Marginal Tubes and Their Signature

A marginally trapped tube is a three-surface assembled from marginal two-surfaces. Its causal signature is not fixed by the condition $\theta_{(\ell)}=0$: the tube may be spacelike, null, timelike, or of mixed signature. Under an outer condition and positive null energy, however, influx and outgoing shear force the physically relevant future tube to be spacelike, with a null limit when the flux vanishes.

Required background. Hypersurface normals and tangential projectors supplies the nested normal geometry and signature conventions.

Helpful background. Black-Hole Geometry and Mechanics develops dynamical, trapping, and isolated horizons; Black-Hole Perturbation Theory supplies null shear; Formulations of General Relativity explains slicing and constraints; and Computational Relativity develops tube tracking.

Signature from preserved marginality

Let $H$ be foliated by smooth closed surfaces $S_v$ with future null normals $\ell\cdot k=-1$ and

$$\theta_{(\ell)}=0,\qquad \theta_{(k)}\lt 0.$$

On a branch whose outgoing coefficient is positive, normalize the vector normal to each leaf but tangent to $H$ as

$$\mathcal V^a=\ell^a-Ck^a.$$

Then $\mathcal V^2=2C$ in the $(-,+,+,+)$ convention: $C\gt 0$ is spacelike, $C=0$ null, and $C\lt 0$ timelike. The evolution must preserve marginality,

$$0=\delta_{\mathcal V}\theta_{(\ell)}=-W+L_{-k}C, \qquad W=\lvert\sigma^{(\ell)}\rvert^2+8\pi G T_{ab}\ell^a\ell^b.$$

Here $\delta_X$ varies the surface and its null normals, rather than differentiating a fixed scalar field. Raychaudhuri supplies $\delta_\ell\theta_{(\ell)}=-W$. Define the normal connection $\omega_A=-k_bq_A{}^a\nabla_a\ell^b$, let $D_A$ and $\mathcal R_S$ be the leaf derivative and scalar curvature, and write $G_{ab}$ for the Einstein tensor. The stability operator for deformation along $-k^a$ is

$$\begin{aligned} L_{-k}C&=-D^AD_AC+2\omega^AD_AC+VC,\\ V&=\frac12\mathcal R_S+D_A\omega^A -\omega_A\omega^A-G_{ab}\ell^ak^b. \end{aligned}$$

The angular derivatives of $C$ matter: moving different points by different amounts changes the surface’s shape. Suppose the chosen normalization has $V=L_{-k}1=-\delta_k\theta_{(\ell)}\gt0$. Under the null energy condition, the maximum principle on a closed connected leaf gives $C\geq0$. If $W$ is nonzero anywhere, $C\gt0$ everywhere on that leaf; if $W$ vanishes everywhere, invertibility gives $C=0$. Positive principal eigenvalue of $L_{-k}$ is a more general sufficient stability condition. The full variation and signature argument are in Booth and Fairhurst 2007, Eqs. (2.23), (3.2)–(3.7), pp. 8, 11–12, PDF.

The pointwise condition $V\gt0$ depends on the chosen null scaling. Under $\ell'=f\ell$, $k'=f^{-1}k$ with smooth $f\gt0$, the connection becomes $\omega'_A=\omega_A+D_A\ln f$ and

$$L'_{-k'}\psi=fL_{-k}(f^{-1}\psi).$$

The operators are conjugate, so their principal eigenvalues agree even though their pointwise potentials need not. This distinguishes a geometric stability test from a test of one normalization.

In spherical symmetry, $C$ is constant on each round leaf, so its derivatives disappear and the equation reduces to

$$C=\frac{\delta_\ell\theta_{(\ell)}} {\delta_k\theta_{(\ell)}}.$$

This quotient is not a pointwise formula for a general distorted MOTS. A patch with zero local flux need not be null when flux elsewhere makes the entire leaf evolve spacelike.

Area rate and normalization

The same decomposition gives

$$\mathcal L_{\mathcal V}dA =\left(\theta_{(\ell)}-C\theta_{(k)}\right)dA =-C\theta_{(k)}\,dA.$$

Thus an outer future spacelike tube has increasing area along $\mathcal V^a$. A timelike tube can have the opposite behavior. The signs are invariant under positive rescaling of the null normals, but $C$ and the parameter $v$ change; a numerical rate $dA/dv$ is meaningless without the normalization.

An equivalent dynamical-horizon normalization uses a unit timelike normal $\hat\tau^a$ to spacelike $H$ and an outward unit spacelike vector $\hat r^a$ tangent to $H$, with $\ell=\hat\tau+\hat r$ and $n=\hat\tau-\hat r$. Then $\ell\cdot n=-2$. The area increase follows directly from $\theta_{(\ell)}=0$ and $\theta_{(n)}\lt 0$. Converting between this normalization and $\ell\cdot k=-1$ changes factors but not the geometric conclusions.

When the sign argument fails

Three hypotheses carry the result. If the null energy condition fails, $W$ can be negative. If the principal eigenvalue reaches zero, the stability operator may cease to be invertible and its positive inverse is no longer available. A zero of $V$ at one point alone does not establish this spectral degeneracy. If the selected branch has $\theta_{(k)}\gt0$, it is not future marginally trapped even though $\theta_{(\ell)}=0$.

Generic merger tubes can display precisely these branch complications. Inner common MOTSs may be timelike, change signature, cease to be future marginally trapped on part of a leaf, or self-intersect. Pook-Kolb and collaborators document signature and expansion diagnostics for a head-on merger in Pook-Kolb et al. 2021, §§II–V, pp. 3–12, PDF. These features do not contradict the spacelike area law, because its outer, future, smooth-branch hypotheses are not all satisfied on the inner tube.

Check your understanding

On a round marginal leaf in spherical symmetry, $\delta_\ell\theta_{(\ell)}=-6/L^2$ and $\delta_k\theta_{(\ell)}=-2/L^2$, while $\theta_{(k)}=-1/L$. Find $C$ and the signed area rate per unit $dA$.

Solution

The signature coefficient is $C=(-6)/(-2)=3\gt 0$, so the tube is spacelike. Its local area rate is

$$\frac{\mathcal L_{\mathcal V}dA}{dA} =-C\theta_{(k)}=\frac{3}{L}\gt 0.$$

The numerical value depends on the scaling of $\mathcal V^a$, but its positive sign does not.

References

• Booth, Ivan, and Stephen Fairhurst. “Isolated, Slowly Evolving, and Dynamical Trapping Horizons: Geometry and Mechanics from Surface Deformations.” Physical Review D 75 (2007): 084019. doi:10.1103/PhysRevD.75.084019.
• Pook-Kolb, Daniel, Ofek Birnholtz, José Luis Jaramillo, Badri Krishnan, and Erik Schnetter. “Horizons in a Binary Black Hole Merger I: Geometry and Area Increase.” Physical Review D 103 (2021): 026005. doi:10.1103/PhysRevD.103.026005.