GHY Approach in Gravitational Theories

• The GHY approach is a method that adds a boundary term to the Einstein–Hilbert action to cancel unwanted normal derivative contributions and secure a well-posed variational principle.
• It supports various boundary conditions, including Dirichlet and Neumann, and is integral to analyses in black hole thermodynamics and gravitational path integrals.
• The formulation is extended to null boundaries, higher-curvature theories, and discrete models, demonstrating its versatility in modern gravitational research.

The Gibbons–Hawking–York approach supplements the Einstein–Hilbert action on a manifold with boundary by a boundary integral of the trace of the extrinsic curvature, so that the total gravitational action has a well-posed variational principle when the induced metric is fixed on the boundary. In standard notation, with induced metric $h_{ab}$, outward unit normal $n^\mu$, and $K=h^{ab}K_{ab}$, the action is

$$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$

with $\epsilon=\pm 1$ for spacelike or timelike boundaries. In current usage, the same approach also denotes the broader program of matching gravitational boundary terms to the chosen boundary data, to higher-curvature dynamics, to null boundaries, and to discrete or reduced formulations of gravity (Lela, 14 Nov 2025, Ahmadain et al., 2024).

1. Variational principle and the standard GHY term

The Einstein–Hilbert action by itself does not admit a well-posed Dirichlet variational principle on a manifold with boundary, because the first variation of $\int_M R\sqrt{|g|}$ contains a boundary term with normal derivatives of $\delta g_{\mu\nu}$. In the conventions stated above,

$$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$

where $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$. The boundary contribution does not vanish under the condition $\delta h_{ab}=0$ on $n^\mu$0, because it contains derivatives of $n^\mu$1 normal to the boundary (Lela, 14 Nov 2025).

The GHY term is engineered to cancel exactly that higher-derivative boundary variation. Its first variation satisfies

$n^\mu$2

The last integral cancels the boundary term from $n^\mu$3, leaving only terms proportional to $n^\mu$4. Hence the full action $n^\mu$5 is stationary under fixed induced metric on $n^\mu$6 (Lela, 14 Nov 2025).

A closely related noncovariant packaging is the Einstein–$n^\mu$7 action. There the boundary quantity $n^\mu$8 decomposes as

$n^\mu$9

so the standard GHY term appears together with terms that depend only on boundary data. Under Dirichlet boundary conditions these additional pieces do not obstruct well-posedness, because they do not reintroduce normal derivatives of $K=h^{ab}K_{ab}$0 (Ahmadain et al., 2024).

2. Geometric data and sign conventions

The boundary geometry is described by the induced metric

$K=h^{ab}K_{ab}$1

with $K=h^{ab}K_{ab}$2 the outward unit normal. The extrinsic curvature and its trace are

$K=h^{ab}K_{ab}$3

In the Riemannian setting one sets $K=h^{ab}K_{ab}$4; in Lorentzian signature, $K=h^{ab}K_{ab}$5 for spacelike and $K=h^{ab}K_{ab}$6 for timelike boundaries (Lela, 14 Nov 2025).

A standard sign convention in the recent literature is the outward normal with $K=h^{ab}K_{ab}$7 on Euclidean spheres. In reflected Fermi coordinates $K=h^{ab}K_{ab}$8, with inward normal coordinate $K=h^{ab}K_{ab}$9, the tube Jacobian expands as

$$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$0

This expansion is used both as a local geometric check on the sign of $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$1 and as a boundary-layer tool in discrete asymptotics (Lela, 14 Nov 2025).

For spacelike or timelike boundaries these definitions are sufficient. They cease to be adequate on null hypersurfaces, where there is no unit normal and the induced three-metric becomes degenerate. In that case the relevant intrinsic geometry is built instead from a null generator $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$2, an auxiliary null vector $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$3 with $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$4, and the induced two-metric

$$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$5

together with the expansion $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$6 and the non-affinity $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$7 (Parattu et al., 2015).

3. Boundary data beyond fixed induced metric

The GHY approach is often presented only for Dirichlet data on $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$8, but the boundary-value problem for gravity is not unique. In the Dirichlet action $$S[g]=S_{EH}[g]+S_{GHY}[g]+S_\Lambda[g] =\frac{1}{16\pi G}\int_M R\sqrt{|g|}\,d^d x +\frac{1}{8\pi G}\int_{\partial M}\epsilon K\sqrt{|h|}\,d^{d-1}x -\frac{\Lambda}{8\pi G}\int_M \sqrt{|g|}\,d^d x,$$9, the momentum density conjugate to the boundary metric is

$\epsilon=\pm 1$0

with $\epsilon=\pm 1$1. A boundary Legendre transform then defines a Neumann action,

$\epsilon=\pm 1$2

which is stationary when $\epsilon=\pm 1$3 is held fixed rather than $\epsilon=\pm 1$4. In $\epsilon=\pm 1$5 the coefficient is one-half the Dirichlet GHY coefficient, whereas in $\epsilon=\pm 1$6 the Neumann boundary term vanishes, so the pure Einstein–Hilbert action already gives the Neumann variational principle (Krishnan et al., 2016).

This distinction has a direct thermodynamic interpretation. The same analysis identifies the Brown–York quasilocal stress tensor as the response to variations of the boundary metric, while the Neumann formulation is interpreted as fixing the momentum or boundary stress tensor density and is described as a natural “microcanonical” path integral for gravity (Krishnan et al., 2016).

Black-hole entropy provides a stringent check on the boundary-condition dependence. For Schwarzschild solutions in GR and in $\epsilon=\pm 1$7 gravity, Dirichlet, Neumann, and mixed boundary conditions can all be paired with appropriate GHY-type terms, yet the entropy remains the same as the value obtained under Dirichlet boundary conditions from the Wald formula or from the semiclassical Euclidean method. In the Noether-charge description, the shift $\epsilon=\pm 1$8 induced by changing the boundary term does not affect the bifurcation surface because $\epsilon=\pm 1$9 there (Khodabakhshi et al., 2020).

The universality of the entropy does not imply that all boundary conditions are equivalent in all settings. In isotropic general relativity, an alternative formulation argues that fixing the induced boundary metric can violate general covariance and can allow the mass of a black hole to vary, because the covariant mass $\int_M R\sqrt{|g|}$0 depends on derivatives of the metric that are not fixed by $\int_M R\sqrt{|g|}$1. That analysis instead imposes the physical boundary conditions $\int_M R\sqrt{|g|}$2 and $\int_M R\sqrt{|g|}$3, with $\int_M R\sqrt{|g|}$4 and $\int_M R\sqrt{|g|}$5 treated as scalar boundary data (Bachlechner, 2018).

4. Generalized boundary terms: null, higher-curvature, and non-Riemannian settings

For null hypersurfaces the standard $\int_M R\sqrt{|g|}$6 expression is not available, because the induced three-metric is degenerate and there is no unit normal. A proposed null analogue is the counterterm

$\int_M R\sqrt{|g|}$7

or equivalently $\int_M R\sqrt{|g|}$8 in adapted coordinates. Here $\int_M R\sqrt{|g|}$9 is the null expansion and $\delta g_{\mu\nu}$0 the non-affinity. This term cancels the variation of normal derivatives on a null boundary in the same structural sense that $\delta g_{\mu\nu}$1 does for non-null boundaries (Parattu et al., 2015).

In $\delta g_{\mu\nu}$2 gravity the situation is formulation-dependent. One result states that the boundary term

$\delta g_{\mu\nu}$3

is appropriate provided the boundary condition

$\delta g_{\mu\nu}$4

is imposed; for $\delta g_{\mu\nu}$5, this is equivalent to $\delta g_{\mu\nu}$6 (Saskowski, 2024). A separate classification shows that $\delta g_{\mu\nu}$7-gravity, in its original higher-derivative form, is not described by a degenerate Lagrangian and that consistent boundary conditions require an Ostrogradsky or Brans–Dicke reformulation with $\delta g_{\mu\nu}$8. In that scalar–tensor representation the Dirichlet, Neumann, and two mixed boundary conditions admit explicit GHY terms, and one mixed class—fixing $\delta g_{\mu\nu}$9 and $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$0—requires no GHY term at all in arbitrary dimension (Khodabakhshi et al., 2018).

Quadratic and Lovelock-type theories require further generalization. For the quadratic family

$$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$1

a local boundary functional built from $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$2, $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$3, and $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$4 is integrable only if

$$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$5

This class includes Gauss–Bonnet gravity, for which the generalized GHY term reduces to the Myers boundary term (Ramirez et al., 2024). In the same spirit, higher-derivative and infinite-derivative theories admit generalized boundary actions expressed in ADM variables and nonlocal form factors $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$6, reducing to the usual GHY term in the Einstein–Hilbert limit and to known quadratic boundary terms in the local limit (Teimouri et al., 2016).

For metric-affine theories the dependence on curvature, torsion, and non-metricity can be disentangled systematically. A universal method based on Lagrange multipliers and an $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$7 split shows that only curvature-dependent terms in the bulk Lagrangian require GHY compensation. Terms polynomial solely in torsion and non-metricity do not require any GHY term for a Dirichlet variational problem (Erdmenger et al., 2022).

This result dovetails with recent work on the generalized geometrical trinity of gravity. In the metric-affine formulation, the boundary term $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$8 that relates the Einstein–Hilbert action to its teleparallel and symmetric teleparallel equivalents is the difference between the GHY term of GR and that of metric-affine gravity. In TEGR and STEGR the correct GHY term must vanish; equivalently, one does not add an extra boundary counterterm to the first-derivative teleparallel action itself (Erdmenger et al., 2023). At the same time, the covariant surface term $$\delta \int_M R\sqrt{|g|}\,d^d x = \int_M (G_{\mu\nu}\delta g^{\mu\nu})\sqrt{|g|}\,d^d x + \int_{\partial M}\epsilon \left(n^\mu\nabla^\nu\delta g_{\mu\nu}-n^\mu\nabla_\mu\delta g\right)\sqrt{|h|}\,d^{d-1}x,$$9 in teleparallelism plays the same variational role as GHY in the Einstein–Hilbert representation and reproduces the correct black-hole entropy from the boundary free energy (Oshita et al., 2017).

5. Reformulations, reductions, and derivations

The GHY structure appears in several nonstandard reformulations of gravity. In the Einstein–$\delta g=g^{\mu\nu}\delta g_{\mu\nu}$0 form, the Ricci scalar differs from a quadratic Christoffel-symbol expression by a total derivative, and the accompanying boundary term decomposes into the standard $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$1 piece plus terms built purely from the metric, the normal vector, and tangential derivatives. This reformulation makes explicit that the covariant GHY term and the Einstein boundary term are variationally equivalent under Dirichlet conditions, even though they are not identical as functionals (Ahmadain et al., 2024).

String-worldsheet derivations sharpen this point. One analysis of the bulk off-shell sphere partition function shows that the noncovariant worldsheet boundary term

$\delta g=g^{\mu\nu}\delta g_{\mu\nu}$2

is precisely the first Einstein boundary term in the $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$3-$\delta g=g^{\mu\nu}\delta g_{\mu\nu}$4 action. When rewritten in terms of $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$5, it carries only one-half of the coefficient required by the standard GHY term for a well-posed Dirichlet variational principle. The missing factor is traced to the absence of the second Einstein boundary term in the worldsheet derivation (Ahmadain et al., 2024).

A complementary linearized calculation in half-space, using the method of images and working to first order in $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$6 and to linear order in the perturbation around flat half-space, recovers the full Einstein–$\delta g=g^{\mu\nu}\delta g_{\mu\nu}$7 boundary structure and the correct relative factor $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$8 between the bulk $\delta g=g^{\mu\nu}\delta g_{\mu\nu}$9 term and the boundary $\delta h_{ab}=0$0 term. In that formulation the total sphere effective action in half-space has a well-posed variational principle under Dirichlet boundary conditions (Ahmadain et al., 2024).

Dimensional reduction preserves the same logic. Under group-manifold and circle Kaluza–Klein reductions with boundaries aligned along the non-compact directions, the higher-dimensional GHY term reduces exactly to the lower-dimensional GHY term in two-derivative gravity. In four-derivative theories, reducing the known higher-dimensional boundary term provides the correct lower-dimensional boundary term, including the Gauss–Bonnet/Myers term, Chern–Simons-modified boundary terms, and the $\delta h_{ab}=0$1 boundary term in the reduced theory (Saskowski, 2024).

6. Discrete, numerical, and broader developments

Recent work shows that the GHY approach is not confined to continuum variational calculus. A diffeomorphism-natural, local MDL-type discrete functional on boundary-fitted, shape-regular meshes can $\delta h_{ab}=0$2-converge to

$\delta h_{ab}=0$3

with $\delta h_{ab}=0$4, $\delta h_{ab}=0$5, and $\delta h_{ab}=0$6 after calibration. The corresponding Carathéodory densities are

$\delta h_{ab}=0$7

so the GHY term emerges as the unique admissible first boundary-layer density compatible with naturality, second-order jets, and the scaling $\delta h_{ab}=0$8 (Lela, 14 Nov 2025).

The same analysis quantifies the difference between interior and boundary error layers. Interior cells contribute

$\delta h_{ab}=0$9

whereas first boundary-layer cells contribute

$n^\mu$00

Summing over boundary-touching cells yields the GHY integral with a global $n^\mu$01 boundary remainder, while the interior aggregate error is $n^\mu$02. Appendix E further gives a reproducible calibration protocol based on a flat torus, closed constant-curvature manifolds, and Euclidean balls; the expected convergence slopes are approximately $n^\mu$03 when a boundary is present and approximately $n^\mu$04 when no boundary is present (Lela, 14 Nov 2025).

GHY-type invariants also appear outside the usual bulk-plus-boundary setting. In Lovelock-type brane cosmology, the geodetic brane worldvolume action includes the odd extrinsic-curvature term $n^\mu$05 and the cubic GHYM-type combination

$n^\mu$06

directly as worldvolume scalars. In that setting these terms are not separate bulk boundary integrals, because the bulk is nondynamical Minkowski space, but they play the same structural role of preserving second-order equations of motion and generating a Friedmann-type equation on the brane (Arroyo et al., 7 Sep 2025).

Across these developments, the common content of the GHY approach is variational rather than merely notational. It identifies the boundary functional that makes the chosen gravitational action differentiable under the chosen boundary data, whether the setting is Einstein–Hilbert gravity, a null hypersurface, a higher-curvature theory, a metric-affine reformulation, a string-worldsheet derivation, a dimensional reduction, or a discrete $n^\mu$07-limit.