Lemaître coordinates

Lemaître coordinates are a particular set of coordinates for the Schwarzschild metric – a spherically symmetric solution to the Einstein field equations in a vacuum – obtained by Monsignor Georges Lemaître in 1932. Changing from Schwarzschild to Lemaître coordinates removes the coordinate singularity at the Schwarzschild radius.

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

where

This metric has a coordinate singularity at the gravitational radius ${\displaystyle r=r_{g}}$.

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

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