Puncture and Excision Black-Hole Data

Black-hole initial data can represent the interior either by analytically isolating a puncture singularity or by removing an inner region and imposing boundary conditions on its surface. Both approaches solve exterior constraints, but they encode topology, regularity, spin, and gauge data differently. The numerical comparison must therefore use invariant exterior quantities—not coordinate radii or puncture locations—and must separate elliptic error from junk radiation generated by the chosen free data.

Required background. Hypersurfaces and Foliations supplies the slice geometry, extrinsic curvature, and horizon-normal projections.

Helpful background. Formulations of General Relativity develops the constraint decompositions, Ordinary Differential Equations supplies nonlinear iteration concepts, and Relativistic Matter and Compact Objects supplies compact-object equilibrium and charge interpretation.

Punctures and inner boundaries

For conformally flat puncture data, use $c=1$ and split the conformal factor as

$$\psi=1+\sum_A\frac{m_A}{2r_A}+u,$$

where $r_A$ is a coordinate length and $m_A=G M_{A,\rm bare}$ is a geometric-length parameter, not the horizon or ADM mass. The singular pieces are known and $u$ is finite at each puncture. Momentum and spin can be supplied through analytic conformal extrinsic curvature. The elliptic solver handles $u$ on a domain that contains the puncture points, using finite-difference stencils or regularized spectral subdomains. Brandt and Brügmann 1997, Eqs. (7)–(11) and Table I gives the multiple-black-hole construction and numerical mass checks.

Excision removes an interior region bounded by a closed surface. The elliptic fields then need inner conditions, often motivated by an apparent horizon in quasi-equilibrium: vanishing outgoing expansion, tangential shift tied to spin, and a condition for normal motion. A lapse boundary condition often fixes slicing rather than physical geometry. Cook and Pfeiffer 2004, §§II–IV, Figs. 1–5 and Table I separates the quasi-equilibrium conditions and examines lapse choices.

The two approaches are not related by merely deleting grid points. Puncture parameters are not directly horizon masses; excision-surface shape and boundary data need not coincide with a stationary horizon. Each representation demands its own regularity and domain checks.

The puncture is a second asymptotic end

For one time-symmetric Schwarzschild hole, $u=0$ and $\psi=1+m/(2r)$ with $m=GM$. The areal radius of a coordinate sphere is

$$R(r)=\psi^2r=r+m+\frac{m^2}{4r}.$$

Its minimum occurs at $r=m/2$, where $R=2m$ and the horizon area is $16\pi m^2$. As $r\to0$, $R\to\infty$: the puncture represents another asymptotically flat end of the initial wormhole slice, not the curvature singularity $R=0$. The inversion $r'=m^2/(4r)$ makes the second end explicit and exchanges the two exterior regions. This explains why a divergent conformal factor need not mean divergent curvature. The later trumpet produced by moving-puncture gauge has a different limiting slice geometry; see Excision and Moving-Puncture Black Holes. The initial construction is described in Baumgarte and Shapiro 2010, §3.1.2, pp. 58–61.

Compare invariant exterior data

Begin with one nonspinning Schwarzschild black hole. For puncture data, recover the analytic isotropic conformal factor. For excision, choose boundary data representing the same mass and asymptotic frame. At three or more resolutions and outer radii, report independent Hamiltonian and momentum norms, ADM mass, apparent-horizon area, and

$$M_H=\sqrt{\frac{A_H}{16\pi G^2}}$$

for zero spin. The extrapolated $M_{\rm ADM}$ and $M_H$ should agree within the combined numerical uncertainty. Their agreement tests normalization and the region between horizon and infinity; a small local residual alone does not.

For spinning data, compare a declared quasi-local spin measure and the Christodoulou relation, not the input puncture spin parameter alone. Vary the excision shape and lapse boundary condition while holding physical target charges fixed. If exterior invariants remain stable but the lapse changes, the variation is gauge evidence. If mass or spin changes, the boundary prescription altered the physical slice or has not converged.

Generic spinning conformally flat puncture data are not an exact stationary Kerr slice. They can contain gravitational radiation that remains after the elliptic constraints converge. A nonzero relaxation burst from such data is therefore not automatically a discretization error; test how it changes with resolution and with the freely chosen initial geometry. Conversely, an exact stationary Kerr benchmark should have no physical outgoing burst. Keep these two tests distinct when assessing initial-data quality.

Evolution as a quality diagnostic

Evolve both isolated data sets with the same formulation, gauge family, resolution hierarchy, and outer boundary. Monitor constraint transients, horizon mass and spin, and spurious gravitational radiation. Stationary Kerr or Schwarzschild data should settle without a physical burst beyond the numerical and slicing transient expected from the mismatch between initial and evolution gauges.

For binaries, compare component horizon charges, ADM energy and angular momentum, eccentricity measures, and junk-radiation energy. The assumptions are asymptotic flatness and the specified conformal ansatz. Near-extremal spins and close separations stress both regularity and quasi-equilibrium assumptions; an isolated benchmark does not establish accuracy there.

Refine elliptic resolution, evolution resolution, outer boundary, and extraction radius independently. Otherwise a cleaner waveform from one data type may simply reflect a more favorable grid representation. Report the numerical uncertainty on the difference between approaches.

Environment and failure criteria

Pin the conformal decomposition, puncture singular fields or excision surfaces, all inner conditions, lapse and shift data, domain maps, solver tolerances, precision, charge integrators, evolution gauge, mesh hierarchy, and code revision. A reproducible interface could be black-hole-data --config isolated-comparison.toml --output run/, emitting fields, constraints, surfaces, charges, refinement sweeps, and evolution diagnostics. It specifies an executable record without claiming that one is included here.

Reject puncture data whose regular remainder loses convergence at the puncture, and reject excision data whose outcome depends on unresolved inner-surface shape. Further failure modes include confusing bare puncture mass with horizon mass, imposing incompatible horizon conditions, interpreting lapse-boundary variation as physical spin, and comparing junk radiation after different waveform alignments. Constraint convergence cannot repair a physically mismatched target.

Diagnostic

Two excision solutions have the same horizon area and constraint residual. Changing only the lapse inner-boundary condition changes the initial lapse strongly but leaves extrapolated ADM mass, horizon spin, and later waveform unchanged within error. Did the physical initial data change?

Solution

The evidence is consistent with a gauge change in the measured exterior observables, but does not establish equality of the full data. Although lapse is not one of the constraint fields $(\gamma_{ij},K_{ij})$, it enters an extended thin-sandwich solve and can indirectly alter them. Check geometry and convergence as well as charges; compare waveforms in one specified asymptotic frame, allowing only a documented common time and phase convention. Report the observed agreement rather than treating an unchanged horizon area as a uniqueness theorem.

References

• Baumgarte, T. W., and Shapiro, S. L. (2010). Numerical Relativity: Solving Einstein’s Equations on the Computer. Cambridge University Press. DOI.
• Brandt, S., and Brügmann, B. (1997). “A simple construction of initial data for multiple black holes.” Physical Review Letters 78, 3606–3609. arXiv.
• Cook, G. B., and Pfeiffer, H. P. (2004). “Excision boundary conditions for black-hole initial data.” Physical Review D 70, 104016. arXiv.