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Tetrads in General Relativity

IX. Spherically Symmetric Spacetimes

PG Coordinates

   The PG-metric, independently found by Paul Painlevé and Allvar Gullstrand in 1921 by solving the vacuum Einstein equations, is given by a line element with a non-diagonal term

9.6

\[d{s^2} = d{t^2} - {\left( {dr + \sqrt {2M/r} dt} \right)^2} - {r^2}d{\Omega ^2}\]

This expression obtains from the general metric (9.2) with the parameter choice

9.7

\[\begin{gathered} &{f_1} = 1{\qquad}{f_2} = 0{\qquad}\quad \to {\quad}f_1^2 - f_2^2 = 1 \hfill \\ &{g_1} = 1{\quad}{g_2} = - \sqrt {2M/r} {\quad} \to {\quad}g_1^2 - g_2^2 = {e^{2\Phi }} \hfill \\ \end{gathered} \]

The PG-metric may be understood as a coordinate transformation of the Schwarzschild metric as shown by Georges Lemaître in 1933. [ Wikipedia: Gullstrand–Painlevé coordinates].

   A striking property of the PG-metric (9.6) is that spacelike surfaces $t = {\text{const }}$, $ dt = 0$, are intrinsically flat, i.e. they have the metric of a Euclidean three-dimensional space in spherical polar coordinates:

9.8

\[ - d{s^2} = d{r^2} + {r^2}(d{\theta ^2} + {\sin ^2}\theta d{\phi ^2})\]

Moreover, these space-like hypersurfaces are orthogonal to the time direction:

9.9

\[{{\mathbf{e}}_t} \cdot {{\mathbf{e}}_r} = {g_{tr}} - \sqrt {\frac{{2M}}{r}} {g_{rr}} = 0\]

The information about the spacetime curvature is entirely encoded in the off-diagonal component of the metric tensor.

   The PG-metric has advantages over the Schwarzschild metric. For one, in PG-coordinates the Schwarzschild metric is regular on the whole domain $r > 0$. So, it reveals that the Schwarzschild radius is a mere coordinate singularity that can be removed by a new choice of coordinates. Only the origin of the spherical coordinates is a real singularity. Moreover, the PG-coordinates are key to a simple physical picture of black holes and the gravitational collapse of stars. Close to the central mass, the spacetime (9.6) describes a black hole with an event horizon at the Schwarzschild radius, while at large distance it reproduces the results of Newtonian gravity.