Lemaître–Tolman metric

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In mathematical physics, the Lemaître–Tolman metric is the spherically symmetric dust solution of Einstein's field equations. It was first found by Lemaître in 1933 and Tolman in 1934 and later investigated by Bondi in 1947. This solution describes a spherical cloud of dust (finite or infinite) that is expanding or collapsing under gravity. It is also known as the Lemaître–Tolman–Bondi metric and the Tolman metric.

Details

The metric is:

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

where:

${\displaystyle \mathrm {d} \Omega ^{2}=\mathrm {d} \theta ^{2}+\sin ^{2}\theta \,\mathrm {d} \phi ^{2}}$
${\displaystyle R=R(t,r)~,~~~~~~~~R'=\partial R/\partial r~,~~~~~~~~E=E(r)>-{\frac {1}{2}}}$

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

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

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

The pressure is zero (hence dust), the density is

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

and the evolution equation is

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

where

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

The evolution equation has three solutions, depending on the sign of ${\displaystyle E}$,

${\displaystyle E>0:~~~~~~~~R={\frac {M}{2E}}(\cosh \eta -1)~,~~~~~~~~(\sinh \eta -\eta )={\frac {(2E)^{3/2}(t-t_{B})}{M}}~;}$
${\displaystyle E=0:~~~~~~~~R=\left({\frac {9M(t-t_{B})^{2}}{2}}\right)^{1/3}~;}$
${\displaystyle E<0:~~~~~~~~R={\frac {M}{2E}}(1-\cos \eta )~,~~~~~~~~(\eta -\sin \eta )={\frac {(-2E)^{3/2}(t-t_{B})}{M}}~;}$

which are known as hyperbolic, parabolic, and elliptic evolutions respectively.

The meanings of the three arbitrary functions, which depend on ${\displaystyle r}$ only, are:

• ${\displaystyle E(r)}$ – both a local geometry parameter, and the energy per unit mass of the dust particles at comoving coordinate radius ${\displaystyle r}$,
• ${\displaystyle M(r)}$ – the gravitational mass within the comoving sphere at radius ${\displaystyle r}$,
• ${\displaystyle t_{B}(r)}$ – the time of the big bang for worldlines at radius ${\displaystyle r}$.

Special cases are the Schwarzschild metric in geodesic coordinates ${\displaystyle M=}$ constant, and the Friedmann–Lemaître–Robertson–Walker metric, e.g. ${\displaystyle E=0~,~~t_{B}=}$ constant for the flat case.