Stability of Marginally Outer Trapped Surfaces

Scrollable formula. Use Left and Right arrow keys to reveal hidden content.

Scrollable table. Use Left and Right arrow keys to reveal additional columns.

Scrollable figure. Use Left and Right arrow keys to inspect hidden content.

Stability turns the equation $\theta_{(\ell)}=0$ from a condition on one surface into a local existence statement for a tube of such surfaces. The relevant operator is elliptic but generally not self-adjoint. Its principal eigenvalue, rather than an energy minimum, decides whether outward deformations make the outgoing expansion positive and whether an implicit-function argument can continue the surface through nearby slices.

Required background. Hypersurface normals and tangential projectors supplies the geometry of a two-surface inside a spacelike foliation.

Helpful background. Black-Hole Geometry and Mechanics develops quasi-local horizon definitions; Black-Hole Perturbation Theory supplies deformation operators; Formulations of General Relativity supplies the slicing and constraints; and Computational Relativity motivates continuation and branch finding for marginal surfaces.

Stability operator and hypotheses

Let $(\mathcal M,g_{ab})$ be a smooth four-dimensional spacetime foliated by smooth spacelike hypersurfaces $\Sigma_t$ . Let $S\subset\Sigma_0$ be a smooth, closed, two-sided marginally outer trapped surface. Write $u^a$ for the future unit normal to $\Sigma_0$ , $s^a$ for the selected outward unit normal to $S$ , and choose $\ell^a=u^a+s^a$ . Thus $\theta_{(\ell)}=0$ on $S$ .

For a normal deformation $\psi s^a$ , define

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

It has the elliptic form

L_S\psi=-\Delta_S\psi+2\omega^A D_A\psi+V\psi,

Set $k=(u-s)/2$ , so $\ell\cdot k=-1$ , and define $\omega_A=-k_bq_A{}^a\nabla_a\ell^b$ . Here $D_A$ is the induced derivative, $\Delta_S=D_AD^A$ , and the potential on a MOTS is

V=\frac12\mathcal R_S+D_A\omega^A-\omega_A\omega^A -\frac12\lvert\sigma^{(\ell)}\rvert^2-G_{ab}\ell^au^b.

The symbols $\mathcal R_S$ , $\sigma^{(\ell)}$ , and $G_{ab}$ denote the intrinsic scalar curvature, outgoing null shear, and Einstein tensor. This is the spatial-deformation operator, with $\ell=u+s$ ; changing the deformation direction changes its potential. Because of the first-derivative term, $L_S$ is not generally self-adjoint. Nevertheless, its principal eigenvalue $\lambda_1$ is real and has an eigenfunction of one sign. The expression translates Andersson, Mars, and Simon 2005, Eq. (1), pp. 2–3, PDF, whose ingoing null vector is twice our $k$ .

The surface is stably outermost if some $\psi\ge0$ , not identically zero, satisfies $L_S\psi\ge0$ . It is strictly stably outermost if the inequality is strict somewhere. Equivalently, $\lambda_1\ge0$ and $\lambda_1\gt 0$ , respectively; see Andersson, Mars, and Simon 2005, Definition 2 and Lemmas 1–2, pp. 2–3, PDF. The word “outward” is part of the hypotheses and need not be globally obvious outside an asymptotically flat end.

A Schwarzschild sign check

On the time-symmetric Schwarzschild bridge, the minimal throat has areal radius $R_0=2GM$ . Its slice extrinsic curvature and null shear vanish, $\omega_A=0$ , and $\mathcal R_S=2/R_0^2$ . Consequently

L_S=-\Delta_S+\frac1{R_0^2}, \qquad \mu_j=\frac{j(j+1)+1}{R_0^2},\quad j=0,1,\ldots.

The principal eigenfunction is constant and $\lambda_1=\mu_0=1/R_0^2\gt0$ . A small outward displacement therefore gives positive outgoing expansion, as expected. In proper distance $\rho$ from the throat, $r=R_0+\rho^2/(4R_0)+O(\rho^4)$ and $\theta_{(\ell)}=2r'/r=\rho/R_0^2+O(\rho^3)$ , independently checking the sign. At this bifurcation sphere both expansions vanish; the MOTS stability statement itself does not require negative ingoing expansion.

Local-existence theorem

Theorem. Under the smoothness, foliation, closure, two-sidedness, and strict-stability hypotheses above, $S$ lies in a smooth three-surface $H$ whose intersections $S_t=H\cap\Sigma_t$ are MOTSs for sufficiently small $\lvert t\rvert$ . The continuation persists while the leaves remain strictly stably outermost. If the null energy condition holds, $H$ is locally achronal. If additionally $G_{ab}\ell^a\ell^b\gt 0$ somewhere on $S$ or the outgoing shear is nonzero somewhere, $H$ is spacelike near $S$ .

This is the content of Theorems 1 and 2 in Andersson, Mars, and Simon 2005, pp. 1 and 4, PDF. It is local in time and tied to the prescribed foliation. It does not say that the tube is the globally outermost marginal tube, extends forever, or coincides with the event horizon.

Proof architecture

First, represent nearby surfaces in $\Sigma_t$ as graphs $r=f(x^A)$ over $S$ . Their outgoing expansion defines a smooth nonlinear map

\Theta:(t,f)\longmapsto\theta_{(\ell)}[S_{t,f}]

between suitable Hölder spaces. Its derivative in the graph direction at $(0,0)$ is $D_f\Theta=L_S$ .

Second, strict stability gives $\lambda_1\gt 0$ . The elliptic operator then has no kernel at zero; the Fredholm alternative makes $L_S:C^{k,\alpha}\to C^{k-2,\alpha}$ invertible. The Banach-space implicit-function theorem produces a unique small graph $f(t)$ satisfying $\Theta(t,f(t))=0$ . Elliptic regularity and patching yield the smooth local tube. This argument is presented explicitly in the proof of Theorem 1 at Andersson, Mars, and Simon 2005, pp. 3–4, PDF.

Third, let a normal vector tangent to $H$ be decomposed into an outgoing null part plus a spatial deformation. Varying the condition $\theta_{(\ell)}=0$ along that vector gives an elliptic equation schematically $L_S\mu=\omega W$ , where

W=\sigma^{(\ell)}_{AB}\sigma_{(\ell)}^{AB} +G_{ab}\ell^a\ell^b\ge0

under the null energy condition. The maximum principle implies $\mu\ge0$ , and strict positivity of $W$ somewhere gives $\mu\gt 0$ . These signs yield achronal and spacelike causal character, respectively.

Failed converse and loss of control

The converse “a smooth MOTS tube implies every leaf is strictly stable” is false without added hypotheses. A tube can pass through a degenerate leaf with $\lambda_1=0$ ; there $L_S$ is not invertible, so this proof supplies neither uniqueness nor a graph over the chosen foliation. Branches can appear, annihilate, or turn backward in slicing time. Likewise, local outermostness implies only $\lambda_1\ge0$ , not strict positivity. The source proves the one-way chain from strict stability to local outermostness and from local outermostness to stability in Proposition 1, pp. 3–4, PDF; the reverse arrows can fail.

Energy conditions do not repair the invertibility failure. They control the tube’s causal signature after local existence has been obtained. Numerically, a small principal eigenvalue warns that continuation is ill-conditioned: a horizon finder may jump between branches even when a smooth collection of MOTSs exists in spacetime.

Check your understanding

Suppose a MOTS has principal eigenvalue $\lambda_1=0$ and a positive principal eigenfunction. Which step of the local-existence proof fails, and what conclusion remains valid?

Solution

The surface is stable but not strictly stable. Since zero lies in the spectrum, $L_S$ is not invertible, so the implicit-function theorem used above cannot solve uniquely for a nearby graph $f(t)$ . The surface may still belong to a smooth marginal tube, but that requires a different argument; existence, local uniqueness, and smooth continuation do not follow from this theorem.

References

Andersson, Lars, Marc Mars, and Walter Simon. “Local Existence of Dynamical and Trapping Horizons.” Physical Review Letters 95 (2005): 111102. doi:10.1103/PhysRevLett.95.111102.