The way the narrator seems to explain it, the geometry of space and time
become so warped in a black hole that time can be "traversed" as it were
a spacial dimension.
A glimpse to the figure was enough for me.
The statement "space and time flip within the event horizon" is a misunderstanding of what really happens. Let me try to explain.
To begin with, it is generally incorrect to speak of "space" and "time" as separate entities within GR. It is one of its distinctive characters that only spacetime is physically meaningful (exceptional cases apart).
Here we are talking about Schwarzschild's geometry of an uncharged, non-rotating black hole (the simplest kind) which has an event horizon. Understanding spacetime below (or beyond?) this horizon is far from simple.
Generally speaking, spacetime is a 4-dimensional manifold, i.e. 4 coordinates are needed to describe it, to single out a point (an event) in it. Even outside the horizon Schwarzschild's geometry is different from SR's, which is also 4-dimensional, because Schwarzschild's is curved, not flat. This means that simple Lorentz-Minkowsky coordinates cannot be used, and that the "distance" (technically, the "metric") between two points has an expression somewhat longer to write and harder to understand.
I will content myself of saying this: the set of coordinates initially used by Schwarzschild and still in use are named $t$, $r$, $\theta$, $\phi$ and their names suggest an almost exact interpretation:
- $t$ is a time coordinate, but this does not mean that $t$ is the time: there is an infinity of other possible time coordinates. It is true that $t$ has a useful property which justifies its common use, but I refrain from entering the subject, for sake of brevity.
- $r$, $\theta$, $\phi$ are spherical coordinates very similar to the ones we use in 3D euclidean space. For instance, if you fix $t$ and $r$ you are left with a 2D spherical surface, of area $4\pi r^2$, which is ok. Nevertheless, $r$ does not measure the distance from the center of the sphere. Actually, there is no such center, as we shall see in a moment. This awkward phenomenon is one of the marks of a curved (non-euclidean) space.
But this is still nothing. The hard thing is that this coordinate system does not work for the whole spacetime: there is an apparent singularity at $r=2GM/c^2$ (the Schwarzschild radius): something goes to infinity in the metric formula for that value of $r$. Moreover, $t$ goes to infinity at the horizon. These unpleasant features were noticed as soon as Schwarzschild published his solution (1916) but a full understanding of their meaning and a satisfactory solution thereof had to wait more than 40 years.
It is now widely known that $r=2GM/c^2$ is not a real singularity of spacetime. Several sets of coordinates have been discovered, having no bad behavior at that $r$-value. A true singularity of Schwarzschild's spacetime actually exists, but for $r=0$: this cannot be avoided by any choice of coordinates.
Two facts I must still recall. The first is that although the Schwarzschild radius is not a singularity proper of spacetime, it represents a horizon. I cannot explain fully what this means, but one thing is known to almost everyone: a body which crosses the horizon cannot come back, out of the horizon. It is doomed to fall into the singularity at $r=0$. Even light emitted by such a body will not come out, but will also fall into the singularity, faster than the body (and this answers your question).
The second is that Schwarzschild coordinates can also be used beyond the horizon, but with a profoundly different meaning for $r$ and $t$ (so profound indeed, that it would be better to use different symbols). For $r$ less than Schwarzschild radius, $t$ is a space coordinate, $r$ a time coordinate. This is wholly another thing from what the narrator says: it is not true that "time can be traversed as it were a spatial dimension", because $t$ is no longer time. I must insist: $t$ is just a symbol, a label. No physical meaning must be attached to it only because we are accustomed that the letter $t$ usually means "time". The physical meaning, if there is one, can only be inferred from its role in the metric, and the metric says that beyond the horizon $t$ acts as a space coordinate. The same (in the reverse) holds for $r$, which is a time coordinate.
I wish to explain this fundamental point in greater detail, giving some examples. If you are outside the horizon, in a spaceship, you can maneuver so that the spaceship stays still, meaning with this that all three space coordinates ($r$, $\theta$, $\phi$) remain constant. This is not strange, as the same happens in Newtonian physics. To accomplish it near our Sun, you only need fire the spaceship's rockets so that their thrust counterbalances Sun's attraction. The same holds near a black hole, until the spaceship remains out of the event horizon. On the other side, there is no way to "stop time": even if the spaceship stands still, time passes and the $t$ coordinate continues to increase.
Now put the spaceship beyond the horizon. Nothing prevents you from keeping the space coordinates constant: $t$, $\theta$, $\phi$. Actually no rockets are required, as this is a special case of free fall. But it is a free fall, and by this two things are meant:
- Time passes, in the sense that the time coordinate $r$ is bound to change.
- It decreases towards zero, the singularity.
So you see that the singularity is not in the center of the black hole, but in its future.