Crossing the Schwarzschild radius

Question

I am struggling to understand where the problem (i.e. impossibility of getting out) at the Schwarzschild radius comes from, solving the equation of motion for a massive particle on a radial geodesic $\sigma$. Brief comment on causal structure at the end of the question.

Equation of motion

4-velocity normalization using proper time gives

$$g(\dot{\sigma}, \dot{\sigma}) = -c^2 = - \left( 1-\frac{R_s}{r} \right) c^2 \dot{t}^2 + \left( 1-\frac{R_s}{r} \right)^{-1}\dot{r}^2$$

The Killing vector $\partial_t$ provides the constant of motion $g(\dot{\sigma},\partial_t)$, so

$$e := \left( 1-\frac{R_s}{r} \right) \dot{t} = \text{const}$$

Combining the two we get a Newtonian-looking equation

$$\dot{r}^2 - \frac{2GM}{r}=e^2c^2-c^2 =: \epsilon = \text{const}$$ Following these lecture notes (eq. 31.4) the equation is solved for the case $\epsilon =0$, which gives a simple power law $\tau \propto \pm r^{3/2}$. The minus signs tells us that nothing strange happens to an observer falling toward the centre when he crosses the Schwarzschild radius; the solution $t(r)$, on the other hands, blows up at $R_s$, so an external observer never sees the crossing. The question is: why can not this argument be used the other way round? From $\tau \propto +r^{3/2}$ the way out seems as smooth as the way in. I tried to plot a solution for the general integral, without the assumption $\epsilon = 0$, but it just looks analytically more complicated but conceptually the same.

Causal structure

I (kind of) understand that the correct setting for this question is the causal structure of spacetime, something like (from this question)

In Schwarzschild coordinates, if you look at the $g_{tt}$ and $g_{rr}$ parts fo the metric, they flip signs at $r=R_s$. Therefore "inside" the $r$ direction is timelike and the $t$ direction is spacelike. The future-timelike light cone of any event inside the horizon points toward smaller values of $r$.

So I would like both:

1. to understand if the "equation of motion" approach is meaningful, and
2. to fully understand the content of the quoted answer.

$\partial_t$ is timelike outside and spacelike inside, $\partial_r$ the opposite, good so far. What is the physical, or geometrical, consequence of this? What does it imply for a direction to be something-like?

This answer makes it all about "light cones tilting", but isn't this a frame dependent statement?

Edit: I think this is the core of the question, speaking of causality: how is the "future" defined, with respect to timelike and spacelike directions? Why is the future (F) "on the left" in these picture?

There are a few questions in the last part that I realize all boil down to the same concept: what does it mean that "time" and "space" switch role?

Answer 1

You're subtly cheating by doing what you're doing -- you can't time-reverse the geodesic without time-reversing the background spacetime.

The time-reversal of the Schwarzschild spacetime in standard coordinates will give the white hole patch of the extended Kruskal spacetime. This is a valid solution of the equations, but is physically different than an object falling out of the black hole.