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Tetrads in General Relativity

VIII. Schwarzschild Solution

Schwarzschild Coordinates

   The coordinates ${\text{\{ }}{x^\mu }\} : = \{ t,r,\theta ,\phi \} $ of the Schwarzschild metric, or any spherically symmetric metric like (8.1), are simply related to chronometric measurements and observations.

  1. The inclination from the axis $\theta $, and the azimuthal angle $\phi $ around the axis are the usual polar angles subtended at the origin.
  2. The coordinate $r$ is the radial distance from the origin. However, in a curved spherically symmetric space, the area of curved successive parallel spheres will not necessarily increase as the square of the distance. The radial Schwarzschild coordinate is defined in such a way that the area of the sphere at radial coordinate $r$ is equal to $4\pi r$, as in flat space, thus, defining the coordinate $r$ as the area distance.

   The ruler distance $l$, i.e. the distance along the shortest geodesic joining two points obtained from the metric (8.1), is different from $r$ in curved space:

8.3

\[  l: = {\int {{e^\lambda }dr} ^{}} \simeq \int\limits_{{r_1}}^{{r_2}} {\left( {1 + \frac{M}{r}} \right)} dr \simeq {r_2} - {r_1} + M\log \frac{{{r_1}}}{{{r_2}}}  \]

In the sun’s gravitational field, the excess of ruler distance over coordinate distance from the sun’s surface to earth is only about 8 km, a discrepancy of one part in $2 \times {10^7}$ .

   In the astronomical praxis other definitions and methods of determining distances are used. These include radar distance, distance from apparent size or luminosity, and distance by parallax, all leading to (slightly) different outcomes.