Lemaître coordinates

Metric

The original Schwarzschild coordinate expression of the Schwarzschild metric, in natural units (c = G = 1), is given as

${\displaystyle d{s}^2=\left(1-\frac{r_s}{r}\right)d{t}^2-\frac{d{r}^2}{1-\frac{r_s}{r}}-r^2\left(d{\theta }^2+{\sin }^2\theta d{\varphi }^2\right),}$

where

${\displaystyle d{s}^2}$ is the invariant interval;

${\displaystyle r_s=\frac{2GM}{c^2}}$ is the Schwarzschild radius;

${\displaystyle M}$ is the mass of the central body;

${\displaystyle t,r,\theta ,\varphi }$ are the Schwarzschild coordinates (which asymptotically turn into the flat spherical coordinates);

${\displaystyle c}$ is the speed of light;

and ${\displaystyle G}$ is the gravitational constant.

This metric has a coordinate singularity at the Schwarzschild radius ${\displaystyle r=r_s}$.

Georges Lemaître 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 Schwarzschild radius. Indeed, inside the Schwarzschild radius everything falls towards the centre and it is impossible for a physical body to keep a constant radius.

A transformation of the Schwarzschild coordinate system from ${\displaystyle \{t,r\}}$ to the new coordinates ${\displaystyle \{\tau ,\rho \},}$

${\displaystyle \begin{array}{c}d\tau =dt+\sqrt{\frac{r_s}{r}}{\left(1-\frac{r_s}{r}\right)}^{-1}dr\\ d\rho =dt+\sqrt{\frac{r}{r_s}}{\left(1-\frac{r_s}{r}\right)}^{-1}dr\end{array}}$

(the numerator and denominator are switched inside the square-roots), leads to the Lemaître coordinate expression of the metric,

${\displaystyle d{s}^2=d{\tau }^2-\frac{r_s}{r}d{\rho }^2-r^2(d{\theta }^2+{\sin }^2\theta d{\varphi }^2)}$

where

${\displaystyle r={\left[\frac{3}{2}(\rho -\tau )\right]}^{2/3}{r}_s^{1/3}.}$

The metric in Lemaître coordinates is non-singular at the Schwarzschild radius ${\displaystyle r=r_s}$. This corresponds to the point ${\displaystyle \frac{3}{2}(\rho -\tau )=r_s}$. There remains a genuine gravitational singularity at the center, where ${\displaystyle \rho -\tau =0}$, which cannot be removed by a coordinate change.

The time coordinate used in the Lemaître coordinates is identical to the "raindrop" time coordinate used in the Gullstrand–Painlevé coordinates. The other three: the radial and angular coordinates ${\displaystyle r,\theta ,\varphi }$ of the Gullstrand–Painlevé coordinates are identical to those of the Schwarzschild chart. That is, Gullstrand–Painlevé applies one coordinate transform to go from the Schwarzschild time ${\displaystyle t}$ to the raindrop coordinate ${\displaystyle t_r=\tau }$. Then Lemaître applies a second coordinate transform to the radial component, so as to get rid of the off-diagonal entry in the Gullstrand–Painlevé chart.

The notation ${\displaystyle \tau }$ used in this article for the time coordinate should not be confused with the proper time. It is true that ${\displaystyle \tau }$ gives the proper time for radially infalling observers; it does not give the proper time for observers traveling along other geodesics.