In physics, the Lemaître–Tolman metric, also known as the Lemaître–Tolman–Bondi metric or the Tolman metric, is a Lorentzian metric based on an exact solution of Einstein's field equations; it describes an isotropic and expanding (or contracting) universe which is not homogeneous,[1][2] and is thus used in cosmology as an alternative to the standard Friedmann–Lemaître–Robertson–Walker metric to model the expansion of the universe.
[3][4][5] It has also been used to model a universe which has a fractal distribution of matter to explain the accelerating expansion of the universe.
[6] It was first found by Georges Lemaître in 1933[7] and Richard Tolman in 1934[1] and later investigated by Hermann Bondi in 1947.
[8] In a synchronous reference system where
and clocks at all points can be synchronized.
For a dust-like medium where the pressure is zero, dust particles move freely i.e., along the geodesics and thus the synchronous frame is also a comoving frame wherein the components of four velocity
The solution of the field equations yield[9] where
is the radius or luminosity distance in the sense that the surface area of a sphere with radius
is just interpreted as the Lagrangian coordinate and subjected to the conditions
is the matter density and finally primes denote differentiation with respect to
that excludes cases resulting in crossing of material particles during its motion.
and its time derivative respectively provides its law of motion and radial velocity.
An interesting property of the solution described above is that when
, the form of these functions plotted for the range
This prediction is evidently similar to the Newtonian theory.
The total mass within the sphere
is given by which implies that Schwarzschild radius is given by
can be obtained upon integration and is given in a parametric form with a parameter
emerges as another arbitrary function.
However, we know that centrally symmetric matter distribution can be described by at most two functions, namely their density distribution and the radial velocity of the matter.
In fact, since no particular selection has been made for the Lagrangian coordinate
[10] For the dust-like medium, there exists another solution where
, although such solution does not correspond to collapse of a finite body of matter.
and therefore the solution corresponds to empty space with a point mass located at the center.
The gravitational collapse occurs when
corresponds to the arrival of matter denoted by its Lagrangian coordinate
, the asymptotic behaviors are given by in which the first two relations indicate that in the comoving frame, all radial distances tend to infinity and tangential distances approaches zero like
, whereas the third relation shows that the matter density increases like
constant where the time of collapse of all the material particle is the same, the asymptotic behaviors are different, Here both the tangential and radial distances goes to zero like
, whereas the matter density increases like