Black Hole Excision and Puncture Methods

Every numerical relativity code that has produced a black-hole waveform has had to answer one blunt question: what do you do with the singularity? A numerical evolution integrates the ADM or BSSN variables forward on a spatial grid, and for a black hole that grid, left alone, marches toward the curvature singularity behind the horizon — for Schwarzschild the point $R=0$ where the Kretschmann scalar $R_{abcd}R^{abcd}=48M^2/R^6$ diverges. Reach it and the metric and its derivatives blow up; the poison spreads outward at light speed and the simulation crashes within a light-crossing time. Yet the physics we want — the gravitational radiation — lives outside the horizon, in a region the interior can never influence. The problem is sharp: advance the exterior faithfully while never evaluating a field at the singularity. Two philosophies answer it, and they are near-opposites. Excision leans on causality: the interior of an apparent horizon cannot signal outward, so delete a chunk of grid inside the horizon and impose no boundary condition — provided the cut is a pure outflow surface, the truncated domain still evolves as a well-posed Cauchy problem. The puncture method instead keeps the singular point but strips out its analytic form: write $\psi=\psi_{\mathrm{BL}}+u$ , let the divergent $1/r$ piece $\psi_{\mathrm{BL}}$ carry the singularity, and solve numerically only for the smooth remainder $u$ . The 2005 moving-puncture breakthrough showed that with BSSN, $1+\log$ slicing, and a $\Gamma$ -driver shift the punctures could drift freely across the grid — no excision boundary, no horizon tracking — and stay stable through merger. Generalized-harmonic codes went the excision route the same year. Both work; both feed the waveform banks that gravitational-wave detectors match against.

Key takeaways

A black hole's curvature singularity, if the grid reaches it, produces diverging fields and destroys the evolution within a light-crossing time; excision and puncture methods keep it out of the computational domain.
Excision deletes a region inside the apparent horizon and imposes no boundary condition, relying on causality — nothing inside a trapped surface can influence the exterior, so an outflow cut needs no data.
The puncture method absorbs each singularity into an analytic factor, $\psi=\psi_{\mathrm{BL}}+u$ , and evolves only the smooth $C^2$ correction $u$ — the puncture is an ordinary grid point.
Moving punctures — the 2005 breakthrough — let the puncture coordinates advect across the grid under BSSN plus $1+\log$ slicing and the $\Gamma$ -driver shift, the lapse collapsing to freeze evolution near the singularity.
A single non-spinning puncture relaxes to a stationary trumpet slice of finite areal radius $R_0\approx1.31M$ , inside the horizon $R=2M$ , with no inner boundary.
Key idea: A black hole's curvature singularity must be kept out of the numerical domain. Excision cuts out a causally disconnected region inside the apparent horizon (no boundary condition required if the surface is outflow); puncture methods instead absorb the singularity into an analytic factor of the conformal field so only smooth quantities are evolved numerically — and with the right BSSN gauge the punctures may move freely.

Where this lives today

Every gravitational wave the LIGO-Virgo-KAGRA network has announced was compared against a bank of templates whose spine is a set of numerical-relativity waveforms — essentially all produced by codes using one of these two strategies. Moving-puncture BSSN codes (Einstein Toolkit, BAM, LazEv) and excision-based spectral codes like SpEC generate the merger waveforms that calibrate the effective-one-body and phenomenological (IMRPhenom) models used in matched filtering; the masses and spins of GW150914 and every event since were inferred against banks rooted in these simulations. The methods are also where the frontier sits: Bowen-York data cannot represent a hole spinning faster than $a/M\approx0.93$ , because conformal flatness cannot encode a near-extremal Kerr slice — so excision with Kerr-Schild data, or superposed-Kerr puncture data, pushes coverage toward the high-spin systems seen in real detections. Both methods lean on apparent-horizon finders every step: to keep the cut inside the moving horizon, and to read masses and spins and confirm both punctures hide behind a common horizon after merger. As Cosmic Explorer, the Einstein Telescope, and LISA demand waveforms accurate to a part in $10^3$ across huge mass ratios, the pressure on these techniques has only grown.

The physical picture

Think of the event horizon as a one-way membrane. Whatever happens at the singularity behind it cannot reach the exterior, so a scheme evolving only the outside need not "know" the interior. Excision deletes the ignorable region. The art is drawing the cut so the difference stencils never call for data from across it: the boundary must sit where every characteristic points inward, a pure outflow surface. Place it inside a trapped surface — where both null congruences converge — and causality does the rest. No boundary condition, no reflected junk; the excised region is simply gone. The puncture picture is geometric and, at first sight, alarming: you keep the singular point. On a maximal slice the Schwarzschild spatial metric is $\mathrm{d}s^2=\psi^4(\mathrm{d}r^2+r^2\mathrm{d}\Omega^2)$ with $\psi=1+M/2r$ , and as $r\to0$ the factor $\psi^4\sim(M/2r)^4$ diverges. But proper distance to $r=0$ is infinite — the puncture is not a place you can reach; it is a second asymptotically flat infinity, the far mouth of an Einstein-Rosen bridge. Factor the analytic $1/r$ out of $\psi$ and only smooth fields remain, so the grid straddles the puncture without noticing it. With moving punctures the gauge does the singularity-avoidance work. The $1+\log$ lapse collapses toward zero wherever the trace $K$ is large and positive — near the throat — so coordinate time freezes and the foliation never advances into the singularity, while the $\Gamma$ -driver shift advects the puncture's coordinate location across the grid like a feature riding a flow. Both strategies agree far from the hole; they differ only in how they treat the unobservable interior — the region general relativity tells us is causally irrelevant to the waves a detector will ever see.

Figure

Two ways to keep the singularity off the grid. Left: excision deletes a region inside the apparent horizon, where every characteristic points inward, so the outflow boundary needs no data. Right: the puncture factors the analytic 1/r spike ψ_BL out of the conformal factor, leaving only the smooth correction u for the solver

The second figure makes the "second infinity" quantitative. On the time-symmetric isotropic slice the areal radius $R=r(1+M/2r)^2$ does not fall to zero as $r\to0$ — it turns around at a minimum $R=2M$ (the throat, exactly the horizon here) and climbs back to infinity, so the coordinate puncture is a second asymptotic end, not the singularity. The $1+\log$ evolution instead relaxes to a stationary trumpet: same exterior, but the coordinate now runs down to a finite areal radius $R_0\approx1.31M$ inside the horizon.

Figure

Areal radius R versus coordinate radius r on two slices of Schwarzschild mass M. The time-symmetric isotropic slice R = r(1+M/2r)² bottoms out at R = 2M (throat = horizon) then climbs back to infinity as r → 0 — the puncture is a second asymptotic end. The 1+log moving-puncture slice instead settles to a finite trumpet radius R₀ ≈ 1.31M inside the horizon

The core equations

The starting point is a single non-spinning black hole in isotropic coordinates on a maximal slice — the geometry every puncture code reduces to for one hole, already containing the puncture as a feature.

\mathrm{d}s^2=\psi^4\big(\mathrm{d}r^2+r^2\,\mathrm{d}\Omega^2\big),\qquad \psi=1+\frac{M}{2r} \qquad (1)

where: $\psi$ — conformal factor relating the physical to the flat 3-metric, $\gamma_{ij}=\psi^4\,\delta_{ij}$ ; $M$ — ADM mass; $r$ — isotropic radial coordinate, horizon at $r=M/2$ ; $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\varphi^2$ . Reading it: $\psi$ is harmonic with a $1/r$ pole, and that pole is the puncture. For several holes one superposes poles. The Brill-Lindquist factor carries all the singular structure analytically; the numerically-solved $u$ absorbs whatever momentum and spin add on top.

\psi=\psi_{\mathrm{BL}}+u,\qquad \psi_{\mathrm{BL}}=1+\sum_{i=1}^{N}\frac{M_i}{2\,|\vec{x}-\vec{x}_i|} \qquad (2)

where: $\psi_{\mathrm{BL}}$ — Brill-Lindquist factor holding every analytic $1/r$ singularity; $u$ — regular $C^2$ correction found on the grid; $M_i$ — bare puncture-mass of hole $i$ ; $\vec{x}_i$ — location of the $i$ -th puncture. Reading it: the singular content lives entirely in $\psi_{\mathrm{BL}}$ , so the solver only touches the smooth $u$ . The correction obeys the Hamiltonian constraint as an elliptic equation whose right-hand side stays bounded even where $\psi_{\mathrm{BL}}\to\infty$ — the reason the method works.

\hat{\nabla}^2 u=-\frac{1}{8}\,\psi_{\mathrm{BL}}^{-7}\,\bar{A}_{ij}\bar{A}^{ij}\,\big(1+\psi_{\mathrm{BL}}^{-1}u\big)^{-7} \qquad (3)

where: $\hat{\nabla}^2$ — flat-space Laplacian; $\bar{A}_{ij}$ — conformally rescaled transverse-traceless extrinsic curvature (typically Bowen-York); $u$ — the regular correction. Reading it: the source $\propto\psi_{\mathrm{BL}}^{-7}$ vanishes at each puncture, so $u$ is smooth there and needs no inner boundary. Excision is governed by the geometry of the cut: the apparent-horizon condition locates the outermost trapped surface, so the boundary can be placed safely inside it.

\Theta=D_i s^i+K_{ij}s^i s^j-K=0 \qquad (4)

where: $\Theta$ — expansion of the outgoing null congruence, zero on the apparent horizon and negative inside; $s^i$ — outward unit normal within the slice; $D_i$ — covariant derivative of $\gamma_{ij}$ ; $K_{ij},K$ — extrinsic curvature and trace. Reading it: $\Theta<0$ marks the trapped region where all light bends inward — the safe place to cut. Moving punctures need no cut but need a gauge that both freezes the singularity and carries the puncture: $1+\log$ slicing and the $\Gamma$ -driver shift.

\partial_t\alpha=-2\alpha K,\qquad \partial_t\beta^i=\tfrac{3}{4}\,\tilde\Gamma^i-\eta\,\beta^i \qquad (5)

where: $\alpha$ — lapse, collapsed near punctures by $1+\log$ ; $\beta^i$ — shift, driven to advect the puncture; $\tilde\Gamma^i$ — BSSN conformal connection functions; $\eta$ — damping parameter of order $1/M$ . Reading it: the lapse kills evolution where $K$ is large, the shift chases the connection so coordinates flow with the hole. To give a boosted or spinning hole its momentum and spin analytically, one adds the Bowen-York extrinsic curvature — an exact conformally-flat solution of the momentum constraint.

\bar{A}^{ij}_{\mathrm{BY}}=\frac{3}{2r^2}\big(P^i n^j+P^j n^i-(\delta^{ij}-n^i n^j)P_k n^k\big)+\frac{3}{r^3}\big(\epsilon^{ikl}S_k n_l n^j+\epsilon^{jkl}S_k n_l n^i\big) \qquad (6)

where: $P^i$ — ADM linear momentum; $S^i$ — spin; $n^i=x^i/r$ — unit radial vector; $\epsilon^{ikl}$ — flat-space Levi-Civita symbol. Reading it: momentum falls off as $1/r^2$ and spin as $1/r^3$ , so $\bar{A}_{ij}\bar{A}^{ij}\sim r^{-6}$ — slower than $\psi_{\mathrm{BL}}^{-7}\sim r^{7}$ shrinks the source. Finally, the quantity that makes the problem unavoidable: the invariant that blows up at the singularity.

R_{abcd}R^{abcd}=\frac{48\,M^2}{R^6} \qquad (7)

where: $R_{abcd}$ — Riemann tensor; $R$ — areal (Schwarzschild) radius; $M$ — mass. Reading it: the Kretschmann scalar is coordinate-invariant, so no chart removes the divergence at $R=0$ — it can only be hidden behind a horizon or factored out.

Derivation AWhy an excision boundary needs no condition

The goal is to demonstrate, from causal structure alone, that an inner boundary placed inside the apparent horizon is a pure outflow surface, so the truncated Cauchy evolution stays well posed with no data imposed on the cut. We combine the characteristic structure of a strongly hyperbolic system near the boundary with the trapped-surface condition $\Theta<0$ that defines a horizon's interior, and find that every characteristic at a well-placed cut leaves the physical domain.

Step 1. Inside the apparent horizon both null congruences converge, $\Theta_{\text{out}}<0$ and $\Theta_{\text{in}}<0$ ; no signal can cross such a trapped surface outward. Why: this is the geometric content of "trapped," the single-slice analogue of the event horizon's one-way property.

Step 2. The boundary characteristic speeds along the outward normal are $\lambda_\pm=-\beta^n\pm\alpha\,c_s$ , shifted from the light-cone values by the normal shift $\beta^n$ . A large inward (superluminal) shift, or a spacelike cut, gives both roots the same sign, pointing into the excised region. Why: the shift is gauge we control, so we tilt the numerical light cones to guarantee outflow even where physical cones would not.

Step 3. If every characteristic leaves the physical domain, no information propagates in and no boundary data are required — the surface is "outflow." Why: a hyperbolic boundary needs data only on incoming characteristics; with none, the interior evolution closes the system.

Step 4. In practice one differences the boundary with one-sided (upwinded) operators referencing only interior points, and relocates the cut each step with a horizon finder so it stays inside the moving horizon. Why: the guarantee holds only if the discretization also refuses to reach across the cut and the cut tracks the horizon.

Go deeper · Derivation B · 1 minExistence of a smooth puncture solution

Derivation BExistence of a smooth puncture solution

The goal is to prove that the Brandt-Bruegmann puncture equation has a unique bounded, twice-differentiable solution $u$ on all of $\mathbb{R}^3$ , including at the punctures themselves, so no inner boundary condition is ever needed. The strategy is to examine the constraint source near a puncture where $\psi_{\mathrm{BL}}\to\infty$ , show it stays bounded, then invoke a maximum-principle argument for existence, uniqueness, and positivity of $u$ .

Step 1. Near the $i$ -th puncture $\psi_{\mathrm{BL}}\sim M_i/(2|\vec{x}-\vec{x}_i|)$ , so $\psi_{\mathrm{BL}}^{-7}\sim r^7\to0$ , while Bowen-York gives $\bar{A}_{ij}\bar{A}^{ij}\sim r^{-6}$ . Their product behaves like $r$ , so the source of $\hat\nabla^2 u$ is bounded — indeed continuous — at the puncture. Why: this cancellation is the whole trick; $\psi_{\mathrm{BL}}$ enters with a high enough negative power to overwhelm the curvature source's divergence.

Step 2. Write $\hat\nabla^2 u=-h(\vec{x},u)$ with $h\ge0$ and $\partial h/\partial u\le0$ . Then $u=0$ is a subsolution, since $\hat\nabla^2 0+h(\vec{x},0)=h\ge0$ ; and the bounded solution $w$ of the linear problem $\hat\nabla^2 w=-h(\vec{x},0)$ is a supersolution, since $w\ge0$ and $\partial h/\partial u\le0$ give $\hat\nabla^2 w+h(\vec{x},w)=h(\vec{x},w)-h(\vec{x},0)\le0$ ; the sub-/super-solution theorem then delivers a unique solution between them. Why: monotone dependence on $u$ makes the problem have exactly one answer — one initial-data set.

Step 3. Far away the source has compact effective support, so $u\sim c/|\vec{x}|+O(|\vec{x}|^{-2})$ , and $c$ feeds the ADM mass $M_{\mathrm{ADM}}=\sum_i M_i+(\text{corrections carried by }u)$ . Why: the $1/r$ tail is how the solved correction contributes to the physical mass.

Step 4. Since $u\in C^2(\mathbb{R}^3)$ everywhere, all singular content sits in the analytic $\psi_{\mathrm{BL}}$ and is never differentiated; the puncture is an ordinary interior grid point. Why: this removes the need for an inner boundary and lets a plain solver march through the puncture.

Go deeper · Derivation C · 1 minSingularity avoidance of one-plus-log slicing

Derivation CSingularity avoidance of one-plus-log slicing

The goal is to show that the $1+\log$ lapse drives $\alpha\to0$ where the trace $K$ is large and positive, freezing coordinate evolution before the foliation reaches the singularity, and that a single puncture relaxes to a finite stationary trumpet slice. We combine the lapse evolution equation with the meaning of $K$ as the fractional rate of change of proper volume, then read off the late-time end state.

Step 1. Proper volume evolves as $\partial_t\sqrt{\gamma}=-\alpha K\sqrt{\gamma}+\partial_i(\sqrt{\gamma}\,\beta^i)$ , so a slice dragged toward a collapsing throat carries large positive $K$ . Why: $K$ literally measures how fast the slice's volume shrinks, which is extreme near a forming singularity.

Step 2. The $1+\log$ condition reads $\partial_t\ln\alpha=-2K$ , integrating to $\alpha\to e^{-2\int K\,\mathrm{d}t}$ ; where $K$ stays large and positive the lapse decays exponentially. Why: the lapse converts coordinate to proper time, so $\alpha\to0$ means clocks near the throat stop advancing.

Step 3. With $\alpha\to0$ near the puncture, proper time ceases to advance: the foliation wraps around the throat and never reaches $R=0$ , so the exterior evolves while the interior is frozen. Why: no boundary condition and no excision are needed because the gauge itself refuses to march into the singular region.

Step 4. For a single non-spinning puncture the gauge relaxes to a time-independent trumpet of limiting areal radius $R_0\approx1.31M$ , well inside the horizon $R=2M$ , on which all fields are finite. Why: a stationary finite end state is what a long stable run needs — the code settles rather than sharpening.

Limiting cases

How the two strategies behave in the characteristic regimes a code actually meets:

Case	Result	Meaning
Single Schwarzschild puncture	$\psi=1+\dfrac{M}{2r},\ u\equiv0$	The correction vanishes; Brill-Lindquist is exact and the constraint holds analytically.
Many holes (Brill-Lindquist)	$\psi_{\mathrm{BL}}=1+\sum_i\dfrac{M_i}{2\,\vert\vec{x}-\vec{x}_i\vert}$	Time-symmetric punctures superpose; ADM mass is $\sum_i M_i$ plus interaction terms in $u$ .
Spinning or boosted holes	$\bar{A}^{ij}_{\mathrm{BY}}\ \text{with}\ P^i,S^i$	Bowen-York supplies momentum and spin analytically; only $u$ is solved numerically.
Excision cut placement	$\Theta<0$ inside $\Theta=0$	The boundary must lie strictly inside the apparent horizon to stay causally outflow.
Late-time single puncture	$R\to R_0\approx1.31M$	The $1+\log$ gauge relaxes to a stationary trumpet inside the horizon — no inner boundary.

Common trapThe puncture is a particle sitting in our space, and excision means throwing away real physics

Both halves are wrong, and both matter. The puncture is not matter and not a point in our universe: it is the image of the spatial infinity of a second asymptotically flat region, reached through an Einstein-Rosen bridge at infinite proper distance. Field values "at" the puncture are not physical, and treating $\psi_{\mathrm{BL}}$ as a Newtonian source on the grid misreads the geometry. Excision, likewise, discards nothing physical: it removes only a region causally disconnected from the exterior, which could never have influenced a wave the detector sees. The real danger is misplacement — putting the cut just outside the horizon, or where characteristics are not purely outflow, injects spurious boundary data that corrupts the exterior. Causality does the work in both methods; break the causal placement and the guarantee evaporates.

Worked examples

1 · Locating the horizon on an isotropic slice (fully worked)

A non-spinning black hole of mass $M$ is written in isotropic coordinates, $\psi=1+M/2r$ . Find the horizon radius, the conformal factor there, and confirm the areal radius equals $2M$ .

Step 1. The areal radius is $R=r\psi^2=r(1+M/2r)^2=r+M+\dfrac{M^2}{4r}$ , since a sphere of coordinate radius $r$ has area $4\pi(r\psi^2)^2$ . The throat is where $R$ is smallest: $\dfrac{\mathrm{d}R}{\mathrm{d}r}=1-\dfrac{M^2}{4r^2}=0$ gives $r=M/2$ . The throat sits at $r=M/2$ .

Step 2. The conformal factor there is $\psi(M/2)=1+\dfrac{M}{2\,(M/2)}=1+1=\mathbf{2}$ , so $\psi^4=16$ and proper lengths near the throat are stretched by four.

Step 3. The areal radius at the throat is $R=(M/2)(2)^2=\mathbf{2M}$ — exactly the Schwarzschild horizon. On this time-symmetric slice throat, minimal surface, apparent horizon, and areal radius $2M$ all coincide; for $r<M/2$ the areal radius climbs again. The isotropic horizon $r=M/2$ maps to the areal horizon $R=2M$ , and the puncture $r\to0$ is a second infinity, not $R=0$ .

2 · Why the grid can straddle the puncture

Take the same slice with $M=1$ . Show that (a) the proper distance to the puncture is infinite, and (b) even with spinning Bowen-York data — spin scale $S\sim M^2$ , so $\bar{A}_{ij}\bar{A}^{ij}\sim C\,r^{-6}$ with $C\approx1$ in units $M=1$ , the most divergent case (pure linear-momentum data falls off only as $r^{-4}$ ) — the elliptic source for $u$ stays finite at the puncture.

Step 1. Proper radial distance uses $\mathrm{d}\ell=\psi^2\,\mathrm{d}r$ , so $\ell(r)=\displaystyle\int_r^{a}\Big(1+\frac{1}{2r'}\Big)^2\mathrm{d}r'$ . Near the puncture the integrand $\to1/(4r'^2)$ , and $\displaystyle\int_r \frac{\mathrm{d}r'}{4r'^2}=\frac{1}{4r}\to\infty$ as $r\to0$ . Proper distance to the puncture diverges — it is a second infinity, not a point on the grid.

Step 2. For the constraint source, near a puncture $\psi_{\mathrm{BL}}^{-7}\sim(2r)^7=128\,r^7$ while $\bar{A}_{ij}\bar{A}^{ij}\sim C\,r^{-6}$ . The product is $\propto r^{7}\cdot r^{-6}=r\to0$ . The elliptic source vanishes linearly, so $u$ is smooth ( $C^2$ ) there.

Step 3. Concretely at $r=0.01M$ the analytic factor is $\psi_{\mathrm{BL}}\approx M/(2r)=50$ , so $\psi_{\mathrm{BL}}^{-7}\approx50^{-7}\approx1.3\times10^{-12}$ — negligible even though $\bar{A}_{ij}\bar{A}^{ij}\sim r^{-6}=10^{12}$ is enormous. A plain finite-difference solver marches through the puncture with no special treatment and no inner boundary.

Physics of the portrait

Excision and punctures are the work of a forty-year relay — from the first two-black-hole data, through the analytic taming of the singularity, to the gauge tricks that let the punctures move. The portrait here is the moving puncture method itself.

Origins · 1 minmoving puncture method and how this was discovered

moving puncture method

the 2005 breakthrough that let black holes advect freely across the grid · numerical relativity of binary black holes

The story starts with the initial-value problem. In 1963 Dieter Brill and Richard Lindquist wrote time-symmetric conformally flat data for several non-spinning black holes as a sum of $1/r$ poles, and around 1980 Jeffrey Bowen and James York added an analytic conformally flat solution of the momentum constraint giving each hole momentum and spin — the ingredients of every puncture data set since. The decisive analytic step came in 1997, when Steven Brandt and Bernd Brügmann showed the singularity could be factored out: split the conformal factor into the analytic Brill-Lindquist piece plus a smooth correction obeying an elliptic equation with a bounded source, so the puncture becomes an ordinary grid point. In parallel, after the failure of the weakly-hyperbolic ADM system, Nakamura, Oohara and Kojima, then Shibata and Nakamura, and Baumgarte and Shapiro built the strongly-hyperbolic BSSN formulation that could evolve such data. The last piece was gauge: for a decade the punctures were pinned to fixed coordinates, distorting the grid as the holes orbited. Then in 2005 two groups independently found that with BSSN, $1+\log$ slicing, and a $\Gamma$ -driver shift the punctures could move with the code stable through merger — Campanelli, Lousto, Marronetti and Zlochower at Brownsville, and Baker, Centrella, Choi, Koppitz and van Meter at Goddard. The same year Frans Pretorius reached merger by the other road, generalized harmonic evolution with excision. Two philosophies, one result: the first stable binary-black-hole inspiral-merger-ringdown waveforms.

History

1963

— Brill and Lindquist write time-symmetric multi-black-hole conformally flat data

1980

— Bowen and York give analytic momentum and spin extrinsic-curvature data

1987

— Nakamura, Oohara and Kojima introduce the conformal-traceless evolution variables

1997

— Brandt and Bruegmann factor the singularity out, making the puncture an ordinary grid point

2005

— Pretorius reaches merger by generalized harmonic evolution with excision

2006

— Campanelli et al. and Baker et al. publish the moving-puncture binary merger

Going further

Texts. Baumgarte & Shapiro, Numerical Relativity (2010), ch. 12-13 (excision, punctures, moving punctures) · Alcubierre, Introduction to 3+1 Numerical Relativity (2008), ch. 10-12 · Misner, Thorne & Wheeler, Gravitation, ch. 31-33 (singularities, Einstein-Rosen bridge) · Wald, General Relativity, ch. 6, 10, 12 · Carroll, Spacetime and Geometry, ch. 5-6 · Hartle, Gravity, ch. 12 · Schutz, A First Course in General Relativity, ch. 10-11. Key results used throughout: the isotropic factor $\psi=1+M/2r$ with horizon $r=M/2$ and areal-radius map $R=r\psi^2$ ; the Kretschmann scalar $48M^2/R^6$ ; the Brill-Lindquist superposition and the Brandt-Bruegmann puncture equation with its $r^7$ source falloff (Brandt & Bruegmann, Phys. Rev. Lett. 78, 3606, 1997); Bowen-York momentum and spin data (Bowen & York, Phys. Rev. D 21, 2047, 1980); the $1+\log$ lapse and $\Gamma$ -driver shift; and the trumpet radius $R_0\approx1.31M$ as the stationary $1+\log$ end state (Hannam et al. 2007). The moving-puncture breakthrough waveforms are from Campanelli, Lousto, Marronetti & Zlochower, Phys. Rev. Lett. 96, 111101 (2006); Baker, Centrella, Choi, Koppitz & van Meter, Phys. Rev. Lett. 96, 111102 (2006); and Pretorius, Phys. Rev. Lett. 95, 121101 (2005).

Prerequisites. ADM 3+1 Decomposition of Spacetime · Hamiltonian and Momentum Constraints · Conformal Method for Constraint Solving · BSSN Formulation · Apparent Horizons and Horizon Finders. Leads to. Numerical Relativity and Binary Black Hole Mergers · Gauge Conditions in Numerical Relativity · Generalized Harmonic Gauge.

Ask the tutor: Ask me to walk through how the $\psi_{\mathrm{BL}}^{-7}$ falloff makes the puncture an ordinary grid point, or how a superluminal shift turns an excision boundary into a pure outflow surface.

Is excision or the puncture method "better"?

Neither dominates — the field kept both on purpose. Excision, paired with generalized-harmonic evolution in spectral codes like SpEC, delivers very high accuracy and handles high spins and large mass ratios gracefully, but demands careful tracking and reshaping of the cut as the horizon moves. Moving punctures with BSSN are robust and simple — no inner boundary, no horizon tracking for the evolution — which is why most finite-difference codes use them, though Bowen-York data limit the initial spin. The strongest validation of numerical relativity is precisely that the two independent methods produce the same waveform for the same binary, so keeping both is a feature.

If the puncture keeps the singularity on the grid, why doesn't it crash the code?

Because the singular part is never differentiated numerically. You split $\psi=\psi_{\mathrm{BL}}+u$ , where $\psi_{\mathrm{BL}}$ is the analytic $1/r$ Brill-Lindquist piece — known in closed form, so its derivatives are exact, not finite-differenced. The solver only touches the smooth correction $u$ , bounded and $C^2$ everywhere including at the puncture, because the elliptic source $\propto\psi_{\mathrm{BL}}^{-7}\bar{A}_{ij}\bar{A}^{ij}$ vanishes like $r$ there. In moving-puncture evolutions the smooth variable $\chi=\psi^{-4}$ passes cleanly through zero at the puncture, and the collapsing lapse freezes the region so no derivative ever explodes.

Does deleting the interior with excision throw away real physics?

No — that is the point of the causal argument. The excised region lies inside a trapped surface, from which no signal can ever reach the exterior, so by the definition of a black hole nothing there can influence the waves that reach a detector; removing it changes no observable. What you must get right is placement: the cut has to sit strictly inside the apparent horizon, where every characteristic points inward, so it is a pure outflow boundary needing no data. Misplace it even slightly outside the horizon and you are imposing artificial data on an inflow boundary, which corrupts the physical exterior.

What actually moves in the "moving puncture" method?

The coordinate location of the puncture moves across the grid, carried by the shift. The gauge does two jobs: the $1+\log$ lapse collapses toward zero near each puncture, freezing coordinate time so the slice never advances into the singularity, while the $\Gamma$ -driver shift $\partial_t\beta^i=\tfrac34\tilde\Gamma^i-\eta\beta^i$ drives the coordinates to flow with the hole, so the puncture drifts through the grid like a feature riding a current. The grid stays fixed; the puncture's coordinate position and the smooth fields around it advect. Before 2005 the punctures were pinned to fixed coordinates, which tore the grid as the holes orbited; letting them move was the breakthrough.

Why can't Bowen-York puncture data make a near-extremal black hole?

Bowen-York extrinsic curvature solves the momentum constraint on a conformally flat slice, and it is a theorem that no conformally flat slice of Kerr exists. So a fast-spinning hole built from conformally flat data plus Bowen-York spin sits on a slice that is not Kerr — it carries extra unphysical "junk" radiation that promptly radiates away, leaving less spin than requested. The maximum for a single Bowen-York puncture is about $a/M\approx0.93$ . Reaching higher spins means abandoning conformal flatness — superposed Kerr-Schild or superposed-Kerr data — one reason excision codes have an edge at high spin.

How do the two methods relate to BSSN versus generalized harmonic?

They pair up naturally, though not by necessity. BSSN is a strongly-hyperbolic formulation that works beautifully with the smooth conformal variable at a puncture, so it is the standard partner of moving punctures — no excision needed. The generalized-harmonic formulation casts Einstein's equations as manifestly wave-like second-order equations and is most naturally evolved with an excised inner boundary, as in Pretorius's 2005 merger and the SpEC code. The formulation (how you write the equations) and the singularity treatment (how you keep it off the grid) are logically separate, but the physics of each has made these two marriages overwhelmingly common.