Lemaître-Tolman-Bondi and Szekeres Families

Lemaître–Tolman–Bondi and Szekeres spacetimes are exact, irrotational, pressureless-dust cosmologies with spatial inhomogeneity. LTB retains spherical symmetry and three radial functions before gauge fixing; Szekeres replaces concentric spheres by angularly varying two-surfaces and generically has no Killing vectors. Their free functions are geometric initial data, constrained by regularity and shell-crossing avoidance rather than arbitrary observable profiles.

Required background. Einstein’s Equation supplies the dust equations; Continuum Kinematics and Balance Laws supplies pressureless flow and density conservation.

Helpful background. The Riemann Curvature Tensor supplies singularity diagnostics; Exact Solutions of Einstein’s Equations supplies the family context; Killing Fields and Isometry Groups supplies the contrast between spherical and nonsymmetric models.

LTB free functions

In comoving proper time, write

$$ds^2=-dt^2+\frac{R'^2(t,r)}{1+2E(r)}dr^2 +R^2(t,r)d\Omega_2^2,$$

where a dot and prime denote $t$ and $r$ derivatives. Let $m(r)$ be the geometrized dust-mass function, excluding the cosmological-constant contribution. Einstein’s equation gives

$$\dot R^2=\frac{2m(r)}{R}+2E(r) +\frac{\Lambda R^2}{3},$$ $$8\pi G\rho=\frac{2m'(r)}{R^2R'}.$$

The geometrized Misner–Sharp mass defined from the areal-radius gradient is

$$m_{\mathrm{MS}}=\frac R2\left(1-g^{ab}\nabla_aR\nabla_bR\right) =m(r)+\frac{\Lambda R^3}{6}.$$

Indeed $g^{ab}\nabla_aR\nabla_bR=-\dot R^2+1+2E$; substitution of the evolution equation gives the result. Thus $m=m_{\mathrm{MS}}$ only when $\Lambda=0$.

Integrating the first equation introduces one time-integration function. On branches that emerge from $R=0$, it is the bang time $t_B(r)$; a bouncing branch instead needs a suitable reference epoch. Together with $m(r)$ and $E(r)$ it describes the family, but a monotone redefinition of $r$ removes one functional degree of freedom. When $\Lambda=0$, $E\gt0$, $E=0$, and $E\lt0$ label hyperbolic, parabolic, and elliptic evolution branches. For nonzero $\Lambda$, the additional $\Lambda R^2/3$ term means that the sign of $E$ alone does not determine the evolution; see Bolejko, Célérier, and Krasiński 2011, §2, arXiv manuscript p. 4, Open PDF. The displayed radial chart assumes $1+2E\gt0$ and $R'\ne0$ away from a center; regular spatial extrema where both factors vanish require a separate limiting analysis.

The FLRW specialization is

$$R(t,r)=a(t)r, \qquad m(r)=m_0r^3, \qquad E(r)=-\frac12kr^2, \qquad t_B=\text{constant}.$$

Substitution reduces the evolution equation to the Friedmann equation with dust density proportional to $a^{-3}$. Bondi’s invariant interpretation of the radial functions is developed in Bondi 1947, pp. 410–425.

Quasi-spherical Szekeres geometry

For the quasi-spherical branch, one convenient form is

$$\begin{aligned} ds^2={}&-dt^2 +\frac{\left(\Phi'-\Phi\mathcal E'/\mathcal E\right)^2} {1-k(r)}dr^2\\ &+\frac{\Phi^2(t,r)}{\mathcal E^2(r,p,q)} (dp^2+dq^2), \end{aligned}$$

where

$$\mathcal E=\frac{(p-P)^2+(q-Q)^2}{2S}+\frac{S}{2},$$

and $S(r)>0$, $P(r)$, and $Q(r)$ control a dipolar variation over each two-sphere. The evolution remains one-dimensional:

$$\dot\Phi^2=\frac{2m(r)}{\Phi}-k(r) +\frac{\Lambda\Phi^2}{3},$$

but the density is angularly dependent,

$$8\pi G\rho= \frac{2\left(m'-3m\mathcal E'/\mathcal E\right)} {\Phi^2\left(\Phi'-\Phi\mathcal E'/\mathcal E\right)}.$$

Constant $S,P,Q$ removes the dipole and recovers LTB after identifying $k=-2E$. More specialized radial functions recover FLRW. Generic Szekeres metrics have no continuous isometries despite being exactly solvable shell by shell. The metrics, function count, and limits are reviewed in Bolejko, Célérier, and Krasiński 2011, §§2–3, Open PDF.

Regularity and shell crossings

At a regular LTB center, a suitable radial gauge gives

$$m=O(r^3), \qquad E=O(r^2), \qquad R(t,r)=O(r),$$

with parity and differentiability conditions on higher coefficients. A shell-focusing singularity occurs when a nontrivial dust shell collapses to $R=0$ or $\Phi=0$ with divergent density or curvature. Zero areal radius alone is not sufficient: the regular center has $R=0$ at every time, with the mass and radius vanishing compatibly. A shell crossing instead occurs when neighboring comoving shells acquire the same areal location: $R'=0$ in LTB, or $\Phi'-\Phi\mathcal E'/\mathcal E=0$ in Szekeres, while the corresponding mass numerator does not vanish compatibly. The density then diverges even though an individual shell radius need not be zero.

Avoidance conditions depend on the evolution branch and domain; monotonic $m$ alone is not sufficient. One must check the denominator throughout the time interval, the center and any extrema, the signature factor, and the density numerator. Hellaby and Lake give branch-sensitive LTB conditions in Hellaby and Lake 1985, pp. 381–387.

The areal radius $R$ or $\Phi$ is geometrically meaningful, but the comoving label $r$ is not. Under $r\mapsto\tilde r(r)$, all free functions compose with the inverse map and their coordinate derivatives change. A comparison of two models should therefore fix a radial gauge—for example $R(t_0,r)=r$ on a regular initial slice—or compare invariant relations such as density versus enclosed mass. Counting three LTB functions without removing this freedom overcounts the data.

These exact geometries do not by themselves establish a realistic cosmology. Light propagation, source evolution, initial-data plausibility, and observational comparison are separate calculations. Radial gauge freedom must also be fixed before interpreting a plotted “radial profile.”

Exercise

Insert the LTB FLRW specialization into the density equation and show that the density is homogeneous.

Solution

$m'=3m_0r^2$, $R=a r$, and $R'=a$. Therefore $8\pi G\rho=2(3m_0r^2)/(a^2r^2a)=6m_0/a^3$, independent of $r$. The evolution equation divided by $a^2r^2$ gives $H^2=2m_0/a^3-k/a^2+\Lambda/3$, the dust Friedmann equation.

References

• Bolejko, Krzysztof, Marie-Noëlle Célérier, and Andrzej Krasiński. “Inhomogeneous Cosmological Models: Exact Solutions and Their Applications.” Classical and Quantum Gravity 28 (2011): 164002. DOI; Open PDF.
• Bondi, Hermann. “Spherically Symmetrical Models in General Relativity.” Monthly Notices of the Royal Astronomical Society 107 (1947): 410–425. DOI.
• Hellaby, Charles, and Kayll Lake. “Shell Crossings and the Tolman Model.” Astrophysical Journal 290 (1985): 381–387; erratum 300 (1986): 461. DOI.
• Szekeres, Peter. “A Class of Inhomogeneous Cosmological Models.” Communications in Mathematical Physics 41 (1975): 55–64. DOI.