Undergrad Strange Spacetime behavior inside Event Horizon

[FallenApple]

This question is regarding classical black holes.

So inside the event horizon, spacetime behaves strangely. Space is now one dimensional and only in the forward direction(that is, into the singularity) and time of events is preceived as forward and backwards since light from a source comes from both the future and the past. Basically, the behavior of time and space got switched.

My question is,

1) why is there an inversion like this?

2) since spacetime behaves so differently inside the event horizon, is this region of spacetime a topologically distinct space, connected merely by the event horizon?

 

[PAllen]

This is incorrect. Spacetime inside the horizon is classically indistinguishable from SR spacetime on small scales. You still experience 3 spatial and one time dimension, and light only reaches you from your past light cone. The misunderstanding you describe stems from the unfortunate labelling of one particular set of coordinates (Schwarzschild coordinates).

Topologically, the whole interior is different from normal (S2 X R2), but this is purely a global feature, not detectable locally.

 

[PeterDonis]

Space is now one dimensional and only in the forward direction

That is not correct, even in Schwarzschild coordinates (which are, as PAllen said, not good ones to use inside the horizon because it is so easy for them to cause confusion). In Schwarzschild coordinates, inside the horizon $r$ is timelike, but $t$, $\theta$, and $\phi$ are all spacelike. So it's obvious that there are still three spacelike dimensions and one timelike dimension.

(For more weirdness in that respect, try Painleve coordinates, in which all four coordinates are spacelike inside the horizon.)

 

[Dale]

Space is now one dimensional

No. The signature of a manifold is the same at every event of the manifold. So in a BH the signature is (-+++) both outside and inside the horizon.

only in the forward direction(that is, into the singularity)

In Schwarzschild coordinates inside the horizon the coordinates t, theta, and phi are all spacelike, while r is timelike.

light from a source comes from both the future and the past

No, each event has a past light cone and a future light cone as normal.

Basically, the behavior of time and space got switched

All that got switched is the label, and only in certain coordinate systems like Schwarzschild coordinates

 

[FallenApple]

No. The signature of a manifold is the same at every event of the manifold. So in a BH the signature is (-+++) both outside and inside the horizon.

In Schwarzschild coordinates inside the horizon the coordinates t, theta, and phi are all spacelike, while r is timelike.

So basically, a person inside an event horizon can move up down, left right, but can only move forward inside.

But what does it mean for time to be spacelike?

 

[FallenApple]

This is incorrect. Spacetime inside the horizon is classically indistinguishable from SR spacetime on small scales. You still experience 3 spatial and one time dimension, and light only reaches you from your past light cone. The misunderstanding you describe stems from the unfortunate labelling of one particular set of coordinates (Schwarzschild coordinates).

Topologically, the whole interior is different from normal (S2 X R2), but this is purely a global feature, not detectable locally.

So if it is indistinguishable on small scales, does that mean a person's range of motions is as usual?

Then if it doesn't hold globally, would that mean that the person inside the event horizon is overall moving forward, even though he/she can move around as usual, while the spacetime patch the person is on is moving forwards into the singularity?

So light only reaches me from my past light cone. Does that mean that my past light cone can only encapsulate events from someone else's past light cone?

 

[Dale]

But what does it mean for time to be spacelike?

It means that we used the letter t for one of the directions in space. It has no physical consequences whatsoever. The letter t does not have any special powers.

Look again at what I said. I did not say that time was spacelike, I said that the t coordinate was spacelike.

 

[Nugatory]

So basically, a person inside an event horizon can move up down, left right, but can only move forward inside.

Inside the horizon you can move left, right, up, down, forwards, backwards, just as you can outside the horizon. You can only move forward in time, but that's just the same as outside the horizon.

But what does it mean for time to be spacelike?

Nothing - that's not what's going on here.

Outside the horizon there are three spacelike Schwarzschild coordinates and one timelike Schwarzschild coordinate. Inside the horizon there are three spacelike Schwarzschild coordinates and one timelike Schwarzschild coordinate. You're just being confused by the naming convention that uses the letter $t$ for the timelike coordinate and $r$ for one of the three spacelike coordinates outside the horizon, and the other way around inside the horizon. This issue is discussed some in this paper; basically the entire problem goes away if we use different letters to label the Schwarzschild coordinates inside the horizon and outside.

 

[Nugatory]

So if it is indistinguishable on small scales, does that mean a person's range of motions is as usual

Yes.

Then if it doesn't hold globally, would that mean that the person inside the event horizon is overall moving forward, even though he/she can move around as usual

Yes. Just as you can "move around as usual" right now, even though you are also moving forward in time. Everyone is always "moving forward" in time.

while the spacetime patch the person is on is moving forwards into the singularity

The spacetime patch isn't moving at all. You and everything else are moving through it, tracing out paths that we call "world lines".

So light only reaches me from my past light cone

Yes, pretty much by definition.

Does that mean that my past light cone can only encapsulate events from someone else's past light cone?

No.
The easiest way to see this is to draw a spacetime diagram in ordinary flat spacetime, no gravity, black holes or general relativity required. You and I are at rest relative to one another and you are standing one light-second to the left of me. Your light cone extends back behind you, my light cone extends back behind me, and they do not completely overlap.