ADM Constraints and Evolution

The ADM split rewrites Einstein’s equation as restrictions on data within each spacelike slice and equations that advance those data between slices. This record applies to four-dimensional metric general relativity with signature $(-,+,+,+)$, constant $\Lambda$, and a spacelike foliation. Every sign below follows from the declared future normal, extrinsic curvature, and matter projections.

Foliation and source conventions

Let $n^a$ be the future-directed unit normal and let $h_{ab}$ be the spatial metric:

$$n^a n_a=-1, \qquad h_{ab}=g_{ab}+n_an_b.$$

The derivative $D_i$ and curvatures ${}^{(3)}R_{ij}$, ${}^{(3)}R$ belong to $h_{ij}$. On tangential slots, define

$$K_{ab} =-h_a{}^c h_b{}^d\nabla_c n_d =-\frac12\mathcal L_n h_{ab}, \qquad K=h^{ij}K_{ij}.$$

For an adapted time coordinate,

$$(\partial_t)^a=\alpha n^a+\beta^a, \qquad n_a\beta^a=0, \qquad \partial_\perp\equiv\partial_t-\mathcal L_\beta.$$

The lapse satisfies $\alpha\gt0$, the shift $\beta^i$ is spatial, and $\partial_\perp=\alpha\mathcal L_n$ after pullback to a slice. The matter projections are

$$\begin{aligned} \rho&=T_{ab}n^an^b,\\ S_i&=-h_i{}^a n^bT_{ab},\\ S_{ij}&=h_i{}^a h_j{}^bT_{ab}, \qquad S=h^{ij}S_{ij}. \end{aligned}$$

Thus positive $S_i$ uses the conventional minus sign in the normal–spatial projection. These foliation and source conventions match Gourgoulhon 2007, §§3.2.2–3.3.5, pp. 40–46; §4.1.2, pp. 52–53.

The linked treatments use $\gamma_{ij}$ for the same spatial metric and $j_i$ for the same momentum density; their notation translates as $\gamma_{ij}=h_{ij}$ and $j_i=S_i$ here.

Constraint equations on each slice

With $G_{ab}+\Lambda g_{ab}=8\pi G T_{ab}$, the normal–normal and normal–spatial projections give

$$\mathcal H \equiv{}^{(3)}R+K^2-K_{ij}K^{ij} -16\pi G\rho-2\Lambda=0,$$ $$\mathcal M_i \equiv D_jK^j{}_i-D_iK-8\pi G S_i=0.$$

These are one Hamiltonian and three momentum constraints. They contain spatial derivatives of the slice data but no normal derivative of $K_{ij}$, so they restrict admissible $(h_{ij},K_{ij})$ and matter data at each time. The cosmological term enters the scalar constraint because $g(n,n)=-1$ and drops out of the mixed projection. The Gauss–Codazzi derivation and Einstein projections are given in Gourgoulhon 2007, §2.5, pp. 34–38; §4.1, pp. 51–54.

Metric and extrinsic-curvature evolution

Once lapse and shift are supplied by a gauge prescription, the spatial fields obey

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

and

$$\begin{aligned} \partial_\perp K_{ij} ={}&-D_iD_j\alpha +\alpha\left({}^{(3)}R_{ij}+KK_{ij}-2K_{ik}K^k{}_j\right)\\ &-8\pi G\alpha \left[S_{ij}-\frac12h_{ij}(S-\rho)\right] -\alpha\Lambda h_{ij}. \end{aligned}$$

Tracing requires differentiating the inverse metric as well as $K_{ij}$. Using the Hamiltonian constraint gives the useful scalar form

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

Gourgoulhon derives the metric and curvature evolution with these signs in Gourgoulhon 2007, §4.3–§4.4.3, pp. 59–64. Lapse and shift are gauge fields, not additional Einstein evolution equations.

Complete opposite-sign translation

Suppose the future normal, lapse, shift, spatial metric, and matter projections are held fixed, but another source defines

$$\widehat K_{ij}=+\frac12\mathcal L_nh_{ij}=-K_{ij}.$$

Then the Hamiltonian constraint is unchanged, while every formula odd under $K_{ij}\mapsto-K_{ij}$ changes consistently:

$$\begin{aligned} 0&={}^{(3)}R+\widehat K^2-\widehat K_{ij}\widehat K^{ij} -16\pi G\rho-2\Lambda,\\ 0&=D_j\widehat K^j{}_i-D_i\widehat K+8\pi G S_i,\\ \partial_\perp h_{ij}&=+2\alpha\widehat K_{ij}. \end{aligned}$$

The evolution equations become

$$\begin{aligned} \partial_\perp\widehat K_{ij} ={}&+D_iD_j\alpha -\alpha\left({}^{(3)}R_{ij}+\widehat K\widehat K_{ij} -2\widehat K_{ik}\widehat K^k{}_j\right)\\ &+8\pi G\alpha \left[S_{ij}-\frac12h_{ij}(S-\rho)\right] +\alpha\Lambda h_{ij},\\ \partial_\perp\widehat K ={}&+D^iD_i\alpha -\alpha\left[\widehat K_{ij}\widehat K^{ij} +4\pi G(\rho+S)-\Lambda\right]. \end{aligned}$$

The map follows by substituting $K_{ij}=-\widehat K_{ij}$ into the sign-fixed system and multiplying the transformed $K_{ij}$ equation by $-1$; the untransformed system is collected in Gourgoulhon 2007, Chapters 4–5, pp. 51–82. Reversing $n^a$ as well would also change the lapse orientation and $S_i$ and is a different translation.

Flat, FRW, and time-symmetric checks

For inertial Minkowski slices, take $\alpha=1$, $\beta^i=0$, $h_{ij}=\delta_{ij}$, $K_{ij}=0$, $T_{ab}=0$, and $\Lambda=0$. Every constraint and evolution term vanishes separately.

For FRW data with $h_{ij}=a^2q_{ij}$, ${}^{(3)}R=6k/a^2$, unit lapse, zero shift, $K_{ij}=-Hh_{ij}$, $S_i=0$, and $S_{ij}=p h_{ij}$, the Hamiltonian and trace equations reduce to

$$3\left(H^2+\frac{k}{a^2}\right)=8\pi G\rho+\Lambda, \qquad \dot H-\frac{k}{a^2}=-4\pi G(\rho+p).$$

This simultaneously checks the $K$ sign, both cosmological terms, and the matter trace. For time-symmetric data $K_{ij}=0$, the constraints require $S_i=0$ and ${}^{(3)}R=16\pi G\rho+2\Lambda$; vacuum data with $\Lambda=0$ are scalar flat, not necessarily fully flat.

Milne slices are the vacuum FRW specialization $k=-1$, $a=t$, $\rho=0$, and $\Lambda=0$. They satisfy both reduced equations while $K_{ij}\ne0$, checking that nonzero extrinsic curvature need not imply four-dimensional curvature.

Constraints and evolution remain distinct: the contracted Bianchi identity and conserved matter propagate vanishing constraints in the exact continuum system, but evolution does not repair arbitrary constraint-violating data or establish well-posedness. For full derivations, continue to The 3+1 Decomposition, Normal and Tangential Projections of Curvature, Extrinsic Curvature and Metric Evolution, ADM Evolution Equations, 3+1 Decomposition of Stress-Energy, and Hamiltonian and Momentum Constraints.

References

• Gourgoulhon, Éric. 3+1 Formalism and Bases of Numerical Relativity. arXiv:gr-qc/0703035v1, 2007, Chapters 2–5, pp. 15–82. Open PDF.