\( \newcommand{\be}{\begin{equation}} \newcommand{\ee}{\end{equation}} \newcommand{\ba}{\begin{array}} \newcommand{\ea}{\end{array}} \newcommand{\bea}{\begin{eqnarray}} \newcommand{\eea}{\end{eqnarray}} \newcommand{\bean}{\begin{eqnarray*}} \newcommand{\eean}{\end{eqnarray*}} \newcommand{\la}{\label} \newcommand{\nn}{\nonumber} \newcommand{\half}{{\scriptstyle \frac{1}{2}}} \newcommand{\third}{{\scriptstyle \frac{1}{3}}} \newcommand{\bli}[2]{\begin{list}{#1}{\itemsep=0.0cm \topsep=0.0cm \partopsep=0.0cm #2}} \newcommand{\eli}{\end{list}} \newtheorem{problem}{Problem}[chapter] \newcommand{\bprob}{\begin{problem}} \newcommand{\eprob}{\end{problem}}\)

Tetrads in General Relativity

VIII. Schwarzschild solution

Red-shift factor

   To determine the physical meaning of the time coordinate, one may consider the line element (8.1) for $r,\theta ,\phi = {\text{const}}$ defining the proper time

8.4

\[  d{\tau ^2} = {e^{2\Phi }}d{t^2}  \]

Appearing here is the gravitational time-dilation factor $\exp - \Phi $ predicted by the equivalence principle. [MTW]

   Physically, this factor may be understood from the photon character of light: the work done by a gravitational field with potential $\Phi $ on a particle of gravitational mass $\varepsilon = h\nu $, as it traverses a potential difference, must equal to the gain in the particle’s energy. Integrating this difference over a finite path from A to B, one finds the gravitational red-shift formula

8.5

\[  \frac{{{\nu _B}}}{{{\nu _A}}} = \sqrt {\frac{{{g_{tt}}(A)}}{{{g_{tt}}(B)}}} = \frac{{{e^{ - {\Phi _B}}}}}{{{e^{ - {\Phi _A}}}}} = {\left( {\frac{{1 - 2M/{r_A}}}{{1 - 2M/{r_B}}}} \right)^{1/2}}  \]

which is an exact result for a stationary metric. It also gives the gravitational time dilation. This effect is observable on earth and has, for example, to be taken into account to ensure accurate operation of the Global Positioning System (GPS). A much greater effect is observed in the absorption lines from the surface of white dwarfs.

   Note that the above argument only determines ${g_{tt}}$, just one component of the metric tensor. This metric component, as well as the other components of the metric tensor, were obtained by Schwarzschild by solving the Einstein equations.