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I have read this question:

Yes, the expansion of space itself is allowed to exceed the speed-of-light limit because the speed-of-light limit only applies to regions where special relativity – a description of the spacetime as a flat geometry – applies. In the context of cosmology, especially a very fast expansion, special relativity doesn't apply because the curvature of the spacetime is large and essential. When this v exceeds c, it means that the two places/galaxies are "behind the horizons of one another" so they can't observer each other anytime soon. But they're still allowed to exist.

Can space expand with unlimited speed?

The event horizon actually isn't a physical thing at all, just like any other border isn't a real thing. Remember that the horizon is just a 'border' which marks where the escape velocity raises above luminal speed.

Are black holes naked singularities for an observer within the event horizon?

So the first one says that space can expand at a speed that exceeds the speed of light, but that is not a violation of SR, because space is not a thing, no information, no particles or objects are actually moving faster then the speed of light. And that SR is not applicable where curvature is large and essential.

Now similarly, the event horizon of a black hole is not a thing, but a boundary, between two regions of spacetime (which similarly are casually disconnected, just like in the case of space expansion), so SR should not be violated if an event horizon would expand faster then the speed of light. Same way, spacetime curvature is large and essential in the case of the event horizon, so SR should not apply.

So there are two things that come to mind:

  1. space is not a thing, and if it expands faster then the speed of light, that does not violate SR, because no information, no particles, no objects are locally moving actually faster then light (although they are moving away from us faster then the speed of light). Same applies for a black hole's event horizon expanding, which is not a physical thing, but a boundary.

  2. SR only applies in regions where spacetime is flat, but if spacetime curvature is large and essential, like in the case of a black hole's event horizon, SR does not apply, so it is not a violation of SR if space expands faster then the speed of light (in the case of cosmology, where curvature is large and essential) or if a black hole's event horizon expands (where curvature is large and essential) faster then the speed of light.

Just to clarify, I am simply asking, as seen from a far away observer (may be inside or outside), how fast can the EH be seen to expand? Can this measured (from far away) speed exceed the speed of light?

Question:

  1. If spacetime can expand faster than the speed of light, then can a black hole do that too?
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    $\begingroup$ What does it mean for spacetime to expand with a linear speed measured in e.g. meters per second? $\endgroup$
    – J. Murray
    Nov 18 '21 at 23:26
  • $\begingroup$ @safesphere That's the average (not current) rate (with respect to the cosmological time) at which the proper distance between a comoving observer on Earth and a comoving observer at the horizon is increasing. The point I was trying to make is that a linear speed does not quantify the expansion of space itself, but rather the rate of increase between two specific objects in it. $\endgroup$
    – J. Murray
    Nov 20 '21 at 0:16
  • $\begingroup$ @safesphere thank you I edited so far everywhere "faster then light" to "faster then the speed of light". $\endgroup$ Nov 20 '21 at 2:39
  • $\begingroup$ @safesphere can you please help me clarify the speed I am referring to as the expansion speed of the EH? I am simply asking, as seen from a far away observer (may be inside or outside), how fast can the EH be seen to expand? Can this measured (from far away) speed exceed the speed of light? $\endgroup$ Nov 20 '21 at 2:40
  • $\begingroup$ Why the downvote? $\endgroup$ Nov 20 '21 at 2:41
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When dealing with a curved spacetime, as in GR, one needs to take care that velocity is a quantity that is only defined in the tangent space of a single point on a manifold. So it's not possible to directly compare velocities$^\star$ between different points in spacetime. Therefore it is not really meaningful to say that "space is expanding faster than light" -- this statement presupposes that you can compare the velocity directly at two different points in space. The same argument applies to a black hole spacetime.

Having said that, there is an analogy between cosmology and black holes. Both have horizons, which are regions where light cannot escape. In the case of a black hole, if a beacon falls beyond the event horizon, it can never send a signal back out to us because light cannot leave the volume contained by the event horizon. The case of cosmology is essentially "inside out" -- if a beacon travels further from us than our "cosmological horizon", then it can never send a signal back to us. Loosely speaking (and I'll admit that these words, unfortunately, contradict the formal point I made above -- the careful statement is the one in the previous sentence), you can say that space is expanding so rapidly that the beacon's light beam cannot travel fast enough to make up the ground and reach us again. Not all spacetimes have a horizon like this, but there is one in an accelerating Universe like the one we currently find ourselves in.


$^\star$ In FRW cosmology, which has a lot of symmetry, you can define a comoving velocity, which is essentially the velocity with respect to the cosmic rest frame. Galaxies expanding with the Universe have, on average, zero comoving velocity.

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  • $\begingroup$ Thank you so much! $\endgroup$ Nov 19 '21 at 2:29

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