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From what I understand relativity predicts that outside observers would never observe external objects actually crossing into the event horizon, due to time dilation.

How does that can reconciled with the fact that the Schwarzschild radius is largely dependent on black hole mass? Then, does this imply that one can never observe an increase in the radius of the event horizon? This seems wrong to me, as there are different size black holes, and I'd like to know what I am missing.

Qmechanic
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Erol Bakkalbasi
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  • Possible duplicates: https://physics.stackexchange.com/q/21319/2451 and links therein. – Qmechanic Aug 31 '23 at 03:28
  • “outside observers would never observe external objects actually crossing into the event horizon” - While technically correct for material objects, there is one interesting caveat. When two black holes merge, their event horizons merge into one very fast from an external perspective. However, material objects just outside each horizon forever remain outside the combined horizon. – safesphere Sep 03 '23 at 21:19

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The issue here is that you are combining ideas from two different spacetimes and arriving at a contradiction because the different spacetimes are different.

The first spacetime is the Schwarzschild spacetime. It has a few important properties:

  1. spherically symmetric
  2. all vacuum
  3. static

Because it is static, it will not grow. It is eternal. And because it is vacuum there is no mass outside it to fall in.

Now, obviously this doesn’t represent the real world, but it can be a reasonable approximation in the space around a large spherical mass. But it is only this idealized case which cannot grow and where the usual coordinates produce infinite time dilation.

For a growing black hole a better example is the Oppenheimer Snyder spacetime. This spacetime is:

  1. spherically symmetric
  2. not vacuum (pressureless dust)
  3. not static

In this spacetime the event horizon grows as matter crosses it. The infalling matter itself alters the shape of the spacetime. As it moves the spacetime is not described by the eternal vacuum solution.

So the presence of the infalling matter means that the spacetime is not described by the static solution. Conclusions based on that spacetime need not apply to a non-static one.

Numerical solutions can be performed without spherical symmetry and with different types of matter besides pressureless dust. The general conclusion of a non-static spacetime and a growing event horizon are quite robust.

Dale
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  • In the Oppenheimer or numerical solutions, do outside observers ever see matter cross the horizon? – RC_23 Aug 31 '23 at 02:54
  • I appreciate the comment about the Swarzschild spacetime being static, but still, even in the non-static solution, do outside observers see matter cross the horizon?

    You state that "in this spacetime, the event horizon grows as matter crosses it," but due to GR, don't we never observe matter crossing it? I still see the contradiction here, can you try to further elaborate on that point?

     – Erol Bakkalbasi Aug 31 '23 at 03:04
  • @RC_23 and Erik no, matter can never be seen to cross the horizon by definition (if you could see it then it wouldn’t be the horizon). But that is a different question from whether the horizon grows and a different question from whether matter crosses it. Don’t confuse these separate issues – Dale Aug 31 '23 at 03:24
  • @RC_23 Dale’s answer would be fine, if not for the critical, but incorrect statement: “In this spacetime the event horizon grows as matter crosses it”. In reality, the horizon grows as matter accumulates in its vicinity and before anything crosses to the inside. Once again, Oppenheimer and Snyder have stated themselves that nothing crosses the horizon in their solution ever from an outside perspective. And yet the black hole grows, as more matter falls to the horizon. – safesphere Sep 01 '23 at 23:18
  • @ErolBakkalbasi “don't we never observe matter crossing it?” - A technically proper statement is that the event of anything crossing the black hole horizon is not in the past lightcone of any external observer. This statement is coordinate independent and thus holds in any coordinate system. Also, the external (non-evaporating) black hole spacetime is globally hyperbolic, meaning the causal order of events is preserved. Therefore, nothing crosses the black hole horizon, static or not, in the coordinates of any external observer, for the eternity of the cosmological time of the universe. – safesphere Sep 01 '23 at 23:32
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From what I understand relativity predicts that outside observers would never observe external objects actually crossing into the event horizon, due to time dilation.

External object has a field around it where gravitational potential is decreased.
Also black hole has a field around it where gravitational potential is decreased.

When gravitational potential is low enough at some place, then light can't escape that place.

So therefore, when external object gets close enough to a black hole, then light can't escape from the external object. In other words external object is behind an event horizon.

stuffu
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