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, is given as
2=\left(1-{r | |
ds | |
s\over |
r}\right)dt2-{dr2\over1-{rs\overr}}-r2\left(d\theta2+\sin2\thetad\phi2\right) ,
ds2
r | ||||
|
M
t,r,\theta,\phi
c
and
G
This metric has a coordinate singularity at the Schwarzschild radius
r=rs
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
\{t,r\}
\{\tau,\rho\},
\begin{align} d\tau=dt+\sqrt{
rs | |
r |
ds2=d\tau2-
rs | |
r |
d\rho2-r2(d\theta2+\sin2\theta d\phi2)
r=\left[ | 3 |
2 |
(\rho-\tau)\right]2/3
1/3 | |
r | |
s |
.
The metric in Lemaître coordinates is non-singular at the Schwarzschild radius
r=rs
3 | |
2 |
(\rho-\tau)=rs
\rho-\tau=0
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
r,\theta,\phi
t
tr=\tau
The notation
\tau
\tau
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. At any point their speed is just equal to the escape velocity from that point.
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
ds2=0
d\tau=\pm\betad\rho
\beta
\beta\equiv\beta(r)=\sqrt{rs\overr}
The two signs correspond to outward-moving and inward-moving light rays, respectively. Re-expressing this in terms of the coordinate
r
dr=\left(\pm1-\sqrt{rs\overr}\right)d\tau
dr<0
r<rs
\tau
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
r>rs
\tau=0
r\to+infty
\tau\to+infty
r\tors
\tau\to-infty
r<rs
r=0
r\tors
\tau\to-infty
r=rs
r=rs
\tau\to-infty