Black-hole punctures and excision

In time-symmetric conformally flat Schwarzschild initial data, the conformal factor has a simple inverse-radius singular term. The zero-coordinate-radius puncture represents another asymptotic end of the initial spatial geometry rather than an ordinary grid point at the physical curvature singularity. Isotropic radius and areal radius are different: the latter includes the square of the conformal factor. Treating the coordinate puncture distance as a physical separation therefore gives a misleading interpretation.

Moving-puncture evolution uses suitably chosen variables, lapse and shift so that puncture coordinates can move while the numerically evolved fields remain manageable. Inverse conformal-factor variables help avoid directly representing a divergent logarithmic conformal factor. Initial regularity, stencil behavior and gauge choices still matter. This method does not simply assume that every field at the puncture is smooth to arbitrary order or that errors inside a coordinate horizon are automatically irrelevant to all evolved modes.

Excision removes an interior region from the numerical domain and places a boundary within the black-hole region. Physical causal intuition motivates this strategy, but a successful PDE boundary also needs the characteristics of the chosen formulation and gauge. With the normal pointing out of the retained domain into the excised region, positive normal speeds flow out of the computational domain. Incoming fields would require appropriate boundary information and invalidate an assumption of pure outflow.

A coordinate apparent horizon is not by itself a guarantee that every gauge or constraint characteristic has the needed sign at the excision surface. The surface can move, so characteristic speeds must be measured relative to its motion. Coordinate control systems and boundary placement are consequently part of a practical method. The toy speeds below illustrate the sign test only; a real first-order reduction may have additional modes and parameters that must all be checked.

An apparent horizon is an outermost closed marginally outer trapped surface on a chosen slice; locating it is an elliptic surface problem and depends on the slicing. An event horizon instead uses global causal information. The area of a located surface and an angular momentum derived with an axial or appropriately approximated symmetry vector can support a quasi-local diagnostic. Finding a surface is therefore different from assigning a coordinate center and radius.

For an uncharged Kerr horizon, or a suitable canonical axisymmetric isolated-horizon mass prescription, area defines an irreducible mass and area together with angular momentum gives a Christodoulou-type mass. The dimensionless spin is angular momentum divided by mass squared. During a distorted dynamical phase, approximate symmetry, horizon finding, resolution and the adopted quasi-local definitions contribute uncertainty. A Kerr formula used diagnostically should not silently become an exact invariant characterization of every evolving surface.