Lemaître coordinates are a particular set of coordinates for the Schwarzschild metric—a spherically symmetric solution to the Einstein field equations in vacuum—introduced by Georges Lemaître in 1932.
[1] 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 Schwarzschild radius
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
(the numerator and denominator are switched inside the square-roots), leads to the Lemaître coordinate expression of the metric, where The metric in Lemaître coordinates is non-singular at the Schwarzschild radius
There remains a genuine gravitational singularity at the center, where
, which cannot be removed by a coordinate change.
The other three: the radial and angular coordinates
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
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.
gives the proper time for radially infalling observers; it does not give the proper time for observers traveling along other geodesics.
The trajectories with ρ constant are timelike geodesics with τ the proper time along these geodesics.
They represent the motion of freely falling particles which start out with zero velocity at infinity.
The Lemaître coordinate system is synchronous, that is, the global time coordinate of the metric defines the proper time of co-moving observers.
The radially falling bodies reach the Schwarzschild radius and the centre within finite proper time.
Radial null geodesics correspond to
d τ = ± β d ρ
is just a short-hand for The two signs correspond to outward-moving and inward-moving light rays, respectively.
Re-expressing this in terms of the coordinate
This is interpreted as saying that no signal can escape from inside the Schwarzschild radius, with light rays emitted radially either inwards or outwards both end up at the origin as the proper time
The Lemaître coordinate chart is not geodesically complete.
This can be seen by tracing outward-moving radial null geodesics backwards in time.
The outward-moving geodesics correspond to the plus sign in the above.
Selecting a starting point
and integrating forward, one arrives at
Thus, one concludes that, although the metric is non-singular at
, all outward-traveling geodesics extend to