\( \newcommand{\be}{\begin{equation}}
\newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}}
\newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}}
\newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}}
\newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label}
\newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}}
\newcommand{\third}{{\scriptstyle \frac{1}{3}}}
\newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm
#2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter]
\newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)
VIII. Schwarzschild solution
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.
- The inclination from the axis $\theta $, and the azimuthal
angle $\phi $ around the axis are the usual polar angles subtended
at the origin.
- 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:
\[ l: = {\int {{e^\lambda }dr} ^{}}
\simeq \int\limits_{{r_1}}^{{r_2}} {\left( {1 + \frac{M}{r}} \right)} dr
\simeq {r_2} - {r_1} + M\log \frac{{{r_1}}}{{{r_2}}} \]
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.