The 3+1 Decomposition

The 3+1 decomposition rewrites a spacetime metric and Einstein’s equation relative to spacelike hypersurfaces. It separates the gauge variables that choose slicing and spatial coordinates from the spatial geometry and its normal velocity; then it separates Einstein’s equations into constraints on each slice and evolution between slices. No physical content is changed, but the data, signs, and mathematical roles of the equations become explicit.

Helpful background. The Einstein Equation supplies the covariant dynamics, Hypersurface Normals and Tangential Projectors supplies the induced geometry, and The Gibbons-Hawking-York Boundary Term supplies the boundary role of the spatial metric and extrinsic curvature.

From a spacetime to spatial data

A time function $t$ with timelike gradient defines spacelike slices $\Sigma_t$. Their future unit normal, spatial metric, and coordinate evolution vector obey

$$n^\mu n_\mu=-1, \qquad \gamma_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu, \qquad (\partial_t)^\mu=\alpha n^\mu+\beta^\mu.$$

The positive lapse $\alpha$ measures normal proper time per unit coordinate time. The spatial shift $\beta^i$ moves coordinate labels within a slice. The extrinsic curvature convention is fixed chapter-wide as

$$K_{ij}=-\frac12\mathcal L_n\gamma_{ij},$$

so an expanding Milne hyperboloid has negative $K$. In adapted coordinates,

$$ds^2=-\alpha^2dt^2 +\gamma_{ij}(dx^i+\beta^idt)(dx^j+\beta^jdt),$$

and the normal derivative is represented by $\partial_\perp=\partial_t-\mathcal L_\beta$. These choices lead to the first evolution equation,

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

The lapse and shift determine how the same spacetime is presented. They are not additional initial gravitational degrees of freedom.

Constraints, evolution, and matter

Projecting curvature with zero, one, and two normal legs gives the Gauss, Codazzi, and Ricci relations. After using Einstein’s equation, the first two become constraints:

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

The two-normal projection supplies the evolution of $K_{ij}$:

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

The sources are the quantities measured by the normal observers:

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

The minus sign in $j_i$ and the negative sign in the definition of $K_{ij}$ are linked to the visible signs of the momentum constraint and the evolution system. They must be translated together when using a source with different conventions.

A route through the chapter

Read the chapter in the following order:

1. Spacelike Foliations and Evolution Vectors defines the time function, future normal, lapse, shift, and distinction between normal advance and tangential relabeling.
2. ADM Metric Decomposition reconstructs the metric, inverse metric, determinant, volume form, and causal character of coordinate threads.
3. Extrinsic Curvature and Metric Evolution derives the sign-fixed relation between embedding geometry and the normal velocity of $\gamma_{ij}$.
4. Normal and Tangential Projections of Curvature identifies Gauss and Codazzi as constraint-producing projections and the Ricci relation as evolution-producing.
5. ADM Evolution Equations assembles the continuum evolution of $(\gamma_{ij},K_{ij})$ with arbitrary lapse, shift, matter, and cosmological constant.
6. 3+1 Decomposition of Stress-Energy reconstructs $T_{\mu\nu}$ from the energy, momentum, and stress measured by normal observers.

The first three pages are kinematic: they apply to any spacetime with the stated foliation. The next two use curvature and Einstein dynamics. The last page completes the matter source map. The following chapter then asks how to solve the four constraints for admissible initial data; the hyperbolicity chapter asks whether a chosen evolution-and-gauge system is well posed.

One spacetime, two decisive checks

Inertial Minkowski slices have

$$\alpha=1, \qquad \beta^i=0, \qquad \gamma_{ij}=\delta_{ij}, \qquad K_{ij}=0.$$

Every vacuum constraint and evolution term vanishes. This is the simplest normalization check, but it cannot expose most relative signs.

Milne coordinates cover the interior of a Minkowski light cone:

$$ds^2=-d\tau^2+\tau^2q_{ij}dx^idx^j,$$

where $q_{ij}$ is the unit hyperbolic metric. They give

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

The Hamiltonian combination vanishes:

$${}^{(3)}R+K^2-K_{ij}K^{ij} =-\frac6{\tau^2}+\frac9{\tau^2}-\frac3{\tau^2}=0.$$

The ADM evolution equation also holds nontrivially. Thus intrinsic curvature and extrinsic curvature can both be nonzero while the four-dimensional Riemann tensor vanishes. This check catches the sign of $K_{ij}$, the Gauss quadratic terms, and the time derivative in the $K_{ij}$ equation at once.

The full construction, including foliation kinematics, curvature projections, Einstein constraints, evolution, and matter variables, is developed in Gourgoulhon 2007, Chs. 2–5, pp. 15–82, PDF.

What the decomposition does not prove

The ADM equations are equivalent to Einstein’s equation only on the constraint-satisfying sector, with compatible matter conservation, regular gauge, and appropriate initial and boundary data. Merely evolving the twelve component equations does not turn constraint-violating fields into an Einstein spacetime.

The split also does not establish well-posedness. Lapse and shift must be specified or evolved, and the principal symbol depends on that gauge system, the chosen variables, and any constraint additions. A formulation may be algebraically equivalent to Einstein dynamics on the constraint surface yet behave differently away from it.

Finally, this chapter is continuum geometry. Discretization, convergence, boundary algorithms, and code performance belong to Computational Relativity.

Review the chapter

In vacuum Minkowski spacetime with $\Lambda=0$, take Milne slices. Verify in one calculation that the metric evolution, Hamiltonian constraint, momentum constraint, and extrinsic-curvature evolution all hold. State which result would change if the opposite convention for $K_{ij}$ were used.

Solution

For $\alpha=1$, $\beta^i=0$, and $\gamma_{ij}=\tau^2q_{ij}$,

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

when $K_{ij}=-\gamma_{ij}/\tau$. Because $K$ is spatially constant and $D_k\gamma_{ij}=0$, $D_jK^j{}_i-D_iK=0$. The scalar calculation displayed above proves $\mathcal H=0$.

For the final evolution equation,

$$\partial_\tau K_{ij}=-\frac1{\tau^2}\gamma_{ij},$$

while

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

Thus all four equations agree. If one defines $K^{\mathrm{opp}}_{ij}=+\tfrac12\mathcal L_n\gamma_{ij}$, then $K^{\mathrm{opp}}_{ij}=+\gamma_{ij}/\tau$. The metric-evolution equation and all formulas linear in $K_{ij}$ change sign, while the Hamiltonian combination $K^2-K_{ij}K^{ij}$ is unchanged. The equations remain consistent only if the translation is made everywhere.

References

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