Lemaître–Tolman metric


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

Details

The metric is:
where:
The matter is comoving, which means its 4-velocity is:
so the spatial coordinates are attached to the particles of dust.
The pressure is zero, the density is
and the evolution equation is
where
The evolution equation has three solutions, depending on the sign of,
which are known as hyperbolic, parabolic, and elliptic evolutions respectively.
The meanings of the three arbitrary functions, which depend on only, are:
Special cases are the Schwarzschild metric in geodesic coordinates
constant, and the Friedmann–Lemaître–Robertson–Walker metric, e.g. constant for the flat case.