Time Slicing and Lapse Conditions

A lapse condition controls how successive spacelike hypersurfaces are separated, not how points are labeled within each hypersurface. Geodesic and harmonic slicing evolve the lapse by transport equations; maximal and constant-mean-curvature slicing determine it from an elliptic equation. Each choice controls a different geometric quantity, and none changes the underlying Einstein spacetime while its coordinate construction remains regular.

Required background. The Einstein Equation supplies the dynamics, while Killing Fields and Isometry Groups supplies the distinction between coordinate evolution and an exact spacetime symmetry.

Helpful background. Hypersurface Normals and Tangential Projectors supplies the normal and induced geometry, and PDEs and Green Functions for Gravitation supplies the hyperbolic–elliptic distinction and the role of boundary data.

Lapse as separation between slices

For an evolution vector

$$t^\mu=\alpha n^\mu+\beta^\mu, \qquad n_\mu\beta^\mu=0,$$

the lapse $\alpha>0$ converts coordinate-time separation into proper-time separation along the future unit normal:

$$d\tau=\alpha\,dt.$$

The shift $\beta^\mu$ moves coordinate labels tangentially and therefore does not select the hypersurfaces. The lapse does more than rescale time globally: its spatial gradient is the acceleration of the normal congruence,

$$a_i\equiv\gamma_i{}^\mu n^\nu\nabla_\nu n_\mu =D_i\ln\alpha.$$

Thus spatially constant lapse makes the normal observers geodesic, whereas a spatially varying lapse bends their worldlines so that the foliation can avoid or approach selected regions.

With the extrinsic-curvature convention

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

the spatial volume density obeys

$$\partial_t\sqrt\gamma-\partial_i(\beta^i\sqrt\gamma) =-\alpha K\sqrt\gamma, \qquad K=\gamma^{ij}K_{ij}.$$

Equivalently, $(\partial_t-\beta^i\partial_i)\ln\sqrt\gamma =-\alpha K+\partial_i\beta^i$. The divergence term is required because $\sqrt\gamma$ is a coordinate density, not a scalar.

This equation provides the common geometric language for slicing conditions: they prescribe how $\alpha$ responds to the local volume expansion $-K$ or constrain $K$ directly.

Hyperbolic lapse conditions

The Bona–Massó family has the continuum form

$$(\partial_t-\beta^iD_i)\alpha =-\alpha^2 f(\alpha)K,$$

with a specified positive function $f$ in the standard hyperbolic regime. Different $f$ produce different coordinate characteristics even though the Einstein solution on the gauge surface is unchanged.

Geodesic slicing takes $\alpha=1$ after a suitable time normalization, usually with zero shift for the simplest construction. Then $a_i=0$, so the normal observers follow timelike geodesics and their proper time is the coordinate time. A geodesic congruence can focus. When neighboring normals cross, the coordinates fail even if curvature remains finite.

Harmonic slicing uses $f=1$:

$$(\partial_t-\beta^iD_i)\alpha=-\alpha^2K.$$

It is the time-coordinate part of $\Box_g t=0$. Combining it with the volume equation gives

$$(\partial_t-\beta^i\partial_i)\ln\frac{\alpha}{\sqrt\gamma} =-\partial_i\beta^i.$$

In coordinates transported along the normals, where $\beta^i=0$, the ratio $\alpha/\sqrt\gamma$ is time independent. In a general spatial coordinate system, shift divergence changes the coordinate volume and must be retained.

For a check, relabel a Minkowski spatial coordinate by $X=a(t)x$, with $a(t)\gt0$, leaving $T=t$, $Y=y$, and $Z=z$. These are still flat $T=\mathrm{constant}$ slices: $\alpha=1$, $K_{ij}=0$, $\gamma_{xx}=a^2$, and $\beta^x=(\dot a/a)x$. Nevertheless $\partial_t\ln\sqrt\gamma=\dot a/a=\partial_i\beta^i$. The changing coordinate-cell volume comes entirely from the moving grid; the harmonic lapse remains constant.

The common $1+\log$ slicing chooses $f=2/\alpha$:

$$(\partial_t-\beta^iD_i)\alpha=-2\alpha K.$$

In normal coordinates with $\beta^i=0$, the volume equation implies $\partial_t\ln\gamma=-2\alpha K=\partial_t\alpha$. Hence $\alpha=\alpha_0+\ln(\gamma/\gamma_0)$; taking $\alpha_0=1$ explains the name. With a general shift the same algebraic formula requires a correction for $2\partial_i\beta^i$ and cannot simply be used at fixed grid labels. In collapsing regions with $K>0$ under the selected sign, the lapse decreases and coordinate time advances more slowly relative to normal proper time. This is singularity avoiding in a coordinate sense; it does not remove a physical singularity or guarantee a globally regular chart. Strong gradients and gauge shocks remain possible.

Gourgoulhon derives the lapse–foliation relations and compares algebraic, harmonic, maximal, and singularity-avoiding choices in Gourgoulhon 2007, §9.2, pp. 152–162, PDF.

Maximal and constant-mean-curvature slicing

The trace of the ADM evolution equation is

$$(\partial_t-\beta^iD_i)K =-D^iD_i\alpha +\alpha\left( K_{ij}K^{ij} +4\pi G(\rho+S) -\Lambda \right),$$

where $\rho=T_{\mu\nu}n^\mu n^\nu$ and $S=\gamma^{ij}S_{ij}$. This formula turns a condition on mean curvature into an elliptic equation for the lapse.

Maximal slicing requires

$$K=0$$

on every slice. Preserving it gives

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

The name refers to stationarity of spatial volume under normal deformations, not necessarily to a global maximum. Boundary conditions are part of the definition: an asymptotically flat end commonly uses $\alpha\to1$, while an inner boundary needs its own regularity or geometric condition.

Constant-mean-curvature slicing requires $D_iK=0$ on each slice but permits $K=K(t)$. Its lapse equation is

$$D^iD_i\alpha -\alpha\left( K_{ij}K^{ij}+4\pi G(\rho+S)-\Lambda \right) =-\frac{dK}{dt}.$$

One must prescribe the time dependence of $K$ or an equivalent normalization. The elliptic operator can have kernels, and solvability depends on geometry, matter, cosmological constant, and boundary data. Writing the equation does not prove that a positive lapse exists.

What each condition controls

Slicing	Defining target	Main advantage	Characteristic limitation
Geodesic	Constant lapse; geodesic normals	Direct normal proper time	Normal congruence can form caustics
Harmonic	Time coordinate satisfies a wave equation	Hyperbolic gauge with metric null principal cone	Residual gauge and coordinate collapse remain
$1+\log$	Lapse responds strongly to positive $K$	Slows coordinate advance in collapse	Gauge steepening and driver-dependent speeds
Maximal	$K=0$	Controls local volume expansion	Elliptic solve and boundary data required
CMC	$K$ spatially constant	Useful geometric foliation and asymptotics	Prescribed $K(t)$ and elliptic solvability required

The table compares continuum gauge conditions, not numerical cost or discretization stability. A lapse condition is also not a complete coordinate choice: a spatial gauge must determine the shift, and residual transformations can preserve both conditions.

Two checks with the same spacetime

Inside the future light cone of Minkowski spacetime, introduce Milne coordinates:

$$T=\tau\cosh\chi, \qquad R=\tau\sinh\chi.$$

The metric is

$$ds^2=-d\tau^2+\tau^2 \left(d\chi^2+\sinh^2\chi\,d\Omega^2\right).$$

Here $\alpha=1$, $\beta^i=0$, and

$$K_{ij}=-\frac1\tau\gamma_{ij}, \qquad K=-\frac3\tau.$$

The normal observers are geodesic, so this is geodesic slicing. It is not harmonic slicing: the harmonic lapse equation would require $\partial_\tau\alpha=3/\tau$. This flat-spacetime check shows that “geodesic” and “harmonic” constrain different geometric properties.

For the standard static Schwarzschild exterior, the constant Schwarzschild-time slices have shift zero, $K_{ij}=0$, and

$$\alpha=\sqrt{1-\frac{2GM}{r}}.$$

They are maximal but not geodesic because $D_i\alpha\ne0$. The lapse approaches zero at the chart horizon while curvature remains finite there. Lapse collapse alone is therefore not evidence of a physical singularity.

Common pitfalls

The lapse is not elapsed proper time everywhere. The relation $d\tau=\alpha dt$ applies along the normal displacement. A coordinate line with nonzero shift is not a normal observer.

Singularity avoiding is not singularity resolving. A lapse can keep a coordinate slice from reaching a high-curvature region in finite coordinate time. It does not extend an inextendible spacetime or regularize divergent curvature.

An elliptic slicing equation needs a domain. Maximal and CMC equations are incomplete without inner, outer, or compact-domain boundary conditions and a statement about kernels and positivity.

Exercise: the static Schwarzschild lapse

On a standard Schwarzschild-time slice in vacuum with $\Lambda=0$, use

$$\gamma_{ij}dx^idx^j =\frac{dr^2}{1-2GM/r}+r^2d\Omega^2, \qquad \alpha=\sqrt{1-\frac{2GM}{r}}$$

to verify directly that $D^iD_i\alpha=0$ for $r>2GM$. Explain why this is the maximal-slicing lapse equation for these data.

Solution

For a radial scalar,

$$D^iD_i\alpha =\frac1{\sqrt\gamma} \partial_r\left(\sqrt\gamma\,\gamma^{rr}\partial_r\alpha\right).$$

Writing $f=1-2GM/r$ gives $\sqrt\gamma=r^2\sin\theta/\sqrt f$, $\gamma^{rr}=f$, and $\partial_r\alpha=GM/(r^2\sqrt f)$. Hence

$$\sqrt\gamma\,\gamma^{rr}\partial_r\alpha =GM\sin\theta,$$

whose radial derivative vanishes. The slice has $K_{ij}=0$ and the region is vacuum, so the right-hand side of the maximal lapse equation also vanishes. The nonconstant harmonic function $\alpha$ is fixed by its asymptotic value and the inner behavior of this exterior domain.

References

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