Lemaître–Tolman metric
In a synchronous reference system where formula_1 and formula_2, the time coordinate formula_3 (we set formula_4) is also the proper time formula_5 and clocks at all points can be synchronized. A comoving frame is appropriate to describe the matter that is collapsing or expanding. For a dust-like medium where the pressure is zero, one can always choose a comoving synchronous coordinates with which the metric is described by
 formula_6
where formula_7 is the "radius" or "luminosity distance" in the sense that the surface area of a sphere with radius formula_7 is formula_9 and
formula_10
The function formula_11 is given in a parametric form with a parameter formula_12 and two functions formula_13 and formula_14 and has three possibilities,
 formula_15
 formula_16
 formula_17
The function formula_18 is related to the matter density formula_19 as follows
formula_20
The above description provides the general solution of the spherically collapsing or expanding dust sphere. Of the three functions formula_21, only two are independent because the coordinate formula_22 can still be subjected to arbitrary transformations. Also, because the centrally symmetric matter distribution is described by at most two functions, namely their density distribution and the radial velocity of the matter. To each formula_22, there corresponds a matter since it is comoving with the frame. Similar to the Newtonian theory, the solution indicates that the property inside a sphere of radius formula_7 is completely determined by the matter within it. The total mass within formula_25 and to some finite value formula_26 is given by
formula_27
which implies that formula_28 is the Schwarzschild radius. When formula_29const., formula_30 and therefore the solution corresponds to empty space with a point mass located at the center. Further by setting formula_31 and formula_32, the solution reduces to Schwarzschild solution expressed in Lemaître coordinates. The gravitational collapse occurs when formula_33 reaches formula_34 with formula_35. The moment formula_36 corresponds to the arrival of matter denoted by its Lagrangian coordinate formula_22 to the center. In all three cases, as formula_38, the asymptotic behaviour is given by
formula_39
indicating that in the comoving frame, all radial distances tend to infinity, whereas tangential distances approaches zero like formula_40. The matter density increases like
formula_41
In the special case formula_42constant where the time of collapse of all the material particle is the same, the asymptotic behaviours are different,
formula_43
Here both the tangential and radial distances goes to zero like formula_44, whereas the matter density increases like formula_45
Details
In mathematical physics, the Lemaître–Tolman metric is the spherically symmetric dust solution of Einstein's field equations. This solution describes a spherical cloud of dust (finite or infinite) that is expanding or collapsing under gravity.