= Solution
A horizon separates regions according to a specified causal, dynamical or geometrical criterion. The criterion must be stated: an <event horizon>, an <apparent horizon>, a <Killing horizon> and a <Cauchy horizon> answer different questions. Some horizons describe a black hole, others an observer's acceleration or the universe's expansion. A vanishing coordinate metric component alone does not establish any of these definitions.
In an asymptotically flat spacetime, choose the relevant <future null infinity> $\mathscr I^+$, the endpoint of escaping outgoing light rays. The <black-hole region> and its future event horizon are
$$
\boxed{\mathcal B=M\setminus J^-(\mathscr I^+),\qquad
\mathcal H^+=\partial J^-(\mathscr I^+)=\partial\mathcal B.}
$$
Here $J^-$ is the <causal past>. Events in $\mathcal B$ cannot send a causal signal to the chosen infinity. This is a global definition: identifying the horizon requires the full future development, so a local measurement need not reveal whether an event lies on it. Smooth portions of an <event horizon> are null hypersurfaces generated by <null geodesics>. Generators may join at past caustics or crease sets, so the horizon need not be smooth everywhere. A past event horizon is defined by the time-reversed escape criterion; in the maximally extended Schwarzschild solution it bounds a white-hole region.
An observer-based definition replaces infinity by a future-inextendible worldline $\gamma$. Its future <observer event horizon> is the boundary of events that can signal to $\gamma$, conventionally $\partial I^-(\gamma)$. A past observer horizon similarly bounds events reachable from its past worldline. Such horizons depend on the observer and need not imply a <black hole>.
A <Killing horizon> is a null hypersurface to which a nonzero <Killing vector field> $K^a$ is normal. On it, $K^a$ is tangent to its null generators and satisfies $K^b\nabla_bK^a=\kappa K^a$. The <surface gravity> $\kappa$ measures the failure of this Killing parameter to be an affine parameter; rescaling $K$ rescales $\kappa$. A <degenerate Killing horizon> has $\kappa=0$. Stationary black-hole horizons often are Killing horizons, whereas a general evolving event horizon need not have any Killing symmetry. The condition $K^aK_a=0$ alone is insufficient: the null vector must also be normal to a null hypersurface.
The <Schwarzschild spacetime> illustrates several of these notions. In units $G=c=1$, let $f=1-2M/r$. Its metric is $ds^2=f\,dt^2-f^{-1}dr^2-r^2d\Omega^2$, and $r=2M$ is the future event horizon of the black-hole region in the appropriate extension. The apparent coordinate singularity disappears in <ingoing Eddington-Finkelstein coordinates>, $v=t+r_*$, $dr_*/dr=f^{-1}$:
$$
ds^2=f\,dv^2-2\,dv\,dr-r^2d\Omega^2.
$$
At $r=2M$ this metric is regular and the normal $dr$ is null. Future outgoing radial rays satisfy $dr/dv=f/2$; inside the horizon their areal radius decreases even on the outgoing family. The stationary <Killing vector field> $\partial_v$ generates the horizon with $\kappa=f'(2M)/2=1/(4M)$ when normalized at infinity. The <Kretschmann scalar> is $48M^2/r^6$, finite at the horizon and divergent at $r=0$. Thus the event horizon is not the curvature singularity. The <photon sphere> at $r=3M$, which supports circular null orbits, is also not the horizon.
Quasi-local horizon notions use <null expansions>. For a closed spacelike two-surface, choose future outgoing and ingoing null normals $\ell,n$, with $\ell\cdot n=1$ in signature $(+---)$. Each expansion is the logarithmic rate of its positive area element along the corresponding null normal. A future <trapped surface> has $\theta_\ell<0$ and $\theta_n<0$: both future light fronts initially shrink. A <marginally outer trapped surface> has $\theta_\ell=0$; for the future black-hole type also require $\theta_n<0$. On a chosen spacelike slice, the outer boundary of the future trapped region is an <apparent horizon>, ordinarily an outermost marginally outer trapped surface when regularity permits this characterization. It depends on the slice, and a slicing need not display an apparent horizon even when the global spacetime contains an event horizon.
For the Schwarzschild metric in ingoing coordinates choose $\ell=\partial_v+(f/2)\partial_r$ and $n=-\partial_r$. The <spherical null expansion in ingoing coordinates> gives
$$
\theta_\ell=\frac{2\ell(r)}r=\frac fr,\qquad
\theta_n=\frac{2n(r)}r=-\frac2r.
$$
The round spheres inside $2M$ are future trapped, while $r=2M$ is marginal. On standard horizon-penetrating slices its sections give the apparent horizon, coinciding with the stationary event and Killing horizon. This coincidence in a symmetric stationary example does not identify the notions in a general spacetime.
A <trapping horizon> is a hypersurface foliated by marginal surfaces. Its future outer type obeys $\theta_\ell=0$, $\theta_n<0$ and $\mathcal L_n\theta_\ell<0$. Unlike an event horizon, this can be diagnosed from local expansion data and their variation. The <null energy condition> explains its causal character. At a marginal surface, the <Null Raychaudhuri equation>, with the present curvature sign and vanishing null twist, gives
$$
\mathcal L_\ell\theta_\ell=-\sigma_{ab}\sigma^{ab}+R_{ab}\ell^a\ell^b
=-\sigma_{ab}\sigma^{ab}-\kappa_g T_{ab}\ell^a\ell^b\le0,
\qquad \kappa_g=8\pi G
$$
in units $c=1$. A possible nonaffinity term vanishes here because it is proportional to the zero expansion. Write a horizon evolution vector as $V=\ell-Cn$. Preserving $\theta_\ell=0$ gives $C=(\mathcal L_\ell\theta_\ell)/(\mathcal L_n\theta_\ell)\ge0$. Hence $V^2=-2C\le0$: the future outer horizon is spacelike or null. Its area expansion is $\theta_V=-C\theta_n\ge0$. These conclusions use the outer condition and the energy condition, not merely marginality.
For the global event horizon, the <future-complete horizon focusing proof of the area theorem> uses the same <Null Raychaudhuri equation>. Under the null energy condition, a negative generator expansion forces a future caustic in finite affine parameter. Future completeness and the usual regularity or predictability assumptions exclude such a caustic on the future event-horizon boundary. Its generator expansion is therefore nonnegative and its classical area cannot decrease. These assumptions matter: this is not an unconditional result for every null surface or every matter model.
Rotating and charged solutions provide further distinctions. A nonextremal <Kerr black hole> has $r_\pm=M\pm\sqrt{M^2-a^2}$, with $a=J/M$. Its outer Killing horizon is generated by $\partial_t+\Omega_H\partial_\phi$, where $\Omega_H=a/(r_+^2+a^2)$, rather than by time translation alone. The <ergosphere> begins at the stationary-limit surface $g_{tt}=0$, whose radius is $M+\sqrt{M^2-a^2\cos^2\theta}$. Away from the axis this lies outside the event horizon: it forbids static observers but does not forbid escape to infinity. In <Reissner-Nordstrom spacetime>, the nonextremal radii are $r_\pm=M\pm\sqrt{M^2-Q^2}$. The outer surface is an event horizon, whereas the inner surface is a Cauchy horizon in the exact analytic extension.
A <Cauchy horizon> concerns predictability from initial data. The future <domain of dependence> $D^+(\Sigma)$ of a partial <Cauchy surface> consists of events for which every past-inextendible causal curve intersects $\Sigma$. Its future Cauchy horizon is
$$
H^+(\Sigma)=\overline{D^+(\Sigma)}\setminus I^-(D^+(\Sigma)).
$$
Beyond it, causal influences can arrive without meeting the specified initial surface; data on that surface no longer determine the solution there. This is distinct from the inability to send signals to infinity. The inner horizon of the exact charged black hole supplies an example, and the definition necessarily specifies which initial surface is being used.
Flat spacetime itself has acceleration horizons. In the right Rindler wedge, set $t=\xi\sinh(a\tau)$ and $x=\xi\cosh(a\tau)$, with $\xi>0$. The <Minkowski spacetime> metric becomes $ds^2=a^2\xi^2d\tau^2-d\xi^2-dy^2-dz^2$. The null boundaries $t=x$ and $t=-x$ are <Rindler horizons> of the boost Killing flow. A uniformly accelerated observer cannot receive signals from one side of its future horizon, even though the curvature vanishes and inertial observers encounter no such boundary. These are observer and Killing horizons, not a black-hole event horizon defined using the full Minkowski null infinity.
In a spatially flat <Friedmann-Lemaître-Robertson-Walker metric> with noncompact spatial sections, radial light rays have $d\chi/dt=\pm1/a(t)$. The <particle horizon> measures how far signals could have travelled since the initial cosmic time; the <cosmological event horizon> measures how far signals emitted now could ever travel toward a comoving observer. Their comoving and proper radii are
$$
\boxed{\chi_{\rm ph}(t)=\int_{t_i}^t\frac{dt'}{a(t')},\qquad
\chi_{\rm eh}(t)=\int_t^{t_f}\frac{dt'}{a(t')},\qquad
R(t)=a(t)\chi(t).}
$$
The respective horizon is present only if its integral is finite. The event-horizon integral uses the actual future endpoint, often $t_f=\infty$, so it is global and observer-dependent. For $a(t)=t^q$ on $0<t<\infty$, a finite particle horizon exists for $0<q<1$, while a finite event horizon exists for $q>1$. Neither integral is finite for $q=1$. In the expanding flat patch of <de Sitter spacetime>, $a=e^{Ht}$ and the proper event-horizon radius is $H^{-1}$. The static form $ds^2=(1-H^2r^2)dt^2-(1-H^2r^2)^{-1}dr^2-r^2d\Omega^2$ identifies it as a cosmological Killing horizon; its surface-gravity magnitude is $H$ with the central static time normalization.
Finally, the <Hubble radius> must not automatically be called an event horizon. For a flat expanding cosmology, the two future radial null expansions of a sphere of proper radius $R$ are $\theta_\pm=2(H\pm1/R)$. At $R=H^{-1}$ the ingoing expansion vanishes, while outside it both expansions are positive: this is the cosmological marginal, past-trapped type rather than a future black-hole trapped surface. A dust universe with $a\propto t^{2/3}$ has such a Hubble sphere but no future event horizon. \b[Global causal boundaries, slice-dependent trapped-surface boundaries, symmetry horizons and initial-data boundaries must therefore be distinguished], even when several coincide in a special example.
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