ADM Evolution Equations

ADM evolution advances the spatial metric $\gamma_{ij}$ and extrinsic curvature $K_{ij}$ once lapse and shift have been chosen. The lapse and shift enter through normal acceleration and spatial Lie transport; they change the coordinate representative, not the covariant Einstein solution. Equivalence to that solution additionally requires the Hamiltonian and momentum constraints on the initial data and their preservation during evolution.

Required background. Hypersurface Normals and Tangential Projectors supplies the projected derivative and extrinsic geometry, while The Einstein Equation supplies the covariant equations being rewritten.

Helpful background. The Gibbons-Hawking-York Boundary Term explains how the same spatial metric and extrinsic curvature enter a boundary-aware variational problem.

The evolved fields and gauge fields

Fix throughout

$$(\partial_t)^\mu=\alpha n^\mu+\beta^\mu, \qquad \gamma_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu, \qquad K_{ij}=-\frac12\mathcal L_n\gamma_{ij}.$$

It is convenient to write

$$\partial_\perp\equiv\partial_t-\mathcal L_\beta.$$

The pair $(\gamma_{ij},K_{ij})$ contains twelve component functions. The definition of $K_{ij}$ supplies six metric-evolution equations; the ten components of Einstein’s equation split into six extrinsic-curvature-evolution equations and four constraints. On initial data, the constraints reduce twelve functions to eight, and four diffeomorphism directions leave four phase-space functions—the two gravitational polarizations and their conjugate momenta. The lapse and shift choose how that gauge freedom is represented during evolution.

Einstein’s equation does not by itself prescribe the lapse or shift. Algebraic, elliptic, or hyperbolic gauge conditions may do so, but the resulting enlarged system must be analyzed separately for well-posedness. The equations below are the continuum ADM identities before discretization.

Evolution of the spatial metric

The first evolution equation is kinematic:

$$\partial_\perp\gamma_{ij}=-2\alpha K_{ij},$$

or in coordinate components,

$$\partial_t\gamma_{ij} =-2\alpha K_{ij}+D_i\beta_j+D_j\beta_i.$$

Normal geometry appears only in the first term. The remaining terms are the change induced by tangential relabeling. If the shift is a Killing field of $\gamma_{ij}$, it can be nonzero while contributing nothing to $\partial_t\gamma_{ij}$.

This equation also evolves the inverse metric:

$$\partial_\perp\gamma^{ij}=2\alpha K^{ij},$$

which follows by differentiating $\gamma^{ik}\gamma_{kj}=\delta^i{}_j$. That sign is needed when tracing the $K_{ij}$ evolution equation.

Evolution of extrinsic curvature

The spatial projection of the four-dimensional Ricci tensor combines the contracted Gauss relation with the two-normal Ricci relation. Solving it for the normal derivative of $K_{ij}$ and using

$$R_{\mu\nu} =8\pi G\left(T_{\mu\nu}-\frac12Tg_{\mu\nu}\right) +\Lambda g_{\mu\nu}$$

gives

$$\begin{aligned} \partial_\perp K_{ij} ={}&-D_iD_j\alpha +\alpha\left( {}^{(3)}R_{ij} +K K_{ij} -2K_{ik}K^k{}_j \right)\\ &-8\pi G\alpha \left[ S_{ij}-\frac12\gamma_{ij}(S-\rho) \right] -\alpha\Lambda\gamma_{ij}. \end{aligned}$$

The matter variables use the fixed signs

$$\rho=T_{\mu\nu}n^\mu n^\nu, \qquad j_i=-\gamma_i{}^\mu n^\nu T_{\mu\nu}, \qquad S_{ij}=\gamma_i{}^\mu\gamma_j{}^\nu T_{\mu\nu}, \qquad S=\gamma^{ij}S_{ij}.$$

The momentum density $j_i$ does not appear explicitly in this spatial projection; it sources the momentum constraint. The Hessian $-D_iD_j\alpha$ is the geometric effect of accelerating the normal congruence. The shift enters only through $\partial_\perp$, again separating gauge transport from normal dynamics.

Gourgoulhon derives these projections and the adapted-coordinate system in Gourgoulhon 2007, §4.1.3, pp. 53–54 and §§4.3–4.4.3, pp. 59–64, PDF.

Constraints and the domain of equivalence

The evolution system is accompanied by

$$\mathcal H ={}^{(3)}R+K^2-K_{ij}K^{ij} -16\pi G\rho-2\Lambda=0,$$

and

$$\mathcal M_i =D_jK^j{}_i-D_iK-8\pi G j_i=0.$$

The contracted Bianchi identity and $\nabla_\mu T^{\mu\nu}=0$ imply a homogeneous subsidiary system for $(\mathcal H,\mathcal M_i)$ when the ADM evolution equations hold. Hence constraint-satisfying initial data remain on the constraint surface in the exact continuum problem, subject to suitable regularity and boundary compatibility.

This does not mean that every unconstrained solution of the twelve evolution equations is an Einstein spacetime. Off the constraint surface, one has an enlarged system. Nor does algebraic equivalence prove that a particular gauge choice gives a strongly hyperbolic Cauchy problem. Those principal-symbol questions belong to the next formulation layer.

Minkowski and Milne checks

In inertial Minkowski coordinates,

$$\alpha=1, \quad \beta^i=0, \quad \gamma_{ij}=\delta_{ij}, \quad K_{ij}=0, \quad T_{\mu\nu}=0, \quad \Lambda=0.$$

Every term in both evolution equations vanishes separately.

For Milne slices of the same spacetime,

$$\gamma_{ij}=\tau^2q_{ij}, \qquad K_{ij}=-\frac1\tau\gamma_{ij}, \qquad K=-\frac3\tau, \qquad {}^{(3)}R_{ij}=-\frac2{\tau^2}\gamma_{ij}.$$

The metric equation gives

$$\partial_\tau\gamma_{ij} =\frac2\tau\gamma_{ij} =-2K_{ij}.$$

For the second equation, $\alpha=1$, $\beta^i=0$, and vacuum imply

$${}^{(3)}R_{ij}+K K_{ij}-2K_{ik}K^k{}_j =\left(-2+3-2\right)\frac1{\tau^2}\gamma_{ij} =-\frac1{\tau^2}\gamma_{ij}.$$

Direct differentiation gives the same result: $\partial_\tau K_{ij}=-\gamma_{ij}/\tau^2$. Thus nontrivial ADM evolution reconstructs a flat four-dimensional metric, confirming every quadratic sign.

Common pitfalls

Lapse and shift do not add physical solutions. They choose how the same Einstein geometry is sliced and labeled, provided the gauge construction stays regular.

Evolution equations alone are not Einstein’s equation. Initial and boundary data must satisfy the constraints, and the subsidiary system must not receive incoming constraint violations.

Algebraic equivalence is not well-posedness. Hyperbolicity depends on the chosen variables, reductions, constraint additions, and gauge equations.

Exercise: trace the curvature evolution

Trace the $K_{ij}$ evolution equation, taking account of the evolution of $\gamma^{ij}$, and use the Hamiltonian constraint to show that

$$\partial_\perp K =-D^iD_i\alpha +\alpha\left[ K_{ij}K^{ij}+4\pi G(\rho+S)-\Lambda \right].$$

Solution

Because $K=\gamma^{ij}K_{ij}$,

$$\partial_\perp K =2\alpha K^{ij}K_{ij} +\gamma^{ij}\partial_\perp K_{ij}.$$

Tracing the displayed evolution equation for $K_{ij}$ gives

$$\begin{aligned} \gamma^{ij}\partial_\perp K_{ij} ={}&-D^iD_i\alpha +\alpha({}^{(3)}R+K^2-2K_{ij}K^{ij})\\ &+4\pi G\alpha(S-3\rho)-3\alpha\Lambda. \end{aligned}$$

Adding the inverse-metric term cancels the explicit $-2K_{ij}K^{ij}$. The Hamiltonian constraint rearranges to

$${}^{(3)}R+K^2 =K_{ij}K^{ij}+16\pi G\rho+2\Lambda.$$

Substitution combines the matter terms into $4\pi G(\rho+S)$ and the cosmological terms into $-\Lambda$, yielding the required trace equation.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. Lecture notes, arXiv:gr-qc/0703035v1 (2007). arXiv.