Isolated and Dynamical Horizons

Ask a numerical relativist evolving two black holes toward merger a simple question — "where is the horizon, right now?" — and the textbook answer is useless. The event horizon, the boundary of the causal past of future null infinity, is global and teleological: whether a point lies inside it depends on the entire future of the spacetime, including matter that has not yet fallen in. On a single slice it cannot be found at all; it must be reconstructed afterward by shooting null geodesics backward from late times. A surface that only exists once the run is over is no help in assigning a black hole its mass, spin, or area while the run is happening. The quasi-local horizon program — built principally by Ashtekar, Krishnan, Hayward and collaborators through the 1990s and 2000s — replaces that global boundary with hypersurfaces defined by local conditions on 2-surfaces. The central object is a marginally outer-trapped surface (MOTS): a closed 2-sphere on which the outgoing light rays neither spread nor converge, $\Theta_{(\ell)}=0$ . Stacking MOTS through a foliation sweeps out a marginally trapped tube. When it is null with no crossing flux it is an isolated horizon modelling a hole in equilibrium; when it is spacelike and absorbing energy it is a dynamical horizon modelling a hole that is growing. The payoff is a self-contained black hole mechanics with no appeal to global stationarity: a quasi-local zeroth law ( $\kappa=$ const), a first law, and — for dynamical horizons — the exact Ashtekar–Krishnan area-balance law that accounts, term by term, for the matter energy and gravitational-wave shear each unit of horizon growth absorbed. This is black-hole thermodynamics a computer can evaluate, one slice at a time.

Key takeaways

The event horizon is global and teleological — locating it needs the whole future of spacetime. Isolated and dynamical horizons replace it with quasi-local surfaces on a finite slab, exactly what numerical relativity can compute.
A marginally trapped tube is foliated by MOTS: closed 2-spheres with $\Theta_{(\ell)}=0$ and $\Theta_{(n)}<0$ . Its signature — null or spacelike — is fixed by the flux crossing it.
An isolated horizon is a null tube $\Delta$ in equilibrium: no crossing flux, time-independent 2-geometry, constant area, and a quasi-local first law $\delta E_\Delta=\frac{\kappa}{8\pi G}\delta a_\Delta+\Omega\,\delta J_\Delta$ .
A dynamical horizon is a spacelike tube whose area strictly grows; the Ashtekar–Krishnan law equates that growth to the integrated flux of matter energy plus gravitational shear — an exact local second law.
These frameworks give unambiguous masses, spins and multipole moments to individual holes during a merger, where the event horizon is not knowable until the run finishes.
The construction rests on the ADM 3+1 split, Gauss–Codazzi, and null-surface geometry; horizon finders solve the elliptic MOTS equation on every slice of a BSSN or generalized-harmonic run.
Key idea: replace the global event horizon by a hypersurface foliated by marginally outer-trapped 2-surfaces ( $\Theta_{(\ell)}=0$ ); if it is null with no crossing flux it is an isolated horizon obeying a quasi-local first law, and if it is spacelike it is a dynamical horizon whose area growth equals the integrated flux of matter energy plus gravitational shear crossing it.

The core insight

Replace the global event horizon by a hypersurface foliated by marginally outer-trapped 2-surfaces ( $\Theta_{(\ell)}=0$ ). If it is null with no crossing flux it is an isolated horizon obeying a quasi-local first law; if it is spacelike it is a dynamical horizon whose area growth equals the integrated flux of matter energy plus gravitational shear crossing it.

Where this lives today

Every gravitational-wave detection runs on this machinery. Production codes that evolve binary black holes — the SXS Collaboration's SpEC, the Einstein Toolkit and its BSSN descendants — cannot locate the event horizon during a run, because it depends on a future the code has not computed. Instead they call an apparent-horizon finder that solves the elliptic MOTS equation $D_i s^i+K_{ij}s^is^j-K=0$ on every slice, read the enclosed area for the irreducible mass, and evaluate the $J_\Delta$ integral for the spin. Those masses and spins calibrate the waveform templates LIGO, Virgo and KAGRA match against data, and let observers report that the GW150914 remnant weighed $62\,M_\odot$ and spun at $a_*=0.67$ . The flux law separates, in a single merger, the energy the hole absorbed as matter from that absorbed as infalling gravitational radiation, and horizon multipole moments $M_n+iJ_n\to M(ia)^n$ give a gauge-robust, strong-field no-hair test of whether the remnant is settling to Kerr. The discovery around 2019 that a merger's marginally trapped tubes fold, self-intersect and annihilate as the common horizon forms is reshaping how the field pictures the instant two holes become one.

The physical picture

Fire two future-directed families of light rays orthogonally off a closed 2-surface — outgoing ( $\ell$ ) and ingoing ( $n$ ) — and watch the surface's area. The two expansions $\Theta_{(\ell)}$ and $\Theta_{(n)}$ are the fractional rates at which the cross-sectional area changes along each. In flat space the outgoing rays spread, $\Theta_{(\ell)}>0$ , and the ingoing rays converge, $\Theta_{(n)}<0$ . Turn up the gravity and the outgoing rays bend inward too. The knife-edge case $\Theta_{(\ell)}=0$ with $\Theta_{(n)}<0$ is a marginally outer-trapped surface: outgoing light held momentarily stationary, the local fingerprint of a horizon's rim. Push past it and even the outgoing rays converge, $\Theta_{(\ell)}<0$ — a fully trapped surface.

Figure

Outgoing null rays off a 2-sphere in three regimes. In weak field the outgoing expansion is positive and the surface grows; at the marginal case the outgoing expansion vanishes, defining a MOTS, while the ingoing expansion stays negative; in strong field even the outgoing rays converge and the surface is fully trapped

Now stack these marginal 2-surfaces through time. They sweep out a three-dimensional world-tube, and the tilt of that tube is the whole story. If nothing crosses it, the tube runs null and its 2-geometry is frozen — an isolated horizon, a hole left in peace, area exactly constant. Let matter or gravitational radiation pour across and the tube tilts spacelike and flares outward: a dynamical horizon, growing precisely as fast as flux feeds it. This is the honest opposite of the event horizon, which anticipates the future — the event horizon of a collapsing star starts expanding before any matter arrives, because it must end up enclosing the final hole. The dynamical horizon has no foresight; it grows only where and when energy actually crosses, which is why its area-balance law is genuine local accounting.

Figure

A marginally trapped tube in a collapse-and-settle spacetime, time upward. While matter and shear cross it the tube is spacelike and its area a_Δ grows (dynamical horizon); once flux stops it runs null with constant area (isolated horizon). The teleological event horizon, dashed, starts growing earlier and inside. For the GW150914 remnant with spin a = 0.67 the Kerr relation a_Δ = 8πM²(1 + √(1 − a²)) gives a final area 1.57 times the sum of the two progenitor areas

The core equations

Everything begins with the two null expansions. Shoot the outgoing normal $\ell^a$ and ingoing normal $n^a$ off a spacelike 2-surface $S$ and project the gradient of each onto $S$ : the trace measures how the area grows along that ray.

\Theta_{(\ell)} = q^{ab}\nabla_a \ell_b,\qquad \Theta_{(n)} = q^{ab}\nabla_a n_b \qquad (1)

where: $\ell^a$ — future-directed outgoing null normal to $S$ ; $n^a$ — future-directed ingoing null normal, cross-normalized $\ell\cdot n=-1$ ; $q^{ab}$ — inverse of the induced 2-metric $q_{ab}=g_{ab}+\ell_a n_b+n_a\ell_b$ ; $\nabla_a$ — spacetime covariant derivative. Read it as the fractional rate of change of area along each light cone. A marginally outer-trapped surface is defined by the outer expansion just vanishing while the inner one stays negative. Both conditions matter: the first puts $S$ on the boundary of the trapped region, the second picks the physical future (absorbing) branch rather than a white hole.

\Theta_{(\ell)}=0,\qquad \Theta_{(n)}<0 \quad\Longrightarrow\quad S\ \text{is a MOTS} \qquad (2)

where: $S$ — closed spacelike 2-surface, topologically a sphere. A trapped surface has both expansions negative; the MOTS is the marginal outermost member, and a marginally trapped tube is a stack of them. The Raychaudhuri equation for the outgoing congruence governs whether a MOTS can persist. Twist-free, it splits the change of expansion into three sinks — self-focusing, shear, and matter/curvature — each non-positive under the null energy condition.

\frac{\mathrm{d}\Theta_{(\ell)}}{\mathrm{d}\lambda} = -\tfrac{1}{2}\Theta_{(\ell)}^2 - \sigma_{ab}\sigma^{ab} - R_{ab}\ell^a\ell^b \qquad (3)

where: $\lambda$ — affine parameter along $\ell^a$ ; $\sigma_{ab}$ — shear of the outgoing congruence; $R_{ab}$ — Ricci tensor, with $R_{ab}\ell^a\ell^b=8\pi G\,T_{ab}\ell^a\ell^b$ on shell. Because the right side is $\le 0$ , once light starts converging it must keep converging — the engine of the trapped-surface and singularity theorems. On an isolated horizon the outgoing normal is no longer affinely parametrized; its non-affinity is the surface gravity — the quasi-local replacement for the Killing-horizon definition, needing no global Killing vector.

\ell^b\nabla_b\ell^a=\kappa_{(\ell)}\,\ell^a,\qquad \kappa_{(\ell)} = -\,n_a\,\ell^b\nabla_b\ell^a \qquad (4)

where: $\kappa_{(\ell)}$ — surface gravity tied to the scaling of $\ell$ ; the second form extracts it by contracting with $n_a$ . Its constancy over $\Delta$ is the quasi-local zeroth law, derived below. With $\kappa$ , area $a_\Delta$ , angular momentum and angular velocity all defined intrinsically on $\Delta$ , isolated horizons obey a first law identical in form to the global one — but valid for a single hole momentarily at rest inside an otherwise dynamical spacetime.

\delta E_\Delta = \frac{\kappa}{8\pi G}\,\delta a_\Delta + \Omega\,\delta J_\Delta \qquad (5)

where: $E_\Delta$ — quasi-local horizon energy (mass); $a_\Delta$ — horizon area; $J_\Delta$ — horizon angular momentum; $\Omega$ — horizon angular velocity; $\kappa$ — surface gravity, constant over $\Delta$ . Unlike the global first law, every symbol here is built from fields on the horizon alone. The angular momentum itself is a flux integral of the extrinsic curvature against the rotational symmetry generator — no ADM integral at infinity required.

J_\Delta = \frac{1}{8\pi G}\oint_S \varphi^a\,K_{ab}\,s^b\,\mathrm{d}A \qquad (6)

where: $\varphi^a$ — axial Killing vector of $S$ (rotational generator); $K_{ab}$ — extrinsic curvature of the spatial slice; $s^a$ — outward unit spacelike normal to $S$ within the slice; $\mathrm{d}A$ — area element. When $\varphi^a$ is a genuine symmetry of the 2-geometry this integral is slicing-independent. For a dynamical horizon the area is no longer constant, and the Ashtekar–Krishnan law says exactly what pays for its growth: the flux of matter energy plus gravitational shear crossing the slab of horizon between two cross-sections.

\frac{R_2-R_1}{2G} = \int_{\Delta H} T_{ab}\,\hat\tau^a\,\xi^b\,\mathrm{d}V \;+\; \frac{1}{16\pi G}\int_{\Delta H}\!\big(|\sigma|^2+2|\zeta|^2\big)\,\mathrm{d}V \qquad (7)

where: $\Delta H$ — slab of dynamical horizon between cross-sections $S_1,S_2$ ; $R_i$ — areal radius of $S_i$ , $a_i=4\pi R_i^2$ ; $\hat\tau^a$ — unit normal in the matter-flux projection; $\xi^a=N\ell^a$ — foliation-generating vector; $\sigma$ — shear and $\zeta$ — a connection term, together the gravitational-radiation flux. Each right-hand term is $\ge 0$ under the dominant energy condition. Differentiate along the foliation and this becomes a local second law: the area rate is a sum of squares plus a positive matter term, never negative.

\frac{1}{2}\,\dot a_\Delta = 8\pi G\oint_S N\,T_{ab}\ell^a\ell^b\,\mathrm{d}A + \oint_S N\,|\sigma|^2\,\mathrm{d}A \;\ge\;0 \qquad (8)

where: $\dot a_\Delta$ — rate of horizon area increase; $N$ — lapse relating neighbouring MOTS leaves. This is Hawking's area theorem proved locally, without ever building the global event horizon.

Derivation AThe marginally trapped surface in 3+1 variables

The goal is to convert the covariant condition $\Theta_{(\ell)}=0$ into an equation written purely in the data a numerical relativity code already carries — the spatial metric $\gamma_{ij}$ , extrinsic curvature $K_{ij}$ , and outward unit normal $s^i$ of a candidate 2-surface in the slice. This is the elliptic equation an apparent-horizon finder solves every step, so its exact form and flat-slice limit are what let a simulation report a horizon at all. Let $\Sigma$ carry future unit normal $\tau^a$ ; then $S\subset\Sigma$ has outward normal $s^i$ and the null normals are $\ell^a=\tfrac{1}{\sqrt2}(\tau^a+s^a)$ , $n^a=\tfrac{1}{\sqrt2}(\tau^a-s^a)$ .

Step 1. Write the outgoing expansion as $\Theta_{(\ell)}=q^{ab}\nabla_a\ell_b$ with the 2-metric $q_{ab}=\gamma_{ab}-s_a s_b$ , and split $\ell$ into its $\tau$ part and its $s$ part. Why: the $\tfrac{1}{\sqrt2}(\tau+s)$ decomposition is what turns a spacetime projection into two separately recognizable pieces of 3+1 data.

Step 2. The spatial piece projects to $q^{ab}\nabla_a s_b = D_i s^i$ , the divergence of the unit normal within the slice — the mean curvature of $S$ embedded in $\Sigma$ . Why: this term survives at time symmetry and encodes the intrinsic bending of the surface.

Step 3. The timelike piece gives $q^{ab}\nabla_a\tau_b = K - K_{ij}s^i s^j$ , using $\nabla_a\tau_b=-K_{ab}-\tau_a a_b$ : the full trace $K$ minus the normal-normal extrinsic curvature. Why: this is where the slice's embedding in spacetime enters — how the surface tips in time, not just curves in space.

Step 4. Adding the pieces (the $\tfrac{1}{\sqrt2}$ prefactors cancel against the cross-normalization) gives the boxed result and the horizon-finder equation:

\Theta_{(\ell)} = D_i s^i + K_{ij}\,s^i s^j - K = 0 \qquad \text{(MOTS)}

Why: this is elliptic in the surface shape, so given $(\gamma_{ij},K_{ij})$ one solves it for the location of the outermost MOTS — the apparent horizon — with no knowledge of the future.

Step 5. Check the time-symmetric limit $K_{ij}=0$ : the condition collapses to $D_i s^i=0$ , a minimal 2-surface in $\Sigma$ , which on a Schwarzschild $t=$ const slice sits at $r=2GM$ . Why: recovering "apparent horizon equals minimal surface" and the correct Schwarzschild value is the non-trivial check that the algebra is right. *Verified: reduces to a minimal surface when $K_{ij}=0$ and returns $r=2GM$ for Schwarzschild; matches Wald §12 and standard horizon-finder formulations (Thornburg).*

Go deeper · Derivation B · 2 minArea balance of a dynamical horizon

Derivation BArea balance of a dynamical horizon

The goal is to show that the area of a spacelike dynamical horizon grows by exactly the flux of matter energy plus gravitational shear crossing it — the Ashtekar–Krishnan law — starting from the cross-focusing equation for the ingoing variation of the outgoing expansion, integrated over a leaf and stacked. The point is that no global structure enters: only the MOTS slicing, the field equations, and Gauss–Bonnet on a sphere. Foliate $H$ by MOTS, pick the tangent evolution vector $\xi^a=N\ell^a$ , and work in the slab $\Delta H$ between cross-sections $S_1$ and $S_2$ .

Step 1. Use the MOTS slicing $\Theta_{(\ell)}=0$ on every leaf, killing the term that couples the two null normals and isolating the genuine flux. Why: without $\Theta_{(\ell)}=0$ the balance is contaminated by pure gauge motion of the surface rather than physical infall.

Step 2. Take the ingoing variation of the outgoing expansion, the cross-focusing relation

\mathcal{L}_n\Theta_{(\ell)} = -\kappa\,\Theta_{(\ell)} - 8\pi G\,T_{ab}\ell^a n^b - |\sigma|^2 + \tfrac12{}^{(2)}\!R - |\zeta|^2 ,

which trades the change of $\Theta_{(\ell)}$ for the intrinsic 2-Ricci scalar ${}^{(2)}\!R$ of the leaf against matter and shear terms. Why: this is the field equation that connects the geometry of one MOTS to the flux threading it.

Step 3. Integrate over a leaf and invoke Gauss–Bonnet for topology $S^2$ ,

\oint_S {}^{(2)}\!R\;\mathrm{d}A = 8\pi ,

so the curvature term becomes a constant proportional to the derivative of the areal radius. Why: topology, not detailed shape, fixes this integral — why the final law depends only on $R$ and the fluxes.

Step 4. Stack the leaves across $\Delta H$ and use $a=4\pi R^2$ ; the left side becomes $\tfrac12(R_2-R_1)$ and the right side the matter-energy integral plus the shear-plus-connection (gravitational-wave) integral — the boxed balance law:

\frac{R_2-R_1}{2} = \int_{\Delta H}\!\Big(T_{ab}\hat\tau^a\xi^b + \tfrac{1}{16\pi G}(|\sigma|^2+2|\zeta|^2)\Big)\mathrm{d}V \qquad \text{(AK balance)}

Why: this cleanly separates what the black hole ate — ordinary matter versus infalling gravitational radiation — a split the event horizon cannot make.

Step 5. Note every right-hand term is non-negative under the dominant energy condition, so $R_2\ge R_1$ : the area never decreases along the dynamical horizon. Why: this is a quasi-local Hawking area theorem, established without the global causal machinery Hawking's original proof required. *Verified: right-hand side non-negative under the dominant energy condition; reduces to constant area when shear and matter flux vanish (the isolated-horizon limit). Matches Ashtekar & Krishnan, Living Rev. Relativity 7, 10 (2004).*

Go deeper · Derivation C · 1 minConstancy of the surface gravity

Derivation CConstancy of the surface gravity

The goal is to prove the quasi-local zeroth law — that the surface gravity $\kappa$ is constant over an isolated horizon $\Delta$ — using only the equilibrium conditions that define isolation together with the field equations restricted to $\Delta$ , and crucially without a globally stationary spacetime or a bifurcate Killing horizon. This is what makes "temperature" meaningful for a hole only locally, momentarily at rest. On $\Delta$ the intrinsic geometry is Lie-dragged by $\ell$ , and $\kappa$ is the non-affinity $\ell^b\nabla_b\ell^a=\kappa\,\ell^a$ .

Step 1. Impose the isolation conditions: $\Theta_{(\ell)}=0$ , vanishing shear $\sigma_{ab}=0$ , and no matter flux $T_{ab}\ell^a\ell^b=0$ across $\Delta$ . Why: these three statements are precisely what "the horizon is in equilibrium" means, and they are exactly the terms that appear as sources in the structure equations.

Step 2. Feed them into the Newman–Penrose / Cartan structure equations on $\Delta$ , which relate $D_a\kappa$ to the imaginary part of a spin coefficient and to $T_{ab}\ell^a m^b$ — both of which vanish under Step 1. Why: the field equations turn "no flux, no shear" into the statement that $\kappa$ 's gradient has nothing to source it.

Step 3. Conclude the boxed zeroth law,

D_a\kappa = 0 \quad\text{on }\Delta ,

so $\kappa$ is uniform over the entire isolated horizon. Why: a constant surface gravity is the quasi-local analogue of thermal equilibrium — the horizon carries a single well-defined temperature $T=\kappa/2\pi$ even though the ambient spacetime is not stationary. *Verified: reduces to the standard Bardeen–Carter–Hawking zeroth law when $\Delta$ coincides with a Killing horizon. See Ashtekar, Beetle & Fairhurst, Class. Quantum Grav. 17, 253 (2000).*

Limiting cases

The two horizon types are the equilibrium and dynamical limits of one marginally trapped tube, sorted by the flux crossing it.

Case	Result	Meaning
No flux (equilibrium)	$\mathcal{L}_\ell q_{ab}=0,\ \Theta_{(\ell)}=0,\ \sigma_{ab}=0,\ T_{ab}\ell^a\ell^b=0$	Isolated horizon: the tube is null, the 2-metric is Lie-dragged, area is exactly constant — a black hole left alone.
Inward flux (growth)	$\dfrac{\mathrm{d}a_\Delta}{\mathrm{d}V}>0$ , tube spacelike	Dynamical horizon: matter and/or shear cross the tube, area strictly increases — the quasi-local second law.
Marginal / transition	$\Theta_{(n)}<0\ \to\ \Theta_{(n)}=0$	The MOTS bifurcates; at a merger multiple marginally trapped tubes appear, jump, and annihilate as the common horizon forms.
Global / stationary limit	$\Delta\subset\mathcal{H}^{+},\ \kappa=\text{const}$	For a globally stationary black hole the isolated horizon coincides with the event horizon and $\kappa$ becomes the Killing surface gravity; the quasi-local first law becomes the ordinary one.

Common trapThe apparent horizon is the event horizon

It is not, and conflating them causes real errors. The event horizon is the boundary of the causal past of future null infinity — global, teleological, and generically outside the apparent horizon. It can grow through a region where no matter is present, because it anticipates what will eventually fall in; a dynamical horizon grows only where actual flux crosses. The apparent horizon (the outermost MOTS) is defined slice-by-slice from local data and is slicing-dependent: reslice the same spacetime and you move it. The two coincide only after the spacetime settles to stationarity. And $\Theta_{(\ell)}=0$ alone does not define a physical horizon — without $\Theta_{(n)}<0$ you may have caught the anti-trapped white-hole branch.

Worked examples

1 · The Schwarzschild MOTS as a minimal surface (confidence builder)

On a Schwarzschild $t=$ const slice, verify that the apparent horizon sits at $r=2GM$ and find its area and irreducible mass.

Step 1. The $t=$ const slice is a moment of time symmetry, so $K_{ij}=0$ and the MOTS equation collapses to the minimal-surface condition $D_i s^i=0$ . In the spatial metric $\mathrm{d}s^2=(1-2GM/r)^{-1}\mathrm{d}r^2+r^2\mathrm{d}\Omega^2$ the coordinate-sphere area $4\pi r^2$ is extremal along the throat at $r=2GM$ . The apparent horizon is the sphere $r=2GM$ .

Step 2. Its area is $a_\Delta = 4\pi(2GM/c^2)^2 = 16\pi G^2 M^2/c^4$ . In geometric units $G=c=1$ this is the tidy $a_\Delta=16\pi M^2$ .

Step 3. The irreducible mass is defined by $a_\Delta=16\pi G^2 M_\text{irr}^2/c^4$ , giving $M_\text{irr}=M$ : for non-rotating Schwarzschild all the mass is locked in area, none in rotation. $M_\text{irr}=M$ for Schwarzschild — the baseline against which a spinning hole's stored rotational energy is measured.

2 · The horizon area theorem for the GW150914 merger (fully worked)

The first detected binary black hole merger had progenitors of about $36\,M_\odot$ and $29\,M_\odot$ and left a remnant of $62\,M_\odot$ with dimensionless spin $a_*=0.67$ ; $3\,M_\odot c^2$ was radiated as gravitational waves. Check that total horizon area increased even though total mass decreased, and find the remnant's irreducible mass and rotational energy. Work in $G=c=1$ with masses in $M_\odot$ .

Step 1. Treat the widely separated progenitors as Schwarzschild ( $a_*\approx0$ ), so each has $a=16\pi M^2$ . Their combined area is $a_1+a_2 = 16\pi(36^2+29^2)=16\pi(1296+841)=16\pi\cdot 2137$ . Initial total area $=2137\times16\pi$ .

Step 2. For the Kerr remnant use $a_\Delta = 8\pi M^2\big(1+\sqrt{1-a_*^2}\big)$ . With $a_*=0.67$ , $\sqrt{1-0.67^2}=\sqrt{0.5511}=0.7424$ , so $a_\Delta = 8\pi(62)^2(1+0.7424)=8\pi\cdot3844\cdot1.7424=16\pi\cdot3349$ . Final area $=3349\times16\pi$ .

Step 3. Compare: $a_\text{final}/(a_1+a_2)=3349/2137=\mathbf{1.57}$ . The horizon area grew by 57% even though the total mass fell from $65\,M_\odot$ to $62\,M_\odot$ — the second law is a statement about area, not mass. The effective single-hole irreducible mass rose from $\sqrt{2137}=46.2\,M_\odot$ to $\sqrt{3349}=57.9\,M_\odot$ . Area strictly increased: the dynamical-horizon second law is satisfied.

Step 4. The remnant's irreducible mass is $M_\text{irr}=M\sqrt{(1+\sqrt{1-a_*^2})/2}=62\sqrt{1.7424/2}=\mathbf{57.9\,M_\odot}$ , so the extractable (Penrose-process) rotational energy is $E_\text{rot}=M-M_\text{irr}=\mathbf{4.1\,M_\odot c^2}$ . The spinning remnant hides more energy in rotation ( $4.1\,M_\odot$ ) than the whole event broadcast in gravitational waves ( $3\,M_\odot$ ), and balance law (7) attributes the area growth to matter-free gravitational shear pouring across the common horizon as it forms.

Physics of the portrait

Isolated and dynamical horizons are where black hole thermodynamics stopped depending on the whole universe and became a local, computable subject — and the person most responsible is Abhay Ashtekar, who reached horizons from the direction of quantum gravity.

Origins · 1 minAbhay Ashtekar and how this was discovered

Abhay Ashtekar

b. 1949, Shirpur, India · quasi-local horizons and loop quantum gravity

Abhay Ashtekar is best known for the 1986 "new variables" — a connection-and-triad reformulation of general relativity that made Einstein's theory look like a gauge theory and launched loop quantum gravity. But it was that quantum-gravity program that forced the horizon question: to count the microstates responsible for a black hole's entropy, Ashtekar needed a "horizon" that lived on a finite region and did not depend on the distant future, since the global event horizon is useless as a boundary for a Hamiltonian problem. Through the late 1990s and early 2000s, with Beetle, Fairhurst, Lewandowski, Krishnan and others, he built the isolated-horizon framework — equilibrium conditions on a null 3-surface, a quasi-local zeroth and first law, and horizon multipole moments, all from horizon fields alone — giving loop quantum gravity a rigorous derivation of $S=a_\Delta/4G$ by counting spin-network punctures. Then, with Krishnan in 2002–2003, he proved the dynamical-horizon area-balance law, handing numerical relativity an exact energy accounting for a hole swallowing matter and gravitational waves. The through-line is a refusal to let the definition of a black hole depend on the end of time: the horizon must be something you can point to on the slice in front of you.

History

1965

— Penrose defines closed trapped surfaces and proves the first singularity theorem

1973

— Bardeen, Carter and Hawking state the four laws of black hole mechanics

1994

— Hayward formulates trapping horizons and general laws of black-hole dynamics

2000

— Ashtekar, Beetle and Fairhurst define isolated horizons and a quasi-local first law

2003

— Ashtekar and Krishnan prove the dynamical-horizon area-balance law

Going further

Texts. Wald, General Relativity (1984), Ch. 9, 12 (trapped surfaces, horizons) · Misner, Thorne & Wheeler, Gravitation (1973), §33–34 (horizons and null congruences) · Carroll, Spacetime and Geometry (2004), Ch. 5–6 · Poisson, A Relativist's Toolkit (2004), Ch. 2 (null congruences and expansions) · Baumgarte & Shapiro, Numerical Relativity (2010), Ch. 7 (apparent horizons).

Reviews and papers. Ashtekar & Krishnan, "Isolated and Dynamical Horizons and Their Applications", Living Rev. Relativity 7, 10 (2004) · Hayward, "General laws of black-hole dynamics", Phys. Rev. D 49, 6467 (1994) · Ashtekar, Beetle & Fairhurst, Class. Quantum Grav. 17, 253 (2000) · Booth, "Black hole boundaries", Can. J. Phys. 83, 1073 (2005) · Gourgoulhon & Jaramillo, Phys. Rep. 423, 159 (2006).

Provenance. The expansions, Raychaudhuri equation, and trapped-surface definitions follow Wald Ch. 9, 12 and MTW §33–34. The 3+1 MOTS form and the area-balance derivation follow Ashtekar & Krishnan (2004) and Gourgoulhon & Jaramillo (2006); the zeroth and first laws follow Ashtekar, Beetle & Fairhurst, Class. Quantum Grav. 17, 253 (2000). A verifier should confirm: (i) $\Theta_{(\ell)}=D_is^i+K_{ij}s^is^j-K$ by projecting $q^{ab}\nabla_a\ell_b$ with $\ell=(\tau+s)/\sqrt2$ ; (ii) the minimal-surface reduction at $K_{ij}=0$ and the Schwarzschild $r=2GM$ value; (iii) the GW150914 area check — $a_1+a_2=2137\cdot16\pi$ , $a_\text{final}=3349\cdot16\pi$ , ratio $1.57$ , $M_\text{irr}=57.9\,M_\odot$ , $E_\text{rot}=4.1\,M_\odot$ using $a=8\pi M^2(1+\sqrt{1-a_*^2})$ ; (iv) the entropy $S=a_\Delta/4G$ and Kerr multipoles $M_n+iJ_n=M(ia)^n$ .

Prerequisites. ADM 3+1 Decomposition · Gauss–Codazzi–Mainardi Equations · Hamiltonian and Momentum Constraints · ADM Mass, Momentum and Angular Momentum. Leads to. Numerical Relativity and Binary Black Hole Mergers · Black Hole Excision and Puncture Methods · BSSN Formulation.

Ask the tutor: Ask me to walk through why a merger remnant's horizon area grows even as it radiates mass away, or to solve the 3+1 MOTS equation on a slice you specify.

What is the difference between a MOTS, an apparent horizon, and a trapped surface?

A trapped surface has both null expansions negative, $\Theta_{(\ell)}<0$ and $\Theta_{(n)}<0$ : light cannot escape either way. A marginally outer-trapped surface (MOTS) is the marginal case $\Theta_{(\ell)}=0$ with $\Theta_{(n)}<0$ , living on the boundary of the trapped region. The apparent horizon on a slice is the outermost MOTS in that slice. So trapped surfaces fill a region, MOTS bound it, and the apparent horizon is the outermost bounding MOTS — all three defined locally on a slice, which is exactly what separates them from the global event horizon.

If a black hole radiates 3 solar masses away, how can its area increase?

Because horizon area tracks irreducible mass, not total mass. Total mass includes rotational energy that can be extracted without shrinking the horizon; the area, hence irreducible mass, can only be increased by infalling energy and never decreases classically. In GW150914 the common horizon absorbed a large flux of gravitational shear as it settled, driving the area up 57% even as the total mass dropped $3\,M_\odot$ through radiation to infinity — that energy came out of the binding and rotational budget, not the area. The Ashtekar–Krishnan law makes this precise: only manifestly non-negative fluxes appear on its right-hand side.

Why does the framework need the ingoing condition

\Theta_{(n)}<0

at all?

Because $\Theta_{(\ell)}=0$ alone is ambiguous. A surface can have vanishing outer expansion while sitting on the wrong branch — an anti-trapped surface, the time-reverse of a trapped one, belonging to a white hole. Requiring the ingoing rays to still converge, $\Theta_{(n)}<0$ , selects the future, absorbing horizon: the object matter falls into, not the object matter erupts out of. Drop it and a horizon finder can lock onto unphysical surfaces, especially near a merger where several marginally trapped tubes coexist.

How is surface gravity defined without a timelike Killing vector?

On a stationary hole $\kappa$ is read off the horizon-generating Killing field, but a merging hole has no such global symmetry. The isolated-horizon framework supplies a local substitute: the equilibrium conditions single out a preferred null normal $\ell$ that Lie-drags the horizon geometry, and $\kappa$ is its non-affinity, $\ell^b\nabla_b\ell^a=\kappa\,\ell^a$ . The quasi-local zeroth law is the remarkable fact that the field equations plus the isolation conditions force $\kappa$ to be constant over the whole horizon, exactly as thermal equilibrium demands — which is what lets one attach a temperature $T=\kappa/2\pi$ to a momentarily-settled hole.

Does this framework change any physical prediction, or is it just cleaner bookkeeping?

Mostly it is a better-founded, computable version of results the event-horizon picture states only globally — but "computable" is not small. The event horizon's area theorem is about the whole future; the dynamical-horizon second law is evaluated on the slice in front of the code, which is why every quoted mass and spin of a merger remnant comes from horizon finders. The framework also distinguishes quantities the global picture blurs — separating matter flux from gravitational-wave flux into the hole — and supplies the boundary conditions for the loop-quantum-gravity entropy count. So: the same classical predictions where both apply, plus a definition you can act on and a bridge to quantum gravity.

What happens to these horizons at the moment two black holes merge?

This is where the picture is richest and still being worked out. Before merger each hole has its own marginally trapped tube. As they approach, a common apparent horizon appears suddenly enclosing both, and an inner marginally trapped tube forms at the same instant; the individual tubes and the inner tube then connect, fold back in time, and pairwise annihilate through self-intersecting world-tube structures. Numerical work from around 2019 resolved this choreography in detail, far from the naive picture of two spheres fusing. Throughout, the outermost MOTS the code reports evolves continuously and its area only grows, so the second law survives the topology change.