Mass distribution behavior of the singularities during black hole mergers

[PeterDonis]

I though that we can transform our general idea how merging process looks like into some general shape diagram (it can have more dimensions).

We can for the region at and outside the horizon; that's what the "pair of trousers" picture is conveying (in a very oversimplified way--a more realistic picture would have the legs of the trousers twisting around each other, to reflect the fact that black hole mergers usually involve two holes in orbits spiraling inward about each other, not just falling straight in).

The region inside the horizon is the problem; we don't have a good way of picturing that (beyond "inside the trousers"), unfortunately.

 

[Tomas Vencl]

It might be worth expanding on this some. Take a look at the diagram shown in the Wikipedia entry on Eddington-Finkelstein coordinates:

https://en.wikipedia.org/wiki/Eddington–Finkelstein_coordinates#Metric

The picture Thorne put in his book is basically a representation of this same diagram, with one angular coordinate put back (so each point in the diagram corresponds to a circle in Thorne's picture--in particular, the line going straight up in the diagram that marks the horizon corresponds to the cylinder in Thorne's picture that marks the horizon).

It's natural to think of the "up" direction in the diagram as pointing towards the future, i.e., in the future timelike direction; but that is only true outside the horizon. The horizon itself is null, yet it goes straight up in the diagram. And inside the horizon, straight up in the diagram is a spacelike direction! So the singularity line at the left of the diagram (which corresponds to the singularity line at the center of the cylinder in Thorne's picture) is a spacelike line, not a timelike one; it represents a moment in time, not a place in space.

This is clear

Now consider two black holes merging, where the horizon now has the "pair of trousers" shape I described. The problem now is, if each leg of the trousers has a "singularity" line in the center, as in Thorne's diagram, that line has to be spacelike (since the singularity is). But the two lines would also have to merge into one in the upper part of the trousers, and the one line would have to keep on going up, so the merged hole looks like the single hole in Thorne's picture. But that's not possible; spacelike lines can't work that way. You can't have two moments of time that merge into one, but that's what two spacelike lines merging into one would describe.

This is interesting, Thank you, I will think about it. (Maybe this tell us something important about contradiction of idea of BH interior and merging (or forming) BH process. But this not belong to this fofum, sorry.)

 

[Tomas Vencl]

More precisely, the "settling down" is a process that occupies a particular region of spacetime.

OK,yes

Undefined, because, as above, it's a region of spacetime that's involved, and there will be pairs of events within that region that have all three possible relationships to each other: timelike, null, and spacelike.

very interesting,will think...

In the "trousers" picture, heuristically, the "settling down" process occupies a region that starts on each leg a little below where they meet, makes the horizon wiggle from that point on each leg up to a little above where the legs meet on the upper part of the trousers, and then quickly narrows down towards the center of the upper part of the trousers, ending on a narrow part of the singularity at the top (this corresponds to the fluctuations in spacetime curvature caused by the merger--the ones that aren't emitted as gravitational waves--falling into the singularity). So someone who falls in well after the merger will not be able to "catch up" with the narrowing "settling down" region; that person will hit the singularity first (somewhere outside the narrow region where the "settling down" region hits it).

So, this is almost picture

 

[PeterDonis]

this is almost picture

It's the best I can do at describing what I understand the numerical simulations to be saying, without making any commitments about the details of the geometry inside the "trousers". Note, in particular, that, while in general the "up" direction in what I've described is the future direction of time, the distortion involved has to be extreme in some parts of the picture.

 

[Tomas Vencl]

It's the best I can do at describing what I understand the numerical simulations to be saying, without making any commitments about the details of the geometry inside the "trousers". Note, in particular, that, while in general the "up" direction in what I've described is the future direction of time, the distortion involved has to be extreme in some parts of the picture.

OK, thank you Peter for interesting discussion.

 

[DarkMattrHole]

Thanks all. I've looked at a number of trousers diagrams now and will dig deeper. The impression I'm getting from this discussion is that there is essentially nothing inside the horizon at all to an outside observer, not even space, space is not part of the inside of a BH, the horizon is a barrier (from outside view) - like an edge in space, but an edge that recedes as you approach it, and any time that is ticking and marking off change, must be taking place in material at or outside the horizon where spacer exists for things to happen or stop happening. How close is that? Also that lower dimensional drawings of BHs, with central singularity indicated are depicting an end to all futures, not a place, and there's no good way to draw a diagram of two BHs merging the way we can draw various diagrams of a single black hole. Things sure get interesting when all four dimensions including time get into play.

I found an interesting trousers model (trousers with pantaloons) here..
..."suggests a possibility of proving the Penrose inequality mathematically for generic astrophysical binary back hole configurations."
https://www.aei.mpg.de/2420822/what-happens-inside-a-black-hole-merger

 

[PeterDonis]

The impression I'm getting from this discussion is that there is essentially nothing inside the horizon at all to an outside observer, not even space

The fact that the outside observer cannot see inside the horizon does not mean the outside observer thinks there is nothing there.

space is not part of the inside of a BH

This is not correct. Spacetime is still 4-dimensional inside the horizon, and at every event one can still find an orthonormal tetrad of one timelike vector and three spacelike vectors, so one can still split spacetime into "space" and "time" locally. The global behavior is what is highly counterintuitive.

the horizon is a barrier (from outside view)

No, it isn't. It's perfectly possible to fall inside the horizon. You just can't get back out again once you're inside.

like an edge in space, but an edge that recedes as you approach it

I don't know where you are getting all this from, but it's wrong.

any time that is ticking and marking off change, must be taking place in material at or outside the horizon where spacer exists for things to happen or stop happening

This is wrong. Time ticks perfectly normally for an observer falling through the horizon and on inside the hole.

lower dimensional drawings of BHs, with central singularity indicated are depicting an end to all futures, not a place

Yes.

there's no good way to draw a diagram of two BHs merging the way we can draw various diagrams of a single black hole

Yes. To put it another way, even the minimum amount of distortion you can have in a diagram of a BH merger is far more distortion than you need to have in a diagram of a single BH.

 

[PeterDonis]

I found an interesting trousers model (trousers with pantaloons) here

Note that what this paper is calling a "horizon" is not precisely the same as what we have been calling a "horizon" in this thread. We have been using the term "horizon" to mean the event horizon--the boundary of the region in spacetime that cannot send light signals to infinity. When I say there is only one horizon, shaped (heuristically) like a pair of trousers, it is the event horizon I mean.

This paper is using the term "horizon" to mean "Marginally Outer Trapped Surface" (or MOTS), which is also referred to as an "apparent horizon". Heuristically, it is a surface at which a spherical radially outgoing wave front of light does not increase in area. In a stationary black hole (i.e., an idealized black hole that isn't merging with another one, and isn't absorbing any matter, and isn't evaporating), the apparent horizon coincides with the event horizon; but in any dynamic situation, like a merger, or a black hole with matter falling into it, or an evaporating black hole (when we take quantum effects into account), the two kinds of horizon no longer coincide. In the case of a merger, the event horizon will in general be outside of any apparent horizons.

The other key difference with apparent horizons is that there can be more than one of them, and they can even overlap and self-intersect, as the paper describes. So you can't really view them as a single "boundary" of a black hole, the way you can view the event horizon. But they are useful in trying to picture what goes on inside the horizon as the merger proceeds and the final hole settles down.