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 (finite or infinite) 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 formula_5 are attached to the particles  of dust. The pressure is zero (hence "dust"), the density is and the evolution equation is where The evolution equation has three solutions, depending on the sign of formula_9, which are known as "hyperbolic", "parabolic", and "elliptic" evolutions respectively. The meanings of the three arbitrary functions, which depend on formula_13 only, are: Special cases are the Schwarzschild metric in geodesic coordinates formula_20 constant (setting formula_21 leads to Schwarzschild metric in Novikov coordinates, while setting formula_22 leads to Schwarzschild metric in Lemaître coordinates), and the Friedmann–Lemaître–Robertson–Walker metric, e.g. formula_23 constant for the flat case. Lemaître–Tolman metric In