LTB dusts

The spherically symmetric dust solution of Einstein's field equations was found by Lemaitre in 1933 and Richard Tolman in 1934. It was again investigated by Bondi in 1947. This solution describes a spherical cloud of dust (finite or infinite) that is expanding or collapsing under gravity.

The metric is:

$\mathrm {d} s^{2}=-\mathrm {d} t^{2}-{\frac {(R')^{2}}{1-2E}}\mathrm {d} r^{2}-R^{2}\,\mathrm {d} \Omega ^{2}$

where:

$\mathrm {d} \Omega ^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}\theta \,\mathrm {d} \phi$

$R=R(t,r)~,~~~~~~~~R'=\partial R/\partial r~,~~~~~~~~E=E(r)$

The matter is comoving, which means its 4-velocity is:

$u^{a}=\delta _{0}^{a}=(1,0,0,0)$

so the spatial coordinates $(r,\theta ,\phi )$ are attached to the particles of dust.

The pressure is zero (hencedust), the density is

$8\pi \rho ={\frac {2M'}{R^{2}\,r'}}$

and the evolution equation is

${\dot {R}}^{2}={\frac {2M}{R}}+2E$

where

${\dot {R}}=\partial R/\partial t$

The evolution equation has three solutions, depending on the sign of $E$,

$E>0:~~~~~~~~R={\frac {M}{2E}}(\cosh \eta -1)~,~~~~~~~~(\sinh \eta -\eta )={\frac {(2E)^{3/2}(t-t_{B})}{M}}~;$

$E=0:~~~~~~~~R=\left({\frac {9M(t-a)^{2}}{2}}\right)^{1/3}~;$

$E>0:~~~~~~~~R={\frac {M}{2E}}(1-\cos \eta )~,~~~~~~~~(\eta -\sin \eta )={\frac {(-2E)^{3/2}(t-t_{B})}{M}}~;$

which are known as hyperbolic, Parabolic, and Elliptic evolutions respectively.

The meanings of the three arbitrary functions, which depend on $r$ only, are $E(r)$ - both a local geometry parameter, and the energy per unit mass of the dust particles at comoving coordinate radius $r$, $M(r)$ - the gravitational mass withing the comoving sphere at radius $r$, $t_{B}(r)$ - the time of the big bang for worldlines at radius $r$

It contains the Schwarzschild metric and the Robertson-Walker metric as special cases.

References

• Lemaitre, G., Ann. Soc. Sci. Bruxelles, A53, 51 (1933).
• Tolman, R.C., Proc. Natl. Acad. Sci. 20, 169 (1934).
• Bondi, H., Mon. Not. R. Astron. Soc. 107, 410 (1947).
• Krasinski, A., Inhomogeneous Cosmological Models, (1997) Cambridge UP, ISBN 0 521 48180 5