Schwarzschild Metric - Alternative (isotropic) Formulations of The Schwarzschild Metric

Alternative (isotropic) Formulations of The Schwarzschild Metric

The original form of the Schwarzschild metric involves anisotropic coordinates, in terms of which the velocity of light is not the same for the radial and transverse directions (pointed out by A S Eddington). Eddington gave alternative formulations of the Schwarzschild metric in terms of isotropic coordinates (provided r ≥ 2GM/c2 ).

In isotropic spherical coordinates, one uses a different radial coordinate, r1, instead of r. They are related by

Using r1, the metric is

c^2 {d \tau}^{2} = \frac{(1-\frac{GM}{2c^2 r_1})^{2}}{(1+\frac{GM}{2c^2 r_1})^{2}} \, c^2 {d t}^2 - \left(1+\frac{GM}{2c^2 r_1}\right)^{4}\left(dr_1^2 + r_1^2 d\theta^2 + r_1^2 \sin^2\theta \, d\varphi^2\right) \,.

For isotropic rectangular coordinates x, y, z, where

and

the metric then becomes

c^2 {d \tau}^{2} = \frac{(1-\frac{GM}{2c^2 r_1})^{2}}{(1+\frac{GM}{2c^2 r_1})^{2}} \, c^2 {d t}^2 - \left(1+\frac{GM}{2c^2 r_1}\right)^{4}(dx^2+dy^2+dz^2) \,.

In terms of these coordinates, the velocity of light at any point is the same in all directions, but it varies with radial distance r1 (from the point mass at the origin of coordinates), where it has the value

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