Event Horizon and Apparent Horizon

A black hole is loosely "the region nothing escapes from," but sharpening that sentence forces a choice between two surfaces that do not agree. The event horizon answers a global question — what can ever reach a distant observer? It is the boundary of the causal past of future null infinity 𝓘⁺, so the black-hole region is $\mathcal{B} = M \setminus J^{-}(\mathscr{I}^{+})$ . Because 𝓘⁺ encodes the entire asymptotic future, the event horizon is teleological: to say where it lies now you must already know the whole fate of the spacetime. That makes it conceptually pristine — slicing-independent, a genuine null hypersurface, the object of the singularity and area theorems — and, in any local sense, uncomputable. The apparent horizon answers the complementary, operational question — where is light already failing to move outward? On a spacelike slice Σ you foliate space into closed 2-spheres and compute the expansion $\theta_{+}$ of the outgoing null normals. A surface with $\theta_{+}=0$ is marginally outer trapped; the apparent horizon is the outermost such surface on that slice. It is quasi-local, readable off a single instant of data — which is exactly why numerical relativity hunts apparent horizons (and the trapping/dynamical horizons near them) to follow black holes through a merger. In equilibrium the two coincide, and with the Killing horizon besides. Out of equilibrium they part: under cosmic censorship and the null energy condition the apparent horizon sits on or inside the event horizon, $r_{\mathrm{AH}}\le r_{\mathrm{EH}}$ . The event horizon can grow before infalling matter reaches it, its position fixed by what the matter is going to do, while the apparent horizon only responds to matter already on the slice, so it jumps. The distinction is not pedantry — Hawking radiation, evaporation, and the information problem all turn on which horizon you track, and for an evaporating hole the global event horizon may never form at all.

Key takeaways

The event horizon is the teleological global boundary $\partial J^{-}(\mathscr{I}^{+})$ of the causal past of future null infinity; locating it requires the entire future evolution of the spacetime.
The apparent horizon is local: the outermost marginally outer trapped surface (MOTS), where the outgoing null expansion vanishes, $\theta_{+}=0$ , with $\theta_{-}<0$ .
In any stationary black hole the event, apparent, and Killing horizons coincide; in dynamical spacetimes $r_{\mathrm{AH}}\le r_{\mathrm{EH}}$ , with equality reached only as the hole settles.
Event-horizon generators are future-endless null geodesics; Raychaudhuri focusing plus the null energy condition forbid $\theta<0$ on them, giving the area theorem $\mathrm{d}A_{\mathrm{EH}}/\mathrm{d}t\ge 0$ .
The event horizon grows acausally — in anticipation of matter not yet crossed — while the apparent horizon is slicing-dependent and can even be absent on a given slice of Schwarzschild.
Key idea: two horizons answer two questions. The event horizon is the global causal boundary $\partial J^{-}(\mathscr{I}^{+})$ — what can ever escape — and is teleological; the apparent horizon is the outermost surface with $\theta_{+}=0$ on a slice — where light is locally trapped — and is quasi-local. They coincide in stationarity; otherwise $r_{\mathrm{AH}}\le r_{\mathrm{EH}}$ .

The core insight

Two horizons answer two questions. The event horizon is the global causal boundary $\partial J^{-}(\mathscr{I}^{+})$ — what can ever escape — and is teleological; the apparent horizon is the outermost surface with vanishing outgoing null expansion $\theta_{+}=0$ on a given slice — where light is locally trapped — and is quasi-local. They coincide in stationarity; otherwise $r_{\mathrm{AH}}\le r_{\mathrm{EH}}$ .

Where this lives today

Every gravitational-wave detection is, in part, an apparent-horizon computation. LIGO–Virgo–KAGRA templates come from numerical-relativity simulations that cannot see the event horizon while they run — it needs the whole future — so the codes solve $\theta_{+}=0$ on each slice, read off the apparent horizons' areas and spins, and stitch the two inspiralling horizons into one. The remnant's mass and spin, quoted to a few percent for events like GW150914, come from the final apparent-horizon area via the irreducible-mass relation of Worked Example 1, and the measured area increase is a direct check of Hawking's theorem (confirmed for GW150914 above 95% confidence in 2021). At the frontier, the event horizon's unsuitability for evaporating and quantum settings has pushed theorists toward quasi-local surrogates: isolated and dynamical horizons carry a local first law and area-increase law (Ashtekar–Krishnan), and the island formula reproducing the Page curve is phrased in terms of quantum extremal surfaces near the apparent horizon, not the global event horizon — which, for a fully evaporating hole, may never have existed.

The physical picture

Picture a swimmer in a river that quickens toward a waterfall. The apparent horizon is the line where the current first matches the swimmer's top speed: past it you can no longer make local headway upstream. The event horizon is the line past which you can never regain the bank ever again — and where it sits depends on whether the river keeps speeding up downstream or eases off. If the flow is steady, the two lines coincide. If the river is still accelerating, the point of no eventual return lies further upstream than the point of no local headway: the event horizon is the outer one. The precise version of "making headway" is the expansion of a bundle of light. A sphere of rays leaving a 2-surface either grows in area ( $\theta_{+}>0$ ) or shrinks ( $\theta_{+}<0$ ). Far outside a hole, outgoing light spreads as usual; deep inside, gravity is so strong that even the rays fired outward are swept to smaller areas — both families converge, the surface is trapped. The knife-edge between them, where outgoing light momentarily neither grows nor shrinks, is $\theta_{+}=0$ : the marginally outer trapped surface that defines the apparent horizon.

Figure

Classifying a closed 2-sphere by its two null expansions. Outside the gravitational radius at r = 3M the outgoing rays still spread, θ₊ = +1/9M greater than 0; at r = 2M the outgoing expansion vanishes, θ₊ = 0, the marginally outer trapped surface; inside at r = 1.5M both expansions are negative and the sphere is trapped. The ingoing expansion θ₋ = −2/r stays negative everywhere

The event horizon's teleology is genuinely strange. A generator can be momentarily expanding and only "learn" later — as more matter falls in — that it was always fated to remain on the boundary. The surface knows the future. This is why a numerical-relativity run cannot mark the event horizon while it computes: only once the spacetime has settled can one shoot null generators backward from the final state and reconstruct where the boundary was all along. During the run the code locates apparent horizons instead, a purely local elliptic problem. The second figure shows the two disagreeing during collapse: a shell of null dust falls in at advanced time $v_{0}$ , doubling the mass, and the apparent horizon jumps while the event horizon glides up to the final size ahead of the shell.

Figure

Vaidya collapse in advanced time v. The apparent horizon r_AH = 2M(v) tracks the instantaneous mass and jumps discontinuously from 2M₁ to 2M₂ = 4M₁ when the null shell arrives at v₀. The event horizon grows smoothly and acausally, climbing from inside 2M₁ up through 2M₁ and reaching 2M₂ before the shell crosses — it already encodes the final mass

Why do the two surfaces reconcile in equilibrium? A stationary horizon is a Killing horizon — a null surface generated by a Killing vector that is null on it. Its generators neither expand nor shear ( $\theta=\sigma=0$ ), so the surface is at once a marginally trapped surface (an apparent horizon) and a non-expanding causal boundary (an event horizon). Motion is what splits them; stillness is what fuses them.

The core equations

The event horizon is defined not by a field value but by a causal relation to infinity: the boundary of the events that can still send a signal to the asymptotic future.

\mathcal{B} = M \setminus J^{-}(\mathscr{I}^{+}), \qquad \mathcal{H} = \partial J^{-}(\mathscr{I}^{+}) \cap M \qquad (1)

where: $M$ — the asymptotically flat spacetime manifold; $\mathscr{I}^{+}$ — future null infinity, where outgoing light rays end; $J^{-}(\mathscr{I}^{+})$ — causal past of 𝓘⁺, events that can signal to infinity; $\mathcal{B}$ — the black-hole region, events that never can; $\mathcal{H}$ — the event horizon, the boundary of $\mathcal{B}$ , a null hypersurface. Reading: every symbol is global — nothing here is measurable near a point. The apparent horizon is built from a strictly local object: how a bundle of light fans out from a 2-surface. The two null normals give two expansions.

\theta_{\pm} = h^{ab}\,\nabla_{a}\, \ell^{\pm}_{b}, \qquad h_{ab} = g_{ab} + \ell^{+}_{a}\ell^{-}_{b} + \ell^{-}_{a}\ell^{+}_{b} \qquad (2)

where: $\ell^{+}_{a}$ — outgoing future-directed null normal to the closed 2-surface 𝒮; $\ell^{-}_{a}$ — ingoing null normal, normalized so $\ell^{+}\!\cdot\ell^{-}=-1$ ; $h_{ab}$ — induced metric on 𝒮 (the projector); $\theta_{\pm}$ — expansion of each congruence, the fractional rate of area change. Reading: $\theta_{+}$ is the local surrogate for "outward" that replaces the global 𝓘⁺. A surface is marginally outer trapped when the outgoing rays momentarily stall while the ingoing rays still converge; the apparent horizon is the outermost such surface on the slice.

\theta_{+}\big|_{\mathrm{AH}} = 0, \qquad \theta_{-}\big|_{\mathrm{AH}} < 0 \qquad (3)

where: $\theta_{+}=0$ — marginally outer trapped: outgoing light neither spreads nor converges; $\theta_{-}<0$ — ingoing light still converges, fixing this as a future (black-hole) horizon, not a past (white-hole) one. Reading: the sign of $\theta_{-}$ carries the time-orientation. Whether a generator can ever have $\theta_{+}<0$ is decided by the null Raychaudhuri equation, the focusing law for light bundles.

\frac{\mathrm{d}\theta_{+}}{\mathrm{d}\lambda} = -\tfrac{1}{2}\theta_{+}^{2} - \sigma_{ab}\sigma^{ab} - R_{ab}\,\ell^{a}\ell^{b} \qquad (4)

where: $\lambda$ — affine parameter along the generators $\ell^{a}=\ell^{+a}$ ; $\sigma_{ab}$ — shear (twist-free for hypersurface-orthogonal generators); $R_{ab}\ell^{a}\ell^{b}$ — null-null Ricci term, $\ge 0$ by the null energy condition. Reading: every right-hand term is $\le 0$ , so gravity only focuses light — the engine behind the area theorem. A non-stationary model where the two horizons visibly separate is ingoing Vaidya: Schwarzschild with a time-dependent mass fed by infalling null dust.

\mathrm{d}s^{2} = -\Big(1-\frac{2M(v)}{r}\Big)\mathrm{d}v^{2} + 2\,\mathrm{d}v\,\mathrm{d}r + r^{2}\,\mathrm{d}\Omega^{2} \qquad (5)

where: $v$ — advanced null coordinate, $v=t+r_{*}$ in the static limit; $M(v)$ — mass function, $\mathrm{d}M/\mathrm{d}v\ge 0$ ; $\mathrm{d}\Omega^{2}=\mathrm{d}\vartheta^{2}+\sin^{2}\!\vartheta\,\mathrm{d}\varphi^{2}$ . Reading: constant $M$ recovers Schwarzschild in ingoing Eddington–Finkelstein form, regular across the horizon. Here the two horizons obey very different laws — one algebraic and local, one an ODE reaching into the future.

r_{\mathrm{AH}}(v) = 2M(v), \qquad \frac{\mathrm{d}r_{\mathrm{EH}}}{\mathrm{d}v} = \tfrac{1}{2}\Big(1-\frac{2M(v)}{r_{\mathrm{EH}}}\Big) \qquad (6)

where: $r_{\mathrm{AH}}$ — apparent horizon, at the instantaneous Schwarzschild radius; $r_{\mathrm{EH}}$ — event horizon, the outgoing radial null ray that neither escapes nor falls in, fixable only with $M(v)$ at all future $v$ . Reading: the apparent horizon needs one slice, the event horizon the whole future. For any stationary black hole the horizon is a coordinate sphere fixed by the conserved charges, and all three notions coincide there.

r_{+} = M + \sqrt{M^{2}-a^{2}-Q^{2}}, \qquad A_{\mathrm{EH}} \ge A_{\mathrm{AH}}, \qquad \frac{\mathrm{d}A_{\mathrm{EH}}}{\mathrm{d}t}\ge 0 \qquad (7)

where: $M$ — ADM mass ( $G=c=1$ ); $a=J/M$ — spin parameter; $Q$ — charge; the horizon exists only for $M^{2}\ge a^{2}+Q^{2}$ , extremal at equality. The inequalities record that the event horizon is the outer surface and its area never shrinks. Reading: area is the master monotone — it grounds black-hole entropy $S=A/4$ .

Derivation AWhere light is marginally trapped, and why Schwarzschild puts it at the gravitational radius

The goal is to convert the abstract condition $\theta_{+}=0$ into a concrete radius, and in doing so to prove that in a static hole the local, slice-dependent apparent horizon lands exactly on the global event horizon at $r=2M$ . We first classify a general 2-sphere by the two signs $(\theta_{+},\theta_{-})$ to see why $\theta_{+}=0$ is the true boundary of the trapped region, then compute both expansions directly in ingoing Eddington–Finkelstein coordinates, where the metric is regular across the horizon and the null normals are simple.

Step 1. Classify a closed 2-sphere by its expansion pair. An ordinary surface has $\theta_{+}>0,\ \theta_{-}<0$ : outgoing light spreads, ingoing light converges, as in flat space. A trapped surface has $\theta_{+}<0,\ \theta_{-}<0$ : both families converge. The marginal case $\theta_{+}=0,\ \theta_{-}<0$ is the boundary between them — the marginally outer trapped surface. Why: the apparent horizon is the outermost boundary of the trapped region, and that boundary is where the outgoing expansion first hits zero.

Step 2. Write the null normals for the metric of equation (5) with $M$ constant. On a sphere of fixed $r$ and $v$ the future-directed null vectors are $\ell^{+}=\partial_{v}+\tfrac12\big(1-\tfrac{2M}{r}\big)\partial_{r}$ and $\ell^{-}=-\partial_{r}$ , normalized so $\ell^{+}\!\cdot\ell^{-}=-1$ . Why: in ingoing coordinates the ingoing rays are pure $-\partial_r$ , so the whole horizon structure sits in the radial part of $\ell^{+}$ .

Step 3. Compute the expansions as area growth rates, $\theta_{\pm}=\ell^{\pm r}\,\partial_{r}\ln(r^{2})$ , with induced metric $h_{ab}\,\mathrm{d}x^{a}\mathrm{d}x^{b}=r^{2}\mathrm{d}\Omega^{2}$ . The ingoing case gives $\theta_{-}=(-1)(2/r)=-2/r<0$ always; the outgoing case gives $\theta_{+}=\tfrac12\big(1-\tfrac{2M}{r}\big)(2/r)=\tfrac{1}{r}\big(1-\tfrac{2M}{r}\big)$ . Why: the fractional rate of change of the area element $r^{2}$ along a null vector is exactly $\ell^{r}\,\partial_r\ln r^2$ for a round sphere.

Step 4. Solve the marginal condition. Setting $\theta_{+}=0$ forces $r=2M$ , where $\theta_{-}=-1/M<0$ confirms a genuine future MOTS. For $r>2M$ , $\theta_{+}>0$ (normal); for $r<2M$ both expansions are negative (trapped). Why: this reproduces the Step 1 classification as a single function of radius, with the sign flip pinned to the gravitational radius.

Step 5. Identify the coincidence. Because the metric is static, $r=2M$ is simultaneously the Killing horizon of $\partial_{t}$ and the event horizon $\partial J^{-}(\mathscr{I}^{+})$ — all three collapse onto one sphere. Why: stationarity removes the time-dependence that would separate a local surrogate from a global boundary, so the definitions cannot disagree. *The result $\theta_{+}=\tfrac1r(1-2M/r)$ vanishes at $r=2M$ with $\theta_{-}=-2/r<0$ , confirming a genuine future MOTS at the gravitational radius.*

Go deeper · Derivation B · 2 minRaychaudhuri focusing and the area theorem for the event horizon

Derivation BRaychaudhuri focusing and the area theorem for the event horizon

The goal is to show that the event horizon's area can never decrease, using only the two ingredients that the horizon is generated by future-endless null geodesics and that gravity focuses light. This is the structural heart of black-hole thermodynamics: it is the classical statement whose quantum echo is the second law $\mathrm{d}(S_{\mathrm{BH}}+S_{\mathrm{matter}})\ge 0$ . We argue by contradiction from a hypothetical negative expansion, drive the focusing to a caustic in finite affine time, and show a caustic is incompatible with a generator that never leaves the horizon.

Step 1. Recall the generator structure. The event horizon $\mathcal{H}=\partial J^{-}(\mathscr{I}^{+})$ is an achronal boundary, hence a null hypersurface ruled by null geodesics, and those generators have no future endpoints — a generator can only enter $\mathcal{H}$ (at a caustic where new generators join), never leave it to the future. Why: leaving outward would place the point strictly inside $J^{-}(\mathscr{I}^{+})$ , and leaving inward would put it inside the black hole — either contradicts membership of the boundary.

Step 2. Suppose, for contradiction, that some generator has $\theta_{+}=\theta_{0}<0$ at parameter $\lambda_{0}$ . In equation (4) every right-hand term is non-positive — $-\tfrac12\theta_{+}^{2}$ , the shear $-\sigma_{ab}\sigma^{ab}\le 0$ , and $-R_{ab}\ell^{a}\ell^{b}\le 0$ by the null energy condition — so $\mathrm{d}\theta_{+}/\mathrm{d}\lambda \le -\tfrac12\theta_{+}^{2}$ . Why: the null energy condition $T_{ab}\ell^{a}\ell^{b}\ge 0$ feeds through Einstein's equations into $R_{ab}\ell^{a}\ell^{b}\ge 0$ , so matter only strengthens the focusing.

Step 3. Integrate. Written as $\mathrm{d}(1/\theta_{+})/\mathrm{d}\lambda\ge \tfrac12$ , this drives $\theta_{+}\to-\infty$ within an affine interval $\Delta\lambda\le 2/|\theta_{0}|$ : the congruence forms a caustic in finite affine time. Why: the $\theta_{+}^{2}$ nonlinearity guarantees a focal point rather than a gentle asymptotic decay.

Step 4. Reach the contradiction. At a caustic neighbouring generators cross, so a horizon generator running into one to the future would have to leave the horizon — impossible for a future-endless generator on $\partial J^{-}(\mathscr{I}^{+})$ . Hence $\theta_{+}\ge 0$ everywhere. Why: global endlessness rules out the focal point that local focusing wants to create.

Step 5. Conclude the area law. Non-negative expansion makes each cross-sectional area element non-decreasing along the generators, so integrating over the cross-section gives $\mathrm{d}A_{\mathrm{EH}}/\mathrm{d}t\ge 0$ , Hawking's area theorem. Why: with generators never leaving and never focusing, area can only grow as new generators join. *The focusing bound $\Delta\lambda\le 2/|\theta_0|$ is the standard Raychaudhuri estimate: a congruence with initial expansion $\theta_0<0$ reaches a caustic within affine interval $2/|\theta_0|$ .*

Go deeper · Derivation C · 2 minAcausal growth of the event horizon under an infalling shell

Derivation CAcausal growth of the event horizon under an infalling shell

The goal is to exhibit, in a fully explicit model, the defining behavioural difference between the two horizons: that the event horizon grows in anticipation of matter not yet arrived while the apparent horizon only reacts to matter already present. We glue Schwarzschild of mass $M_{1}$ to Schwarzschild of mass $M_{2}>M_{1}$ across a single ingoing null shell at advanced time $v_{0}$ — the Vaidya limit with $M(v)=M_{1}+(M_{2}-M_{1})\,\Theta(v-v_{0})$ — and track each horizon through the transition.

Step 1. Locate the apparent horizon slice by slice. From equation (6), $r_{\mathrm{AH}}(v)=2M(v)$ equals $2M_{1}$ for $v<v_{0}$ and $2M_{2}$ for $v>v_{0}$ ; it jumps by $2(M_{2}-M_{1})$ at the shell. Why: the marginal condition $\theta_{+}=0$ depends only on the instantaneous metric, so at a step in $M$ the trapped region enlarges in a single instant.

Step 2. Set up the event-horizon ODE. The event horizon is the outgoing radial null generator that neither escapes nor falls in, obeying $\mathrm{d}r_{\mathrm{EH}}/\mathrm{d}v=\tfrac12\big(1-2M(v)/r_{\mathrm{EH}}\big)$ — an ODE, not an algebraic condition, selected by a future boundary condition. Why: only one member of the outgoing null congruence stays on the boundary forever, and finding it requires integrating forward.

Step 3. Fix the future boundary condition. For $v>v_{0}$ the spacetime is Schwarzschild $M_{2}$ , whose only future-endless marginal generator is the static sphere $r=2M_{2}$ ; continuity forces $r_{\mathrm{EH}}(v_{0})=2M_{2}$ . Why: a generator above $2M_{2}$ escapes to 𝓘⁺ and one below falls into the singularity, so the single survivor pins the value at the shell.

Step 4. Integrate backward. For $v<v_{0}$ the generator lives in Schwarzschild $M_{1}$ yet must arrive at $r=2M_{2}>2M_{1}$ by $v_{0}$ ; the ODE with $M=M_{1}$ has $\mathrm{d}r_{\mathrm{EH}}/\mathrm{d}v>0$ for $r>2M_{1}$ , so the horizon starts inside $2M_{1}$ and expands monotonically through $2M_{1}$ up to $2M_{2}$ . Why: every radius above the old horizon is expanding outward, which is how the boundary climbs to meet the incoming mass.

Step 5. Read off the anticipation. The event horizon reaches its final size $2M_{2}$ at the instant the shell arrives, having grown through the whole interval before it — while the apparent horizon registers the mass only at $v_{0}$ . With $M_{2}=2M_{1}$ the apparent horizon jumps $2M_{1}\to4M_{1}$ while the event horizon has smoothly covered the same span in advance; they agree only as $v\to\infty$ . Why: the event horizon's position is fixed by the entire future mass, so it must move before the cause reaches it — acausality is a feature of a global surface, not a violation of physics. *The ODE sign $\mathrm{d}r_{\mathrm{EH}}/\mathrm{d}v>0$ for $r>2M_1$ is immediate from equation (6): every radius above the old horizon is expanding outward, driving the boundary up to meet the incoming mass.*

Limiting cases

Case	Result	Meaning
Stationary (Schwarzschild / Kerr / Kerr–Newman)	$r_{\mathrm{EH}} = r_{\mathrm{AH}} = r_{+}$	Event, apparent, and Killing horizons coincide; for Kerr–Newman $r_{+}=M+\sqrt{M^{2}-a^{2}-Q^{2}}$ in geometric units.
Dynamical collapse / accretion	$r_{\mathrm{AH}} \le r_{\mathrm{EH}}$	Under the null energy condition and cosmic censorship the apparent horizon never lies outside the event horizon; equality is reached only as the hole settles.
Vaidya infalling null dust	$r_{\mathrm{AH}}(v)=2M(v),\ \dot r_{\mathrm{EH}}\ge 0$ acausally	The apparent horizon tracks the instantaneous mass and is null or spacelike; the event horizon grows in anticipation and is always null.
Fully evaporating hole (semiclassical)	event horizon may never form; AH appears and disappears	If everything eventually reaches 𝓘⁺, then $J^{-}(\mathscr{I}^{+})$ is all of spacetime and $\mathcal{B}=\varnothing$ — yet trapped surfaces and an apparent horizon existed transiently.
de Sitter / cosmological	$r_{\mathrm{cosmo}}=\sqrt{3/\Lambda}$	Here 𝓘⁺ is spacelike; observer-dependent cosmological event horizons exist and apparent horizons are defined per worldline.

Common trapThe apparent horizon is a property of the black hole alone, like its mass.

It is not. The apparent horizon depends on the choice of spacelike slicing, not on the spacetime alone. There exist perfectly good slicings of the same Schwarzschild geometry — certain spherically symmetric slices threading the interior — on which no apparent horizon appears at all, even though the event horizon remains rigidly at $r=2M$ . So "the black hole has no apparent horizon on this slice" is a statement about your foliation, not about the hole. The mirror-image error is just as common: treating the event horizon as something a local instrument could find. It is defined by 𝓘⁺, the entire asymptotic future, so no measurement confined to a neighbourhood or a single instant can ever detect it — an infalling observer crosses $r=2M$ of a large hole with no local signal whatsoever, feeling only mild tidal forces. One surface is slice-dependent but local; the other is slice-independent but global; neither is "the" horizon in isolation.

Worked examples

1 · Rotational energy locked behind a Kerr horizon (fully worked)

A Kerr black hole of mass $M$ spins at $a=0.9M$ . Find its horizon radius, its irreducible mass, and the fraction of the total mass–energy that could in principle be extracted by spinning it down. Then repeat for a maximal (extremal) hole, $a=M$ .

Step 1. Horizon radius from equation (7) with $Q=0$ : $r_{+}=M+\sqrt{M^{2}-a^{2}}=M\big(1+\sqrt{1-0.81}\big)=M(1+\sqrt{0.19})=\mathbf{1.436\,M}$ . The horizon has shrunk from $2M$ (non-spinning) toward $M$ as spin is added.

Step 2. Horizon area. For Kerr, $A=4\pi(r_{+}^{2}+a^{2})=8\pi M r_{+}=8\pi M(1.436M)=\mathbf{11.49\,\pi M^{2}}$ , down from the Schwarzschild value $16\pi M^{2}$ at the same mass — spin costs area.

Step 3. Irreducible mass, the mass of the Schwarzschild hole with the same area: $M_{\mathrm{irr}}=\sqrt{A/16\pi}=\sqrt{11.49\pi M^{2}/16\pi}=\sqrt{0.718}\,M=\mathbf{0.847\,M}$ . Because the area theorem forbids $A$ (hence $M_{\mathrm{irr}}$ ) from decreasing, this is the floor no classical process can breach.

Step 4. Extractable rotational energy: $M-M_{\mathrm{irr}}=(1-0.847)M=\mathbf{0.153\,M}$ , about 15% of the total mass–energy, available via the Penrose process or superradiance before the hole reaches Schwarzschild.

Step 5. Extremal case $a=M$ : $r_{+}=M$ , $A=8\pi M^{2}$ , $M_{\mathrm{irr}}=M/\sqrt2=0.707M$ , so the extractable fraction rises to $1-1/\sqrt2=\mathbf{29.3\%}$ — the famous ceiling on energy extraction from a maximally spinning black hole. A spinning hole stores up to ~29% of its mass as extractable rotational energy, all guarded by the non-decreasing horizon area.

2 · The area theorem as an audit of a black-hole merger

Two black holes of masses $36\,M_\odot$ and $29\,M_\odot$ (each effectively Schwarzschild before merger) coalesce into a final Kerr hole of mass $62\,M_\odot$ and dimensionless spin $a/M=0.67$ , radiating roughly $3\,M_\odot$ in gravitational waves — the parameters of GW150914. Check that the total horizon area still increased, as the area theorem demands.

Step 1. Initial area, the sum of two Schwarzschild horizons ( $A=16\pi M^{2}$ each, in units where $G=c=1$ ): $A_{i}=16\pi(36^{2}+29^{2})=16\pi(1296+841)=16\pi(2137)=\mathbf{3.42\times10^{4}\,\pi\,M_\odot^{2}}$ .

Step 2. Final horizon radius: $r_{+}=M_{f}\big(1+\sqrt{1-(a/M)^{2}}\big)=62\big(1+\sqrt{1-0.449}\big)=62(1+0.742)=\mathbf{108.0\,M_\odot}$ .

Step 3. Final area: $A_{f}=8\pi M_{f}r_{+}=8\pi(62)(108.0)=\mathbf{5.36\times10^{4}\,\pi\,M_\odot^{2}}$ .

Step 4. Compare: $A_{f}/A_{i}=5.36/3.42=\mathbf{1.57}$ . The horizon area grew by 57% even though the system lost $3\,M_\odot$ of mass–energy to radiation. Area, not mass, is the monotone: the merger obeys the area theorem, and the same inequality caps the radiated energy — you cannot emit so much that the final area drops below the sum of the initial two.

Physics of the portrait

The modern understanding of horizons rests on one person's cast of ideas: conformal infinity, the trapped surface, the causal diagram. Before the 1960s a black hole was a static curiosity and "the region nothing escapes" had no rigorous, dynamical meaning; it was the geometrization of causality that turned the surface at $r=2M$ into a global, physical boundary.

Origins · 1 minRoger Penrose and how this was discovered

Roger Penrose

1931– · global causal structure of spacetime and the trapped-surface concept

Roger Penrose reshaped general relativity by importing topology and global causal analysis into a subject dominated by explicit metrics. His 1964 introduction of conformal infinity — attaching the ideal boundary 𝓘⁺ where outgoing light ends — gave the event horizon its permanent definition as $\partial J^{-}(\mathscr{I}^{+})$ , and the "Penrose diagram" that compactifies a whole spacetime onto a finite page remains the field's basic picture. In 1965 he defined the closed trapped surface — a 2-sphere whose outgoing and ingoing light both converge — and used it to prove the first singularity theorem: once a trapped surface forms, under the null energy condition, a singularity is inevitable. That idea is the seed of the apparent horizon, the outermost marginally trapped surface on a slice. He went on to state the cosmic censorship conjecture — precisely what keeps the apparent horizon hidden inside the event horizon, $r_{\mathrm{AH}}\le r_{\mathrm{EH}}$ — and to devise the Penrose process behind Worked Example 1. His argument that black holes are a robust prediction of general relativity, not an artefact of symmetry, was recognized with the 2020 Nobel Prize. Nearly every object in this article — 𝓘⁺, the trapped surface, the causal boundary — is his.

History

1956

— Rindler names the "event horizon" as a causal boundary of no return

1965

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

1968

— Hawking and Ellis formalize the apparent horizon on a Cauchy slice

1971

— Hawking proves the event-horizon area theorem

2004

— Ashtekar and Krishnan develop the dynamical-horizon framework

Going further

Texts. Misner, Thorne & Wheeler, Gravitation, §33–34 · Wald, General Relativity, §9.2, §12.1–12.2 · Hawking & Ellis, The Large Scale Structure of Space-Time, §8–9 · Carroll, Spacetime and Geometry, §5.6, Ch. 6 · Hartle, Gravity, Ch. 12 · Poisson, A Relativist's Toolkit, §5 · Ashtekar & Krishnan, Isolated and Dynamical Horizons, Living Rev. Rel. 7, 10 (2004). Note on key results. Achronal boundaries are null where differentiable and their generators are future-endless. The Schwarzschild/Vaidya expansion $\theta_{+}=\tfrac1r(1-2M/r)$ with $\theta_{-}=-2/r$ follows in ingoing Eddington–Finkelstein coordinates from $\ell^{+}=\partial_v+\tfrac12(1-2M/r)\partial_r$ . The Kerr numbers are exact: $r_+=1.436M$ , $A=11.49\pi M^2$ , $M_{\mathrm{irr}}=0.847M$ at $a=0.9M$ , and $M_{\mathrm{irr}}=M/\sqrt2$ with $29.3\%$ extractable at $a=M$ (Christodoulou irreducible mass; $M_{\mathrm{irr}}=\sqrt{A/16\pi}$ ). The GW150914 audit uses $A=16\pi(36^2+29^2)\to 8\pi(62)(108)$ , giving $A_f/A_i=1.57$ ; the final spin $0.67$ and $\sim3\,M_\odot$ radiated are the published LIGO values. Kerr–Newman $r_+=M+\sqrt{M^2-a^2-Q^2}$ is the standard horizon root.

Prerequisites. Trapped Surfaces and Closed Trapped Regions · Schwarzschild Black Hole Structure · Killing Horizons and Surface Gravity. Leads to. Hawking Area Theorem · Four Laws of Black Hole Mechanics · No-Hair Theorem and Black Hole Uniqueness · Kerr Metric and Rotating Black Holes.

Ask the tutor: Want to see the two horizons pulled apart in a live Vaidya or numerical-collapse example, or walk through why cosmic censorship is what keeps the apparent horizon tucked inside the event horizon?

What is the single sharpest difference between the event and apparent horizons?

The time direction of the information they require. The apparent horizon is read off one spacelike slice — present data only — by solving $\theta_{+}=0$ , so it is computable on the spot but depends on how you sliced spacetime. The event horizon is read off the entire future, being the boundary $\partial J^{-}(\mathscr{I}^{+})$ of what can ever reach infinity, so it is slicing-independent but cannot be known until the whole history is in. Everything else — that one jumps and the other cannot, that one is slice-dependent and the other global, that one grows acausally — flows from this contrast between present-data and full-future definitions.

Why does the event horizon grow before the infalling matter reaches it?

Because its location is fixed by the total future mass, not the present mass. The event horizon is the one outgoing light ray that manages to hover forever — never escaping to infinity, never falling in. Which ray does that depends on how much mass will eventually be inside, so as soon as you know a shell is on its way, the hovering ray is already the larger one appropriate to the final mass. It therefore expands through the region ahead of the shell. Nothing travels faster than light here: no local signal outraces the matter. It is only that a globally defined surface reflects a global fact, and the global fact includes the future infall.

If an evaporating black hole has no event horizon, was there ever a black hole?

There was certainly a trapped region and an apparent horizon — real, locally detectable structure that formed in the collapse and shrank away as the hole evaporated. What may be absent is the global event horizon: if every ray, including all the Hawking radiation, ultimately reaches infinity, then nothing was permanently cut off from 𝓘⁺, so $\mathcal{B}=\varnothing$ . This is one reason quantum-gravity work leans on quasi-local horizons: they capture the operational reality of "a black hole was here" even in situations where the textbook global definition quietly evaporates along with the hole.

How does a numerical-relativity code find a horizon if the event horizon needs the future?

It does not look for the event horizon during the run. On each time slice it solves an elliptic equation for the outermost surface with $\theta_{+}=0$ — the apparent horizon — which needs only the data on that slice. Those apparent horizons give the masses, spins, and areas the simulation reports in real time. The event horizon, when wanted, is reconstructed afterward: once the spacetime has settled to a stationary remnant, one seeds a bundle of null geodesics on the final horizon and integrates them backward, and the surface they sweep out is the event horizon. It can only ever be drawn in hindsight.

What role does the null energy condition play in ordering the two horizons?

It is what forbids the apparent horizon from poking outside the event horizon. The null energy condition, $T_{ab}\ell^{a}\ell^{b}\ge 0$ , makes gravity purely focusing in the Raychaudhuri equation, which both drives the area theorem and — together with cosmic censorship — guarantees that any marginally trapped surface lies within the causal boundary, so $r_{\mathrm{AH}}\le r_{\mathrm{EH}}$ . Violate it, as quantum fields do during Hawking evaporation, and the ordering can loosen: the apparent horizon can move outward or the trapped region can shrink, which is exactly the regime where the tidy classical picture breaks down and quasi-local horizons take over.

Is the Killing horizon a third, independent kind of horizon?

Not independent — it is the label the same surface earns when the spacetime is stationary. A Killing horizon is a null surface generated by a Killing vector that becomes null on it, such as $\partial_t+\Omega_H\partial_\varphi$ for Kerr. In equilibrium this surface is the event horizon and is the apparent horizon simultaneously, because its generators have zero expansion and zero shear. The Killing-horizon description is the one that makes thermodynamics clean: surface gravity $\kappa$ is constant on it (the zeroth law), it carries the temperature $T=\kappa/2\pi$ , and it is where the four laws of black-hole mechanics are stated. Out of equilibrium there is no timelike Killing vector, so the Killing horizon simply ceases to exist, and only the apparent and event horizons remain, now distinct.