Emergence of Marginal Tubes in Collapse

When collapse first produces marginally outer trapped surfaces, successive leaves may assemble into a three-dimensional marginally trapped tube. The tube need not be null, unique, or smoothly represented by the outermost surface on every slice. Its signature records a balance between matter or gravitational influx and the outer stability of its leaves, while its apparent “birth” and jumps can depend strongly on the spacetime foliation.

Required background. Hypersurface normals and tangential projectors supplies the normal geometry of the leaves and of the tube.

Helpful background. Black-Hole Geometry and Mechanics develops quasi-local horizon classes; Black-Hole Perturbation Theory describes shear and deformations; Formulations of General Relativity supplies slicing and constraint data; and Computational Relativity explains marginal-surface tracking.

From one marginal surface to a tube

Let $H$ be a smooth hypersurface foliated by closed spacelike surfaces $S_\lambda$. Choose future null normals with $\ell\cdot k=-1$ and impose

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

on every leaf. This is a future marginally trapped tube. If $H$ is spacelike, it is a dynamical horizon. A null portion is nonexpanding; an isolated horizon additionally requires time independence of its intrinsic connection Ashtekar and Krishnan 2004, §2.1.1, pp. 9–11, PDF. More general trapping-horizon terminology also specifies the sign of the transverse surface variation $\delta_k\theta_{(\ell)}$.

On a branch with positive outgoing coefficient, a vector tangent to $H$ and normal to its leaves can be normalized as

$$\mathcal V^a=\ell^a-Ck^a, \qquad \mathcal V^2=2C.$$

Thus $C\gt0$, $C=0$, and $C\lt0$ give spacelike, null, and timelike portions. Preservation of marginality is a surface-deformation equation,

$$L_{-k}C=W, \qquad L_{-k}C\equiv\delta_{-Ck}\theta_{(\ell)},$$

where Raychaudhuri’s equation gives

$$W=-\delta_\ell\theta_{(\ell)} =\sigma^{(\ell)}_{ab}\sigma_{(\ell)}^{ab} +8\pi G T_{ab}\ell^a\ell^b.$$

The elliptic operator $L_{-k}$ contains angular derivatives of $C$. Under the null energy condition and strict outer stability, its maximum principle gives $C\geq0$. Nonzero $W$ somewhere makes the whole connected leaf spacelike; $W=0$ everywhere gives the null case. In spherical symmetry the derivative terms disappear, leaving $C=\delta_\ell\theta_{(\ell)}/\delta_k\theta_{(\ell)}$. Using that quotient pointwise for an arbitrary distorted leaf would omit its shape response. See Booth and Fairhurst 2007, Eqs. (3.2)–(3.7), pp. 11–12, PDF and the complete operator in Marginal Tubes and Their Signature.

Spherical collapse as a signature check

The ingoing Vaidya metric provides a transparent model,

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

Round marginal surfaces lie at $r=2Gm(v)$. Restricting the metric to their tube, where $dr=2Gm'(v)\,dv$, gives

$$ds_H^2=4Gm'(v)\,dv^2+(2Gm)^2d\Omega^2.$$

Positive ingoing energy has $m'(v)\gt 0$, so the tube is spacelike and its area grows. When accretion stops, $m'=0$ and it is null. Choosing $m'\lt 0$ in this ingoing model makes the tube timelike but violates the corresponding positive-energy condition. This calculation checks both the signature sign and the isolated limit.

This monotone Vaidya example has one round marginal radius at each $v$; more general spherical matter profiles can already have several marginal branches. In a generic slicing, a pair of MOTSs can appear at a degenerate leaf and then separate into outer and inner branches. The outermost MOTS selected on successive slices can jump from one branch to another even when a smooth marginal tube connects the relevant spacetime regions. Andersson, Mars, and Simon explain how strict stability gives a smooth local tube and how a foliation can make outermost surfaces jump in Andersson, Mars, and Simon 2005, pp. 1–4, PDF.

What emergence does not establish

The first coordinate time at which a horizon finder returns a MOTS is slicing dependent. It need not be the first spacetime point on a marginal tube, and it is not the formation time of the event horizon. Likewise, the area of the outer branch can grow while an inner branch shrinks; treating both as the same monotone object loses the orientation and stability data.

A robust account records the leaf expansions, principal stability eigenvalue, tube signature, area, matter flux, outgoing shear, and branch identity. It follows interior as well as outermost solutions when possible. A principal eigenvalue approaching zero makes the elliptic continuation ill-conditioned. In the spherical quotient this appears as a vanishing denominator, but a pointwise zero of the outer variation is not generally the same as spectral degeneracy. Energy conditions constrain the sign only after orientation and stability have been checked.

Check your understanding

On a round marginal leaf in spherical symmetry, suppose $\delta_\ell\theta_{(\ell)}=-2/L^2$ and $\delta_k\theta_{(\ell)}=-5/L^2$. Compute $C$ and classify the tube.

Solution

The angular derivatives vanish, so $0=\delta_\ell\theta_{(\ell)}-C\delta_k\theta_{(\ell)}$ gives

$$C=\frac{\delta_\ell\theta_{(\ell)}}{\delta_k\theta_{(\ell)}} =\frac{2}{5}\gt 0.$$

Therefore $\mathcal V^2=2C\gt 0$ in the $(-,+,+,+)$ convention and the tube is spacelike.

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.
• Ashtekar, Abhay, and Badri Krishnan. “Isolated and Dynamical Horizons and Their Applications.” Living Reviews in Relativity 7 (2004): 10. doi:10.12942/lrr-2004-10.
• 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.