Null Expansions and Trapped Surfaces

Null expansion measures the fractional rate at which the area of a light-front changes as it leaves a spacelike two-surface. Its signs distinguish ordinary, trapped, and marginal surfaces without assuming stationarity. The construction is local to the surface, but its interpretation depends on a declared choice of future and outward directions and on keeping normalization-dependent values separate from normalization-invariant signs.

Required background. Hypersurface normals and tangential projectors supplies the induced metrics, normals, and extrinsic curvature used below.

Helpful background. Black-Hole Geometry and Mechanics develops global and quasi-local horizons; Black-Hole Perturbation Theory describes deformations of marginal surfaces; Formulations of General Relativity supplies the $3+1$ variables; and Computational Relativity explains how expansions become horizon-finding equations.

The two null expansions

Let $S$ be a smooth, closed, spacelike two-surface in a four-dimensional spacetime. Choose future-directed null normals $\ell^a$ and $k^a$ with $\ell\cdot k=-1$, with $\ell^a$ designated outward. The induced metric and area element are

$$q_{ab}=g_{ab}+\ell_a k_b+k_a\ell_b, \qquad dA=\sqrt{q}\,d^2x.$$

The expansions are traces of the two null second fundamental forms,

$$\theta_{(\ell)}=q^{ab}\nabla_a\ell_b, \qquad \theta_{(k)}=q^{ab}\nabla_a k_b.$$

If the null congruence is parametrized so that its deformation carries $S$ to nearby cross-sections, then $\mathcal{L}_{\ell}dA=\theta_{(\ell)}dA$. Thus positive, zero, and negative expansion mean local increase, stationarity, and decrease of cross-sectional area along that congruence.

The boost rescaling $\ell^a\mapsto f\ell^a$, $k^a\mapsto f^{-1}k^a$, for smooth $f\gt 0$ on $S$, preserves $\ell\cdot k=-1$ and gives

$$\theta_{(\ell)}\mapsto f\theta_{(\ell)}, \qquad \theta_{(k)}\mapsto f^{-1}\theta_{(k)}.$$

Their numerical magnitudes are normalization dependent, but their signs and zero sets are not. Andersson, Mars, and Simon use $\ell\cdot k_{\rm source}=-2$; setting $k=k_{\rm source}/2$ translates their ingoing expansion to the present normalization Andersson, Mars, and Simon 2005, pp. 1–2, PDF.

Trapped, outer trapped, and marginal

A future trapped surface has $\theta_{(\ell)}\lt 0$ and $\theta_{(k)}\lt 0$ everywhere: both future light-fronts converge. A future outer trapped surface requires only $\theta_{(\ell)}\lt 0$. A marginally outer trapped surface, or MOTS, has $\theta_{(\ell)}=0$; in black-hole applications one usually also requires $\theta_{(k)}\lt 0$, but that extra sign should be stated rather than silently built into “MOTS.” Weak inequalities define weakly trapped variants.

These are properties of an embedded surface, not of a spacetime point. A point can lie on many surfaces with different expansions. Nor is a MOTS automatically an event horizon: an event horizon is a global null boundary determined by the entire future, whereas a MOTS is found on a chosen spacelike slice. The outermost MOTS on one slice is often called that slice’s apparent horizon, so its location can change with the slicing.

For a round sphere of areal radius $r$ in Minkowski spacetime, choose $\ell=(\partial_t+\partial_r)/\sqrt2$ and $k=(\partial_t-\partial_r)/\sqrt2$. Then

$$\theta_{(\ell)}=\frac{\sqrt2}{r}, \qquad \theta_{(k)}=-\frac{\sqrt2}{r},$$

so the sphere is untrapped. The opposite signs are the essential check; their factors change under a boost rescaling.

A round-sphere trapping test

For a symmetry sphere with area $4\pi r^2$, logarithmically differentiating the area gives $\theta_{(\ell)}=2\ell(r)/r$ and $\theta_{(k)}=2k(r)/r$. Since the normal two-metric has inverse $-\ell^ak^b-k^a\ell^b$,

$$\theta_{(\ell)}\theta_{(k)} =-\frac{2}{r^2}\,g^{ab}\nabla_a r\nabla_b r =-\frac{2}{r^2}\left(1-\frac{2Gm_H}{r}\right),$$

where the last equality defines the physical spherical Hawking mass $m_H$. In a collapsing region with $\theta_{(k)}\lt0$, the sphere is future trapped precisely when $2Gm_H/r\gt1$. Without that sign assumption a positive product could instead describe a past-trapped sphere. The mass identity therefore needs the time orientation as well as compactness to decide future trapping.

For ingoing Vaidya, $f=1-2Gm(v)/r$, $\ell=\partial_v+(f/2)\partial_r$, and $k=-\partial_r$ give $\theta_{(\ell)}=f/r$ and $\theta_{(k)}=-2/r$. This checks the zero at $r=2Gm(v)$ directly and links the definition to marginal-tube emergence.

Expansion in a spatial slice and focusing

Let $u^a$ be the future unit normal to a spacelike slice $\Sigma$, let $s^a$ be the outward unit normal to $S$ within $\Sigma$, and adopt $K_{ij}=h_i{}^a h_j{}^b\nabla_a u_b$. With $\ell=u+s$ and $k=(u-s)/2$,

$$\theta_{(\ell)}=D_i s^i+K-K_{ij}s^is^j,$$

while $2\theta_{(k)}=-D_i s^i+K-K_{ij}s^is^j$. The equation $\theta_{(\ell)}=0$ is a nonlinear elliptic equation for the shape of $S$ in the slice. Gourgoulhon and Jaramillo derive the same $3+1$ decomposition with the opposite extrinsic-curvature sign convention in Gourgoulhon and Jaramillo 2006, § 10.2.1, Eqs. (10.5)–(10.6), pp. 142–143. Thus, if one adopts their convention, every $K_{ij}$ term in the displayed formula changes sign.

Along an affinely parametrized, twist-free null congruence, Raychaudhuri’s equation gives

$$\frac{d\theta_{(\ell)}}{d\lambda} =-\frac12\theta_{(\ell)}^2-\sigma_{ab}\sigma^{ab} -8\pi G T_{ab}\ell^a\ell^b.$$

Under the null energy condition, shear and matter cannot increase the expansion. This focusing mechanism explains why a negative expansion tends toward a caustic, but focusing alone does not prove that a closed trapped surface forms from arbitrary initial data.

Check your understanding

A surface has expansions $(\theta_{(\ell)},\theta_{(k)})=(0,-3/L)$ in one null normalization. After the boost $f=4$, classify it and compute the new pair.

Solution

It is a future MOTS with negative ingoing expansion. The boosted pair is $(4\times0,4^{-1}(-3/L))=(0,-3/(4L))$. Its classification is unchanged because a positive boost preserves each sign and the vanishing of the outgoing expansion.

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.
• Gourgoulhon, Éric, and José Luis Jaramillo. “A 3+1 Perspective on Null Hypersurfaces and Isolated Horizons.” Physics Reports 423 (2006): 159–294. doi:10.1016/j.physrep.2005.10.005.