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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? enter image description here

enter image description here

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?

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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.

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  • $\begingroup$ Quite brief, it took my a few days to elaborate it, but now it makes sense, thanks. $\endgroup$ – DavideL Jul 26 '17 at 17:34

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