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Tetrads in General Relativity

IX. Spherically Symmetric Spacetimes

Schwarzschild Black Hole

   Photons with frequency 4-vector ${\mathbf{k}} = (\nu ,\vec k)$, $\nu = \left| {\vec k} \right|$, follow null trajectories ${{\mathbf{k}}^2} = 0$ specified by the geodesic equation ${\mathbf{k}} \cdot D{\mathbf{k}} = 0$. A radial null trajectory has tangent

9.14

\[{\mathbf{k}} = \nu ({{\mathbf{e}}_t} \mp {{\mathbf{e}}_r}){\qquad}\nu : = {\mathbf{k}} \cdot {{\mathbf{e}}_t}\]

The minus sign applies to a light ray towards the center; the plus sign to an outward light ray. The frequency $\nu $ is measured by a  radially free-falling observer (at rest at infinity).

   From (9.3) with (9.7), one derives the tangent in coordinate representation

9.15

\[{\mathbf{k}} = \nu {{\mathbf{g}}_t} - \nu \left( { \pm 1 + \sqrt {2M/r} } \right) {{\mathbf{g}}_r}\]

It follows that the velocity along the photon path is

9.16

\[\frac{{dr}}{{dt}} = \pm 1 - \sqrt {\frac{{2M}}{r}} \]

This result could have been directly obtained from the PG-metric (9.6.) by setting $ds = 0$, $d\theta = 0$, $d\phi = 0$.

   At the event horizon, the speed of light (9.16) vanishes for rays shining outward $(+)$: $dr/dt = 0$. More strangely, inside the event horizon, $r < {r_{\text{S}}}$, this light ray moves toward the center, implying that light cannot escape. It gets stuck at the event horizon which marks the boundary between the interior region that cannot signal to the outer region. Thus, everything inside the event horizon is hidden from the outside world: it is a Black Hole.

   Matter can only move inwards from the event horizon. If any object collapses to within its event horizon, it must carry on collapsing to form a central singularity. There is no force capable of preventing the collapse. This is because matter is always constrained to follow timelike paths, and if the entire future light-cone points inwards towards the singularity, no matter can escape. All paths for in-falling matter terminate on the singularity.