Lapse and Shift Conditions

Lapse and shift determine how numerical coordinates thread an already determined spacetime. They can avoid singular regions, move punctures across a grid, and control coordinate stretching, but they can also create gauge shocks, zero-speed modes, or severe characteristic speeds. A gauge should therefore be judged by both coordinate regularity and invariant observables. This page compares common numerical conditions through matched Schwarzschild and binary tests; it does not assign physical meaning to coordinate tracks.

Helpful background. Formulations of General Relativity develops lapse, shift, and gauge hyperbolicity, Ordinary Differential Equations supplies stability and damping concepts, Computational Relativity supplies shared evidence standards, and Black-Hole Dynamics and Collapse supplies the physical spacetimes being foliated.

Lapse conditions control slicing

Use $K_{ij}=-\tfrac12\mathcal L_n\gamma_{ij}$, so positive $K$ corresponds to contraction of the normal volume element. A widely used singularity-avoiding condition is advective $1+\log$ slicing,

$$(\partial_t-\beta^i\partial_i)\alpha=-2\alpha K,$$

or a generalized Bona–Massó form with a specified function of $\alpha$. In moving-puncture evolutions the lapse collapses near the puncture, limiting advance of proper time there, while exterior slices approach a stationary trumpet geometry. Harmonic slicing instead ties the coordinates to wave equations and is natural in generalized-harmonic systems.

Record advective terms, initial lapse, source coefficients, and any floors. Omitting advection defines a different coordinate system and can produce stationary gauge structures in the wrong place. Lapse collapse is not evidence of physical horizon formation; check apparent horizons and curvature invariants separately.

The name has a simple derivation when the shift vanishes. The metric equation gives $\partial_t\ln\gamma=-2\alpha K$, so the lapse condition integrates at each fixed spatial coordinate to

$$\alpha(t,x)=\alpha(0,x)+\ln\frac{\gamma(t,x)}{\gamma(0,x)}.$$

For initially unit lapse this is literally “one plus a logarithm.” The ratio makes the logarithm dimensionless. This algebraic relation is not valid without modification when a nonzero shift changes coordinate volumes; the advective differential equation is the definition used in moving-puncture evolution. The more general Bona–Massó choice is $(\partial_t-\beta^i\partial_i)\alpha=-\alpha^2f(\alpha)K$, with harmonic slicing $f=1$ and $1+\log$ slicing $f=2/\alpha$. See Baumgarte and Shapiro 2010, §4.3, Eqs. (4.47)–(4.52), pp. 113–114.

Shift conditions move coordinates

A common hyperbolic Gamma-driver uses

$$(\partial_t-\beta^j\partial_j)\beta^i=\mu_S B^i, \qquad (\partial_t-\beta^j\partial_j)B^i= (\partial_t-\beta^j\partial_j)\tilde\Gamma^i-\eta B^i.$$

Coefficients, dimensions, advective choices, and damping profile $\eta$ materially change coordinate speeds. van Meter et al. 2006, §§II–IV and Figures 1–3 diagnoses gauge choices for moving punctures, including modes that remain near their point of creation. A shift that transports punctures but leaves zero-speed gauge or constraint contamination can eventually spoil a long run.

Harmonic, damped-harmonic, minimal-distortion-inspired, and moving-puncture gauges optimize different goals. Compare them on coordinate stretching, characteristic speeds, constraint behavior, horizon resolution, and cost—not on puncture tracks alone. No gauge is uniformly best across excision, puncture, matter, and wave-zone domains.

A matched Schwarzschild benchmark

Evolve one Schwarzschild black hole with the same formulation, spatial and temporal discretization, mesh, and outer boundary while changing only the gauge family or parameter. Use at least three resolutions. Track lapse and shift profiles, proper distance to the horizon, trumpet areal-radius relation where applicable, Hamiltonian and momentum constraints, horizon area, and the Kretschmann scalar at geometrically identified locations.

The coordinate fields may settle to different profiles, while horizon mass and invariant curvature should agree within numerical uncertainty. Repeat with varied $\eta$ and boundary location; a coordinate pulse that reflects from the boundary should shift in arrival time. Campanelli et al. 2006, Figure 3 and Table I demonstrates long-term black-hole motion with the moving-puncture approach and provides a historical strong-field benchmark.

For a binary, add waveform phase and amplitude, but compare them after extraction into one asymptotic frame rather than aligning away arbitrary gauge effects. Gauge changes can alter coordinate merger time without changing the invariant waveform. The domain of the benchmark is the tested formulation and parameter range; a stable isolated trumpet does not establish robust gauge behavior through merger.

Environment and failure criteria

Pin the complete lapse and shift equations, all advective terms, initial gauge fields, coefficient profiles, formulation, stencil, dissipation, integrator, CFL factor, mesh hierarchy, boundary conditions, precision, and code revision. A clean rerun interface could be compare-gauge --config schwarzschild.toml --output run/, emitting coordinate fields, characteristic speeds, constraints, invariants, horizon data, refinement sweeps, and environment metadata. This specifies the experiment without claiming that a program ran here.

Reject a gauge if gradients steepen without convergence, characteristic speeds violate the numerical boundary design, constraints grow at a zero-speed mode, or invariant horizon quantities drift despite apparently stationary coordinates. A failure mode can be hidden by lapse collapse that freezes evolution at under-resolved points. Excessive shift damping can also pin coordinate distortions; insufficient damping can drive oscillations. Vary parameters rather than tuning one resolution by eye.

Archive the coordinate transformation or geometric markers needed to compare runs. Without them, a pointwise difference between metric components mixes true numerical error with an expected relabeling of events and cannot support a convergence claim.

Diagnostic

Two gauges give different puncture trajectories and coordinate merger times, but their horizon masses and waveforms in the same asymptotic retarded time agree under refinement. Which result indicates different physics?

Solution

Neither coordinate trajectory nor coordinate merger time alone indicates different physics. The agreement of horizon and radiative invariants supports two coordinate descriptions of the same spacetime. Report the gauge-dependent tracks as numerical diagnostics, retain the invariant comparison, and check that constraints and boundary contamination are controlled for both gauges.

References

• Baumgarte, T. W., and Shapiro, S. L. (2010). Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press. DOI.
• Campanelli, M., Lousto, C. O., Marronetti, P., and Zlochower, Y. (2006). “Accurate evolutions of orbiting black-hole binaries without excision.” Physical Review Letters 96, 111101. arXiv.
• van Meter, J. R., Baker, J. G., Koppitz, M., and Choi, D.-I. (2006). “How to move a black hole without excision: Gauge conditions for the numerical evolution of a moving puncture.” Physical Review D 73, 124011. arXiv.