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Once you start studying black holes, one of the first things you'll probably hear is that from an outsider's perspective objects falling into the black hole take an infinite time to do so due to time dilation. The object will asymptotically approach, but never reach the event horizon.

As far as I understand the typical description of this, people are referring to the event horizon of the black hole as it was before the approach of the new object. However, my understanding is also that you can calculate a Schwarzschild radius for the total mass $M+m$ (where $M$ is the mass of the black hole and $m$ is the mass of the falling object) which would constitute the radius for the event horizon formed by the combined mass.

This new radius would be only very slightly larger, but positively so. Initially, this new event horizon would be inside the total mass, just like the old one. However, if the assumption here is that the object is approaching the old event horizon, then eventually in finite time, the object would reach the new radius. At that point, the total mass would be inside its Schwarzschild radius, so it would effectively be a black hole of increased mass compared to the original one, while the new event horizon would be exposed outwards.

As such, this would imply that from an outside perspective the falling object would eventually vanish entirely in finite time and the black hole would appear larger.

Some potential flaws in my reasoning that I can see, but I'm not sure about are the following:

  1. The total mass of the black hole + the object is not perfectly spherical or symmetrical and therefore the Schwarzschild radius is not necessarily an appropriate way to describe or approximate the situation.
  2. The way the mass is distributed inside an event horizon may not matter for the gravitational effects outside it, but it might matter for the effects inside it. Thus the region between the two event horizons does not necessarily behave like the inside of a black hole and could still be visible to outsiders.
  3. Even if we account for the new event horizon, the object actually approaches that and not the old horizon, therefore it still only asymptotically approaches it.

If I'm right about any of these doubts I'd appreciate if someone confirmed that, but if there's something else going on that'd be even more interesting.

It's also worth noting that is the strong version of my syllogism where only one object is enough to allow itself to cross the horizon. Even if we conclude that this is not the case, there is a weaker version where we study the effect of multiple objects approaching the black hole. If one assumes a constant inflow of matter (unrealistic, I know, but bear with the hypothetical for a moment) then eventually (given enough time) the area near the event horizon would get dense enough and the expanded horizon wide enough that some, if not all, the approaching matter would end up inside it. If this is correct, then it might also be generalised for cases where the inflow isn't constant, but it's large enough for a long enough time for the effect to kick in.

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5 Answers 5

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In the animation below, every colored Kugelblitz shell has an energy equivalent of 1Mc², so together they have a total of 5Mc². The innermost shell will asymptotically, but never quite reach r→2GM/c², while the outermost will converge to r→10GM/c².

The inner shells don't know about the outer shells due to the shell theorem, but the outer ones are affected by the inner ones. So in the frame of a stationary observer far away whose proper time is the Schwarzschild coordinate time t, nothing ever crosses r≤10GM/c².

The innermost shell, representing the core of a collapsed star, does of course end up inside the horizon generated by the outer layers, but still outside the horizon that would be generated if the outer shells were not present:

enter image description here

The gray background doesn't have mass of its own, it just represents the different radii. If the situation were not spherically symmetric it would be a little bit more complicated, but the principle would most likely be the same, although I haven't done the math for the latter case.

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  • $\begingroup$ This is a great animation visualising the ideas discussed in the other answers very well. Concerning this quote, though "nothing ever crosses r≤10GM/c²": is that including the shells or are you referring to everything beyond the shells? The former doesn't seem quite right to me. I would imagine after a certain point the only light from the blue shell that would reach the observer would be what already reached the purple shell, but until then, one would be able to watch the blue shell shrink beyond the radius the purple shell reached. If that's what you meant, can you clarify a bit further? $\endgroup$
    – Giorgos G
    Commented May 20 at 17:09
  • $\begingroup$ @Giorgos G - the shells are infinitely thin in this example, and they themselves only converge to, but never reach the 2Gm/c², with m being all the Ms within its own radius, including itself's M, see the numeric display on the top left. $\endgroup$
    – Yukterez
    Commented May 20 at 19:12
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    $\begingroup$ @safesphere - your comment is wrong, the extra time dilation caused by the potential is already taken into account in my calculations. $\endgroup$
    – Yukterez
    Commented May 21 at 9:01
  • $\begingroup$ @safesphere wrote: "During the second collapse the first shell expands from r=2 to r=4" - I'm afraid that's not the general consensus, but feel free to publish your expanding shells as an answer to see if anyone agrees. In my book everything within the horizon can only move inwards, even outgoing lightrays. The time dilation never becomes negative as it does with an eternal black hole that has a central singularity, it only approaches infinity with the collapsing shells, so nothing moves out. $\endgroup$
    – Yukterez
    Commented May 21 at 18:59
  • $\begingroup$ @safesphere You've been saying this in a lot of the comments, but I'm not sure how you suggest that the horizon gradually grows as an object approaches. To my understanding, the event horizon's radius does not depend on the object's distance from the horizon, so its smooth motion is not satisfactory justification for the effect. Each infalling object sees its own event EH, I believe, ignoring what comes after. But what does the faraway observer see? $\endgroup$
    – Giorgos G
    Commented May 22 at 9:34
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No, in the coordinate system of an external observer the infalling object never crosses the event horizon even allowing for the horizon growing outwards to meet the object.

This is a rather arm waving explanation, but I think it's still rigorous. The definition of the horizon is the set of points from which light will never escape to infinity no matter how long we wait. In the case of the infalling object the horizon becomes time dependent, but the definition still applies. That means the light ray that leaves the object as the object crosses the horizon cannot escape to infinity.

This is where it gets arm waving: if the light ray never escapes that must mean for the observer at infinity the time dilation at the horizon is infinite i.e.

$$ \frac{dt}{d\tau}(r = r_s) = \infty $$

where $t$ is the observer's coordinate time and $\tau$ is the proper time. Then for the external observer the elapsed time for the fall to horizon will be something like:

$$ \Delta t = \int_{\tau_0}^{\tau_1} \left(\frac{dt}{d\tau}\right) d\tau $$

where $\tau_0$ and $\tau_1$ are the proper times the object started falling and crossed the horizon. But since $\frac{dt}{d\tau}(\tau_1) = \infty$ the integral diverges and we get an infinite elapsed coordinate time.

I guess the objection is that an infinite derivative does not necessarily mean the integral diverges, but we expect that locally (in space and time) the time dependent metric will look like a Schwarzschild metric and we know the integral diverges for the Schwarzschild metric. Hence we expect it diverges in this case as well.

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  • $\begingroup$ Comments have been moved to chat; please do not continue the discussion here. Before posting a comment below this one, please review the purposes of comments. Comments that do not request clarification or suggest improvements usually belong as an answer, on Physics Meta, or in Physics Chat. Comments continuing discussion may be removed. $\endgroup$
    – ACuriousMind
    Commented May 22 at 19:32
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Once you start studying black holes, one of the first things you'll probably hear is that from an outsider's perspective objects falling into the black hole take an infinite time to do so due to time dilation. The object will asymptotically approach, but never reach the event horizon.

Inprecise statements like this are unfortunately common. They build on concepts that have an intuitive meaning for most people, but do not have a precise well-defined meaning.

The more accurate statement is that an outside observer will never observe anything crossing an event horizon. However, this is also something of a tautology, since the event horizon is itself defined as part of a region of spacetime from which signals can never reach outside observers.

Unlike special relativity, general relativity does not have a unique canonical notion of "the time at event A according to an observer at event B". The best B can do is "the time at which they observe A" (and even that is not necessarily uniquely defined for a far away observer since signals from A to B may follow inequivalent paths and arrive at different times.

So, whether or not object A ever crosses the event horizon according observer B depends on a choice. In general, we can construct both "global times according to B" that will answer in the negative and in the positive. Hyperboloidal slicings are a popular example of the latter, which play and important role in modern numerical relativity.

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  • $\begingroup$ I don't think this answers the question very well. It mentions a choice that can yield either a negative or a positive answer for observer B seeing the event, but does not demonstrate how that choice works. It brings up the example of Hyperboloidal slicings, but does not explain what they are and how they're relevant. Looking them up online does not reveal results that would be particularly helpful to the average person who doesn't already know the answer to the original question This answer contradicts the one by John Rennie that came first, but does not attempt to explain the discrepancy $\endgroup$
    – Giorgos G
    Commented May 20 at 13:09
  • $\begingroup$ While I believe it's true that different paths from the event of crossing the event horizon to observer B can be defined, it's my understanding that there is only one shortest path (for a given spacetime). As long as one can prove that the shortest path is infinite (due to infinite time dilation at the horizon as John Rennie shows), then any other path would take longer to get there, rather than less time. I'm not sure what insight the use of hyperboloidal slicings could bring that would change this conclusion. $\endgroup$
    – Giorgos G
    Commented May 20 at 13:17
  • $\begingroup$ If the question is "can B ever 'see' A crossing the event horizon?" Then it is answered by the second paragraph.. $\endgroup$
    – TimRias
    Commented May 20 at 13:17
  • $\begingroup$ I guess, but then I don't see what point you're trying to make with the rest of your answer. Maybe I'm misunderstanding and it actually makes sense, but I think it mostly just causes confusion. $\endgroup$
    – Giorgos G
    Commented May 20 at 13:20
  • $\begingroup$ The main point of the rest of the answer is that any question beyond what B can see and when is not physically uniquely defined. $\endgroup$
    – TimRias
    Commented May 20 at 13:29
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All of the following statements can be simultaneously true:

  1. An observer outside the event horizon will never observe a photon (or any other signal) emitted from inside the event horizon.

  2. As an object falls into an event horizon, the time it takes for photons emitted by the falling object to reach an external observed tends to infinity.

  3. An observer outside the event horizon can observe photons that appear to have been emitted from a point in space (however you define that) which at some later time will be inside the event horizon.

Statements 1 and 2 together form the basis of the claim that "an external observer never sees anything fall through the event horizon." In particular, statement 1 is basically just a restatement of what the term "event horizon" means, while statement 2 says that an external observer can in principle forever continue to receive a stream of (increasingly faint, sparse and redshifted) photons from the object falling into the event horizon.

(In practice, of course, any light emitted by the falling object will fade out and quickly become impossible for any external observer to detect, even if the last few photons emitted before the object crosses the event horizon could in principle take an arbitrarily long time to reach the external observer.)

However, remember that an event horizon is a boundary in spacetime, not just a line in space. In particular, when we say something happens inside or outside the event horizon, what we mean is that it happens at a point in space that is inside or outside the event horizon at the time when it happens.

In particular, you can observe light emitted from any point in spacetime outside the event horizon (and inside your own past light cone), even if that same spatial location (however you define "same" for this purpose) at some later time might be inside the event horizon.

In other words, it's perfectly possible for you to see an object fall towards (and arbitrarily close to) an event horizon of radius $r$, and then later (indirectly) observe the event horizon expanding to a radius $R > r$. That does not mean that the light from the falling object would've reached you from inside the event horizon, since the object was (by definition!) outside the event horizon when it emitted the light that you saw.

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I'm not completely certain of this but I think there's a fundamental flaw here in that real world objects always have size. No object can fly close enough to the black hole to lift the event horizon to cover it without said object having hit the event horizon and not coming back.

The only object without a "size" is another black hole--but I believe in this case when the event horizons touch they're not coming apart again.

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