ADM Lagrangian and the 3+1 Decomposition
Decomposing spacetime into space and time: from the Gauss–Codazzi equations to the Hamiltonian formulation and its Ricci flow analogue
1. The 3+1 Metric Decomposition
The ADM (Arnowitt–Deser–Misner) formalism begins by foliating spacetime into a one-parameter family of spacelike hypersurfaces $\Sigma_t$. The line element decomposes as:
$$ds^2 = -N^2\,dt^2 + h_{ij}(dx^i + N^i\,dt)(dx^j + N^j\,dt)$$
Here $N$ is the lapse function, measuring the proper time elapsed between adjacent hypersurfaces; $N^i$ is the shift vector, measuring the displacement of spatial coordinates between slices; and $h_{ij}$ is the induced 3-metric on each $\Sigma_t$.
The unit normal to $\Sigma_t$ is $n_\mu = (-N, 0, 0, 0)$, with $n^\mu = (1/N, -N^i/N)$. The full 4-metric and its inverse are:
$$g_{\mu\nu} = \begin{pmatrix} -N^2 + N_kN^k & N_j \\ N_i & h_{ij} \end{pmatrix}, \qquad g^{\mu\nu} = \begin{pmatrix} -1/N^2 & N^j/N^2 \\ N^i/N^2 & h^{ij} - N^iN^j/N^2 \end{pmatrix}$$
The determinant factorizes: $\sqrt{-g} = N\sqrt{h}$, where $h = \det(h_{ij})$.
2. Extrinsic Curvature in ADM Variables
The extrinsic curvature of each slice $\Sigma_t$ can be expressed in terms of the ADM variables as the time derivative of the spatial metric, corrected by the shift:
$$K_{ij} = \frac{1}{2N}\left(\dot{h}_{ij} - D_iN_j - D_jN_i\right)$$
where $\dot{h}_{ij} = \partial_t h_{ij}$ and $D_i$ is the covariant derivative compatible with $h_{ij}$. The trace is:
$$K = h^{ij}K_{ij} = \frac{1}{2N}\left(h^{ij}\dot{h}_{ij} - 2D_iN^i\right) = \frac{1}{N}\left(\frac{\partial_t\sqrt{h}}{\sqrt{h}} - D_iN^i\right)$$
This shows that $K$ measures the expansion rate of the spatial volume element as seen by normal observers, corrected for the shift of the coordinate system.
3. The Gauss–Codazzi Equations
The Gauss–Codazzi equations relate the 4-dimensional Riemann tensor to the intrinsic and extrinsic geometry of the hypersurfaces. The Gauss equation gives:
$$\phantom{.}^{(3)}R_{ijkl} = h_i^{\ \mu} h_j^{\ \nu} h_k^{\ \rho} h_l^{\ \sigma}\,\phantom{.}^{(4)}R_{\mu\nu\rho\sigma} + K_{ik}K_{jl} - K_{il}K_{jk}$$
Contracting, the four-dimensional Ricci scalar decomposes as:
$$R^{(4)} = R^{(3)} + K_{ij}K^{ij} - K^2 + 2\nabla_\mu(n^\mu K - n^\nu\nabla_\nu n^\mu)$$
The last term is a total divergence involving the normal vector. When integrated over $\mathcal{M}$, it becomes a boundary term that is absorbed by the GHY term. The Codazzi equation provides the momentum constraint:
$$D_j K^j_{\ i} - D_i K = 8\pi G\,n^\mu T_{\mu i}$$
4. The ADM Lagrangian
Substituting the Gauss–Codazzi decomposition into the Einstein–Hilbert action (with GHY term to cancel the divergence), the gravitational Lagrangian density becomes:
$$\mathcal{L}_{ADM} = \frac{\sqrt{h}\,N}{16\pi G}\left(R^{(3)} + K_{ij}K^{ij} - K^2\right)$$
The action is $S = \int dt\int_{\Sigma_t}\mathcal{L}_{ADM}\,d^3x$. Notice that$\mathcal{L}_{ADM}$ is first-order in time derivatives (through $K_{ij}$) and the lapse $N$ and shift $N^i$ appear without time derivatives — they are Lagrange multipliers enforcing constraints.
The kinetic term $K_{ij}K^{ij} - K^2$ is the DeWitt supermetric acting on velocities$\dot{h}_{ij}$. Explicitly:
$$K_{ij}K^{ij} - K^2 = G^{ijkl}K_{ij}K_{kl}, \qquad G^{ijkl} = \frac{1}{2}(h^{ik}h^{jl} + h^{il}h^{jk}) - h^{ij}h^{kl}$$
where $G^{ijkl}$ is the DeWitt supermetric on the space of Riemannian metrics. This supermetric has indefinite signature, reflecting the hyperbolic nature of Einstein's equations.
5. Canonical Momenta and Constraints
The canonical momentum conjugate to $h_{ij}$ is obtained by differentiating $\mathcal{L}_{ADM}$with respect to $\dot{h}_{ij}$:
$$\pi^{ij} = \frac{\partial\mathcal{L}_{ADM}}{\partial\dot{h}_{ij}} = \frac{\sqrt{h}}{16\pi G}\left(K^{ij} - h^{ij}K\right)$$
Since $N$ and $N^i$ have no time derivatives in $\mathcal{L}_{ADM}$, their conjugate momenta vanish identically: $\pi_N = 0$, $\pi_i = 0$. The Euler–Lagrange equations for $N$ and $N^i$ then give the constraint equations.
The Hamiltonian constraint (from variation with respect to $N$):
$$\mathcal{H} = \frac{16\pi G}{\sqrt{h}}\left(\pi^{ij}\pi_{ij} - \frac{1}{2}\pi^2\right) - \frac{\sqrt{h}}{16\pi G}R^{(3)} \approx 0$$
where $\pi = h_{ij}\pi^{ij}$. The momentum constraint (from variation with respect to $N^i$):
$$\mathcal{H}_i = -2D_j\pi^j_{\ i} \approx 0$$
The symbol $\approx$ denotes weak equality (Dirac constraint). These four constraints per spatial point reduce the 6 components of $h_{ij}$ to the 2 physical gravitational degrees of freedom per point.
6. Wick Rotation and Perelman Dissipation
The deep connection between the ADM formalism and Ricci flow emerges upon Wick rotation $t \to i\tau$. Under this analytic continuation, the Lorentzian action $iS$ becomes the Euclidean action$-S_E$, and the kinetic term transforms as:
$$K_{ij}K^{ij} - K^2 \quad \longrightarrow \quad -\left(\tilde{K}_{ij}\tilde{K}^{ij} - \tilde{K}^2\right)$$
where $\tilde{K}_{ij} = \frac{1}{2N}(\partial_\tau h_{ij} - D_iN_j - D_jN_i)$ is the Euclidean extrinsic curvature. Setting the shift to zero and the lapse to unity (Euclidean gauge fixing), and identifying $\partial_\tau h_{ij} = -2R_{ij} - 2\nabla_i\nabla_j f$ from the gradient Ricci flow:
$$\tilde{K}_{ij}\tilde{K}^{ij} - \tilde{K}^2 = |R_{ij} + \nabla_i\nabla_j f|^2 - (R + \Delta f)^2$$
The first term is precisely Perelman's dissipation rate for the $\mathcal{F}$-functional. Along the gradient Ricci flow:
$$\frac{d\mathcal{F}}{d\tau} = 2\int_M |R_{ij} + \nabla_i\nabla_j f|^2\,e^{-f}\,d\mu_g \geq 0$$
This monotonicity formula is the Ricci flow analogue of the positive-definite kinetic energy in the ADM Hamiltonian. The key correspondence is:
$$\underbrace{K_{ij}K^{ij} - K^2}_{\text{ADM kinetic term}} \quad \xleftarrow{\text{Wick rotation}} \quad \underbrace{|R_{ij} + \nabla_i\nabla_j f|^2}_{\text{Perelman dissipation rate}}$$
Off shell, the ADM kinetic term contains both the physical gravitational degrees of freedom and the gauge (constraint) degrees. On shell (imposing the constraints), only the transverse-traceless modes survive. The Perelman dissipation rate similarly contains both the Ricci flow evolution and the diffeomorphism gauge freedom (encoded in $\nabla_i\nabla_j f$); on the gradient flow (the analogue of on shell), these combine into the monotone quantity $d\mathcal{F}/d\tau$.
This correspondence reveals that the ADM kinetic term and Perelman's dissipation rate are two faces of the same geometric structure: the DeWitt supermetric norm of the velocity in the space of Riemannian metrics, evaluated on Lorentzian vs. Euclidean sections respectively.
7. The Hamilton–Jacobi Equation
The ADM Hamiltonian is a sum of constraints, $H = \int_\Sigma(N\mathcal{H} + N^i\mathcal{H}_i)\,d^3x$, which vanishes on shell. The Hamilton–Jacobi equation for the gravitational field takes the functional form:
$$\frac{16\pi G}{\sqrt{h}}G_{ijkl}\frac{\delta S}{\delta h_{ij}}\frac{\delta S}{\delta h_{kl}} + \frac{\sqrt{h}}{16\pi G}R^{(3)} = 0$$
where $G_{ijkl} = \frac{1}{2}(h_{ik}h_{jl} + h_{il}h_{jk}) - h_{ij}h_{kl}$ is the inverse DeWitt supermetric. The Hamilton principal function $S[h_{ij}]$ is the on-shell action evaluated on a solution with boundary data $h_{ij}$, and its functional derivatives give the canonical momenta:$\pi^{ij} = \delta S/\delta h_{ij}$.
In the Ricci flow analogue, the Hamilton–Jacobi equation becomes the equation for Perelman's$\mu$-functional. The minimizer $f$ of $\mathcal{W}(g, f, \tau)$ satisfies:
$$\tau(2\Delta f - |\nabla f|^2 + R) + f - n = \mu(g, \tau)$$
This is a nonlinear elliptic equation whose solution determines the optimal dilaton profile, just as the Hamilton–Jacobi equation determines the optimal trajectory in classical mechanics.
8. The Wheeler–DeWitt Equation
Canonical quantization of the ADM Hamiltonian promotes the constraints to operator equations acting on a wave functional $\Psi[h_{ij}]$. The Hamiltonian constraint becomes the Wheeler–DeWitt equation:
$$\left(-\frac{16\pi G\,\hbar^2}{\sqrt{h}}G_{ijkl}\frac{\delta^2}{\delta h_{ij}\,\delta h_{kl}} + \frac{\sqrt{h}}{16\pi G}R^{(3)}\right)\Psi[h_{ij}] = 0$$
This is a functional Schrodinger equation with no explicit time — the "problem of time" in canonical quantum gravity. The wave functional $\Psi$ depends on the 3-geometry (the equivalence class of $h_{ij}$ under spatial diffeomorphisms), and the momentum constraint $\mathcal{H}_i\Psi = 0$ enforces diffeomorphism invariance.
Under Wick rotation, the Wheeler–DeWitt equation maps to a functional heat equation on the space of Riemannian metrics. The Ricci flow provides a natural time parameter for this heat equation: the $\tau$-parameter of the shrinking soliton becomes the WKB time. In the semiclassical limit $\Psi \sim e^{iS/\hbar}$, the Hamilton–Jacobi equation is recovered, and trajectories in superspace correspond to Ricci flow lines in the Euclidean sector.
9. Counting Physical Degrees of Freedom
The ADM formalism provides a transparent counting of gravitational degrees of freedom. The spatial metric $h_{ij}$ has 6 independent components. The 4 constraint equations ($\mathcal{H} \approx 0$ and $\mathcal{H}_i \approx 0$) and the 4 gauge freedoms (lapse and shift choices) remove $4 + 4 = 8$ degrees of freedom from the phase space count of $2 \times 6 = 12$, leaving:
$$\text{Physical DOF} = 2 \times 6 - 4 - 4 = 4 \quad \text{(per spatial point in phase space)}$$
This corresponds to 2 configuration-space degrees of freedom per point: the two polarizations ($+$ and $\times$) of gravitational waves. In the linearized theory, these are the transverse-traceless components of the metric perturbation.
In the Ricci flow analogue, the DeTurck gauge fixing removes the diffeomorphism redundancy (3 components in 3 dimensions), and the conformal factor is fixed by the volume-normalized condition$\int e^{-f}d\mu = \text{const}$. The remaining degrees of freedom are the traceless-transverse deformations of the metric, which propagate along the Ricci flow as damped modes. The dissipation rate $|R_{ij} + \nabla_i\nabla_j f|^2$ restricted to these physical modes is strictly positive, ensuring monotonicity of $\mathcal{F}$.
Simulation: ADM Decomposition of Schwarzschild
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