Excision and Moving-Puncture Black Holes

Black-hole interiors need not supply physical information to an exterior evolution, but two numerical strategies exploit that fact differently. Excision removes an interior region across whose boundary no characteristic field enters the computational domain. Moving-puncture evolution retains grid points and uses singularity-avoiding coordinates that compress the unresolved interior into a trumpet geometry. Their coordinate pictures are not directly comparable; horizon invariants, constraints, and radiation provide the meaningful axes.

Helpful background. Formulations of General Relativity supplies characteristic fields and gauge systems, Ordinary Differential Equations supplies evolution stability, Computational Relativity supplies common evidence standards, and Black-Hole Dynamics and Collapse supplies the physical horizon and merger setting.

Excision is a causal boundary design

Place the excision surface inside an apparent horizon. Let $s^i$ point outward from the excised region into the retained computational domain, and define each characteristic speed relative to the moving surface in that orientation. Absence of incoming characteristics requires

$$v_{\hat\alpha}^{(s)}\leq0.$$

Strictly negative speeds give a useful safety margin for propagating fields. Identically zero-speed fields can instead be tangent to the boundary and need no incoming data when the formulation’s characteristic boundary analysis permits it; the metric field of the common generalized-harmonic system with $\gamma_1=-1$ is one example at a stationary boundary. A positive speed enters the computational domain and invalidates excision without boundary data. Gauge and constraint modes can travel differently from light, so geometric placement inside the horizon alone is insufficient. Codes using the outward normal of the computational domain reverse the signs; the recorded orientation and inequality must remain consistent.

The surface moves and changes shape. Spectral implementations often use mapped domains whose inner boundary tracks the horizon; finite-volume or finite-difference schemes use moving masks and one-sided stencils. Record the shape controller, safety margin, characteristic speeds, interpolation, and failure threshold. Scheel et al. 2006, §§II–V and Figures 1–13 demonstrates dual-frame excision and horizon tracking for several binary orbits; it does not demonstrate merger in that paper.

No physical boundary condition is imposed on a pure-outflow surface, but numerical extrapolation or penalty flux still affects stability. Test the discrete operation on a stationary black hole and vary the excision radius while it remains safely causal.

For a concrete sign check, use ingoing Kerr–Schild Schwarzschild coordinates with areal radius $r$ and $m=GM$. The radial metric coefficients are $\alpha=(1+2m/r)^{-1/2}$, $\beta^r=(2m/r)/(1+2m/r)$, and $\gamma_{rr}=1+2m/r$. Radial light rays therefore have coordinate speeds

$$\frac{dr}{dt}=-\beta^r\pm\frac{\alpha}{\sqrt{\gamma_{rr}}} =\left\{\frac{r-2m}{r+2m},\;-1\right\}.$$

At a stationary excision radius $r=1.5m$, both are negative: $-1/7$ and $-1$. Light moves toward the excised region as intended. At $r=2.5m$ the outgoing speed is $1/9\gt 0$, so inner boundary data would be needed. This checks the physical null sector only; all gauge and constraint characteristics of the chosen formulation must pass the same test, and a moving surface requires subtracting its speed. The horizon is $r=2m$ in this chart, unlike the isotropic initial-data coordinate radius $m/2$.

Moving punctures compress the interior

Puncture data factor an analytic singularity initially, but under advective $1+\log$ slicing and Gamma-driver shift the evolved slice approaches a trumpet. Grid points do not reach the physical singularity in finite coordinate time, while the conformal variable remains numerically manageable. No inner boundary is introduced.

Resolution near the puncture remains critical: steep gradients, upwind advection, refinement-box motion, and algebraic projections interact. The puncture coordinate path is gauge, not the black hole’s invariant trajectory. Campanelli et al. 2006, Figure 3 and Table I provides an early binary demonstration with convergent radiation and horizon quantities.

Unlike excision, puncture grids include an unresolved interior coordinate region. Stability relies on causal and gauge properties plus numerical dissipation, not on resolving physical fields through the singularity. Do not interpret values at the puncture as geometric observables.

Matched benchmarks

First evolve the same stationary Schwarzschild or Kerr exterior with each strategy. Match physical mass and spin, evolution duration, wave-zone resolution, and outer boundary while allowing the gauges appropriate to each method. Use at least three resolutions. Compare horizon mass and spin, exterior curvature invariants at geometrically identified radii, constraints outside the horizon, and spurious radiation. Vary excision margin and puncture near-zone resolution independently.

Then compare a binary case through inspiral and merger. Extract waveforms in one asymptotic frame and estimate phase and amplitude error without using arbitrary alignment to hide method differences. Check remnant horizon charges and radiated energy–angular-momentum balance. Excision trajectories and puncture tracks need not agree; waveform and invariant charge agreement within independent uncertainties is the central test.

The assumptions include censorship of the excised singular region by a horizon, causal outflow for all formulation modes, and adequate exterior resolution. Horizon finding is slicing dependent, so keep an explicit buffer and monitor it continuously. Moving-puncture success in BSSN does not automatically transfer to a formulation with different characteristic speeds.

Environment and failure criteria

Pin formulation and gauge, excision map and controller or puncture variable, characteristic-speed convention, stencil or flux, dissipation, integrator, CFL factor, mesh hierarchy, horizon finder, outer boundary, precision, and code revision. A clean interface could be black-hole-evolve --config isolated-method-compare.toml --output run/, emitting horizons, characteristic margins, constraints, invariants, waveforms, refinement sweeps, and environment metadata. This is a reproducibility specification rather than a claim that a solver was executed here.

Reject excision when any mode becomes incoming, the surface crosses the horizon, or map control loses convergence. Reject puncture evolution when gradients remain under-resolved, constraint contamination leaks outside the horizon, or results depend on arbitrary dissipation. A shared failure mode is a stable coordinate evolution with drifting horizon mass. Comparing coordinate tracks instead of invariants is not evidence of physical disagreement.

Diagnostic

An excision surface lies inside the apparent horizon, but one gauge characteristic has positive speed into the computational domain for a short interval. The metric remains smooth. May the run be accepted because no instability appeared?

Solution

No. With the stated normal pointing from the excised region into the retained domain, that positive speed is incoming and the excision surface is not pure outflow for the implemented formulation. The discrete scheme lacks justified incoming data and can silently contaminate the exterior. Move or reshape the surface, adjust gauge or map control, and maintain a resolved negative-speed safety margin. Then verify exterior constraints and invariants across resolution.

References

• 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.
• Scheel, M. A., Pfeiffer, H. P., Lindblom, L., et al. (2006). “Solving Einstein’s equations with dual coordinate frames.” Physical Review D 74, 104006. arXiv.