In physics, theLemaître–Tolman metric, also known as theLemaître–Tolman–Bondi metric or theTolman metric, is aLorentzian metric based on an exact solution ofEinstein's field equations; it describes anisotropic andexpanding (or contracting)universe which is nothomogeneous,^[1][2] and is thus used incosmology as an alternative to the standardFriedmann–Lemaître–Robertson–Walker metric to model theexpansion of the universe.^[3][4][5] It has also been used to model a universe which has afractal distribution of matter to explain theaccelerating expansion of the universe.^[6] It was first found byGeorges Lemaître in 1933^[7] andRichard Tolman in 1934^[1] and later investigated byHermann Bondi in 1947.^[8]

In asynchronous reference system where${\displaystyle g_{00}=1}$ and${\displaystyle g_{0\alpha }=0}$, the time coordinate${\displaystyle x^0=t}$ (we set${\displaystyle G=c=1}$) is also theproper time${\displaystyle \tau =\sqrt{g_{00}}{x}^0}$ 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${\displaystyle u^i=d{x}^i/ds}$ are${\displaystyle u^0=1,u^{\alpha }=0}$. The solution of the field equations yield^[9]

${\displaystyle d{s}^2=d{\tau }^2-e^{\lambda (\tau ,R)}d{R}^2-r^2(\tau ,R)(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)}$

where${\displaystyle r}$ is theradius orluminosity distance in the sense that the surface area of a sphere with radius${\displaystyle r}$ is${\displaystyle 4\pi r^2}$ and${\displaystyle R}$ is just interpreted as theLagrangian coordinate and

${\displaystyle e^{\lambda }=\frac{r^{\prime 2}}{1+f(R)},{\left(\frac{\mathrm{\partial }r}{\mathrm{\partial }\tau }\right)}^2=f(R)+\frac{F(R)}{r},4\pi r^2\rho =\frac{F^{\prime }(R)}{2r^{\prime }}}$

subjected to the conditions${\displaystyle 1+f>0}$ and${\displaystyle F>0}$, where${\displaystyle f(R)}$ and${\displaystyle F(R)}$ are arbitrary functions,${\displaystyle \rho }$ is the matter density and finally primes denote differentiation with respect to${\displaystyle R}$. We can also assume${\displaystyle F^{\prime }>0}$ and${\displaystyle r^{\prime }>0}$ that excludes cases resulting in crossing of material particles during its motion. To each particle there corresponds a value of${\displaystyle R}$, the function${\displaystyle r(\tau ,R)}$ and its time derivative respectively provides its law of motion and radial velocity. An interesting property of the solution described above is that when${\displaystyle f(R)}$ and${\displaystyle F(R)}$ are plotted as functions of${\displaystyle R}$, the form of these functions plotted for the range${\displaystyle R\in [0,R_0]}$ is independent of how these functions will be plotted for${\displaystyle R>R_0}$. This prediction is evidently similar to the Newtonian theory. The total mass within the sphere${\displaystyle R=R_0}$ is given by

${\displaystyle m=4\pi {\int }_0^{r(\tau ,R_0)}\rho r^{2}dr=4\pi {\int }_0^{R_0}\rho r^{\prime }{r}^{2}dR=\frac{F(R_0)}{2}}$

which implies thatSchwarzschild radius is given by${\displaystyle r_s=2m=F(R_0)}$.

The function${\displaystyle r(\tau ,R)}$ can be obtained upon integration and is given in a parametric form with a parameter${\displaystyle \eta }$ with three possibilities,

${\displaystyle f>0:r=\frac{F}{2f}(\cosh \eta -1),{\tau }_0-\tau =\frac{F}{2f^{3/2}}(\sinh \eta -\eta ),}$

${\displaystyle f=0:r={\left(\frac{9F}{4}\right)}^{1/3}({\tau }_0-\tau {)}^{2/3}.}$

where${\displaystyle {\tau }_0(R)}$ 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. This means that of the three functions${\displaystyle f,F,{\tau }_0}$, only two are independent. In fact, since no particular selection has been made for the Lagrangian coordinate${\displaystyle R}$ yet that can be subjected to arbitrary transformation, we can see that only two functions are arbitrary.^[10] For the dust-like medium, there exists another solution where${\displaystyle r=r(\tau )}$ and independent of${\displaystyle R}$, although such solution does not correspond to collapse of a finite body of matter.^[11]

When${\displaystyle F=r_s=}$const.,${\displaystyle \rho =0}$ and therefore the solution corresponds to empty space with a point mass located at the center. Further by setting${\displaystyle f=0}$ and${\displaystyle {\tau }_0=R}$, the solution reduces toSchwarzschild solution expressed inLemaître coordinates.

The gravitational collapse occurs when${\displaystyle \tau }$ reaches${\displaystyle {\tau }_0(R)}$ with${\displaystyle {\tau }_0^{\prime }>0}$. The moment${\displaystyle \tau ={\tau }_0(R)}$ corresponds to the arrival of matter denoted by its Lagrangian coordinate${\displaystyle R}$ to the center. In all three cases, as${\displaystyle \tau \to {\tau }_0(R)}$, the asymptotic behaviors are given by

${\displaystyle r\approx {\left(\frac{9F}{4}\right)}^{1/3}({\tau }_0-\tau {)}^{2/3},e^{\lambda /2}\approx {\left(\frac{2F}{3}\right)}^{1/3}\frac{{\tau }_0^{\prime }}{\sqrt{1+f}}({\tau }_0-\tau {)}^{-1/3},4\pi \rho \approx \frac{F^{\prime }}{3F{\tau }_0^{\prime }({\tau }_0-\tau )}}$

in which the first two relations indicate that in the comoving frame, all radial distances tend to infinity and tangential distances approaches zero like${\displaystyle \tau -{\tau }_0}$, whereas the third relation shows that the matter density increases like${\displaystyle 1/({\tau }_0-\tau ).}$ In the special case${\displaystyle {\tau }_0(R)=}$constant where the time of collapse of all the material particle is the same, the asymptotic behaviors are different,

${\displaystyle r\approx {\left(\frac{9F}{3}\right)}^{1/3}({\tau }_0-\tau {)}^{2/3},e^{\lambda /2}\approx {\left(\frac{2}{3}\right)}^{1/3}\frac{F^{\prime }}{2F^{2/3}\sqrt{1+f}}({\tau }_0-\tau {)}^{2/3},4\pi \rho \approx \frac{2}{3({\tau }_0-\tau {)}^2}.}$

Here both the tangential and radial distances goes to zero like${\displaystyle ({\tau }_0-\tau {)}^{2/3}}$, whereas the matter density increases like${\displaystyle 1/({\tau }_0-\tau {)}^2.}$

• Lemaître coordinates
• Introduction to the mathematics of general relativity
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• Metric tensor (general relativity)
• Relativistic angular momentum
• inhomogeneous cosmology

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