# Causality and Event Horizon

So in GR, for a classical black hole, if A is approaching the event horizon, to an observer far away, let say B, B would never observe A crossing the event horizon as B would observe A's time slow down in the limit to 0 and A's length contract in the limit to 0. In fact, according to B, A never crossed the event horizon.

But A in it's own reference frame crossed the event horizon.

So then in this case, would A and B disagree on causality since both A and B completely disagree on what even happened? Then would that mean that causality is relative?

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PeterDonis
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B would observe A's time slow down in the limit to 0
B would see A's clock appear to slow down. But that is a distortion because of the curvature of spacetime between A and B. It is not because of anything happening to A's clock.

and A's length contract in the limit to 0
This is not correct.

according to B, A never crossed the event horizon
No. B cannot see A cross the horizon, but that does not mean B can say that A never crossed the horizon. All B can say is that he does not see A crossing the horizon. It's no different than an object going below your horizon when you're standing somewhere on Earth; the object doesn't cease to exist just because you can no longer see it.

would A and B disagree on causality since both A and B completely disagree on what even happened?
No. A and B don't disagree on what happened. B cannot see everything that A can see; but B is perfectly capable of calculating what A sees using the laws of physics.

Buzz Bloom and FallenApple
B would see A's clock appear to slow down. But that is a distortion because of the curvature of spacetime between A and B. It is not because of anything happening to A's clock.

This is not correct.

No. B cannot see A cross the horizon, but that does not mean B can say that A never crossed the horizon. All B can say is that he does not see A crossing the horizon. It's no different than an object going below your horizon when you're standing somewhere on Earth; the object doesn't cease to exist just because you can no longer see it.

No. A and B don't disagree on what happened. B cannot see everything that A can see; but B is perfectly capable of calculating what A sees using the laws of physics.

Ok got it. So B simply cannot detect A falling experimentally with observation? This is due to increasing redshift?

This leads me to a followup question. It's actually based on what you posed. I found it when searching for info on black holes.

You stated: "The classical (GR) model says (not "assumes"--it's derived from the Einstein Field Equation) that the black hole is vacuum inside. The mass that originally collapsed to form the hole reaches the singularity and disappears."

So according to your earlier post in this thread, and your statement in the linked thread I just posted, an observer can conclude that the matter from a stellar collapse would cross the event horizon and then converge to nothingness in finite time.

Then would that not violate conservation of mass and energy? I mean, the black hole would be massless.

Also, as an aside, spacetime is still curved, inside and outside the event horizon, which should not be possible since the presence of mass and energy is required to explain curved spacetime. Unless, the blackhole became unstable and dissapears in finite time, due to lacking mass.

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PeterDonis
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So B simply cannot detect A falling experimentally with observation?
Yes.

This is due to increasing redshift?
It's because the region at and beneath the event horizon cannot send light signals to anywhere outside the horizon.

would that not violate conservation of mass and energy?
No. The mass/energy is still there, stored in the curvature of spacetime, and the black hole still acts as a source of gravity the same way the original object did.

It is true that the black hole is a vacuum, but that does not mean "zero mass/energy". It means "zero stress-energy". See below.

the black hole would be massless
No, it wouldn't.

the presence of mass and energy is required to explain curved spacetime.
The correct term is "stress-energy". But there is stress-energy in the spacetime; the object that originally collapsed to form the black hole has it. A curved spacetime does not have to have stress-energy at every single event where there is curvature. It just has to have stress-energy somewhere in the past light cone of an event where there is curvature.

No. The mass/energy is still there, stored in the curvature of spacetime, and the black hole still acts as a source of gravity the same way the original object did.

It is true that the black hole is a vacuum, but that does not mean "zero mass/energy". It means "zero stress-energy". See below.
So the substance that possesses the stress energy disappears but leaves behind the stress energy inside the curved spacetime. Makes sense since conservation must be preserved.

So the center spacial point can be said to possess mass even if there is no substance there?

The correct term is "stress-energy". But there is stress-energy in the spacetime; the object that originally collapsed to form the black hole has it. A curved spacetime does not have to have stress-energy at every single event where there is curvature. It just has to have stress-energy somewhere in the past light cone of an event where there is curvature.
Is this similar as the following situation?

If I waved a magic wand and made the sun disappear, we would only feel the effects 8 min later because of the speed limit of causality.

the instant the sun disappears, the spacial warping seems to propagate outwards and becomes more dilute.

For the blackhole, it's similar since the mass going into the event horizon is like a magic wand( the horizon shields the universe from the inner mass.) But is it different from the disappearing Sun situation since the curvature does not change and propagate outwards? What causes this?

PeterDonis
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So the substance that possesses the stress energy disappears but leaves behind the stress energy inside the curved spacetime.
No. Remember we are talking about spacetime. Spacetime doesn't "change"; it just is. And there is a portion of it that contains stress-energy, and a portion that does not. Nothing "disappears"; it's just different regions of spacetime with different properties.

So the center spacial point can be said to possess mass even if there is no substance there?
As I have already said, "mass" is a property of the spacetime geometry as a whole. It isn't a property of any particular point.

If I waved a magic wand and made the sun disappear
Then you would be violating the conservation of energy and the laws of GR. So this is a meaningless hypothesis.

For the blackhole, it's similar since the mass going into the event horizon is like a magic wand( the horizon shields the universe from the inner mass.)
No, it isn't. Once more: the mass is a property of the spacetime geometry as a whole. It is not "located" inside the horizon.