Lagrangian Connections
Einstein-Hilbert, Gibbons-Hawking-York, ADM, Perelman, Bondi-Sachs, and Mabuchi as facets of one variational structure
1. The Einstein-Hilbert Action
The bulk gravitational action in $D$ dimensions is:
$$S_{\rm EH} = \frac{1}{16\pi G}\int_{\mathcal{M}} R\,\sqrt{-g}\,d^D x$$
Variation yields the Einstein equations $G_{\mu\nu} = 8\pi G\,T_{\mu\nu}$, but the action principle requires a boundary term because $R$ contains second derivatives of the metric. The Gibbons-Hawking-York boundary term completes the variational principle:
$$S_{\rm GHY} = \frac{1}{8\pi G}\int_{\partial\mathcal{M}} K\,\sqrt{|h|}\,d^{D-1}x$$
where $K = h^{ab}K_{ab}$ is the trace of the extrinsic curvature and $h_{ab}$ is the induced metric on the boundary. The total gravitational action is $S_{\rm grav} = S_{\rm EH} + S_{\rm GHY}$.
2. The ADM Hamiltonian
The Arnowitt-Deser-Misner (ADM) decomposition writes the spacetime metric in terms of lapse $N$, shift $N^i$, and spatial metric $\gamma_{ij}$:
$$ds^2 = -N^2\,dt^2 + \gamma_{ij}(dx^i + N^i\,dt)(dx^j + N^j\,dt)$$
The conjugate momentum to $\gamma_{ij}$ is $\pi^{ij} = \frac{\sqrt{\gamma}}{16\pi G}(K\gamma^{ij} - K^{ij})$, and the ADM Hamiltonian is a sum of constraints:
$$H_{\rm ADM} = \int_\Sigma \bigl(N\,\mathcal{H} + N_i\,\mathcal{H}^i\bigr)\,d^3x + \oint_{\partial\Sigma} B\,d^2x$$
where $\mathcal{H} \approx 0$ is the Hamiltonian constraint and $\mathcal{H}^i \approx 0$ the momentum constraint. The boundary term $B$ gives the ADM mass when evaluated at spatial infinity:
$$\boxed{M_{\rm ADM} = \frac{1}{16\pi G}\oint_{S^2_\infty} (\partial_j \gamma_{ij} - \partial_i \gamma_{jj})\,dS^i}$$
3. Perelman’s $\mathcal{F}$ as Gravitational Action
Perelman’s $\mathcal{F}$-functional on a Riemannian 3-manifold $(\Sigma, g)$ with scalar field $f$:
$$\mathcal{F}(g, f) = \int_\Sigma (R + |\nabla f|^2)\,e^{-f}\,dV$$
is structurally parallel to the Einstein-Hilbert action with a dilaton coupling. Under the identification $e^{-f}\,dV = d\mu$ (fixed measure), the gradient flow of $\mathcal{F}$ produces the modified Ricci flow:
$$\partial_t g_{ij} = -2(R_{ij} + \nabla_i\nabla_j f)$$
The critical points satisfy $R_{ij} + \nabla_i\nabla_j f = 0$, which are gradient Ricci solitons, the Riemannian analog of Einstein metrics with a cosmological term.
4. The Bondi-Sachs Framework
At null infinity, the gravitational dynamics is encoded in the Bondi-Sachs metric with expansion parameter $1/r$:
$$ds^2 = -\left(1 - \frac{2m_B}{r}\right)du^2 - 2\,du\,dr + r^2\gamma_{AB}\,dx^A dx^B + r\,C_{AB}\,dx^A dx^B + \ldots$$
The symplectic structure on the phase space at $\mathscr{I}^+$ yields the Bondi mass-loss formula as a monotonicity statement:
$$\boxed{\frac{dM_B}{du} = -\frac{1}{8}\oint N_{AB}N^{AB}\,d\Omega \;\leq\; 0}$$
The inequality $M_{\rm ADM} \geq M_B(u) \geq M_B(+\infty)$ connects the ADM and Bondi formulations: the ADM mass is the total energy, while $M_B$ tracks what remains after radiation escapes.
5. Mabuchi K-Energy
On a Kahler manifold $(M, \omega)$ with $[\omega] = c_1(M)$, the Mabuchi K-energy is:
$$\mathcal{K}(\varphi) = -\int_0^1\int_M \dot{\varphi}_s\,(R_{\varphi_s} - \underline{R})\,\omega_{\varphi_s}^n\,ds$$
where $\underline{R}$ is the average scalar curvature. Under the Kahler-Ricci flow $\partial_t \omega = -\mathrm{Ric}(\omega) + \omega$, the K-energy is monotonically non-increasing:
$$\frac{d\mathcal{K}}{dt} = -\int_M |R_{i\bar{j}} - g_{i\bar{j}}|^2\,\omega^n \;\leq\; 0$$
This parallels the Perelman monotonicity and connects complex geometry to gravitational dynamics. The critical points are Kahler-Einstein metrics.
6. The Unified Variational Chain
All five Lagrangians are connected through dimensional reduction, boundary terms, and limit procedures:
$$S_{\rm EH} + S_{\rm GHY} \;\longrightarrow\; H_{\rm ADM} \;\longrightarrow\; M_{\rm ADM}$$
$$S_{\rm EH}\big|_{\mathscr{I}^+} \;\longrightarrow\; S_{\rm Bondi} \;\longrightarrow\; M_B(u)$$
$$S_{\rm EH}\big|_{\Sigma} + \text{dilaton} \;\longrightarrow\; \mathcal{F}(g,f)$$
$$\mathcal{F}\big|_{\text{Kahler}} \;\longrightarrow\; \mathcal{K}(\varphi)$$
Each monotone quantity $(\mathcal{F} \uparrow, \mathcal{W} \uparrow, M_B \downarrow, M_{\rm ADM} = \text{const}, \mathcal{K} \downarrow)$ reflects the same underlying principle: spacetime geometry evolves toward equilibrium, dissipating information through radiation at null infinity.
For the full derivation of these connections, see GR Part VIII: The Master Lagrangian.
Simulation: Five Monotone Quantities
We visualize the five monotone quantities simultaneously, demonstrating how each evolves under its respective flow. The key observation: all point in the same thermodynamic direction.
The master equation chain: 5 monotone quantities
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