Mathematical Analysis of Blackhole

Question

Mathematically, for black holes old enough that the stellar material has collapsed all the way into the singularity, the region between the horizon and the singularity is occupied by a spacetime where the time and space coordinates are reversed from those of the outside world. What this means in terms of what you experience is unknown. Other more complex conditions can occur of the black hole is rotating. In that case the singularity becomes a ring around the center of the black hole. You can pass through the center, but the tidal gravitational field would be lethal in all likelihood. In nearly all cases there would be gravitational radiation rattling about, and this would cause distortions in spacetime that would probably lead to spectacular optical distortions.

My question is Black hole absorbs everything. Is there any mathematical proof regarding this?

Answer 1

The Schwarzschild metric can be written as $$ c^2 d\tau^2 = \left( 1 - \frac{r_s}{r}\right)\ dt^2 - \left(1 - \frac{r_s}{r}\right)^{-1}\ dr^2 - ...,$$ where $r$ is the radial coordinate, $t$ is the coordinate time, $\tau$ is the proper time (that measured on an observer's own clock) and $r_s = 2GM/c^2$ is the Schwarzschild radius. I have left out the angular terms on the right hand side which contribute a further negative term independent of whether $r$ is greater or less than $r_s$.

For an observer with mass, $d\tau>0$; for a massless particle $d\tau=0$ (e.g. a photon).

When $r<r_s$ the first term on the RHS is negative, while the second term becomes positive. In order for the LHS to be $\geq 0$, then $$ \left(\frac{r_s}{r}-1 \right)^{-1}\ dr^2 \geq \left(\frac{r_s}{r}-1\right)\ dt^2 + ...$$ $$ \left| \frac{dr}{dt}\right| \geq \left(\frac{r_s}{r} -1\right)$$

What this means is that $dr/dt$ can never be zero, which means the direction of radial travel can never reverse. ie. Anything that enters a black hole (i.e. for which $r<r_s$) and has $dr/dt<0$, can never have $dr/dt >0$.

There is a slightly more satisfactory "proof" using Eddington-Finkelstein coordinates, that shows that all future light cones point inwards and that $dr<0$ when $r<r_s$.