ij}$, not the normal derivative of its variation, so that term remains. Varying the GHY integral supplies the opposite normal-derivative term. Their sum leaves a boundary contribution proportional to $(K_{ij}-Kh_{ij})\delta h^{ij}$, which vanishes when the induced metric is fixed.

Onward connection

The next chapter, Diffeomorphisms, Symmetry, and Conservation, interprets infinitesimal diffeomorphisms as field variations, derives the off-shell Noether identity, and distinguishes gauge transformations from boundary symmetries. Those arguments depend on having retained the surface potential here rather than treating every divergence as zero.

References

• Brill, Dieter R., and Geoffrey Hayward. “Is the Gravitational Action Additive?” Physical Review D 50 (1994): 4914–4919. DOI; arXiv.
• Dadhich, Naresh, and Josep M. Pons. “On the Equivalence of the Einstein–Hilbert and the Einstein–Palatini Formulations of General Relativity for an Arbitrary Connection.” General Relativity and Gravitation 44 (2012): 2337–2352. DOI; arXiv.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94 (2016): 084046. DOI; arXiv.
• Reall, Harvey S. General Relativity. Cambridge Part III lecture notes (2013). PDF.
• Sotiriou, Thomas P., and Stefano Liberati. “Metric-Affine $f(R)$ Theories of Gravity.” Annals of Physics 322 (2007): 935–966. DOI; arXiv.