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Tetrads in General Relativity

VIII. Schwarzschild solution

Static observers

   The geometric interpretation of the connection coefficients (8.10) may be obtained by analyzing the parallel-transport of the tetrad base defined in (8.8) in the time direction. The time-like connection coefficient gives

8.12

\[  \begin{gathered}{{\mathbf{e}}_t} \cdot D{{\mathbf{e}}_t} = {e^{ - \Phi }}{{\mathbf{g}}_t} \cdot D{{\mathbf{e}}_t} = {e^{ - \Phi }}{{\mathbf{\omega }}_t} \cdot {{\mathbf{e}}_t} \\ = - {e^{ - \Phi }}\frac{M}{{{r^2}}}{{\mathbf{e}}_t} \wedge {{\mathbf{e}}_r} \cdot {{\mathbf{e}}_t} = g{{\mathbf{e}}_r} \end{gathered}  \]

   The gravitational acceleration $g$ is as defined in (8.11). It matches with $g$ being the gradient of the relativistic potential measured with respect to the radial ruler distance (8.3):

8.13

\[  g = \left| {\nabla \Phi } \right| : = \frac{{d\Phi }}{{dl}} = \frac{{d\Phi }}{{dr}}\frac{{dr}}{{dl}} = \frac{{M/{r^2}}}{{\sqrt {1 - 2M/{r^2}} }}  \]

This is the Newtonian value multiplied by the gravitational dilatation factor, which becomes increasingly large as the Schwarzschild horizon is approached.

   The acceleration vector $g{{\mathbf{e}}_r}$ in (8.12) is radially outward pointing since the static frame $\{{{\mathbf{e}}_t},{{\mathbf{e}}_r},{{\mathbf{e}}_\theta },{{\mathbf{e}}_\phi }\} $ needs to accelerate away from the central object to avoid falling in. The thrust required to maintain position is given by the magnitude of the acceleration vector (8.13). This is a constant, since the position is fixed for these observers. It is only possible to remain at rest outside the horizon.

   The parallel transport of the spatial vectors gives

8.14

\[  \begin{gathered} {{\mathbf{e}}_t} \cdot D{{\mathbf{e}}_r} = {e^{ - \Phi }}{{\mathbf{g}}_t} \cdot D{{\mathbf{e}}_r} = g{{\mathbf{e}}_r} \wedge {{\mathbf{e}}_t} \cdot {{\mathbf{e}}_r}\\ {\quad} {{\mathbf{e}}_t} \cdot D{{\mathbf{e}}_\theta } = 0 {\quad} {{\mathbf{e}}_t} \cdot D{{\mathbf{e}}_\phi } = 0 \end{gathered}  \]

With the notation ${\mathbf{u}}: = {{\mathbf{e}}_t} = {e^{ - \Phi }}{{\mathbf{g}}_t}$ and ${\mathbf{\dot u}}: = g{{\mathbf{e}}_r} = (M/{r^2}){{\mathbf{g}}_r}$, equations (8.12,14) are recognized as an instance of the FW-transport equation (7.7). So, the static frames $\left\{ {{{\mathbf{e}}_t},{{\mathbf{e}}_r},{{\mathbf{e}}_\theta },{{\mathbf{e}}_\phi }} \right\}$ may be identified as FW-transported non-spinning observer tetrads.