Trapping Horizons and Marginal Tubes

A marginal tube is a three-dimensional hypersurface foliated by marginal two-surfaces. Expansion signs classify whether its leaves are future or past, while the change of the vanishing expansion in the other null direction classifies outer or inner behavior. These labels carry more information than “apparent horizon,” but they still refer to a chosen marginal foliation and do not turn the tube into a global event horizon.

Helpful background. Exact Solutions of Einstein’s Equations supplies spherical checks; Causality and Global Structure supplies causal boundaries; The Einstein Equation turns focusing into a signature statement; Timelike Congruences and Spatial Projection supplies Raychaudhuri evolution; Hypersurface Normals and Tangential Projectors supplies the normal decomposition; and Black-Hole Geometry and Mechanics supplies the taxonomy used below.

Marginal tubes and their four sign classes

Let $\mathcal H$ be a smooth hypersurface foliated by closed spacelike two-surfaces $S_v$. Choose future null normals $\ell^a$ and $n^a$ with $\ell\!\cdot n=-1$, and choose $\ell^a$ as the outward direction. A future marginally trapped tube satisfies

$$\theta_{(\ell)}=0, \qquad \theta_{(n)}\lt0$$

on every leaf. Reversing the second sign gives a past marginal tube. The additional derivative

$$\mathcal L_n\theta_{(\ell)}$$

distinguishes outer from inner behavior: it is negative for an outer horizon and positive for an inner horizon. The equality case is degenerate and needs separate analysis. Thus a future outer trapping horizon, the black-hole-like case, obeys

$$\theta_{(\ell)}=0, \qquad \theta_{(n)}\lt0, \qquad \mathcal L_n\theta_{(\ell)}\lt0.$$

The derivative condition asks what happens when the marginal surface is deformed inward along $n^a$. For an outer horizon the outward expansion becomes negative, so nearby inner surfaces are trapped. Positive rescalings $\ell'^a=e^\alpha\ell^a$ and $n'^a=e^{-\alpha}n^a$ preserve all three signs: on $\theta_{(\ell)}=0$, $\mathcal L_{n'}\theta_{(\ell')}=\mathcal L_n\theta_{(\ell)}$. Booth gives the trapping-horizon and marginal-tube hierarchy in Booth 2005, §3, pp. 1083–1088.

The word “outer” is not supplied by the equations alone. It presupposes a choice of the outward null direction, usually fixed by an asymptotic end or a known untrapped exterior. In cosmology, compact universes, or multiple-horizon geometries, that choice must be stated rather than inferred from a radial coordinate.

Signature from focusing

Choose the vector normal to each leaf and tangent to $\mathcal H$ in the form

$$h^a=\ell^a-Cn^a.$$

Its norm is

$$h^ah_a=2C,$$

so $C\gt0$, $C=0$, and $C\lt0$ correspond respectively to spacelike, null, and timelike portions of the tube. Because $h^a$ transports one marginal leaf to the next,

$$0=\mathcal L_h\theta_{(\ell)} =\mathcal L_\ell\theta_{(\ell)} -C\mathcal L_n\theta_{(\ell)},$$

and hence, where the denominator is nonzero,

$$C= \frac{\mathcal L_\ell\theta_{(\ell)}} {\mathcal L_n\theta_{(\ell)}}.$$

For hypersurface-orthogonal outgoing null normals, the Raychaudhuri equation on a marginal leaf reduces in four-dimensional Einstein gravity to

$$\mathcal L_\ell\theta_{(\ell)} =-\sigma^{(\ell)}_{ab}\sigma_{(\ell)}^{ab} -8\pi G T_{ab}\ell^a\ell^b,$$

provided $\ell^a$ is affinely parametrized at the leaf; the possible nonaffinity term is proportional to the vanishing expansion and therefore drops out there. Under the null energy condition the numerator is nonpositive. A future outer trapping horizon has a negative denominator, so $C\geq0$: it is spacelike when gravitational shear or matter flux is present and null when both vanish. This is a conditional result. If the energy condition fails, the regularity assumptions fail, or the outer derivative passes through zero, a timelike or signature-changing segment is not excluded.

Area change and physical interpretation

The area element evolves along $h^a$ according to

$$\mathcal L_h(dA) =\left(\theta_{(\ell)}-C\theta_{(n)}\right)dA =-C\theta_{(n)}\,dA.$$

For a future outer horizon under the preceding conditions, $C\geq0$ and $\theta_{(n)}\lt0$, so its area is nondecreasing in the direction of $h^a$. Strict growth accompanies nonzero flux or shear. On a null equilibrium portion $C=0$, the area is constant. This local balance is more informative about ongoing accretion than the event-horizon area theorem, but it applies to the selected marginal tube and is not a statement about every possible foliation.

For a future inner trapping horizon, the denominator is positive. With the same focusing assumption one obtains $C\leq0$, so the tube is timelike or null and its area is nonincreasing in the corresponding orientation. Inner black-hole horizons and cosmological examples require careful attention to which null normal is called outward; the sign table should be reconstructed from the geometry rather than memorized without its orientation.

Ashtekar and Krishnan derive the spacelike dynamical-horizon specialization and its area law in Ashtekar and Krishnan 2004, §§2.2–3. Detailed flux mechanics is developed later in this volume; here the calculation is used only to connect expansion signs to causal signature and area direction.

Vaidya as a complete sign check

For ingoing Vaidya, set

$$f(v,r)=1-\frac{2Gm(v)}r, \qquad \ell^a=\partial_v+\frac f2\partial_r, \qquad n^a=-\partial_r.$$

The round-sphere expansions are

$$\theta_{(\ell)}=\frac fr, \qquad \theta_{(n)}=-\frac2r.$$

On the marginal tube $r_H=2Gm(v)$,

$$\left.\mathcal L_n\theta_{(\ell)}\right|_{r_H} =-\frac{1}{r_H^2}\lt0,$$

so it is outer as well as future. At the same surface,

$$\left.\mathcal L_\ell\theta_{(\ell)}\right|_{r_H} =-\frac{2G\,dm/dv}{r_H^2}, \qquad C=2G\frac{dm}{dv}.$$

Positive-energy accretion has $dm/dv\gt0$, giving a spacelike growing tube. If $dm/dv=0$, the tube is null and nonexpanding. A negative mass derivative would make this spherical tube timelike, but in classical Einstein gravity it also violates the radial null energy condition; it cannot be smuggled into the classical area conclusion without changing the hypotheses.

Smooth tubes, jumps, and changing character

A marginally trapped tube is the broad object defined by a foliation of MOTSs. A future outer trapping horizon adds sign and stability information. A dynamical horizon in the original Ashtekar–Krishnan usage adds the requirement that the tube be spacelike. These classes overlap but are not synonyms.

In a generic evolution, marginal surfaces may appear in pairs, merge, change stability, or cease to form one smooth tube. A numerical apparent-horizon finder can then jump from an inner branch to an outer branch between slices. Calling the resulting sequence “the horizon” suppresses exactly the branch and foliation data needed to interpret it. The sign conditions classify each regular portion; they do not prove a unique continuation through every transition.

Check your understanding

For the Vaidya marginal tube, suppose $G\,dm/dv=0.03$. Find $C$, state the causal signature, and determine the sign of the area change along $h^a$.

Solution

The calculation above gives $C=2G\,dm/dv=0.06\gt0$, so $h^2=2C=0.12\gt0$ and the tube is spacelike. Since $\theta_{(n)}=-2/r_H\lt0$, $\mathcal L_h(dA)=-C\theta_{(n)}dA\gt0$. The area grows. No conclusion about the location of the event horizon follows from these local signs alone.

Next, Nonexpanding and Isolated Horizons asks which additional null boundary data characterize equilibrium without requiring a stationary exterior.

References

• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004), article 10. DOI.
• Booth, Ivan. “Black-Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. arXiv:gr-qc/0508107.