The Gibbons-Hawking-York Boundary Term

The Einstein–Hilbert action is not differentiable under ordinary Dirichlet boundary data by itself. Its variation contains a normal derivative of the metric variation, while fixing the induced boundary metric sets only $\delta h_{ab}=0$. On a smooth timelike or spacelike boundary, the Gibbons–Hawking–York (GHY) term cancels that normal-derivative piece and leaves a well-defined metric variational problem.

Required background. We use variational boundary terms, curvature, and hypersurface projectors and normals.

Helpful background. Special relativity, tensor fields, and classical field actions provide local and variational comparisons.

Boundary and extrinsic-curvature conventions

Let a smooth non-null boundary segment $\Sigma$ have outward-pointing unit normal $n^a$ and

$$\varepsilon=n^an_a=\begin{cases} +1,&\Sigma\text{ timelike},\\ -1,&\Sigma\text{ spacelike}. \end{cases}$$

The induced metric and the convention used here for extrinsic curvature are

$$h_{ab}=g_{ab}-\varepsilon n_an_b, \qquad K_{ab}=h_a{}^ch_b{}^d\nabla_cn_d, \qquad K=h^{ab}K_{ab}.$$

Reversing $n^a$ reverses $K_{ab}$ and $K$. For a spacelike initial or final surface, “outward” fixes whether the unit normal is past- or future-directed. Stating the normal only as future-directed is insufficient for a closed region.

With these conventions the completed action on smooth non-null segments is

$$S_{\mathrm{EH+GHY}} =\frac{1}{16\pi G}\int_{\mathcal M}\sqrt{-g}(R-2\Lambda)\,d^4x +\frac{1}{8\pi G}\int_{\Sigma}\varepsilon\sqrt{|h|}\,K\,d^3y.$$

Lehner and collaborators use the same projected derivative convention and track the timelike/spacelike orientation signs explicitly Lehner et al. 2016, § II B, pp. 7–10, Eqs. (2.6)–(2.32), PDF.

The offending normal derivative

Variation of the bulk curvature density yields the surface term

$$\delta S_{\mathrm{EH}}\big|_\Sigma =\frac{1}{16\pi G}\int_\Sigma d^3y\,\sqrt{|h|}\, \varepsilon n_a \left[ \nabla^a\!\left(g_{cd}\delta g^{cd}\right) -\nabla_b\delta g^{ab} \right].$$

Even when the tangential metric variation vanishes at the boundary, normal derivatives such as $n^c\nabla_c\delta h_{ab}$ remain. Requiring both $\delta h_{ab}$ and its normal derivative to vanish would overconstrain a second-order Dirichlet problem. Writing the contraction explicitly avoids confusing it with the determinant variation $\delta(\det g_{ab})$ used in the metric identities.

Gaussian normal coordinates expose the term. Near $\Sigma$, write

$$ds^2=\varepsilon\,dn^2+h_{ij}(n,y)dy^i dy^j, \qquad K_{ij}=\frac12\partial_nh_{ij}$$

Here $n$ increases in the direction of the chosen outward normal. For a local demonstration, use adapted variations with $\delta g_{nn}=\delta g_{ni}=0$ in this collar. The derivative part of the bulk surface variation is

$$\delta S_{\mathrm{EH}}\big|_{\partial_n\delta h} =-\frac{\varepsilon}{16\pi G} \int_\Sigma\sqrt{|h|}\,h^{ij}\partial_n\delta h_{ij}\,d^3y.$$

It is precisely a variation of the normal derivative encoded in $K$; setting $\delta h_{ij}=0$ at the surface does not set this slope to zero.

Cancellation by the GHY term

Since $K=\tfrac12h^{ij}\partial_nh_{ij}$, the derivative part of $\delta K$ is $\tfrac12h^{ij}\partial_n\delta h_{ij}$. The factor of two between the GHY and Einstein–Hilbert prefactors then gives

$$\delta S_{\mathrm{GHY}}\big|_{\partial_n\delta h} =+\frac{\varepsilon}{16\pi G} \int_\Sigma\sqrt{|h|}\,h^{ij}\partial_n\delta h_{ij}\,d^3y,$$

canceling the offending term exactly. Including the algebraic variations of the measure and inverse induced metric, the total boundary variation takes the Dirichlet form

$$\delta(S_{\mathrm{EH}}+S_{\mathrm{GHY}})\big|_\Sigma =\frac{1}{16\pi G}\int_\Sigma d^3y\,\sqrt{|h|}\, \varepsilon\left(K_{ij}-Kh_{ij}\right)\delta h^{ij},$$

with the outward normal and $\varepsilon$ already fixed above. Tangential divergences integrate to zero on a closed smooth segment, or for variations supported away from its edges; otherwise their edge terms belong to the joint problem. Fixing $h_{ij}$ makes the displayed integral vanish. The remaining coefficient is closely related to the Brown–York boundary momentum; the present purpose is only the differentiability of the action.

The cancellation and the remaining momentum term, with segment signs kept explicit, are derived in Lehner et al. 2016, § II B, pp. 7–10, Eqs. (2.27)–(2.32), PDF.

The GHY term does not cancel the bulk Einstein tensor. It cancels the derivative of the boundary variation. Stationarity then yields the same interior Einstein equation as compactly supported variation.

Corners and the domain of the result

The formula assumes a smooth non-null segment. If two segments meet with discontinuous normals, their extrinsic curvature develops a distributional contribution in a smoothing limit, and an explicit joint term is required for additivity. A null boundary has no unit normal distinct from its tangent, so neither $h_{ab}$ as a nondegenerate induced metric nor the above trace $K$ provides the correct data. Null segments require a generator, parameterization, screen metric, and their own surface and joint terms.

Original formulations by York and by Gibbons and Hawking established the boundary role in complementary settings York 1972, pp. 1082–1085 and Gibbons and Hawking 1977, §§ II–III, pp. 2752–2754. Their sign conventions must be translated before combining formulas.

Exercises

Why does fixing $h_{ij}$ on $\Sigma$ not imply $\partial_n\delta h_{ij}=0$ there?

Solution

Dirichlet data specify the value of the tangential metric at the boundary, so $\delta h_{ij}|_\Sigma=0$. They do not specify how the varied metric approaches that value from the interior. Different variations can vanish at $n=0$ while having different slopes, so $\partial_n\delta h_{ij}|_\Sigma$ is unrestricted. The GHY term removes precisely this slope dependence from the surface variation.

References

• Gibbons, G. W., and S. W. Hawking. “Action Integrals and Partition Functions in Quantum Gravity.” Physical Review D 15 (1977): 2752–2756. DOI.
• Lehner, Luis, Robert C. Myers, Eric Poisson, and Rafael D. Sorkin. “Gravitational Action with Null Boundaries.” Physical Review D 94 (2016): 084046. DOI; arXiv.
• York, James W., Jr. “Role of Conformal Three-Geometry in the Dynamics of Gravitation.” Physical Review Letters 28 (1972): 1082–1085. DOI.