Apparent Horizons and Marginally Trapped Surfaces

An apparent horizon is extracted from a chosen spacelike slice by finding the outer boundary of its trapped region. Its basic building blocks are closed two-surfaces whose outward null expansion vanishes. This makes apparent horizons computable from one time slice and valuable in numerical relativity, but also makes them slicing dependent: a different slice of the same spacetime can contain different marginal surfaces, or none with the same outermost status.

Helpful background. Exact Solutions of Einstein’s Equations supplies the Schwarzschild and Vaidya checks; Causality and Global Structure supplies trapped-region implications; The Einstein Equation relates focusing to matter; Timelike Congruences and Spatial Projection supplies optical scalars; Hypersurface Normals and Tangential Projectors supplies the slice decomposition; and Black-Hole Geometry and Mechanics supplies the surrounding horizon taxonomy.

Null expansions of a closed two-surface

Let $S$ be a smooth, closed, orientable spacelike two-surface. Choose future-directed null normals $\ell^a$ and $n^a$, with $\ell^a$ designated outward and

$$\ell\!\cdot n=-1.$$

The induced metric and expansions are

$$q_{ab}=g_{ab}+\ell_a n_b+n_a\ell_b, \qquad \theta_{(\ell)}=q^{ab}\nabla_a\ell_b, \qquad \theta_{(n)}=q^{ab}\nabla_a n_b.$$

If a small surface element has area $dA$, then along either null congruence $\theta=(dA)^{-1}\mathcal L_k(dA)$. Thus a future trapped surface has $\theta_{(\ell)}\lt0$ and $\theta_{(n)}\lt0$: both future-directed light fronts converge. A marginally outer trapped surface, or MOTS, satisfies $\theta_{(\ell)}=0$; for a future marginal surface one also requires $\theta_{(n)}\lt0$.

The residual boost

$$\ell'^a=e^\alpha\ell^a, \qquad n'^a=e^{-\alpha}n^a$$

preserves $\ell'\!\cdot n'=-1$ and gives $\theta_{(\ell')}=e^\alpha\theta_{(\ell)}$ and $\theta_{(n')}=e^{-\alpha}\theta_{(n)}$. Positive rescaling therefore preserves zero and sign conditions. The numerical values of nonzero expansions are normalization dependent, but the MOTS classification is not. Booth states these definitions and their causal interpretation in Booth 2005, §2.2, pp. 1078–1081.

The apparent horizon on a slice

Now embed $S$ in a spacelike hypersurface $\Sigma$. Let $u^a$ be the future-directed unit normal to $\Sigma$ and $s^a$ the outward unit normal to $S$ within $\Sigma$. A convenient normalized pair is

$$\ell^a=\frac{u^a+s^a}{\sqrt2}, \qquad n^a=\frac{u^a-s^a}{\sqrt2}.$$

To make the sign explicit, define the slice extrinsic curvature by

$$K_{ab}=-h_a{}^c h_b{}^d\nabla_cu_d, \qquad h_{ab}=g_{ab}+u_au_b.$$

Writing $D_i$ for the derivative compatible with $h_{ij}$ and $K=h^{ij}K_{ij}$, projection onto $S$ yields

$$\theta_{(\ell)} =\frac{1}{\sqrt2} \left(D_i s^i+K_{ij}s^is^j-K\right),$$ $$\theta_{(n)} =\frac{1}{\sqrt2} \left(-D_i s^i+K_{ij}s^is^j-K\right).$$

The MOTS equation is therefore a nonlinear elliptic equation for the shape of $S$ inside the initial data $(\Sigma,h_{ij},K_{ij})$. It can be solved without knowing the future spacetime. That is its operational advantage over event- horizon finding.

The traditional apparent horizon is the boundary in $\Sigma$ of the union of trapped regions. Under appropriate asymptotic, regularity, and barrier hypotheses, a smooth outer component of that boundary is an outermost MOTS. Numerical-relativity usage often calls the outermost MOTS itself the apparent horizon. The two formulations should not be conflated when the trapped-region boundary is nonsmooth, has several components, or the existence hypotheses fail. Ashtekar and Krishnan discuss the slice-global character of apparent horizons in Ashtekar and Krishnan 2004, §2.2.1.

Round spheres in Vaidya spacetime

For ingoing Vaidya,

$$ds^2=-\left(1-\frac{2Gm(v)}r\right)dv^2 +2\,dv\,dr+r^2d\Omega^2,$$

take

$$\ell^a=\partial_v+\frac12 \left(1-\frac{2Gm(v)}r\right)\partial_r, \qquad n^a=-\partial_r.$$

Because the area of a round sphere is $4\pi r^2$, its expansions follow directly from the rate of change of $r$:

$$\theta_{(\ell)} =\frac{2\ell(r)}r =\frac1r\left(1-\frac{2Gm(v)}r\right), \qquad \theta_{(n)}=\frac{2n(r)}r=-\frac2r.$$

The round spheres at $r=2Gm(v)$ are future marginal: the outgoing expansion vanishes and the ingoing expansion is negative. Spheres with $r\lt2Gm(v)$ are future trapped. The marginal spheres foliate the tube $r=2Gm(v)$; this classifies a family of two-surfaces but does not yet make any leaf an apparent horizon. On a specified spherically symmetric spacelike slice that crosses the tube and has a regular untrapped exterior, the intersection is the outer boundary of the round trapped region. Identifying it as the slice’s apparent horizon still requires outermost status among all candidate surfaces on that slice.

This marginal tube is not generally the event horizon. For $dm/dv\gt0$, the marginal tube is spacelike, while the event horizon is null and lies outside it during accretion. If $m(v)$ becomes exactly constant after a finite advanced time and there is no later infall, the round marginal tube is then null and coincides with the event horizon. If $m(v)$ only tends asymptotically to a constant, the two surfaces instead approach one another asymptotically. The calculation tests both the signs and the distinction between instantaneous marginality and ultimate escape.

Stability and outermost status

The linear response of the outgoing expansion to an outward deformation $\psi s^a$ defines a MOTS stability operator,

$$\delta_{\psi s}\theta_{(\ell)}=L_S\psi.$$

This is a linear elliptic operator on $S$, generally not self-adjoint when the normal bundle carries rotation. On a compact connected $S$, its principal eigenvalue $\lambda_0$ is nevertheless real and has an everywhere-positive eigenfunction. With the chosen outward direction, the MOTS is stable when $\lambda_0\geq0$ and strictly stable when $\lambda_0\gt0$; a negative principal eigenvalue signals an outward instability of that marginal branch. These statements concern the specified deformation direction; choosing a different normal deformation defines a different stability operator.

A locally outermost MOTS is stable under the standard smoothness assumptions, but stability alone does not make a surface globally outermost on the slice. Strict stability also supplies a local barrier: nearby surfaces just outside and inside have outgoing expansions of opposite sign. The spectral facts and these implications are proved in Andersson, Mars, and Simon 2008, Lemma 4.1, Proposition 5.1, and Theorem 7.1, pp. 861, 866–867, and 871–872. Stability is therefore the local deformation test used to continue a MOTS smoothly, whereas “apparent horizon” remains a slice-global selection rule. This distinction becomes decisive when a common outer and inner pair is born during a merger.

Why slicing dependence is unavoidable

Changing $\Sigma$ changes $u^a$, $s^a$, $h_{ij}$, and $K_{ij}$ in the MOTS equation, so it changes the candidate surfaces available on the slice, the trapped region, and which component is outermost. A new slice also usually intersects a candidate marginal tube in a different two-surface.

This is geometric dependence, not coordinate dependence, but it does not make the marginality of one fixed embedded surface slice dependent. If the same $S$ lies in two spacelike slices and the same outward direction is retained, the two induced pairs of future null normals are related by a positive boost. The expansions rescale reciprocally, so their zeros and signs agree. What the new slice can change is which surface is selected and whether it is outermost there. Likewise, the existence of a MOTS does not by itself identify a unique world tube: neighboring MOTSs may join into several tubes, undergo jumps, or change causal character. Those questions require stability information and a sequence of slices.

Two common shortcuts fail. First, “$\theta_{(\ell)}=0$” identifies a MOTS, not automatically an apparent horizon; outermost and trapped-region conditions still matter. Second, an apparent horizon is not a locally invariant boundary of the four-dimensional spacetime. It is a geometrically defined surface relative to specified initial data.

Check your understanding

On a time-symmetric initial-data slice, $K_{ij}=0$. Show what the MOTS equation becomes, and explain why a closed minimal surface on that slice is marginal in both future null directions rather than future trapped.

Solution

With $K_{ij}=0$, the formulas reduce to $\theta_{(\ell)}=D_is^i/\sqrt2$ and $\theta_{(n)}=-D_is^i/\sqrt2$. A minimal surface has $D_is^i=0$, hence both expansions vanish. It is a MOTS for either choice of outward direction, but it does not satisfy the strict future condition $\theta_{(n)}\lt0$. The bifurcation sphere on the standard time-symmetric Schwarzschild slice is the canonical example.

Next, Trapping Horizons and Marginal Tubes follows marginal surfaces through spacetime and adds the derivative condition that distinguishes outer from inner behavior.

References

• Andersson, Lars, Marc Mars, and Walter Simon. “Stability of Marginally Outer Trapped Surfaces and Existence of Marginally Outer Trapped Tubes.” Advances in Theoretical and Mathematical Physics 12 (2008): 853–888. DOI.
• 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.