Gibbons–Hawking–York Boundary Term
The extrinsic curvature boundary supplement that renders the gravitational action well-posed under Dirichlet conditions
1. The Need for a Boundary Term
Recall from the previous section that the variation of the Einstein–Hilbert action produces a boundary integral involving normal derivatives of $\delta g_{\mu\nu}$ that does not vanish under Dirichlet boundary conditions $\delta g_{\mu\nu}|_{\partial\mathcal{M}} = 0$. The complete gravitational action requires an additional boundary term:
$$S_{EH+GHY} = \frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{-g}\,R\,d^4x + \frac{1}{8\pi G}\int_{\partial\mathcal{M}}\sqrt{|h|}\,K\,d^3y$$
The second integral is the Gibbons–Hawking–York (GHY) boundary term, where $K$ is the trace of the extrinsic curvature of the boundary, $h$ is the determinant of the induced metric on $\partial\mathcal{M}$, and $d^3y$ is the coordinate volume element on the boundary. The factor of $1/(8\pi G)$ (twice the bulk prefactor) is fixed by the cancellation requirement.
2. Extrinsic Curvature
Let $n^\mu$ be the outward-pointing unit normal to $\partial\mathcal{M}$, with$\epsilon = n_\mu n^\mu = \pm 1$ ($+1$ for a timelike boundary, $-1$ for spacelike). The induced metric on $\partial\mathcal{M}$ is:
$$h_{\mu\nu} = g_{\mu\nu} - \epsilon\,n_\mu n_\nu$$
The extrinsic curvature tensor measures how the boundary is embedded in the ambient spacetime. It is defined as the Lie derivative of the induced metric along the normal:
$$K_{ij} = -\frac{1}{2}\mathcal{L}_n h_{ij} = -h_i^{\ \mu} h_j^{\ \nu} \nabla_\mu n_\nu$$
In adapted coordinates $(y^i, n)$ where $n$ is the proper distance along the normal, the extrinsic curvature takes the form:
$$K_{ij} = -\frac{1}{2}\frac{\partial h_{ij}}{\partial n}$$
The trace is $K = h^{ij}K_{ij} = -\nabla_\mu n^\mu$, which measures the divergence of the normal vector field and thus the rate at which the boundary volume element is changing.
3. Gaussian Normal Coordinates
To perform the cancellation calculation explicitly, it is convenient to work in Gaussian normal coordinates near $\partial\mathcal{M}$. In these coordinates, the metric takes the form$ds^2 = \epsilon\,dn^2 + h_{ij}(n, y)\,dy^i\,dy^j$, so that $n_\mu = (\epsilon, 0, 0, 0)$.
The Christoffel symbols at the boundary simplify, and the connection variation projected along the normal becomes:
$$n_\mu\delta\Gamma^\mu_{ij} = -\delta K_{ij} + \frac{1}{2}h_{ij}h^{kl}\delta K_{kl} + \text{tangential derivatives}$$
Tracing over the boundary indices with $h^{ij}$:
$$n_\alpha V^\alpha\big|_{\partial\mathcal{M}} = h^{ij}\partial_n(\delta g_{ij}) - h^{ij}h_{ij}\,g^{kl}\partial_n(\delta g_{kl}) = -2\,\delta K + \text{tangential}$$
The tangential derivative terms vanish when $\delta g_{\mu\nu} = 0$ on the boundary. Thus the problematic boundary contribution from $\delta S_{EH}$ is proportional to $\delta K$.
4. Variation of the GHY Term
Now we vary the GHY boundary term. The variation of $K\sqrt{|h|}$ has two contributions:
$$\delta\!\left(K\sqrt{|h|}\right) = \sqrt{|h|}\,\delta K + K\,\delta\sqrt{|h|}$$
Using $\delta\sqrt{|h|} = \frac{1}{2}\sqrt{|h|}\,h^{ij}\delta h_{ij}$ and the relation$\delta h_{ij} = \delta g_{ij}|_{\partial\mathcal{M}}$, the second term vanishes under Dirichlet conditions $\delta g_{ij}|_{\partial\mathcal{M}} = 0$. The surviving contribution is:
$$\delta S_{GHY} = \frac{1}{8\pi G}\int_{\partial\mathcal{M}}\sqrt{|h|}\,\delta K\,d^3y$$
This is precisely the term needed to cancel the boundary contribution from $\delta S_{EH}$, since $n_\alpha V^\alpha = -2\,\delta K$ on shell.
5. Proof of Boundary Term Cancellation
Combining the bulk and boundary contributions, the total variation of the gravitational action is:
$$\delta S_{EH+GHY} = \frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{-g}\,G_{\mu\nu}\,\delta g^{\mu\nu}\,d^4x + \frac{1}{16\pi G}\oint_{\partial\mathcal{M}}\sqrt{|h|}\left(n_\alpha V^\alpha + 2\,\delta K\right)d^3y$$
The boundary integrand $n_\alpha V^\alpha + 2\,\delta K = 0$ under Dirichlet conditions, leaving:
$$\delta S_{EH+GHY} = \frac{1}{16\pi G}\int_{\mathcal{M}}\sqrt{-g}\,G_{\mu\nu}\,\delta g^{\mu\nu}\,d^4x$$
This is the desired result: the variational principle is now well-posed, and the equation of motion$G_{\mu\nu} = 0$ (or $G_{\mu\nu} = 8\pi G\,T_{\mu\nu}$ with matter) follows from stationarity without any uncontrolled boundary terms.
6. Perelman Analogue: Non-collapsing as Positive Energy
Perelman's $\mu$-functional provides a Ricci flow analogue of the GHY term and the positive energy theorem. Define the log-Sobolev functional:
$$\mathcal{W}(g, f, \tau) = \int_M \left[\tau(R + |\nabla f|^2) + f - n\right](4\pi\tau)^{-n/2}e^{-f}\,d\mu_g$$
with $\mu(g, \tau) = \inf\{\mathcal{W}(g, f, \tau) : \int_M (4\pi\tau)^{-n/2}e^{-f}d\mu_g = 1\}$. The functional $\mu(g, \tau)$ plays the role of the Hamilton principal function: it is the value of the action evaluated on the optimal path (the minimizing $f$).
Perelman's non-collapsing theorem states that if $\mu(g_0, \tau) \geq -A$ at the initial time, then the Ricci flow remains $\kappa$-non-collapsed at all subsequent times. This is the precise analogue of the positive energy theorem in GR:
$$\mu(g, \tau) \geq -C(A, T) \quad \longleftrightarrow \quad E_{ADM} \geq 0$$
Both results prevent geometric collapse: the positive energy theorem prevents gravitational collapse to negative mass singularities, while non-collapsing prevents the volume ratio from degenerating along the Ricci flow. The GHY term ensures a well-posed variational problem whose solutions obey the positive energy theorem; analogously, the $|\nabla f|^2$ term in $\mathcal{W}$ ensures monotonicity of $\mu$ along the flow.
7. Corner Terms and Joints
When the boundary $\partial\mathcal{M}$ is not smooth but consists of multiple segments meeting at corners (joints), additional terms are required. Consider two boundary segments meeting at a codimension-2 surface $\mathcal{C}$ with normals $n_1^\mu$ and $n_2^\mu$. The joint contribution is:
$$S_{\text{corner}} = \frac{\eta}{8\pi G}\int_{\mathcal{C}}\sqrt{\sigma}\,\Theta\,d^2z$$
where $\sigma$ is the determinant of the induced metric on $\mathcal{C}$,$\Theta$ is the boost angle between the normals defined by $\cosh\Theta = -\epsilon\,n_1 \cdot n_2$for timelike normals, and $\eta = \pm 1$ depends on the orientation convention. For the common case of a spacetime region bounded by two spacelike surfaces $\Sigma_1$, $\Sigma_2$and a timelike surface $\mathcal{B}$, the full action is:
$$S = S_{EH} + S_{GHY}[\Sigma_1] + S_{GHY}[\Sigma_2] + S_{GHY}[\mathcal{B}] + S_{\text{corner}}[\mathcal{C}_1] + S_{\text{corner}}[\mathcal{C}_2]$$
These corner terms are essential for the path integral formulation of quantum gravity, where the amplitude for a transition between spatial geometries requires a well-defined action on regions with corners. They also appear in holographic entanglement entropy calculations, where the entangling surface creates a corner in the bulk region.
8. Euclidean Action and Black Hole Thermodynamics
The GHY term plays a crucial role in black hole thermodynamics through the Euclidean path integral. For a Schwarzschild black hole of mass $M$, the Euclidean section has a conical singularity unless the Euclidean time is periodic with period $\beta = 8\pi GM$. The on-shell Euclidean action evaluates to:
$$I_E = -S_{EH+GHY}^{(\text{Eucl})} = \frac{\beta^2}{16\pi G} = 4\pi GM^2$$
The entire contribution comes from the GHY boundary term at large radius (the bulk term vanishes on shell since $R = 0$ in vacuum). The thermodynamic partition function $Z = e^{-I_E}$ gives the free energy $F = I_E/\beta = M/2$, and the entropy follows from the standard relation:
$$S_{BH} = \beta\frac{\partial I_E}{\partial\beta} - I_E = \frac{\beta^2}{16\pi G} = \frac{A}{4G}$$
where $A = 16\pi G^2 M^2$ is the horizon area. This derivation of the Bekenstein–Hawking entropy relies entirely on the GHY term, underscoring its physical importance beyond mere mathematical well-posedness.
In the Ricci flow analogue, Perelman's $\mathcal{W}$-entropy evaluated at the shrinking soliton plays the role of $I_E$, and the non-collapsing constant $\kappa$is the analogue of the Bekenstein–Hawking entropy: both measure the minimal volume ratio (geometric entropy) that a solution can achieve.
9. Null Boundaries and the Bondi Limit
When $\partial\mathcal{M}$ includes null segments (as at null infinity $\mathscr{I}$), the GHY formalism must be generalized. On a null boundary with tangent vector $k^\mu$and auxiliary null vector $\ell^\mu$ satisfying $k \cdot \ell = -1$, the boundary term involves the expansion $\theta = q^{AB}\nabla_A k_B$ and shear$\sigma_{AB}$ of the null generators:
$$S_{\text{null}} = \frac{1}{8\pi G}\int_{\mathcal{N}}\sqrt{q}\,\left(\theta + \kappa\right)d\lambda\,d^2\theta$$
where $\kappa$ is the non-affinity parameter defined by $k^\mu\nabla_\mu k^\nu = \kappa\,k^\nu$,$q$ is the determinant of the 2-metric on cross-sections, and $\lambda$ is the parameter along the generators. This null boundary term connects directly to the Bondi–Sachs formalism: at future null infinity, $\theta$ encodes the news tensor and the expansion of $\mathscr{I}^+$, providing the variational bridge between the bulk Einstein equations and the asymptotic BMS charges.