Mathematical Analysis of Blackhole
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, $d\tau&gt;0$; for a massless particle $d\tau=0$ (e.g. a photon).
When $r&lt;r_s$ 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 $r&lt;r_s$) and has $dr/dt&lt;0$, can never have $dr/dt &gt;0$.
There is a slightly more satisfactory &quot;proof&quot; using Eddington-Finkelstein coordinates, that shows that all future light cones point inwards and that $dr&lt;0$ when $r&lt;r_s$.