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?