Lemaitre metric

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Lemaitre metric is a spherically symmetric solution to vacuum Einstein equation apparently^[1]^[2] obtained by Georges Lemaître in 1938 by a coordinate transformation which removed the singularity at the gravitational radius in the Schwarzschild metric.

The Schwarzschild metric in units $c = G = 1$ is given as

$ds^2=(1-{r_g\over r})dt^2-{dr^2\over 1-{r_g\over r}} - r^2(d\theta^2+\sin^2\theta d\phi^2) \;,$

where $d s 2$ is the invariant interval, $r g = 2 M$ is the gravitational radius, $M$ is the mass of the central body, and $t, r,θ,φ$ are the Schwarzschild coordinates which asymptotically turn into the flat spherical coordinates. This metric has an apparent singularity at the gravitational radius $r = r g$ .

Georges Lemaitre was the first to show that this is not a real physical singularity but simply a manifestation of the fact that the static Schwarzschild coordinates cannot be realized with material bodies inside the gravitational radius. Indeed inside the gravitational radius everything falls towards the centre and it is impossible for a physical body to keep a constant radius.

A transformation in the Schwarzschild metric from ${t, r}$ to new coordinates ${τ,ρ}$ ,

$d\tau = dt + \sqrt{\frac{r_{g}}{r}}\frac{1}{(1-\frac{r_{g}}{r})}dr~,~~ d\rho = dt + \sqrt{\frac{r}{r_{g}}}\frac{1}{(1-\frac{r_{g}}{r})}dr$

leads to the Lemaitre metric,

$ds^{2} = d\tau^{2} - \frac{r_{g}}{r} d\rho^{2} - r^{2}(d\theta^{2} +\sin^{2}\theta d\phi^{2})$

where

$r=[\frac{3}{2}(\rho-\tau)]^{2/3}r_{g}^{1/3} \;.$

In Lemaitre coordinates there is no singularity at the gravitational radius, which corresponds to the point $\frac{3}{2}(\rho-\tau)=r_g$ . However there is a genuine singularity at the centrum, where $ρ - τ = 0$ .

The Lemaitre metric is synchronous, that is the bodies which are at rest in Lemaitre coordinates are actually free falling in the gravitational field of the central mass. The radially falling bodies reach the gravitational radius and the centre within finite proper time.

Along the tragectory of a radial light ray

$dr=\left(\pm 1 - \sqrt{r_g\over r}\right)d\tau ,$

therefore no signal can escape from inside the Schwarzschild radius, where always $d r < 0$ and the light rays emitted radially inwards and outwards both end up at the origin.

[edit] References

^ L.D.Landau and E.M.Lifshitz Course of Theoretical Physics, vol. 2: "The Classical Theory of Fields".
^ Andre Gsponer, More on the early interpretation of the Schwarzschild solution, physics/0408100