Future Outer Trapping Horizons

• Future outer trapping horizons (FOTHs) are quasi-local black-hole boundaries defined by a vanishing outgoing null expansion, a negative ingoing expansion, and a negative ingoing derivative of the outgoing expansion.
• They are typically spacelike in dynamical regimes, obeying local area-increase and flux laws that make them observable without knowledge of the entire future causal structure.
• Numerical and analytical studies reveal that under strong perturbations multiple marginally trapped surfaces can form, with FOTHs transitioning to isolated or Killing horizons in equilibrium.

A future outer trapping horizon (FOTH) is a quasi-locally defined hypersurface foliated by marginally outer trapped spacelike two-surfaces on which the outgoing null expansion vanishes, the ingoing null expansion is negative, and the ingoing variation of the outgoing expansion is negative. In the notation used across the literature surveyed here, this is written as

$$\theta_{+}\big|_{H}=0,\qquad \theta_{-}\big|_{H}<0,\qquad \mathcal{L}_{n}\theta_{+}\big|_{H}<0,$$

or equivalently as $\theta_{(\ell)}=0$, $\theta_{(n)}<0$, and $\delta_{\beta n}\theta_{(\ell)}<0$ for some positive function $\beta$ on each leaf (Visser, 2014). FOTHs supply a local or quasi-local notion of black-hole boundary in dynamical spacetimes, in contrast with event horizons, which depend on the entire future causal structure. In equilibrium a FOTH can degenerate into a null isolated or Killing horizon, while in genuinely dynamical regimes it is generically spacelike and obeys local area and flux laws (Booth, 2012).

1. Definition and taxonomic position

Let $S$ be a smooth, closed spacelike 2-surface, often taken to have topology $S^2$, with future-directed null normals chosen as outgoing and ingoing directions. The expansions are defined either as $\theta_{\pm}=\nabla_a \ell_\pm^a$ or, with the induced metric $q_{ab}$ on $S$, as $\theta_{(\ell)}=0$0 and $\theta_{(\ell)}=0$1 (Visser, 2014). The literature summarized here uses both normalizations $\theta_{(\ell)}=0$2 and $\theta_{(\ell)}=0$3; the classification statements are the same once the convention is fixed (Kobakhidze et al., 2021).

A trapping horizon is a hypersurface foliated by marginal surfaces on which one expansion vanishes. Taking the outgoing expansion to vanish, the standard Hayward-style classification separates horizons into future or past according to the sign of the ingoing expansion, and into outer or inner according to the sign of the ingoing derivative of the outgoing expansion (Visser, 2014). In the convention most commonly used in these sources, a FOTH satisfies $\theta_{(\ell)}=0$4, $\theta_{(\ell)}=0$5, and $\theta_{(\ell)}=0$6. In a deformation-based formulation one requires $\theta_{(\ell)}=0$7 for some positive function $\theta_{(\ell)}=0$8 on each leaf (Booth, 2012).

Type	Defining signs
Future-outer	$\theta_{(\ell)}=0$9
Future-inner	$\theta_{(n)}<0$0
Past-outer	$\theta_{(n)}<0$1
Past-inner	$\theta_{(n)}<0$2

The outer condition expresses that a small inward displacement makes the outgoing null congruence converging, so the marginal surface is the outer boundary of a trapped region (Visser, 2014). Some double-null treatments quoted in this literature write the outer condition with the opposite sign, $\theta_{(n)}<0$3, reflecting a different sign convention for the transverse direction (Svitek, 2013).

The physically relevant black-hole case is the future-outer one. In an astrophysical or accreting black hole, outgoing null rays just fail to expand on the boundary while ingoing null rays are already converging, and immediately inside the boundary the outgoing congruence also converges. This is precisely the FOTH configuration (Visser, 2014).

2. Signature, area growth, and local mechanics

Under the null energy condition, a FOTH is generically spacelike or null rather than timelike. In the formulation of slowly evolving and dynamical horizons, one introduces an evolution vector field $\theta_{(n)}<0$4 on the horizon; then the null energy condition implies $\theta_{(n)}<0$5, and the area element $\theta_{(n)}<0$6 of each leaf evolves according to

$\theta_{(n)}<0$7

This is the local second law or area-increase law for FOTHs (Booth, 2012).

The dynamical signature can also be characterized directly through the null Raychaudhuri equation. On a marginal surface,

$\theta_{(n)}<0$8

or, in vacuum-plus-matter form used in numerical relativity,

$\theta_{(n)}<0$9

If this quantity is strictly negative somewhere, nullness is ruled out and the marginally trapped tube is spacelike; nullness occurs only in the isolated-horizon limit (Chu et al., 2010). In the distorted Kerr evolutions studied numerically, all marginally trapped tube sections formed during the dynamical phase are spacelike, whereas at very early and very late times the shear tends to zero and the horizon approaches null equilibrium (Chu et al., 2010).

A useful subextremality diagnostic is the extremality parameter

$\delta_{\beta n}\theta_{(\ell)}<0$0

which is found to satisfy $\delta_{\beta n}\theta_{(\ell)}<0$1 on the dynamical horizons discussed in the numerical study, confirming their subextremal and spacelike character (Chu et al., 2010).

Local mechanics on a dynamical FOTH can be expressed through flux laws. For the irreducible mass or horizon energy, the Ashtekar–Krishnan flux law is written in the geometric form

$\delta_{\beta n}\theta_{(\ell)}<0$2

where $\delta_{\beta n}\theta_{(\ell)}<0$3 is the shear of the outgoing null normal and $\delta_{\beta n}\theta_{(\ell)}<0$4 is the normal connection one-form (Chu et al., 2010). In the numerical Kerr calculation, the dominant contribution arises from $\delta_{\beta n}\theta_{(\ell)}<0$5, consistent with a gravitational-radiation flux into the hole (Chu et al., 2010).

Angular momentum also obeys a dynamical-horizon flux law. In Gourgoulhon’s form,

$\delta_{\beta n}\theta_{(\ell)}<0$6

with $\delta_{\beta n}\theta_{(\ell)}<0$7 chosen as an approximate axial Killing vector on each leaf (Chu et al., 2010). In that setting, the net change in $\delta_{\beta n}\theta_{(\ell)}<0$8 is small, but the dominant term is the one involving the outgoing shear, indicating angular-momentum transport by ingoing gravitational radiation (Chu et al., 2010).

3. Dynamical behavior and multiple marginally trapped tubes

In fully dynamical spacetimes, a FOTH need not be represented by a single smooth world tube at each coordinate time. Numerical simulations of a Kerr black hole perturbed by an ingoing pulse of gravitational radiation show that strong perturbations can produce up to five concentric marginally outer trapped surfaces (MOTSs) on the same Cauchy slice (Chu et al., 2010). In that study the black-hole boundary is located quasi-locally by searching for MOTSs on each slice $\delta_{\beta n}\theta_{(\ell)}<0$9 and tracking them in time.

The computational setup employs spectral trial surfaces

$\beta$0

with roots of $\beta$1 found by a fast pseudospectral flow method; when that fails for inner, unstable MOTSs, a slower minimization of $\beta$2 is used (Chu et al., 2010). Once $\beta$3 and $\beta$4 are confirmed everywhere, the outer condition is checked through

$\beta$5

with $\beta$6 the Ricci scalar of the leaf, $\beta$7 the normal connection one-form, and $\beta$8 the intrinsic covariant derivative (Chu et al., 2010).

For sufficiently strong pulses, specifically for amplitudes $\beta$9–$S$0 in the units of that calculation, a new pair of MOTSs is nucleated outside the original apparent horizon (Chu et al., 2010). One branch rapidly shrinks and annihilates the older MOTS, while the sibling branch survives until the next pair-creation event. Because the surfaces always appear and disappear in pairs, the total number of MOTSs present at any instant remains odd: first one, then three, then five, then back to three, then one (Chu et al., 2010). The branches are labeled MTT1 for the original tube, MTT3 and MTT5 for the long-lived outer branches, and MTT2 and MTT4 for the inner short-lived branches (Chu et al., 2010).

This multiplicity affects the apparent horizon, understood as the outermost MOTS on a given slice. In plots of the irreducible mass

$S$1

each branch traces out a smooth curve, but the apparent horizon jumps discontinuously from MTT1 to MTT3 and then to MTT5 as the outermost branch changes (Chu et al., 2010). This behavior shows that the quasi-local horizon structure of a strongly dynamical black hole can be richer than a single tube.

The event horizon in the same evolution behaves very differently. It is determined only after the full simulation is complete, by integrating outgoing null geodesics backward in time from the late-time apparent horizon. It remains a smooth null surface that never bifurcates, its area begins growing well before the dynamical trapping horizons form, and no new generators ever enter it during the evolution (Chu et al., 2010).

4. Relation to event horizons and physical observability

A central distinction in this literature is between quasi-local horizons such as FOTHs and global event horizons. Event horizons are generically not physically observable, whereas apparent horizons and trapping horizons are generically physically observable in the sense that they can be detected by observers working in finite-size regions of spacetime (Visser, 2014). This is the principal reason advanced for using FOTHs in evolving black-hole physics.

The quasi-local observability claim is operational. A finite-size laboratory can measure tidal fields, namely components of the Riemann tensor, reconstruct the local area-radius $S$2 and mass aspect $S$3, and infer the sign of the outgoing expansion from the relation $S$4 (Visser, 2014). No knowledge of the entire future of the spacetime is required. By contrast, the location of an event horizon depends on the global causal structure all the way into the infinite future and therefore cannot in principle be detected by any finite-region experiment (Visser, 2014).

This contrast is especially sharp in dynamical settings such as accretion, merger, or Hawking evaporation, where FOTHs adapt naturally to the evolution while event horizons remain teleological (Visser, 2014). The numerical Kerr example illustrates the point concretely: the FOTH structure tracks black-hole growth during the highly dynamical regime, whereas the event horizon is a smooth null boundary whose area starts increasing before the gravitational-wave pulse reaches the hole (Chu et al., 2010).

The stationary limit is the exceptional case in which the distinction collapses. In Schwarzschild or Kerr, the Killing horizon, the event horizon, and the unique marginally trapped surface coincide. In that special situation the event horizon becomes physically observable only because it is degenerate with the apparent and trapping horizons (Visser, 2014). Outside strict stationarity, however, the FOTH need not be null and in general deviates from the event horizon by a finite spacetime separation (Visser, 2014).

5. Spherical symmetry, Kodama foliation, and cosmological black holes

In spherically symmetric dynamical spacetimes, the FOTH framework becomes especially explicit. For the Thakurta metric, adopting the preferred foliation associated with Kodama time yields a Schwarzschild-like form

$S$5

with Misner–Sharp mass

$S$6

and areal radius $S$7 (Kobakhidze et al., 2021). In horizon-regular Painlevé–Gullstrand–type coordinates, the null normals can be chosen so that their expansions are

$S$8

The apparent horizon is where $S$9, equivalently $S^2$0, and on that surface

$S^2$1

Using the Einstein equations and the horizon evolution equation, one further obtains $S^2$2 under the null-energy condition and $S^2$3, so the Thakurta horizon is indeed a FOTH (Kobakhidze et al., 2021).

The same analysis defines a Kodama–Hayward surface gravity,

$S^2$4

which is non-negative when $S^2$5, again matching the outer-horizon character (Kobakhidze et al., 2021). The physical interpretation drawn there is foliation-sensitive: Kodama observers identify a black-hole horizon, whereas cosmological observers using comoving time may classify the same surface differently. The stated conclusion is that the Thakurta metric describes a genuine cosmological black hole in the Kodama foliation (Kobakhidze et al., 2021).

Collapse models provide a complementary spherical setting. In a flat-FRW collapse spacetime one finds

$S^2$6

so that $S^2$7 implies $S^2$8 and $S^2$9 with Misner–Sharp mass $\theta_{\pm}=\nabla_a \ell_\pm^a$0 (Baier et al., 2015). The ingoing derivative satisfies

$\theta_{\pm}=\nabla_a \ell_\pm^a$1

where the fluid equation of state is $\theta_{\pm}=\nabla_a \ell_\pm^a$2. Hence the horizon is future outer for $\theta_{\pm}=\nabla_a \ell_\pm^a$3, equivalently $\theta_{\pm}=\nabla_a \ell_\pm^a$4, and in that case it is spacelike (Baier et al., 2015).

In the Robertson–Walker class, the causal character of marginally trapped tubes can be expressed in terms of the equation-of-state parameter $\theta_{\pm}=\nabla_a \ell_\pm^a$5 defined by $\theta_{\pm}=\nabla_a \ell_\pm^a$6. The horizon-character function is

$\theta_{\pm}=\nabla_a \ell_\pm^a$7

so the horizon is spacelike for $\theta_{\pm}=\nabla_a \ell_\pm^a$8 or $\theta_{\pm}=\nabla_a \ell_\pm^a$9, timelike for $q_{ab}$0, and null for $q_{ab}$1 or $q_{ab}$2 (Sherif et al., 2019). For Lemaitre–Tolman–Bondi dust collapse, one obtains $q_{ab}$3, so the marginally trapped tube is spacelike everywhere in regular collapse; the associated stability eigenvalue is $q_{ab}$4, and the same monotonicity condition that ensures this stability also excludes shell crossing (Sherif et al., 2019).

6. Hawking effect, slowly evolving horizons, and near-equilibrium structure

In dynamical spherical symmetry, the Hayward–Kodama surface gravity supplies the natural local quantity entering tunneling derivations of Hawking radiation. One geometric definition is

$q_{ab}$5

where $q_{ab}$6 is the metric on the two-dimensional normal space and $q_{ab}$7 on the horizon (Helou, 2015). For a future-outer horizon, the outer condition implies $q_{ab}$8 (Helou, 2015).

In advanced Eddington–Finkelstein–Bardeen coordinates,

$q_{ab}$9

the Hamilton–Jacobi method yields an outgoing mode with radial momentum

$S$0

where $S$1 is the Kodama energy (Giavoni et al., 2020). Near the horizon, $S$2, and the action acquires the imaginary part

$S$3

The tunneling rate therefore takes the thermal form

$S$4

corresponding to

$S$5

for emission into the future (Giavoni et al., 2020). In the static limit this reproduces the standard Schwarzschild and Reissner–Nordström temperatures (Giavoni et al., 2020).

The sign structure of thermal effects depends on the horizon class. The Hayward–Kodama analysis summarized in this literature states that future-outer and past-inner trapping horizons exhibit particle emission, whereas future-inner and past-outer horizons correspond to absorption (Giavoni et al., 2020). In the FRW collapse model analyzed with a null tunneling path $S$6, the local temperature can be written as

$S$7

but the observability of radiation depends on whether the tunneling path actually escapes the trapped region (Baier et al., 2015). The conclusion there is that only radiation from a spacelike future outer trapping horizon has a chance to be observed by an external observer (Baier et al., 2015).

Near equilibrium, the relevant refinement is the slowly evolving horizon, defined as a nearly isolated FOTH. Introducing a length scale $S$8 and a smallness parameter $S$9, a slowly expanding horizon is required to satisfy bounded intrinsic curvature and mixed fluxes, controlled surface gradients, and the slow-expansion condition

$\theta_{(\ell)}=0$00

with an $\theta_{(\ell)}=0$01-compatible scaling $\theta_{(\ell)}=0$02 (Booth, 2012). Additional bounds on $\theta_{(\ell)}=0$03, $\theta_{(\ell)}=0$04, $\theta_{(\ell)}=0$05, and $\theta_{(\ell)}=0$06 then define a slowly evolving horizon (Booth, 2012).

These assumptions recover near-equilibrium mechanics. The zeroth law becomes approximate constancy of the surface gravity analogue,

$\theta_{(\ell)}=0$07

while the first law takes the integrated form

$\theta_{(\ell)}=0$08

with separate gravitational-wave and stress-energy flux terms (Booth, 2012). A further result is that there is an event-horizon candidate in close proximity to any slowly evolving horizon (Booth, 2012).

7. Conformal-frame issues, failure modes, and algebraic structure

The standard FOTH definition is not conformally invariant. Under a conformal transformation

$\theta_{(\ell)}=0$09

one finds

$\theta_{(\ell)}=0$10

Hence the locus $\theta_{(\ell)}=0$11 is shifted, and a marginally trapped surface in one frame need not remain marginal in the conformal frame (Faraoni et al., 2011). This is the starting point for an entropy-based modification of the trapping-horizon concept in scalar–tensor and $\theta_{(\ell)}=0$12 gravity.

In that construction one introduces an entropy 2-form

$\theta_{(\ell)}=0$13

and defines the quasi-local horizon by the conditions

$\theta_{(\ell)}=0$14

In the Einstein limit, where $\theta_{(\ell)}=0$15 is constant, these reduce to the usual FOTH conditions; in non-Einstein frames they track the Einstein-frame trapping horizon and therefore remain invariant under conformal rescaling (Faraoni et al., 2011). This construction is used to prove an entropy-increase law for quasi-local horizons in scalar–tensor and $\theta_{(\ell)}=0$16 gravity (Faraoni et al., 2011).

A distinct limitation arises from strong geometric distortion. In Weyl-distorted Schwarzschild spacetimes, the null hypersurface at $\theta_{(\ell)}=0$17 is an isolated horizon with positive surface gravity, but for sufficiently large quadrupole distortion it ceases to be a FOTH (Pilkington et al., 2011). Specializing to a pure quadrupole distortion parameter $\theta_{(\ell)}=0$18, the theorem stated there is that if $\theta_{(\ell)}=0$19, then no differentiable foliation function can make the ingoing expansion negative everywhere; consequently the isolated horizon is neither a marginally trapped surface nor a FOTH (Pilkington et al., 2011). The stated implication is that staticity and axisymmetry alone do not guarantee the existence of a foliation by future outer trapping surfaces (Pilkington et al., 2011).

The local curvature algebra at a spacelike FOTH is also constrained. In four-dimensional vacuum relativity, for a spacelike FOTH admitting a smooth double-null foliation and a frame in which the shears and twist vanish on each leaf, the Newman–Penrose Weyl scalars satisfy

$\theta_{(\ell)}=0$20

while generically $\theta_{(\ell)}=0$21 (Svitek, 2013). The horizon is therefore algebraically special of Petrov type II, although the bulk spacetime away from the horizon is generically type I (Svitek, 2013).

Taken together, these results show that the FOTH concept is neither merely a reformulation of the event horizon nor a universally rigid geometric structure. It is a quasi-local horizon notion with local observability, local flux and area laws, and a clear role in dynamical black-hole physics, yet it is sensitive to foliation, sign conventions, conformal frame, and strong geometric distortions (Visser, 2014).