research38 min studyLesson 823 in Physics

Trapped surfaces and singularity-theorem assumptions

Compute null expansions on spheres and distinguish slice-based trapping from global incompleteness and event-horizon claims.

Before you begin

On this page

The question for today

What you’ll understand

Classify a round sphere using its two future null expansions.
State the separate hypotheses and conclusion of the Penrose theorem.
Explain why a marginally trapped tube need not be a null event horizon.

A trapped surface is a compact spacelike surface for which both future-directed families of orthogonal light rays initially converge. The condition captures strong gravitational collapse without requiring exact spherical symmetry. It is local in the geometry of the surface and its null normals, while an event horizon is a global causal boundary. A singularity theorem connects trapping to an incomplete geodesic only after adding curvature and global causality assumptions. These are three related but different statements, and none should silently replace another.

The signs of two future expansions carry the information

Use ingoing coordinates with signature (−,+,+,+) and G=c=1. For a spherical metric with mass function m(v), choose outgoing null normal ell and ingoing null normal n with inner product minus one. A sphere has area four pi r squared, so either expansion is twice the normal derivative of r divided by r. This gives one positive and one negative expansion outside r=2m, both negative inside, and a vanishing outward expansion on that radius.

The numerical expansion depends on null-normal normalization, but its sign does not change under a positive rescaling preserving future orientation. The chosen outgoing normal need not be affine, so its expansion is not to be inserted unmodified into the affine focusing equation. A marginally outer trapped surface has outward expansion zero and inward expansion negative. Identifying an apparent horizon as an outermost such surface additionally refers to a chosen spatial slice and appropriate regularity; not every marginal sphere is globally the outermost one.

ds^2=-f\,dv^2+2\,dv\,dr+r^2d\Omega^2,\quad f=1-\frac{2m(v)}r,\quad \ell=\partial_v+\frac f2\partial_r,\quad n=-\partial_r,\qquad \theta_\ell=\frac f r,\quad\theta_n=-\frac2r

v,r,m have length units, ell·n=−1, and both normals are future directed in the chosen ingoing chart. The formulas are for round symmetry spheres. A varying m(v) describes a Vaidya-type geometry rather than a stationary vacuum solution.↔ Scroll wide equations horizontally.

A singularity theorem gives incompleteness, with hypotheses

One standard four-dimensional Penrose theorem assumes a globally hyperbolic spacetime with a noncompact Cauchy surface, a closed future trapped surface, and the null convergence condition. In Einstein gravity the null energy condition is a sufficient way to obtain that curvature condition. Focusing limits how long null generators can remain on a causal boundary. Global causal and topological arguments then contradict future null completeness under the stated assumptions.

The conclusion is the existence of an incomplete future null geodesic, not a theorem that every infaller reaches one specified point of infinite density. It does not itself identify a diverging curvature invariant or prove that an event horizon hides the incomplete region. Further assumptions about extensions, predictability, and cosmic censorship enter such interpretations. Different singularity theorems use different global and energy hypotheses; the conditions of this version should not be replaced by a vague statement that gravity is attractive.

\theta_\ell<0,\ \theta_n<0\quad\text{on a closed surface},\qquad R_{ab}k^ak^b\geq0\quad\text{for every null }k^a,\qquad \text{noncompact Cauchy surface}\ \Longrightarrow\ \text{future null incompleteness}

The displayed implication summarizes the specified Penrose theorem with the smoothness and global-hyperbolicity assumptions stated in the text. Closed means compact without boundary. Incompleteness means finite affine extent of an inextendible null geodesic within the spacetime under discussion.↔ Scroll wide equations horizontally.

A trapping condition is not an event-horizon finder

An event horizon is the boundary of the causal past of future null infinity and depends on the entire future geometry. A local equation for zero outward expansion cannot determine that global boundary during arbitrary evolution. In a spherical stationary Schwarzschild geometry, the familiar marginal sphere and horizon occur at the same areal radius. In an accreting geometry the marginal tube can be spacelike, whereas the regular event horizon is a null causal boundary.

For the Vaidya-type example, the tube r=2m(v) has tangent radial speed twice the mass derivative. At that tube the outgoing null speed from the metric is zero, so a positive mass derivative separates the two directions. Evaluating the norm of the tube's normal verifies that it is spacelike. This is a concrete reason to label the object being computed rather than calling every zero of a convenient metric function an event horizon. Slice changes create additional ambiguity for apparent-horizon locations.

Work it through

Classify three round spheres

Set m=1L at a chosen advanced time. Evaluate both future null expansions at r=3L, r=2L, and r=1L in the displayed normalization. Classify each sphere and identify which conclusion requires information beyond these local numbers.

At r=3L, $f=1-2/3=1/3$ , so $\theta_\ell=1/(9L)$ and $\theta_n=-2/(3L)$ . The outward family expands and the inward family converges.
At r=2L, f=0, giving $\theta_\ell=0$ and $\theta_n=-1/L$ . The sphere is marginally outer trapped in this spherical example.
At r=1L, f=−1, so $\theta_\ell=-1/L$ and $\theta_n=-2/L$ . It is future trapped. The local calculation alone neither identifies the full event horizon nor verifies every global assumption of a singularity theorem.

The result

The spheres are respectively untrapped, marginally outer trapped, and future trapped; event-horizon and incompleteness conclusions need additional global information.

The living notebook

Explore the model

Interactive lab

Interactive lab · root-finding

Solving a local equation does not supply a causal history

For each fresh comparison, Reset and set BOTH Initial upper bound U=2 and Target c. Compare c=1.00 with c=2.25, predicting positive roots 1 and 1.5. Use Step until each exact bracket collapses. This instrument bisects x²−c on [0,U]; it is not the null-expansion equation or a horizon solver. A successful algebraic root illustrates only the limited information supplied by a local condition.

Quadratic / midpointCurrent bracketPositive root

Midpoint after 0 refinements: 1.5
Bracket error bound: 1.5
Computed residual · x² − c: 0.25

Target c2Initial upper bound U3

t = 0 s

Bisection of x² − c on [0,U], capped at 50 iterations for floating-point resolution. The error bound is the maximum distance from the displayed midpoint to either bracket endpoint (half the width in exact arithmetic). Compensated square evaluation preserves the bracket when ordinary multiplication rounds a residual to zero. A truly exact endpoint or midpoint collapses the bracket. This is one monotone quadratic, not a general root solver. Parameter changes pause playback and recompute the model at the current elapsed time. Reset restores both parameters and returns time to zero. Playback starts paused. Adjust either parameter, then choose Play or Step.

Look forNumerically locating a zero is different from proving that it represents a particular global causal boundary.

Pause & reason

Check your understanding

Interactive checks need JavaScript. You can still read the explanation.

Read the checkpoint explanation

The theorem establishes future null incompleteness under trapping, null convergence, and global hypotheses. Curvature divergence and an event horizon require further analysis.

Your turn

Try it yourself

At v=0 take m(v)=1L+0.050v. For the tube r=2m(v), compute dr/dv and compare it with the outgoing null slope there. Using the inverse metric, the squared norm of the normal to F=r−2m(v) is f−4m′. Determine the tube's causal character at f=0.

Take a moment to make a prediction or work it out on paper. Open a hint when you need one.

Give me a hint

A negative squared norm is timelike for a normal in signature (−,+,+,+), which means the hypersurface it is normal to is spacelike.

Compare with the solution

The marginal tube has $dr/dv=2m'=0.100$ . The outgoing null direction at f=0 has $dr/dv=f/2=0$ , so the tube is not that null hypersurface. Its normal has squared norm $-4(0.050)=-0.200$ , hence is timelike and the tube is spacelike. The instantaneous mass and its derivative locate this marginal tube but do not fix the event horizon without the future mass history and boundary conditions. This comparison concerns the spherical trapping tube, not a claim that all apparent horizons have one universal causal character.

Keep the idea

Take it with you

Trapped-surface signs are local geometric data with a declared future normalization.
Penrose incompleteness follows only after the theorem’s curvature and global hypotheses are supplied.
Apparent, trapping, event, and Killing horizons are different concepts during evolution.

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