Exploring Geometric Meaning of Time and Space at a Black Hole's Event Horizon

[scope]

hello,

what does exactly mean geometrically that time and space switch roles at the event horizon of a nonrotating black hole?. I understand that the - for time becomes a + and the + for space becomes -, but how to interpret it geometrically?
also I want to know if after the event horizon, inside the black hole, (its nearly the same question) whether the infalling object is still in exactly the same 4-dimensionnal spacetime as the observer or if the object has exited it.
thank you!

 

[phyzguy]

One way to look at it is that outside the event horizon you can only move forward in time, not backward. Similarly inside the event horizon you can only move to smaller r, you can't move to larger r no matter what you do. Your second question is more philosophical. The spacetime inside the event horizon is causally disconnected from the spacetime outside the event horizon, so what is inside the event horizon cannot influence what is outside. Does this mean it is not "the same spacetime"? I'm not sure.

 

[scope]

thanks, but for the second question i was just wondering if the content of the black hole inside the event horizon is exactly contained in the same 4 spacetime dimensions of the observer. I mean: are only 4 spacetime dimensions required to contain the behavior outside of the event horizon and inside of the even horizon(without talking about the singularity) for a Schwarzschild black hole, mathematically (Schwarzschild metric)?
thanks

 

[phyzguy]

Yes, a Schwarzschild metric has just 4 spacetime dimensions, both inside and outside the event horizon.

 

[PeterDonis]

The spacetime inside the event horizon is causally disconnected from the spacetime outside the event horizon, so what is inside the event horizon cannot influence what is outside. Does this mean it is not "the same spacetime"? I'm not sure.

What's inside the horizon can't causally influence what's outside, but what's outside *can* influence what's inside. So the region inside the horizon is still part of the same spacetime. The reason causal influences can't propagate from inside the horizon to outside is simply that to do so, they would have to travel faster than light, and causal influences can't travel faster than light.

 

[Passionflower]

what does exactly mean geometrically that time and space switch roles at the event horizon of a nonrotating black hole?.

Time and space do not switch roles at the EH.

I think you are confused by the behavior of Schwarzschild coordinates outside and beyond the event horizon where the r looks like t and the t like r. But this is a coordinate effect not something physical.

And observer free falling through the EH will not notice anything related to switching space and time.

 

[scope]

Time and space do not switch roles at the EH.

I think you are confused by the behavior of Schwarzschild coordinates outside and beyond the event horizon where the r looks like t and the t like r. But this is a coordinate effect not something physical.

And observer free falling through the EH will not notice anything related to switching space and time.

of course i was referring about the coordinate effect: how the spacetime inside the event horizon behaves mathematically in the observer reference frame. in such case, time and space switch roles

 

[Passionflower]

of course i was referring about the coordinate effect: how the spacetime inside the event horizon behaves mathematically in the observer reference frame. in such case, time and space switch roles

Spacetime behaves locally the same everywhere. I am still not convinced if you fully understand that this 'switching' is simply a coordinate chart effect and nothing physically.