Since at the Event Horizon, time stops completely, how do two black holes merge together? Shouldn't they should stop moving due to time dilation when they get closer to each other's Schwarzschild radius?
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2 Answers
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The "event horizon" is defined as the point (or surface) from within which light rays can never (ever) reach a distant observer. To find the location of the event horizon implies that you must know everything about the future of the black hole - so in practice what is referred to is often the event horizon of a Schwarzschild black hole, which is static and eternal (or a Kerr black hole if it is spinning). i.e. It never changes and can be calculated.
When black holes merge, they cannot be considered as Schwarzschild (or even Kerr) black holes. It is a dynamic situation. In practice, what is done (in numerical computations) is to define the surface of an apparent horizon, from within which, light rays appear not to be making their way outwards towards a distant observer.
The location of this surface (or surfaces when the black holes are well separated) must be calculated dynamically, and it changes as the merger progresses. After the merger it settles down to approximate the event horizon of an eternal Kerr black hole (a merger remnant will always have some spin).
However, the root of your question is the apparent paradox around the simpler situation of how anything can fall into a black hole if time dilation slows this process infinitely at any (apparent) event horizon. There is no need to try to resolve this paradox (it isn't a paradox, because there is no one "truth" of what happens in relativity, only what different observers observe) because the apparent horizon is dynamic (it moves) and objects that get close to the horizon become unobservable to a distant observer.
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If you have two black holes, then the event horizons are distorted.
As event horizons are regions of spacetime (not just spheres in space) there is no issue with "time stopping". That is to say, while there is infinite time dilation for an observer at the event horizon, relative to an observer at great distance from the event horizon, the event horizons themselves are not "things". The shape of the event horizons is determined by the solutions of the General relativity equations, and the event horizons for two black holes are not spherical.
So the event horizons merge in space time, and the merged blackhole rapidly (over a few hundreds of milliseconds) settles down to a state that is asymptotically the same as a single rotating, "Kerr" black hole.
The key confusion here is thinking that event horizons are "things in spacetime" which are therefore subject to Relativity rules like time dilation. They are not. By way of (weak) analogy (and only to illustrate that relativistic rules only apply to "things") consider how the dot of a laser pointer can move faster than light.
In order to understand without analogy, you need to solve (numerically) the GR equations. Look at this picture of merging black holes. And remember, that the surfaces you see are merely 2d representations embedded in flat 3d, of a horizon that is actually embedded in curved 4d spacetime.
(image credit to Ligo:
converted to gif by [user@uhoh])
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$\begingroup$ Suppose you were falling into the black hole some time before the merger occurred... would your remnants experience the merger before they reached the singularity? $\endgroup$– MichaelCommented Jan 8, 2023 at 16:41
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2$\begingroup$ Not sure what you mean by "experience the merger". But the event horizon will be in your past, you won't be able "experience" the event horizon after you fall in. I think it is an interesting question that if you fell in black hole A, some light fell in black hole B, then A and B merged, could the light reach you. (but it a new question and I won't try to answer it here, not least because I don't know.) And you won't see a "singularity" because the singularity is always in your future. $\endgroup$– James KCommented Jan 8, 2023 at 18:05
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$\begingroup$ @Michael there's no guarantee that you'll ever reach a singularity (either the two original or a possible merged one), only that you will never leave the region inside the (dynamic) event horizon. $\endgroup$ Commented Jan 9, 2023 at 10:29
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1$\begingroup$ On the contrary, In the GR model, reaching the singularity is certain and unavoidable. Quantum gravity may have something to say about that, but if it does, it is on a very small scale. $\endgroup$– James KCommented Jan 9, 2023 at 17:51
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$\begingroup$ @JamesK I guess my "experience the merger" was more along the lines of "which singularity will you reach" i.e. since time is passing more slowly for you, will the singularities combine before you reach one of them and if not is it possible to reach the "new" singularity. I suppose though that even if the merging singularity is approaching the original one such that you are right in between you'll still reach the original one first? $\endgroup$– MichaelCommented Jan 10, 2023 at 15:46
Since at the Event Horizon, time stops completely: This is an incorrect but very common popular science portrayal. Suppose you have a spacecraft that takes you to the supermassive black hole at the center of the galaxy, and you decide to commit suicide by diving straight into it. Time will not stop for you. You will enter the black hole's event horizon in a finite amount of time as measured by you and shortly later you will die (possibly painfully) by spaghettification. OTOH, an external observer will see your spacecraft appear to grind to a halt at the event horizon. $\endgroup$