Lemaître–Tolman–Bondi Collapse

Lemaître–Tolman–Bondi spacetimes describe the most general spherically symmetric, comoving, pressureless dust flow before shells intersect. Relative to homogeneous Oppenheimer–Snyder collapse, two freely specifiable radial functions let different shells carry different masses and energies. That freedom reveals three logically separate events: shell focusing at zero areal radius, shell crossing when neighboring labels meet, and trapping when the Misner–Sharp compactness reaches unity. The construction originates in Lemaître 1933, pp. 51–85 and Tolman 1934, pp. 169–176, with its physical interpretation developed systematically in Bondi 1947, pp. 410–425.

Required background. Use The Einstein Equation for the relation between the dust stress tensor and the radial evolution equations.

Helpful background. Black-Hole Geometry and Mechanics supplies horizon geometry, Black-Hole Perturbation Theory supplies the contrast with linear stability, Formulations of General Relativity supplies initial-data language, Relativistic Matter and Compact Objects explains the dust approximation, and Computational Relativity treats evolutions once an exact dust description breaks down.

Inhomogeneous dust data

Use comoving proper time $\tau$ and a radial shell label $r$. Before shell crossing, the metric can be written

$$ds^2=-d\tau^2+ \frac{R'(\tau,r)^2}{1+2E(r)}dr^2+R(\tau,r)^2d\Omega^2,$$

where a dot and a prime denote $\partial_\tau$ and $\partial_r$. With $T_{ab}=\rho u_a u_b$ and $u^a=(\partial_\tau)^a$, Einstein’s equations become

$$\dot R^2=\frac{2Gm(r)}{R}+2E(r), \qquad \rho(\tau,r)=\frac{m'(r)}{4\pi R^2R'}.$$

The first equation is an energy equation for each shell. The second identifies $m(r)$ as the Misner–Sharp mass inside that shell. One radial relabeling remains; a convenient initial gauge is $R(0,r)=r$. The physical initial data are then a mass profile $m(r)$ and an energy profile $E(r)$, subject to $1+2E>0$, nonnegative density, and regularity.

At a regular center,

$$m(r)=m_3r^3+O(r^5), \qquad E(r)=E_2r^2+O(r^4),$$

so $m/R^3$ and the spatial metric remain finite. The homogeneous dust ball is recovered when $m\propto r^3$, $E\propto-r^2$, and all shells share the same collapse time. The displayed even-power remainder assumes a smooth spherical center: smooth radial scalars extend as even functions of Cartesian radius. Lower-regularity center data allow different expansions and must be identified separately. The homogeneous limit is an essential check on signs and normalizations.

Marginally bound collapse

The case $E=0$ makes the distinct singular mechanisms transparent. On the collapsing branch,

$$R(\tau,r)= \left(\frac{9Gm(r)}{2}\right)^{1/3} [\tau_s(r)-\tau]^{2/3},$$

where $\tau_s(r)$ is the time at which shell $r$ focuses at $R=0$. In the gauge $R(0,r)=r$,

$$\tau_s(r)=\frac{2r^{3/2}}{3\sqrt{2Gm(r)}}.$$

A decreasing outward density can make $\tau_s(r)$ increase with $r$: central shells then focus before outer ones. This time delay is the basic reason that inhomogeneous dust has causal structures absent from the homogeneous model.

The density formula exposes a different pathology. A shell-focusing singularity occurs when a material shell with nonzero initial radius reaches $R=0$; the permanently regular center also has $R=0$, so that equation alone is not a singularity test there. A shell-crossing singularity has $R'=0$ while $R>0$ and $m'\neq0$, so distinct comoving labels occupy the same areal radius and the dust density diverges. Shell crossing normally signals failure of the single-stream, zero-pressure matter model; it should not be silently treated as the central strong-curvature endpoint. Conditions such as $R'>0$ throughout the regular region must therefore be checked before interpreting the exact solution.

For this marginally bound branch, logarithmic differentiation provides a useful sufficient no-crossing test:

$$\frac{R'}{R} =\frac{m'}{3m} +\frac{2\tau_s'}{3(\tau_s-\tau)}.$$

Thus $m'\gt0$ and $\tau_s'\geq0$ imply $R'\gt0$ before focusing. This is a branch-specific sufficient condition, not the complete shell-crossing criterion for all bound and unbound LTB solutions.

Trapping and possible null escape

The invariant mass identity is

$$1-\frac{2Gm}{R}=g^{ab}\nabla_aR\nabla_bR.$$

Consequently a round marginally outer trapped sphere on the collapsing branch satisfies $R=2Gm$. For marginally bound collapse, substituting this value into the exact solution gives

$$\tau_{\mathrm{AH}}(r) =\tau_s(r)-\frac{4Gm(r)}{3}.$$

Every noncentral shell is therefore trapped before it focuses. At the center, however, $m(r)\sim r^3$, so the interval $4Gm/3$ vanishes as $r\to0$. Whether an outgoing radial null geodesic can emerge from the central limiting event depends on the first nontrivial terms of the initial profiles. The geodesic equation is

$$\frac{d\tau}{dr}=\frac{R'}{\sqrt{1+2E}}.$$

The endpoint $(\tau_s(0),0)$ is a singular coordinate limit, so $d\tau/dr$ need not approach a finite nonzero slope there. Instead choose $u=r^\alpha$, with $\alpha\geq1$ fixed by the leading nonhomogeneous term in the initial profiles, and define $X=R/u$. After substituting the central expansions into the null equation, a locally escaping ray requires a positive root

$$X_0=\lim_{r\to0}\frac{R}{u} =\lim_{r\to0}\frac{dR}{du}>0$$

of the resulting tangent equation, with the ray lying outside the trapped region Miyamoto, Jhingan, and Harada 2013, §2.3, Eqs. (15)–(17). This is a local visibility test. Reaching a regular exterior or future null infinity is a stronger global test, and survival under a stated class of nonspherical or matter perturbations is stronger again.

Exact LTB families contain data with covered central focusing and restricted families with locally or globally visible central behavior. They also contain shell crossings. These examples disprove any claim that spherical dust profiles have one universal causal endpoint, but they do not by themselves settle generic collapse or weak cosmic censorship. The data topology, differentiability, energy conditions, matching surface, and allowed perturbations must accompany every such statement; a detailed review is given by Joshi and Malafarina 2011, §§2–4.

A smooth decreasing-density example

One can see the local visibility mechanism without leaving the regular-center class. Let

$$m(r)=m_3r^3[1-\mu r^2+O(r^4)], \qquad m_3\gt0,\quad \mu\gt0,$$

in a sufficiently small neighborhood of the center, with $E=0$ and $R(0,r)=r$. Then

$$\rho(0,r)=\frac{m_3}{4\pi}[3-5\mu r^2+O(r^4)], \qquad \tau_s(r)=\tau_0\left[1+\frac{\mu r^2}{2}+O(r^4)\right],$$

where $\tau_0=2/(3\sqrt{2Gm_3})$. Density is positive and decreases outward near the center, and $\tau_s'\gt0$ excludes a local shell crossing before focusing. At $\tau=\tau_0$, neighboring shells are still regular.

For a possible outgoing ray set $\tau(r)=\tau_0+A r^\alpha+\cdots$. The exact solution can also be written $R=r(1-\tau/\tau_s)^{2/3}$. If $\alpha\gt2$, its leading term along the ray is

$$R\simeq\left(\frac{\mu}{2}\right)^{2/3}r^{7/3}.$$

Computing the partial derivative $R'$ at fixed $\tau$ and then evaluating it on the ray gives $(7/3)(\mu/2)^{2/3}r^{4/3}$ to leading order. Matching $d\tau/dr=R'$ therefore yields the self-consistent positive tangent

$$\alpha=\frac73,\qquad A=\left(\frac{\mu}{2}\right)^{2/3}.$$

Moreover,

$$\tau_{\mathrm{AH}}-\tau_0 =\frac{\tau_0\mu}{2}r^2-\frac{4Gm_3}{3}r^3+O(r^4),$$

so the candidate ray, whose time delay is $O(r^{7/3})$, lies before the trapping curve for sufficiently small $r\gt0$. This supplies an explicit local outgoing asymptotic with $R/(2Gm)\to\infty$. Establishing its continuation uses the null-geodesic equation in the regular neighboring region; global visibility additionally depends on the outer profile and matching. The tangent test and its scope are developed in Miyamoto, Jhingan, and Harada 2013, §2.3, Eqs. (15)–(17).

Diagnostic checks

Four calculations prevent most interpretive errors. Recompute $m$ from the invariant mass identity rather than from coordinates alone. Verify $R'>0$ before the endpoint to exclude shell crossing. Locate $R=2Gm$ and compare its time with $\tau_s(r)$. Finally, integrate or asymptotically solve the outgoing-null equation near the center; a delayed trapping curve without an escaping null solution is not evidence of visibility.

The dust assumptions are consequential. Pressure gradients can remove shell crossings, introduce sonic structure or shocks, and alter the collapse threshold. Rotation and nonspherical modes can radiate or redistribute angular momentum. LTB is therefore a controlled exact laboratory for radial inhomogeneity—not an astrophysical closure model.

Exercises

In marginally bound collapse, show directly that a noncentral shell becomes marginally trapped $4Gm(r)/3$ in proper time before it focuses. Explain why this result alone does not decide whether the central singularity is locally visible.

Solution

Raise the exact solution to the $3/2$ power:

$$R^{3/2}=\frac{3}{2}\sqrt{2Gm}\,[\tau_s-\tau].$$

At a marginal sphere $R=2Gm$, division by $\sqrt{2Gm}$ gives $2Gm=(3/2)(\tau_s-\tau_{\mathrm{AH}})$, hence $\tau_s-\tau_{\mathrm{AH}}=4Gm/3$. For every fixed $r>0$ with $m(r)>0$, trapping precedes focusing. At a regular center $m(r)\sim r^3$, so this interval tends to zero. One must compare the leading approach of the trapping curve with solutions of $d\tau/dr=R'/\sqrt{1+2E}$; the noncentral inequality alone does not determine the central null tangent.

References

• Bondi, Hermann. “Spherically Symmetrical Models in General Relativity.” Monthly Notices of the Royal Astronomical Society 107 (1947): 410–425. doi:10.1093/mnras/107.5-6.410.
• Joshi, Pankaj S., and Daniele Malafarina. “Recent Developments in Gravitational Collapse and Spacetime Singularities.” International Journal of Modern Physics D 20 (2011): 2641–2729. doi:10.1142/S0218271811020792. Open PDF.
• Lemaître, Georges. “L’Univers en expansion.” Annales de la Société Scientifique de Bruxelles A 53 (1933): 51–85. English translation.
• Miyamoto, Umpei, Sanjay Jhingan, and Tomohiro Harada. “Weak Cosmic Censorship in Gravitational Collapse with Astrophysical Parameter Values.” Progress of Theoretical and Experimental Physics 2013 (2013): 053E01. doi:10.1093/ptep/ptt027.
• Tolman, Richard C. “Effect of Inhomogeneity on Cosmological Models.” Proceedings of the National Academy of Sciences 20 (1934): 169–176. doi:10.1073/pnas.20.3.169.