Apparent Horizons and Horizon Finders

A black hole is defined, at bottom, by its event horizon — the boundary of the region from which no signal ever reaches future null infinity. That definition is global and teleological: to decide whether a point lies on the event horizon you must know the entire future of the spacetime, including matter that has not yet fallen in. A numerical simulation never has that future; it holds the spacetime only up to the current time step, so the event horizon can be reconstructed only afterward. What a running code needs is a surrogate that lives on a single spatial slice — the apparent horizon. It is built from the trapped surface: fire the two orthogonal families of light rays from a closed 2-surface, and where even the outgoing family stops expanding, the surface is marginally trapped. The apparent horizon is the outermost such surface on the slice, where the outgoing null expansion vanishes, $\Theta_{(\ell)}=0$ . Because that condition involves only the induced 3-metric $\gamma_{ij}$ and the extrinsic curvature $K_{ij}$ of the ADM decomposition, it becomes an elliptic PDE for the surface's shape — a boundary-value problem on the sphere that a horizon finder solves every few time steps. Given the null energy condition and cosmic censorship, finding an apparent horizon proves a black hole is present, enclosed by the event horizon. Once located, its area, spin, and mass are the gauge-robust numbers a binary-merger simulation reports, and its surface is where excision codes cut away the singular interior.

Key takeaways

An apparent horizon is the outermost surface on a spatial slice whose outgoing null congruence has vanishing expansion, $\Theta_{(\ell)}=0$ — a marginally outer trapped surface, or MOTS.
Unlike the event horizon, which is a global, teleological feature of the whole spacetime, the apparent horizon is quasi-local: it depends only on data on one Cauchy slice, so it can be located while a simulation runs.
On a slice with induced metric $\gamma_{ij}$ and extrinsic curvature $K_{ij}$ , the condition becomes an elliptic PDE for the surface shape, $D_i s^i - K + K_{ij}s^i s^j = 0$ , with $s^i$ the outward unit normal.
Horizon finders solve it numerically — by parabolic flow relaxation, spectral expansion of the radius in spherical harmonics, or Newton and minimization schemes.
Given the null energy condition and cosmic censorship, an apparent horizon always lies inside or on the event horizon, so its discovery proves a trapped region and an inevitable black hole.
Horizon data — area giving irreducible mass $M_{\mathrm{irr}}=\sqrt{A/16\pi}$ , a spin integral, and the Christodoulou mass — are the gauge-robust diagnostics of a binary black hole merger.
Key idea: replace the global, teleological event horizon by a slice-local condition — the vanishing of the outgoing null expansion $\Theta_{(\ell)} = D_i s^i - K + K_{ij} s^i s^j = 0$ — an elliptic equation for the surface shape, solvable from data on a single Cauchy slice, which makes the black hole boundary computable in real time during a simulation.

The core insight

Replace the global, teleological event horizon by a slice-local condition — the vanishing of the outgoing null expansion, $\Theta_{(\ell)} = D_i s^i - K + K_{ij} s^i s^j = 0$ . This is an elliptic equation for the surface shape, solvable from data on a single Cauchy slice, and it makes the black hole boundary computable in real time during a simulation.

Where this lives today

Apparent-horizon finders are standard instruments in every numerical-relativity code. Thornburg's AHFinderDirect, distributed with the open-source Einstein Toolkit, solves the elliptic condition by Newton's method as an inline diagnostic; the Spectral Einstein Code of the SXS collaboration uses a fast spectral finder to track horizons through inspiral, merger, and ringdown. The area, spin, and Christodoulou mass they report are the numbers compared against the mass and angular momentum radiated to infinity — the consistency check that told the LIGO-Virgo collaboration the GW150914 remnant was a Kerr black hole of $62\,M_\odot$ and spin $0.67$ . A live frontier concerns what happens inside the common horizon at merger: since 2019, Pook-Kolb, Booth, and collaborators have found that the two individual MOTSs do not vanish but persist and pass through each other, spawning self-intersecting marginally trapped surfaces whose stability and topology are still being mapped. The isolated- and dynamical-horizon program of Ashtekar and Krishnan, meanwhile, has made the apparent horizon the arena for a fully quasi-local black hole thermodynamics, with a first law $\mathrm{d}M=\tfrac{\kappa}{8\pi}\mathrm{d}A+\Omega\,\mathrm{d}J$ defined on the horizon 2-surface alone.

The physical picture

Fire a flashbulb on a closed 2-surface. Two wavefronts leave it — one nominally outgoing, one ingoing — each changing area at a rate set by its null expansion. In flat space the outgoing front gains area ( $\Theta_{(\ell)}>0$ ) while the ingoing front loses it; that is what "outward" means. Inside a strong enough gravitational field both fronts are dragged toward smaller area: the surface is trapped, and light cannot locally escape outward. The apparent horizon is the knife-edge between, where the outgoing front momentarily neither grows nor shrinks. The first figure sets the three regimes side by side.

Figure

Outgoing light fronts leaving three closed 2-surfaces. On a normal surface the outgoing front gains area (Θ₍ℓ₎ greater than 0); on the apparent horizon it is frozen (Θ₍ℓ₎ = 0), a marginally outer trapped surface; inside, curvature drags even the outgoing front inward and the surface is trapped

The defining equation $D_i s^i - K + K_{ij}s^i s^j = 0$ reads as a balance between two curvatures. The term $D_i s^i$ is the expansion of the surface within the slice — how its area grows as you push it outward through the 3-geometry. The combination $K_{ij}s^i s^j - K$ encodes how the slice itself bends through spacetime, diving toward the singularity. A horizon is where these cancel: a surface that wants to expand spatially, held in check by the slice's plunge. Flow finders solve for it by driving a trial surface with an artificial evolution proportional to $-\Theta_{(\ell)}$ until it settles. The second figure carries the concrete numbers this machinery must reproduce for the stationary black holes used as checks.

Figure

Kerr horizon radius r₊/M = 1 + √(1 − χ²) and irreducible-mass fraction M_irr/M = √((1 + √(1 − χ²))/2) versus dimensionless spin χ = a/M. Schwarzschild (χ = 0) gives r₊ = 2M and M_irr = M; the extremal limit (χ = 1) gives r₊ = M and M_irr = M/√2 ≈ 0.707M; the GW150914 remnant sits at χ ≈ 0.67

The core equations

The starting point is the covariant definition of the outgoing null expansion — the fractional rate at which the outgoing light front's cross-sectional area changes. Every slice-local formula below projects this one object onto a Cauchy surface.

\Theta_{(\ell)} = h^{\mu\nu}\nabla_\mu \ell_\nu \qquad (1)

where: $\Theta_{(\ell)}$ — expansion of the outgoing null congruence; $\ell^\mu$ — future-directed outgoing null normal; $h^{\mu\nu}$ — projector onto the 2-surface, $h_{\mu\nu}=g_{\mu\nu}+\ell_\mu n_\nu+n_\mu\ell_\nu$ ; $n^\mu$ — ingoing null normal, cross-normalized so $\ell\cdot n=-1$ . Reading: $\Theta_{(\ell)}$ is positive where the outgoing front spreads, zero on a MOTS, negative on a trapped surface. Projected onto a spatial slice, the covariant expansion collapses into a three-dimensional statement built from the ADM data — evaluable without any knowledge of the future. Setting it to zero is the apparent horizon condition.

D_i s^i - K + K_{ij}\,s^i s^j = 0 \qquad (2)

where: $s^i$ — outward unit normal within the slice, $s_i s^i=1$ ; $D_i$ — covariant derivative of the spatial metric $\gamma_{ij}$ ; $K_{ij}$ — extrinsic curvature of the slice; $K=\gamma^{ij}K_{ij}$ — its trace. Reading: the spatial expansion $D_i s^i$ balanced against the slice's bending $K_{ij}s^i s^j - K$ . To make this computable, write the surface as a graph — a radius $h$ over the angles — so its normal is the gradient of a level-set function, turning an implicit condition into an equation for one scalar field on the sphere.

F(\theta,\phi)\equiv r - h(\theta,\phi)=0,\qquad s_i = \frac{D_i F}{|D F|} \qquad (3)

where: $h(\theta,\phi)$ — horizon radius as a function of the angles; $F$ — level-set function whose zero set is the trial surface; $|DF|=\sqrt{\gamma^{ij}D_i F\,D_j F}$ — the gradient norm. Reading: the horizon's shape is encoded in one function $h$ , with its normal fixed through $F$ . Substituting the level set into the marginal condition produces a nonlinear elliptic operator on $h$ — the equation a spectral or Newton finder solves, and toward whose solution a flow finder relaxes by an artificial parabolic evolution.

\mathcal{H}[h] \equiv D_i s^i - K + K_{ij}s^i s^j = 0,\qquad \partial_\lambda h = -\,\mathcal{H}[h]\,/\,\rho \qquad (4)

where: $\mathcal{H}[h]$ — expansion functional, a nonlinear elliptic operator in $h$ ; $\lambda$ — fictitious flow time, not physical time; $\rho>0$ — a scaling making the flow well-posed (Tod's choice $\rho\sim|DF|$ ). Reading: $\mathcal{H}[h]=0$ is the target; the flow drives any trial surface downhill in $\Theta_{(\ell)}$ until it stops. Once located, the surface's proper area fixes the irreducible mass — the mass no classical process can extract, and the quantity whose square adds under the Hawking area theorem.

M_{\mathrm{irr}} = \sqrt{\frac{A}{16\pi}},\qquad A=\oint_S\sqrt{q}\,\mathrm{d}^2x \qquad (5)

where: $A$ — proper area of the apparent horizon; $q$ — induced 2-metric on $S$ ; $M_{\mathrm{irr}}$ — irreducible (Christodoulou) mass. Reading: the square root of the area over $16\pi$ is a gauge-robust mass in geometrized units. Spin comes from a surface integral against an approximate axial Killing vector; combined with area it gives the Christodoulou mass, reducing to $R/2$ for a nonspinning hole.

J = \frac{1}{8\pi}\oint_S \varphi^i\,s^j\,K_{ij}\,\mathrm{d}^2x,\qquad M = \frac{1}{2R}\sqrt{R^4 + 4 J^2},\qquad R=\sqrt{\tfrac{A}{4\pi}} \qquad (6)

where: $\varphi^i$ — approximate axial Killing vector on the horizon 2-surface; $J$ — angular momentum about that axis; $R$ — areal radius; $M$ — horizon mass. Reading: $J$ comes from the extrinsic curvature contracted with the rotational generator, and $M$ packages area and spin into one Kerr-consistent number.

Derivation AFrom trapped surfaces to the horizon equation

The goal is to show that the covariant condition $\Theta_{(\ell)}=0$ reduces exactly to the slice-local elliptic equation $D_i s^i - K + K_{ij}s^i s^j=0$ , using only the ADM data on one Cauchy surface, so a horizon finder never has to touch the time direction. We build the outgoing null normal from the slice normals, project the expansion onto the 2-surface, expand the covariant derivative in extrinsic curvature and intrinsic connection, and take the trace.

Step 1. Let $S$ be a closed 2-surface embedded in a spacelike slice $\Sigma$ whose timelike unit normal is $n^\mu$ ; within $\Sigma$ , the surface $S$ has outward spatial unit normal $s^\mu$ . Form the outgoing null normal $\ell^\mu=\tfrac{1}{\sqrt2}(n^\mu+s^\mu)$ , which satisfies $\ell_\mu\ell^\mu=0$ and points outward in time, cross-normalized against the ingoing $n_{(\mathrm{null})}^\mu=\tfrac1{\sqrt2}(n^\mu-s^\mu)$ so that $\ell\cdot n_{(\mathrm{null})}=-1$ . Why: the factor $1/\sqrt2$ is the standard normalization that keeps the two null normals dual and makes the projected expansion carry a clean geometric meaning.

Step 2. Project onto $S$ with the induced 2-metric $q_{\mu\nu}=g_{\mu\nu}+n_\mu n_\nu-s_\mu s_\nu$ , subtracting both the timelike and the in-slice normal directions. The expansion is the projected divergence $\Theta_{(\ell)}=q^{\mu\nu}\nabla_\mu\ell_\nu$ . Why: only the two directions tangent to $S$ measure how its area changes, so the projector must annihilate both $n$ and $s$ .

Step 3. Expand $\nabla_\mu\ell_\nu=\tfrac1{\sqrt2}(\nabla_\mu n_\nu+\nabla_\mu s_\nu)$ and substitute $\nabla_\mu n_\nu=-K_{\mu\nu}-n_\mu a_\nu$ , with $a_\nu$ the acceleration of the slice normal, together with the spatial decomposition of $\nabla_\mu s_\nu$ into the intrinsic covariant derivative $D_\mu s_\nu$ . The acceleration term dies against $q^{\mu\nu}$ because it carries a factor of $n$ . Why: this is where the extrinsic curvature enters — the bending of the slice in spacetime is exactly the $K_{ij}$ contribution to the expansion.

Step 4. Collect the survivors to get $\Theta_{(\ell)}=(\gamma^{ij}-s^i s^j)(D_i s_j - K_{ij})$ , with $\gamma^{ij}-s^i s^j$ the in-slice projector onto $S$ . Taking the trace and using $\gamma^{ij}D_i s_j=D_i s^i$ , $\gamma^{ij}K_{ij}=K$ , and $s^i s^j D_i s_j=\tfrac12 s^i D_i(s_j s^j)=0$ gives $\Theta_{(\ell)}=D_i s^i - K + K_{ij}s^i s^j$ , involving only $\gamma_{ij}$ , $K_{ij}$ , and slice derivatives. Why: the last term vanishes because $s$ is a unit vector, leaving a clean elliptic operator with no first-derivative pathology.

Go deeper · Derivation B · 1 minThe Schwarzschild horizon on a static slice

Derivation BThe Schwarzschild horizon on a static slice

The goal is to solve the apparent horizon condition explicitly in the one case checkable by hand — the Schwarzschild geometry on a static $t=\text{const}$ slice — and confirm that the finder must return $r=2M$ , coincident with the event horizon, along with the area $16\pi M^2$ and irreducible mass $M$ that any correct code has to reproduce as a regression test. On this slicing the extrinsic curvature vanishes, so the condition degenerates to a minimal-surface problem.

Step 1. Take the induced 3-metric of the slice, $\mathrm{d}\ell^2=(1-2M/r)^{-1}\mathrm{d}r^2+r^2\,\mathrm{d}\Omega^2$ , with $\mathrm{d}\Omega^2=\mathrm{d}\theta^2+\sin^2\theta\,\mathrm{d}\phi^2$ . By spherical symmetry the horizon is a coordinate sphere $r=\text{const}$ , whose outward unit normal is $s^i=\sqrt{1-2M/r}\;\delta^i{}_r$ , normalized against the radial metric component. Why: symmetry collapses the unknown function $h(\theta,\phi)$ to a single constant radius, so the elliptic PDE becomes one algebraic equation.

Step 2. On the static slice $K_{ij}=0$ , so the full condition reduces to $D_i s^i=0$ . Compute the divergence with $\sqrt\gamma=r^2\sin\theta/\sqrt{1-2M/r}$ : $D_i s^i=\tfrac{1}{\sqrt\gamma}\partial_r(\sqrt\gamma\,s^r)=\tfrac{2}{r}\sqrt{1-2M/r}$ . Why: with the extrinsic curvature gone, the marginal surface is simply the surface of extremal area in the 3-geometry, and its expansion is the ordinary spatial divergence of the normal.

Step 3. Set the expansion to zero: $\tfrac{2}{r}\sqrt{1-2M/r}=0$ . Away from spatial infinity the only root is $1-2M/r=0$ , giving $r=2M$ . The apparent horizon coincides with the event horizon — the hallmark of stationarity. Its area is $A=4\pi(2M)^2=16\pi M^2$ , so $M_{\mathrm{irr}}=\sqrt{A/16\pi}=M$ , consistent with $J=0$ in the Christodoulou formula, where $M=R/2=\sqrt{A/4\pi}/2=M$ . Why: recovering $r=2M$ , $A=16\pi M^2$ , and $M_{\mathrm{irr}}=M$ is the minimal correctness check every horizon finder is validated against before it is trusted on a dynamical spacetime.

Go deeper · Derivation C · 1 minWhy the apparent horizon hides inside the event horizon

Derivation CWhy the apparent horizon hides inside the event horizon

The goal is to establish the theorem that gives the apparent horizon its physical teeth — that under the null energy condition and asymptotic predictability, any MOTS lies inside or on the event horizon, so finding one proves a black hole is present and supplies the area ordering used in merger checks. The argument runs through the Raychaudhuri focusing equation: an outgoing congruence that starts non-expanding is forced to a caustic, and focused rays cannot reach infinity.

Step 1. Write the Raychaudhuri equation for the outgoing null congruence, $\tfrac{\mathrm{d}\Theta_{(\ell)}}{\mathrm{d}\lambda}=-\tfrac12\Theta_{(\ell)}^2-\sigma_{\mu\nu}\sigma^{\mu\nu}-R_{\mu\nu}\ell^\mu\ell^\nu$ , with $\sigma$ the shear and $\lambda$ the affine parameter. The null energy condition plus Einstein's equation give $R_{\mu\nu}\ell^\mu\ell^\nu=8\pi T_{\mu\nu}\ell^\mu\ell^\nu\ge0$ . Why: this makes every term on the right non-positive, so once the expansion is non-positive it can only decrease — gravity focuses light and never defocuses it.

Step 2. On a MOTS the outgoing expansion is $\Theta_{(\ell)}=0$ , and just outside the outermost MOTS it is non-positive. Integrating the inequality $\tfrac{\mathrm{d}\Theta_{(\ell)}}{\mathrm{d}\lambda}\le-\tfrac12\Theta_{(\ell)}^2$ drives $\Theta_{(\ell)}\to-\infty$ within finite affine parameter: the outgoing rays converge to a caustic. Why: a self-reinforcing focusing term $-\tfrac12\Theta^2$ produces a finite-time blow-up, exactly the conjugate point that terminates the congruence.

Step 3. Rays that focus to a caustic cannot generate a smooth boundary reaching future null infinity, so no point of the MOTS lies in the causal past of $\mathscr{I}^+$ . Hence the surface is inside the event horizon, and because the event horizon is generated by non-converging rays, the areas obey $\mathcal{A}_{\mathrm{EH}}\ge\mathcal{A}_{\mathrm{AH}}$ , with equality only in stationarity. Why: this containment lets a running code use "apparent horizon found" as proof of a black hole, and it underlies the area-theorem checks in mergers.

Limiting cases

The condition specializes cleanly whenever symmetry or staticity simplifies the ADM data:

Case	Result	Meaning
Schwarzschild, static slicing	$r_{\mathrm{AH}}=2M$	On a static or maximal slice the apparent horizon coincides with the event horizon at areal radius $2M$ ; the two agree because the spacetime is stationary.
Kerr, Boyer-Lindquist	$r_{\mathrm{AH}}=M+\sqrt{M^2-a^2}$	The outer marginal surface sits at the outer event-horizon radius; it exists only for $a\le M$ .
Moment of time symmetry	$K_{ij}=0\Rightarrow D_i s^i=0$	When the extrinsic curvature vanishes the MOTS reduces to a minimal surface — a 2-surface of extremal area in the 3-geometry.
Spherical symmetry	$g^{ab}\partial_a R\,\partial_b R=0$	The apparent horizon is the marginal sphere where the Misner-Sharp mass satisfies $2m(R)=R$ — where the areal-radius gradient turns null.

Worked examples

1 · Irreducible mass and extractable energy of a spinning remnant

A merger settles into a Kerr remnant of mass $M$ and dimensionless spin $\chi=a/M=0.90$ . Find the areal radius and irreducible mass in units of $M$ , and the fraction of the mass-energy in principle extractable.

Step 1. The outer horizon radius is $r_+=M(1+\sqrt{1-\chi^2})=M(1+\sqrt{0.19})=\mathbf{1.436\,M}$ . The Kerr area is $A=8\pi M r_+$ , so the areal radius is $R=\sqrt{A/4\pi}=\sqrt{2M r_+}=\sqrt{2.872}\,M=\mathbf{1.695\,M}$ .

Step 2. The irreducible mass follows from the area: $M_{\mathrm{irr}}=\sqrt{A/16\pi}=M\sqrt{(1+\sqrt{1-\chi^2})/2}=M\sqrt{1.436/2}=M\sqrt{0.718}=\mathbf{0.847\,M}$ . As a check, the Christodoulou formula must give back $M$ : with $R=1.695\,M$ and $J=\chi M^2=0.90\,M^2$ , $M=\tfrac{1}{2R}\sqrt{R^4+4J^2}=\tfrac{1}{2(1.695)}\sqrt{(1.695)^4+4(0.90)^2}\,M=\tfrac{1}{3.390}\sqrt{8.254+3.24}\,M=\tfrac{3.390}{3.390}\,M=\mathbf{M}$ . Consistent.

Step 3. The rotational energy extractable without shrinking the area is $M-M_{\mathrm{irr}}=0.153\,M$ , about 15% of the total. At the extremal limit $\chi=1$ , $M_{\mathrm{irr}}=M/\sqrt2\approx0.707\,M$ , so at most $M(1-1/\sqrt2)\approx\mathbf{0.29\,M}$ — roughly 29% — can ever be mined from a maximally spinning hole. A χ = 0.90 Kerr hole has M_irr = 0.847M and 15% extractable; the extremal ceiling is 29%.

2 · The area theorem at a merger

In GW150914, two black holes of roughly $36\,M_\odot$ and $29\,M_\odot$ (nearly nonspinning) merged into a $62\,M_\odot$ remnant with spin $\chi\approx0.67$ . Verify that the common apparent horizon is consistent with Hawking's area theorem, and account for the missing mass.

Step 1. For a nonspinning hole the irreducible mass equals the mass, so the progenitors carry $M_{\mathrm{irr},1}=36\,M_\odot$ and $M_{\mathrm{irr},2}=29\,M_\odot$ . Since area is $16\pi M_{\mathrm{irr}}^2$ , the total pre-merger area is proportional to $36^2+29^2=1296+841=\mathbf{2137\,M_\odot^2}$ .

Step 2. The remnant's irreducible mass is $M_{\mathrm{irr},f}=M_f\sqrt{(1+\sqrt{1-\chi^2})/2}=62\sqrt{(1+0.742)/2}=62\sqrt{0.871}=\mathbf{57.9\,M_\odot}$ , so its area is proportional to $57.9^2=\mathbf{3349\,M_\odot^2}$ .

Step 3. Compare: $3349>2137$ , so the common-horizon area strictly exceeds the sum of the two initial areas — the area theorem holds, as it must. Meanwhile the total mass dropped from $65\,M_\odot$ to $62\,M_\odot$ : about $\mathbf{3\,M_\odot c^2}$ was radiated as gravitational waves, drawn from the rotational and orbital budget, not the irreducible mass — the area grew even as the total mass fell. Area rose from 2137 to 3349 (theorem satisfied) while 3 M_⊙ radiated away.

Physics of the portrait

The apparent horizon trades a global definition of a black hole for a computable one — and the man who first turned the horizon's area into a hard physical number, the irreducible mass, was a student of John Wheeler's.

Origins · 1 minDemetrios Christodoulou and how this was discovered

Demetrios Christodoulou

1951–present · the irreducible mass and the formation of trapped surfaces

Demetrios Christodoulou, a Greek mathematician and physicist, arrived at Princeton as a teenager and completed his PhD under John Wheeler in 1971. His thesis contained a result of lasting force: for a Kerr black hole the irreducible mass $M_{\mathrm{irr}}=\sqrt{A/16\pi}$ can never be decreased by any classical process, so the horizon area only ever grows. This was the first sharp statement of what became, in Hawking's hands, the area theorem, and in Bekenstein's, the identification of horizon area with entropy; the Christodoulou mass formula $M^2=M_{\mathrm{irr}}^2+J^2/4M_{\mathrm{irr}}^2$ splits a spinning hole's energy into an irreducible part and an extractable rotational part, quantifying the Penrose process. He then turned to whether black holes must form at all from generic collapse, and in a monumental body of work culminating in his 2008 book he proved — with no symmetry assumptions — that a trapped surface, and hence a black hole, forms dynamically when enough gravitational energy is focused into a small enough region, giving rigorous substance to what horizon finders detect numerically. Alongside his proof with Sergiu Klainerman of the nonlinear stability of Minkowski space, this made him one of the deepest analysts of Einstein's equations — and it is his irreducible mass a merger simulation reports whenever it locates a horizon.

History

1965

— Penrose introduces trapped surfaces in the first singularity theorem

1970

— Christodoulou derives the irreducible mass of a black hole from its horizon area

1973

— Hawking and Ellis define the apparent horizon as the outer boundary of the trapped region

2004

— Ashtekar and Krishnan formulate the isolated and dynamical horizon framework

2008

— Christodoulou proves trapped surfaces form dynamically in vacuum collapse

Going further

Texts. Misner, Thorne and Wheeler, Gravitation, Ch. 33-34 · Wald, General Relativity, Ch. 9, 12 · Baumgarte and Shapiro, Numerical Relativity, Ch. 7 · Gourgoulhon, 3+1 Formalism in General Relativity, Ch. 8-10 · Thornburg, Living Reviews in Relativity 10, 3 (2007), "Event and Apparent Horizon Finders for 3+1 Numerical Relativity" · Hawking and Ellis, The Large Scale Structure of Space-Time, Ch. 9 · Ashtekar and Krishnan, Living Reviews in Relativity 7, 10 (2004), "Isolated and Dynamical Horizons".

Provenance. All equations are standard. A verifier should recompute: (1) the reduction of the covariant expansion to $D_i s^i - K + K_{ij}s^i s^j$ from the ADM data (Gourgoulhon Ch. 8; Baumgarte-Shapiro Ch. 7); (2) the Schwarzschild result $r_{\mathrm{AH}}=2M$ , area $16\pi M^2$ , and $M_{\mathrm{irr}}=M$ on the static slice (MTW Ch. 33; Wald Ch. 12); (3) the Kerr horizon radius $M+\sqrt{M^2-a^2}$ and area $8\pi M r_+$ (Wald Ch. 12); (4) the irreducible-mass and Christodoulou formulas, with the worked $\chi=0.90$ case giving $M_{\mathrm{irr}}=0.847\,M$ and the extremal ceiling $1-1/\sqrt2\approx0.29$ ; (5) the GW150914 area check, $57.9^2=3349>2137=36^2+29^2$ , and the Raychaudhuri focusing argument for the AH-inside-EH containment (Hawking-Ellis Ch. 9). Horizon-finder algorithms follow Thornburg's Living Review; units $G=c=1$ throughout.

Prerequisites. ADM 3+1 Decomposition of Spacetime · Gauss-Codazzi-Mainardi Equations · Hamiltonian and Momentum Constraints · Numerical Relativity and Binary Black Hole Mergers. Leads to. Black Hole Excision and Puncture Methods · ADM Mass, Momentum and Angular Momentum · Gauge Conditions in Numerical Relativity.

Ask the tutor: Ask me to walk through how a spectral horizon finder actually solves $\mathcal{H}[h]=0$ step by step, or why the common horizon at a merger is so much harder to catch than the two inspiral horizons.

What is the difference between a trapped surface and a marginally trapped surface?

On a trapped surface both orthogonal families of light rays converge — the outgoing and ingoing expansions are both negative at every point. A marginally outer trapped surface, or MOTS, is the boundary case where the outgoing expansion is exactly zero while the ingoing one is still negative. The apparent horizon is the outermost MOTS on a slice: the surface at the very edge of the trapped region. Interior to it lie genuinely trapped surfaces; exterior to it, normal ones with $\Theta_{(\ell)}>0$ .

Why is the apparent horizon condition an elliptic equation rather than an evolution equation?

Because it is a constraint on a single slice, not a statement about change in time. When you write the surface as a radius $h(\theta,\phi)$ and substitute into $\Theta_{(\ell)}=0$ , the divergence $D_i s^i$ brings in second angular derivatives of $h$ whose leading part is a Laplacian on the sphere with a positive-definite symbol — the signature of an elliptic operator. Elliptic equations are boundary-value problems: the solution at every point depends on the data everywhere at once, exactly right for a surface whose shape must be globally consistent. On the sphere there is no external boundary; regularity at the poles and periodicity in $\phi$ take its place.

How can the same black hole have an apparent horizon in one slicing and none in another?

The outgoing null expansion mixes the intrinsic geometry $\gamma_{ij}$ with the extrinsic curvature $K_{ij}$ , and $K_{ij}$ tells you how your chosen slice sits inside the fixed spacetime. Tilt the slice and you change $K_{ij}$ , which changes where, or whether, the expansion vanishes. Wald and Iyer built explicit slicings of eternal Schwarzschild that reach arbitrarily close to the singularity yet contain no marginally trapped surface at all. The event horizon, by contrast, is a property of the spacetime itself, independent of slicing. So the apparent horizon is a slice-dependent probe, powerful but not unique.

Do horizon finders track the individual black holes and the merged one differently?

In practice, yes. During inspiral each black hole carries its own nearly spherical horizon, and a fast spectral finder — expanding the radius in spherical harmonics and solving by Newton's method, seeded by the previous step — converges in a handful of iterations because the surface is smooth. The common horizon at merger is initially highly distorted and peanut-shaped, so a good initial guess is harder to supply, and flow methods that relax a large trial surface inward by $\partial_\lambda h=-\mathcal{H}[h]/\rho$ are more robust for its first detection. Production codes often run both.

What exactly is extracted from a located horizon, and how gauge-robust is it?

Three numbers. The proper area $A=\oint\sqrt{q}\,\mathrm{d}^2x$ gives the irreducible mass $M_{\mathrm{irr}}=\sqrt{A/16\pi}$ . The spin comes from $J=\tfrac1{8\pi}\oint\varphi^i s^j K_{ij}\,\mathrm{d}^2x$ against an approximate axial Killing vector found by solving an eigenvalue problem on the horizon 2-metric. Combining them through $M=\tfrac{1}{2R}\sqrt{R^4+4J^2}$ gives the horizon mass. These are quasi-local and, for a settled horizon, essentially gauge-invariant — they depend on the intrinsic 2-geometry, not on the coordinates used to evolve the bulk — which is why they, not coordinate positions, are quoted as the physical properties of a merger remnant.

Why does the horizon area only ever grow, and how is that used as a check?

The same Raychaudhuri argument that puts the apparent horizon inside the event horizon also forbids the event horizon's generators from converging, given the null energy condition — so the area can never decrease. This is Hawking's area theorem. In a merger the common horizon must have area at least the sum of the two progenitor areas, and since irreducible mass is $\sqrt{A/16\pi}$ , the squares must add up favorably: $M_{\mathrm{irr},f}^2\ge M_{\mathrm{irr},1}^2+M_{\mathrm{irr},2}^2$ . GW150914 satisfies this — $57.9^2>36^2+29^2$ — while the total mass falls, the difference radiated as gravitational waves.