Einstein-Hilbert action and the boundary terms

The Einstein–Hilbert action looks deceptively simple,

$$S_{\rm EH}[g]=\frac{1}{16\pi G}\int_M d^{d+1}x\,\sqrt{-g}\,R,$$

but it is not functionally differentiable on a manifold $M$ with boundary $\partial M$ under the usual Dirichlet boundary condition “hold the induced metric fixed”. The reason is that $R$ contains second derivatives of the metric, so $\delta S_{\rm EH}$ contains boundary terms with normal derivatives of $\delta g_{\mu\nu}$. The GHY term is exactly the boundary functional that cancels those “$\partial_n\delta g$” terms.

On this page we derive that statement carefully and end with the standard result (for non-null boundaries):

$$S_{\rm grav}=S_{\rm EH}+S_{\rm GHY},\qquad S_{\rm GHY}=\frac{\varepsilon}{8\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\,K,$$

with $\varepsilon=n_\mu n^\mu=\pm1$ the sign of the unit normal’s norm.

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1. Boundary geometry: $\gamma_{\mu\nu}$, $n^\mu$, and $K_{\mu\nu}$

Assume $\partial M$ is non-null (time-like or space-like). Let $n^\mu$ be the outward-pointing unit normal, normalized as

$$n_\mu n^\mu=\varepsilon,\qquad \varepsilon= \begin{cases} +1 & \text{(space-like normal, i.e. time-like boundary)},\\ -1 & \text{(time-like normal, i.e. space-like boundary)}. \end{cases}$$

The induced metric (first fundamental form) is the tangential projector

$$\gamma_{\mu\nu}=g_{\mu\nu}-\varepsilon n_\mu n_\nu,\qquad \gamma^\mu{}_\nu=\delta^\mu{}_\nu-\varepsilon n^\mu n_\nu.$$

The extrinsic curvature (second fundamental form) is

$$K_{\mu\nu}=\gamma_\mu{}^{\rho}\gamma_\nu{}^{\sigma}\nabla_\rho n_\sigma,\qquad K=\gamma^{\mu\nu}K_{\mu\nu}.$$

All boundary indices will be denoted by $a,b,\dots$ when we use coordinates intrinsic to $\partial M$.

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2. Varying the Einstein–Hilbert action: where the boundary term comes from

Start from

$$\delta(\sqrt{-g}R)=\delta(\sqrt{-g})\,R+\sqrt{-g}\,\delta R.$$

We will use $\delta g=gg^{\mu\nu}\delta g_{\mu\nu}=-gg_{\mu\nu}\delta g^{\mu\nu}$ and $\delta R=R_{\mu\nu}\delta g^{\mu\nu}+g^{\mu\nu}\delta R_{\mu\nu}$. Two standard identities are:

1. Variation of the determinant:

$$\delta\sqrt{-g}=-\frac12\sqrt{-g}\,g_{\mu\nu}\,\delta g^{\mu\nu}.$$

2. The Palatini identity:

$$\delta R_{\mu\nu}=\nabla_\rho\,\delta\Gamma^\rho_{\mu\nu}-\nabla_\nu\,\delta\Gamma^\rho_{\mu\rho},$$

with

$$\delta\Gamma^\rho_{\mu\nu} =\frac12 g^{\rho\sigma}\left( \nabla_\mu \delta g_{\nu\sigma}+\nabla_\nu \delta g_{\mu\sigma}-\nabla_\sigma \delta g_{\mu\nu} \right).$$

Using these, one can show that

$$g^{\mu\nu}\delta R_{\mu\nu}=\nabla_\rho(g^{\mu\nu}\delta\Gamma^\rho_{\mu\nu}-g^{\rho\nu}\delta\Gamma^\mu_{\mu\nu})=\nabla_\mu v^\mu,$$

where

$$v^\mu\equiv \nabla_\nu \delta g^{\mu\nu}-g_{\rho\sigma}\nabla^\mu \delta g^{\rho\sigma}.$$

Putting everything together gives the key decomposition:

$$\delta(\sqrt{-g}R)=\sqrt{-g}\,G_{\mu\nu}\,\delta g^{\mu\nu}+\sqrt{-g}\,\nabla_\mu v^\mu, \qquad G_{\mu\nu}\equiv R_{\mu\nu}-\frac12 g_{\mu\nu}R.$$

Therefore,

$$\delta S_{\rm EH} =\frac{1}{16\pi G}\int_M d^{d+1}x\,\sqrt{-g}\,G_{\mu\nu}\,\delta g^{\mu\nu} +\frac{1}{16\pi G}\int_M d^{d+1}x\,\sqrt{-g}\,\nabla_\mu v^\mu.$$

Using Stokes’ theorem,

$$\int_M \sqrt{-g}\,\nabla_\mu v^\mu =\varepsilon\int_{\partial M} d^d x\,\sqrt{|\gamma|}\,n_\mu v^\mu,$$

so the boundary term from Einstein–Hilbert is

$$\delta S_{\rm EH}\Big|_{\partial M} =\frac{\varepsilon}{16\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\, n_\mu v^\mu.$$

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3. Why Dirichlet data on $\gamma_{ab}$ does not kill the Einstein–Hilbert boundary term

The usual “Dirichlet” boundary condition in gravity is:

Fix the induced metric on the boundary, i.e. $\delta\gamma_{ab}=0$ on $\partial M$.

However, the boundary term above contains normal derivatives of $\delta g_{\mu\nu}$, so $\delta\gamma_{ab}=0$ is not enough.

The cleanest way to see this is to adopt Gaussian normal coordinates near $\partial M$:

$$ds^2=\varepsilon\,d\rho^2+\gamma_{ab}(\rho,y)\,dy^a dy^b, \qquad \partial M:\ \rho=0.$$

In these coordinates,

$$n^\mu\partial_\mu=\partial_\rho,\qquad K_{ab}=\frac12\,\partial_\rho\gamma_{ab},\qquad K=\frac12\,\gamma^{ab}\partial_\rho\gamma_{ab}.$$

If we impose $\delta\gamma_{ab}(\rho=0,y)=0$, then $\partial_\rho\delta\gamma_{ab}$ is still free. One finds that the Einstein–Hilbert boundary variation contains a term of the schematic form

$$\delta S_{\rm EH}\Big|_{\partial M}\supset -\frac{\varepsilon}{16\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\, \gamma^{ab}\,\partial_\rho(\delta\gamma_{ab}),$$

which does not vanish when $\delta\gamma_{ab}=0$. This is the precise obstruction: the Einstein–Hilbert action wants you to fix both $\gamma_{ab}$ and its normal derivative if you do not modify the action.

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4. The GHY boundary term and its variation

To cancel the unwanted normal-derivative piece, add the GHY functional

$$S_{\rm GHY}=\frac{\varepsilon}{8\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\,K.$$

4.1 Variation of $\sqrt{|\gamma|}K$ in Gaussian normal coordinates

In Gaussian normal coordinates,

$$K=\frac12\gamma^{ab}\partial_\rho\gamma_{ab}.$$

Vary:

$$\delta K=\frac12\,\delta\gamma^{ab}\,\partial_\rho\gamma_{ab} +\frac12\,\gamma^{ab}\,\partial_\rho(\delta\gamma_{ab}).$$

Also,

$$\delta\sqrt{|\gamma|}=\frac12\sqrt{|\gamma|}\,\gamma^{ab}\delta\gamma_{ab}.$$

Therefore

$$\delta(\sqrt{|\gamma|}K) =\sqrt{|\gamma|}\biggl[ \frac12\,\gamma^{ab}\partial_\rho(\delta\gamma_{ab}) -\biggl(K^{ab}-\frac12 K\gamma^{ab}\biggr)\delta\gamma_{ab} \biggr].$$

The key point is the first term: it contains exactly the normal derivative $\partial_\rho(\delta\gamma_{ab})$ that we need to cancel.

Multiplying by the coefficient in $S_{\rm GHY}$ gives

$$\delta S_{\rm GHY}\supset +\frac{\varepsilon}{16\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\, \gamma^{ab}\,\partial_\rho(\delta\gamma_{ab}),$$

which cancels the problematic piece in $\delta S_{\rm EH}$.

4.2 The combined boundary variation

After the cancellation, the total boundary variation of $S_{\rm EH}+S_{\rm GHY}$ takes the standard form

$$\delta(S_{\rm EH}+S_{\rm GHY})\Big|_{\partial M} =\frac{\varepsilon}{16\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\, \left(K_{ab}-K\gamma_{ab}\right)\delta\gamma^{ab} \quad(+\ \text{possible corner/joint terms}).$$

So, with Dirichlet boundary conditions $\delta\gamma_{ab}=0$, we get $\delta S=0$ on-shell: the variational principle is now well-posed.

What exactly did we fix?

We fixed the intrinsic boundary geometry $\gamma_{ab}$, not its normal derivative (which is encoded in $K_{ab}$). The role of $S_{\rm GHY}$ is precisely to ensure the action depends on the boundary metric without forcing us to also fix $\partial_n\gamma_{ab}$.

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5. Final answer (including a cosmological constant)

With a cosmological constant, the full gravitational action (for non-null boundaries) is

$$S_{\rm grav} =\frac{1}{16\pi G}\int_M d^{d+1}x\,\sqrt{-g}\,(R-2\Lambda) +\frac{\varepsilon}{8\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\,K.$$

Its variation is

$$\delta S_{\rm grav} =\frac{1}{16\pi G}\int_M d^{d+1}x\,\sqrt{-g}\,\left(G_{\mu\nu}+\Lambda g_{\mu\nu}\right)\delta g^{\mu\nu} +\frac{\varepsilon}{16\pi G}\int_{\partial M} d^d x\,\sqrt{|\gamma|}\, \left(K_{ab}-K\gamma_{ab}\right)\delta\gamma^{ab} \quad(+\ \text{corner/joint terms}).$$

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6. Bonus viewpoint: “GHY removes second derivatives” (ADM/Gauss–Codazzi)

Another way to “see” why the same boundary term must appear is to note that $R$ contains second derivatives of the metric. Using a foliation by constant-$\rho$ hypersurfaces, one can rewrite (schematically)

$$R = {}^{(d)}\!R + \varepsilon\left(K_{ab}K^{ab}-K^2\right) - 2\varepsilon\,\nabla_\mu(\cdots),$$

where ${}^{(d)}\!R$ is the intrinsic Ricci scalar of the hypersurface. Integrating the total divergence produces a boundary integral proportional to $K$. Adding $S_{\rm GHY}$ precisely cancels that divergence term, so the bulk Lagrangian becomes “first-derivative” in time (or radial) derivatives, which is what you want for canonical (Hamiltonian/ADM) formulations.

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7. Beyond smooth non-null boundaries: joints (corners) and null boundaries

The discussion above assumed:

1. $\partial M$ is smooth, so it has no “boundary of a boundary”, and
2. $\partial M$ is non-null, so a unit normal $n^\mu$ exists.

In Lorentzian gravity—especially in holography—these assumptions often fail. The good news is that the needed extensions are known and fit naturally with the GHY logic: add whatever boundary functional cancels the leftover boundary-of-boundary terms in $\delta S$ and makes the action additive under gluing.

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8. Joint (corner) terms for piecewise smooth non-null boundaries (Hayward term)

8.1 Geometry of a joint

Suppose the boundary is a union of smooth segments,

$$\partial M=\bigcup_i \Sigma_i,$$

and two segments $\Sigma_1$ and $\Sigma_2$ meet along a codimension-2 surface

$$\mathcal{J}=\Sigma_1\cap\Sigma_2.$$

Let $\sigma_{AB}$ be the induced metric on $\mathcal{J}$ (often called the “corner metric”), with volume element $\sqrt{|\sigma|}$.

Each segment $\Sigma_i$ has its own (outward) unit normal $n_i^\mu$ and its own GHY term

$$S_{\rm GHY}[\Sigma_i]=\frac{\varepsilon_i}{8\pi G}\int_{\Sigma_i} d^d x\,\sqrt{|\gamma^{(i)}|}\,K^{(i)}.$$

When you vary the total action, the cancellation that produced Section 4 happens separately on each segment, but there is still a leftover contribution localized at $\mathcal{J}$ because the normal (and therefore the adapted Gaussian coordinate) jumps when moving from $\Sigma_1$ to $\Sigma_2$.

Concretely, the “$\partial_\rho(\delta\gamma)$” terms in $\delta S_{\rm EH}$ and $\delta S_{\rm GHY}$ cancel on each $\Sigma_i$, but in integrating by parts on $\Sigma_i$ you produce a boundary term on $\partial\Sigma_i$. For a smooth boundary, $\partial\Sigma_i$ is empty; for a piecewise boundary, $\partial\Sigma_i$ contains the joints.

8.2 What the leftover variation looks like

Under Dirichlet boundary conditions on each $\Sigma_i$ (fix $\gamma_{ab}^{(i)}$), the remaining joint variation takes the universal form

$$\delta S\Big|_{\mathcal{J}} =\frac{1}{8\pi G}\int_{\mathcal{J}} d^{d-1}x\,\sqrt{|\sigma|}\,\delta\eta,$$

where $\eta$ is the “angle” (Euclidean) or “boost parameter” (Lorentzian) measuring the relative orientation between the normals $n_1$ and $n_2$ in the 2D normal plane of $\mathcal{J}$.

This immediately tells you what to add: a joint term whose variation produces $-\delta\eta$.

8.3 The Hayward joint term

The required joint functional is

$$S_{\rm joint}[\mathcal{J}] =\frac{1}{8\pi G}\int_{\mathcal{J}} d^{d-1}x\,\sqrt{|\sigma|}\,\eta.$$

How to compute $\eta$

• Euclidean signature: $n_1$ and $n_2$ are unit vectors in a Euclidean normal plane, so the dihedral angle is

  $$\eta=\arccos(n_1\cdot n_2).$$

• Lorentzian signature: $\eta$ is a rapidity (boost parameter) in the normal plane. The precise “cosh/sinh” relation depends on whether each segment is time-like or space-like. A practical way to record it is case-by-case:

  • If $\Sigma_1$ and $\Sigma_2$ are both space-like (so $n_1^2=n_2^2=-1$ are time-like), define

    $$\cosh\eta = -\,n_1\cdot n_2\qquad (\text{for consistently oriented time-like normals}).$$

  • If $\Sigma_1$ and $\Sigma_2$ are both time-like (so $n_1^2=n_2^2=+1$ are space-like), define

    $$\cosh\eta = |n_1\cdot n_2|.$$

  • If one segment is time-like and the other space-like (the most common “time-slice meets radial wall” corner), define the rapidity by

    $$\sinh\eta = n_{\rm timelike}\cdot n_{\rm spacelike}.$$

  The joint term depends on the signed $\eta$ (orientation matters). In practice, if you only need the on-shell value (not the detailed variation), choosing a consistent outward/future convention and taking the principal value of $\eta$ is usually sufficient.

Why the joint term is unavoidable

Even if you never vary the action, you need $S_{\rm joint}$ for additivity: if you glue two manifolds along a boundary segment, the action of the union should be the sum of the actions. Without the joint term, the GHY contributions double-count or miss the “corner angle” at the seam.

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9. Null boundaries: boundary terms, joint terms, and reparametrization invariance

Null boundaries require special care because the normal becomes null and cannot be normalized. Moreover, on a null hypersurface the normal is also tangent, so we must keep track of the choice of null generator.

9.1 Setup for a null boundary

Let $\mathcal{N}$ be a null hypersurface ruled by null generators with tangent vector $k^\mu$:

$$k^\mu k_\mu=0,\qquad k^\mu \partial_\mu=\frac{d}{d\lambda},$$

where $\lambda$ is a parameter along the generators. Choose coordinates $(\lambda,\theta^A)$ on $\mathcal{N}$, where $\theta^A$ label the generators. Each constant-$\lambda$ slice has induced (Riemannian) metric $\gamma_{AB}$ and area element $\sqrt{\gamma}$.

Define the non-affinity $\kappa$ by

$$k^\nu\nabla_\nu k^\mu=\kappa\,k^\mu.$$

If $\lambda$ is an affine parameter along the null geodesics, then $\kappa=0$.

The natural “extrinsic data” on $\mathcal{N}$ is encoded in the null extrinsic curvature

$$B_{AB} \equiv \nabla_A k_B\quad\text{projected to the cross-sections}, \qquad \Theta \equiv \gamma^{AB}B_{AB}=\frac{1}{\sqrt{\gamma}}\frac{d\sqrt{\gamma}}{d\lambda},$$

where $\Theta$ is the expansion.

9.2 The null boundary term (analogue of GHY)

For a region whose boundary includes a null piece $\mathcal{N}$, a standard choice of boundary term that yields a well-posed Dirichlet variational principle (fixing the intrinsic geometry $\gamma_{AB}$ on $\mathcal{N}$) is

$$S_{\rm null}[\mathcal{N}] =\frac{1}{8\pi G}\int_{\mathcal{N}} d\lambda\,d^{d-1}\theta\ \sqrt{\gamma}\,\kappa.$$

• If you choose $\lambda$ affine, then $\kappa=0$ and this term vanishes.
• For general parametrizations, this term cancels the $\partial_\lambda(\delta\gamma_{AB})$-type variation that plays the same role as $\partial_n(\delta\gamma_{ab})$ on a non-null boundary.

Parametrization dependence

Under a reparametrization $\lambda\to\lambda'(\lambda)$, the generator rescales $k^\mu\to k'^\mu = (d\lambda/d\lambda')\,k^\mu$, and $\kappa$ shifts by a total derivative. As a result, $S_{\rm null}$ is not invariant under arbitrary reparametrizations unless you fix a preferred parameterization (e.g. affine).

9.3 Joint terms involving null boundaries

Just like non-null segments meet at joints, null segments also meet other segments at codimension-2 surfaces. The required joint terms are logarithmic because rescaling a null normal is physical at the level of the action.

Let $\mathcal{J}$ be a codimension-2 joint with induced metric $\sigma_{AB}$ and area element $\sqrt{\sigma}$.

• Null–non-null joint: if a null boundary with generator $k^\mu$ meets a non-null boundary with unit normal $n^\mu$, then a standard joint term is

  $$S_{\rm joint}^{(k,n)}[\mathcal{J}] =\frac{1}{8\pi G}\int_{\mathcal{J}} d^{d-1}\theta\ \sqrt{\sigma}\, a,\qquad a\equiv \ln|k\cdot n|.$$

• Null–null joint: if two null boundaries with generators $k^\mu$ and $\bar{k}^\mu$ meet, then a standard choice is

  $$S_{\rm joint}^{(k,\bar{k})}[\mathcal{J}] =\frac{1}{8\pi G}\int_{\mathcal{J}} d^{d-1}\theta\ \sqrt{\sigma}\, a,\qquad a\equiv \ln\left|\frac{k\cdot \bar{k}}{2}\right|.$$

The exact placement of absolute values and additive constants corresponds to conventions (and to the freedom to add local counterterms on $\mathcal{J}$), but the logarithmic dependence is robust.

9.4 A useful counterterm: restoring reparametrization invariance on null boundaries

In many holographic applications (notably the “complexity = action” proposal), one wants the action to be independent of how one parametrizes the null generators. A widely used fix is to add the null counterterm

$$S_{\rm ct}^{\rm null} =\frac{1}{8\pi G}\int_{\mathcal{N}} d\lambda\,d^{d-1}\theta\ \sqrt{\gamma}\, \Theta\,\ln\!\big(|\ell_{\rm ct}\,\Theta|\big),$$

where $\ell_{\rm ct}$ is an arbitrary length scale. This term:

• is intrinsic to $\mathcal{N}$ (depends only on $\gamma_{AB}$ and its $\lambda$-derivative),
• does not affect the bulk equations of motion,
• shifts the action by quantities that behave like boundary “scheme choices”, and
• makes the total null-boundary contribution invariant under $\lambda\to f(\lambda)$ in the settings where it is used.

What data is “fixed” on a null boundary?

A null hypersurface has a degenerate induced metric, so “fixing the metric on $\mathcal{N}$” means fixing the geometry of the spatial cross-sections $\gamma_{AB}$ and the location of $\mathcal{N}$ in spacetime (equivalently, fixing the embedding and the null direction up to rescaling). Different choices lead to slightly different but equivalent boundary terms, related by adding intrinsic counterterms.

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10. Applications

Lorentzian path integrals and Hamiltonian evolution in a finite region: When a spacetime region is bounded by two space-like “initial/final time” slices together with a time-like wall (a finite box or radial cutoff), codimension-2 corners appear where each slice meets the wall. The Hayward joint term contributes there and is needed for additivity under gluing regions in time, obtaining the correct Hamiltonian/ADM interpretation of the action, and keeping the variational principle well-posed when you fix the induced metric on each boundary segment separately.

Wheeler–DeWitt patches and holographic complexity (CA proposal): A Wheeler–DeWitt patch is typically bounded by null hypersurfaces shot inward from specified boundary times. The gravitational action for such a region generically requires null boundary terms (often evaluated with an affine choice so that $\kappa=0$), logarithmic null joint terms where null sheets intersect each other and/or a regulator surface, and frequently the reparametrization-invariance counterterm $S_{\rm ct}^{\rm null}$. The dominant time dependence often comes from the joint terms at the intersections.

These boundary and joint terms also appear in actions of causal diamonds and entanglement wedges bounded partly by null surfaces, gravitational thermodynamics in Lorentzian signature when regulating spacetime regions with mixed (time-like/space-like) cutoffs, and any computation where actions of regions are compared after cutting and gluing along non-smooth hypersurfaces.

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Quick reference: the full “good” gravitational action for generic boundaries

For a spacetime region $M$ in $(d+1)$ dimensions with bulk metric $g_{\mu\nu}$ and (possibly mixed) boundary, a standard “good” gravitational action is

$$\begin{aligned} S = {} & \frac{1}{16\pi G}\int_M d^{d+1}x\,\sqrt{-g}\,(R-2\Lambda) +\frac{1}{8\pi G}\sum_i \varepsilon_i\int_{\Sigma_i} d^d x\,\sqrt{|\gamma|}\,K \\ & +\frac{1}{8\pi G}\sum_j \int_{\mathcal{N}_j} d\lambda\,d^{d-1}\theta\ \sqrt{\gamma}\,\kappa +\frac{1}{8\pi G}\sum_\alpha \int_{\mathcal{J}_\alpha} d^{d-1}\theta\ \sqrt{|\sigma|}\,\eta_\alpha \;+\;S_{\rm ct}^{\rm null}\ (\text{optional}). \end{aligned}$$

Here:

• $\Sigma_i$ are non-null boundary pieces with induced metric $\gamma_{ab}$ and extrinsic curvature $K$ (this term is the usual GHY term).
• $\mathcal{N}_j$ are null boundary pieces with generator $k^\mu=d/d\lambda$ and non-affinity $\kappa$.
• $\mathcal{J}_\alpha$ are non-null joints (Hayward term) with “angle/boost” $\eta$.
• $\mathcal{J}_\beta$ are joints involving at least one null boundary with logarithmic integrand $a$.
• $S_{\rm ct}^{\rm null}$ is the common null counterterm used to restore reparametrization invariance (see Section 9.4). Signs and corner conventions vary by author; this page keeps the most common “outward normal” conventions and flags where choices enter.

For Euclidean gravity with smooth boundaries, only the first line matters.