Null Shells and Vaidya Collapse

The ingoing Vaidya spacetime replaces a material star by radial null dust whose total mass changes with advanced time. It is simple enough that null expansions and horizon generators can be calculated exactly, yet dynamic enough to display the decisive difference between a quasi-local marginal tube and a global event horizon. In the thin-shell limit, the event horizon extends into a region that is exactly flat before the shell arrives.

Required background. Use The Einstein Equation to connect the changing mass function to null stress-energy.

Helpful background. Black-Hole Geometry and Mechanics supplies event and trapping horizons, Black-Hole Perturbation Theory supplies the contrast with small disturbances of Schwarzschild, Formulations of General Relativity supplies junction and initial-value language, Relativistic Matter and Compact Objects clarifies why null dust is an idealization, and Computational Relativity treats horizon location in generic data.

Ingoing Vaidya geometry

Let $v$ be advanced time and $m(v)$ the physical mass. The ingoing Vaidya line element is

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

Einstein’s equation gives a single nonzero flux component,

$$T_{vv}=\frac{1}{4\pi r^2}\frac{dm}{dv}.$$

Thus $dm/dv\geq0$ is the null-energy condition for accretion. A constant $m$ gives ingoing Eddington–Finkelstein Schwarzschild; $m=0$ gives flat spacetime in advanced coordinates. Vaidya’s original outgoing radiating solution, under time reversal, supplies this accreting advanced-time form Vaidya 1951, pp. 264–276. The model has a regular stress tensor for smooth $m(v)$ away from $r=0$, but its matter consists of radial massless streams and carries no pressure transverse to the rays.

Choose future null normals to a round sphere as

$$\ell^a=(\partial_v)^a+\frac{f}{2}(\partial_r)^a, \qquad n^a=-(\partial_r)^a, \qquad \ell\cdot n=-1.$$

Their expansions are

$$\theta_{(\ell)}=\frac{f}{r}, \qquad \theta_{(n)}=-\frac{2}{r}.$$

The spherical marginally outer trapped tube is therefore

$$r_{\mathrm{MTT}}(v)=2Gm(v).$$

Its tangent is $t^a=(\partial_v)^a+2G(dm/dv)(\partial_r)^a$, whose squared norm on the tube is $t^at_a=4G\,dm/dv$. It is spacelike while positive-energy flux crosses it and null when the flux vanishes. This local response is reviewed in Booth 2005, §3.3, pp. 1084–1085.

Outgoing generators and the event horizon

An outgoing radial null curve satisfies

$$\frac{dr}{dv}=\frac{1}{2} \left(1-\frac{2Gm(v)}{r}\right).$$

The marginal tube is obtained algebraically from the instantaneous mass. The event horizon is instead the particular outgoing solution that asymptotes to the final stationary horizon. To locate it, one must know the future mass history and integrate backward. If accretion later increases, the event horizon generally lies outside $2Gm(v)$ in anticipation of the future flux; no local measurement at that earlier event determines this global selection.

This is not matter or information propagating backward in time. Each horizon generator is an ordinary future-directed null geodesic. What uses the future is the classification of which generator fails to reach future null infinity.

A collapsing thin null shell

Take a shell at $v=v_0$ with

$$m(v)=M\,\Theta(v-v_0).$$

Inside the shell’s past, $v<v_0$, the geometry is flat. To its future, it is Schwarzschild with mass $M$. The distributional derivative of the step function supplies the shell stress-energy; integrating $4\pi r^2T_{vv}$ across $v_0$ gives the mass jump $M$. A full junction treatment keeps the induced geometry continuous and accounts for this null surface layer.

In the Schwarzschild region the final event horizon is $r=2GM$. Its generator meets the shell at $(v_0,2GM)$. Trace the same outgoing generator backward through the flat region. There $m=0$, so $dr/dv=1/2$ and

$$r_{\mathrm{EH}}(v)=2GM+\frac{v-v_0}{2}, \qquad v_0-4GM\leq v<v_0.$$

The generator begins at the regular center at $v=v_0-4GM$, even though the geometry there is locally Minkowski and the shell has not arrived. By contrast, there is no spherical marginally trapped surface in the flat region; after the shell, the marginal tube is $r=2GM$. The two boundaries meet once the accretion is complete.

The sharp shell is a useful limit, not ordinary material. Smoothing the step into a monotone mass ramp smooths the stress tensor and the marginal tube. Backward integration still places the event horizon outside the instantaneous marginal tube during accretion, so the qualitative lesson does not depend on the distributional idealization.

Checks and limits of the model

The mass normalization can be checked from the invariant relation $1-2Gm/r=g^{ab}\nabla_ar\nabla_br$. The expansions then reproduce $r=2Gm$ independently. The null-energy condition fixes the marginal tube’s non-timelike signature for increasing mass. Finally, the outgoing geodesic equation reduces to slope $1/2$ in flat space and to a constant generator at $r=2GM$ in the final Schwarzschild region.

Outgoing Vaidya, described with retarded rather than advanced time, represents emission and carries different sign conventions. Mixing the two forms can reverse the inferred energy condition. Nor should $r=2Gm(v)$ automatically be called “the event horizon”: it is the spherical marginal tube, while the event horizon comes from the complete causal future. These exact distinctions underlie horizon-finding practice in more general collapse.

Exercises

For the thin shell above, verify the event-horizon starting time and show that a radial light ray emitted from the center just before that time escapes, while one emitted just after it does not.

Solution

In the flat region every outgoing radial ray has $r=(v-v_e)/2$, where $v_e$ is its center-emission time. At the shell $v=v_0$, its radius is $(v_0-v_e)/2$. It lies outside the final Schwarzschild horizon and escapes precisely when

$$\frac{v_0-v_e}{2}>2GM,$$

or $v_e<v_0-4GM$. Equality selects the event-horizon generator, so $v_{\mathrm{start}}=v_0-4GM$. A ray emitted later meets the shell at $r<2GM$ and remains in the black-hole region. The comparison uses the future matching, even though every ray in the pre-shell region follows the same local flat-space equation.

References

• Booth, Ivan. “Black Hole Boundaries.” Canadian Journal of Physics 83 (2005): 1073–1099. doi:10.1139/p05-063. Open PDF.
• Vaidya, P. C. “The Gravitational Field of a Radiating Star.” Proceedings of the Indian Academy of Sciences, Section A 33 (1951): 264–276. doi:10.1007/BF03173260.