- #1
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- TL;DR Summary
- What is the appropriate coordinate change?
According to Schutz, the line element for large r in Schwarzschild is
$$ ds^2 \approx - ( 1 - \frac {2M} {r}) dt^2 + (1 + \frac {2M} {r}) dr^2 + r^2 d\Omega^2 $$
and one can find coordinates (x, y, z) such that this becomes
$$ ds^2 \approx - ( 1 - \frac {2M} {R}) dt^2 + (1 + \frac {2M} {R}) (dx^2+dy^2+dz^2) $$
where ## R \equiv (x^2 + y^2 + z^2)^{1/2} ##
This makes sense, as it is the same as the weak field metric with ## \phi = -M/r ## (Newtonian gravitational field).
However, if I try to go from the first to the second using the usual spherical to Cartesian transformation, I get something different. Am I missing a trick here?
$$ ds^2 \approx - ( 1 - \frac {2M} {r}) dt^2 + (1 + \frac {2M} {r}) dr^2 + r^2 d\Omega^2 $$
and one can find coordinates (x, y, z) such that this becomes
$$ ds^2 \approx - ( 1 - \frac {2M} {R}) dt^2 + (1 + \frac {2M} {R}) (dx^2+dy^2+dz^2) $$
where ## R \equiv (x^2 + y^2 + z^2)^{1/2} ##
This makes sense, as it is the same as the weak field metric with ## \phi = -M/r ## (Newtonian gravitational field).
However, if I try to go from the first to the second using the usual spherical to Cartesian transformation, I get something different. Am I missing a trick here?