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Tetrads in General Relativity

IX. Spherically Symmetric Spacetimes

PG Observers

   The substitution of the parameters (9.7) into (9.4) gives the PG connection bivectors

9.10

\[\begin{gathered} {{\mathbf{\omega }}_t} = - \frac{M}{{{r^2}}}{{\mathbf{e}}_t} \wedge {{\mathbf{e}}_r} {\qquad \qquad} {{\mathbf{\omega }}_\theta } = \left( {{{\mathbf{e}}_r} + \sqrt {\frac{{2M}}{r}} {{\mathbf{e}}_t}} \right) \wedge {{\mathbf{e}}_\theta } \hfill \\ {{\mathbf{\omega }}_r} = \sqrt {\frac{M}{{2{r^3}}}} {{\mathbf{e}}_r} \wedge {{\mathbf{e}}_t} {\qquad}{{\mathbf{\omega }}_\phi } = \sin \theta \left( {{{\mathbf{e}}_r} + \sqrt {\frac{{2M}}{r}} {{\mathbf{e}}_t}} \right) \wedge {{\mathbf{e}}_\phi } + \cos \theta {{\mathbf{e}}_\theta } \wedge {{\mathbf{e}}_\phi } \hfill \\ \end{gathered} \]

   A PG-observer is an observer tied to the tetrad $\left\{ {{{\mathbf{e}}_t},{{\mathbf{e}}_r},{{\mathbf{e}}_\theta },{{\mathbf{e}}_\phi }} \right\}$. From (9.3), with the parameter values (9.7), the observer velocity in the coordinate basis is seen to be

9.11

\[{\mathbf{u}}: = {{\mathbf{e}}_t} = {{\mathbf{g}}_t} - {{\mathbf{g}}_r}\sqrt {2M/r} \]

The observer velocity is then found to satisfy the geodesic equation

9.12

\[{\mathbf{u}} \cdot D{\mathbf{u}} = \left( {{{\mathbf{\omega }}_t} - {{\mathbf{\omega }}_r}\sqrt {2M/r} } \right) \cdot {\mathbf{u}} = 0\]

Thus, the PG observer frame is free-falling and non-spinning because also the spatial frame is parallel transported along the worldline of the observer: ${{\mathbf{e}}_t} \cdot D{{\mathbf{e}}_j} = 0$, $j = r,\theta ,\phi $.

   In PG-coordinates ${\text{\{ }}{x^\mu }\} : = \left\{ {t,r,\theta ,\phi } \right\}$, the velocity (9.11), affinely parametrized by $\tau$, has  the components

9.13

\[{u^\mu } = (\dot t,\dot r,\dot \theta ,\dot \phi ) = \left( {1, - \sqrt {2M/r} ,0,0} \right)\]

with respect to the coordinate base; notation $\dot t: = dt/d\tau $ etc. Thus, the free-falling observer goes radially inward with 3-velocity $\dot r = \sqrt {2M/r}$, which equals the negative of the Newtonian escape velocity. At the event horizon, the observer reaches the velocity of light $c=1$. There is no discontinuity or singularity at the event horizon.

   The value of the time component $\dot t = dt/d\tau = 1$ implies that the PG time-coordinate $t$ is the proper time of the geodesic observer. The GP-coordinates could have been defined by this requirement.