The Schwarzschild metric is

$$ ds^2 = -\left(1-\frac{2M}{r}\right)dt^2 + \frac{dr^2}{1 - 2M/r} + r^2(d\theta^2 + sin^2\theta d\phi^2) $$

and to make it isotropic we'd like to get it into the form:

$$ ds^2 = -A^2(r') dt^2 + B^2(r')\left(dr'^2 + r'^2(d\theta^2 + sin^2\theta d\phi^2)\right) $$

This can be done with the coordinate transformation:

$$ r = r'\left(1 + \frac{M}{2r'}\right)^2 $$

Is there a good way to physically interpret this new $r'$ coordinate?

It is no longer related to the circumference in a simple way like $r$ was. Nor does it appear to be simply related to the distance from the spherical mass this space-time is outside of.


2 Answers 2


I can tell you that much that as when JPL and others are calculating the orbits of celestial objects in the solar system they are using the first order of the "post-Newtonian expansion". They take a first order expansion of the Schwarzschild metric in isotropic coordinates from which they get relativistic "correction terms" that they add to the classical Newtonian gravitational acceleration. This is explained in their official documentation, "Formulation for Observed and Computed Values of Deep Space Network Data Types for Navigation"

So basically NASA/JPL interprets the new $r'$ coordinate as an ordinary euclidian distance in order to find relativisic correction terms to add to the classical Newtonian gravitation to account for relativistic effects in the weak fields of our solar system.

This interpretation might introduce errors, but that is what they do.


Is there a good way to physically interpret this new r′ coordinate?

In the regular Schwarzschild metric, the speed of light is not isotropic. To the Schwarzschild observer at 'infinity' the speed of light is not isotropic and lower down in the gravitational field, the speed of light appears to slower to this observer. In flat spacetime a multidirectional flash of light forms a perfect expanding sphere, but to the Schwarzschild observer, the light forms an sort of expanding oblate sphere, more like an egg shape. For example if a flash occurs at $r = 5 R_s$, it reaches $6 R_s$ before it reaches $4 R_s$ according to this distant observer.

Isotropic coordinates seek to correct this anisotropic behaviour of light in the Schwarzschild metric, by defining a new r coordinate that allows a light flash to be measured as expanding in a perfect sphere, just like in flat space, from the point of view of the distant observer.

In the regular Schwarzschild metric, the radar distance measured by sending a signal going down from $r=5 R_s$ to $4 R_s$ and back up to $5 R_s$ is longer than that measured for a signal going up to $6 R_s$ and back down to $5 R_s$. In the new coordinate system a local observer measures equal radar distances up and down.

However, the new system is not completely perfect. (You knew that was coming, right?) A local observer at say $4 R_s$ will measure a shorter radar distance from $4 R_s$ to $5 R_s$ than a local observer at $5 R_s$ measures for the distance from $5 R_s$ to $4 R_s$.


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