[David Cooper (OP)]

Picture a situation where two black holes are moving in opposite directions at just a fraction under the speed of light. Their event horizons are length contracted so much that they practically become discs rather than spheres. The two black holes pass each other in such a way that at the point of closest approach the edges of the two discs pass through each other. (Picture one moving towards you and the other moving away, and imagine them side by side as they pass each other - both should be deflected sideways by each other's pull, and the amount of that deflection will be related to how fast they're moving.) Now, if we have a photon moving perpendicular to that action (you should visualise this as moving straight upwards) such that it passes through the edge of both discs exactly at the moment when they pass through each other, it won't be directed towards either black hole and should just continue on up the same path as it was on before, exiting both event horizons simultaneously.

[Eternal Student]

Hi.

The two black holes pass each other in such a way that at the point of closest approach the edges of the two discs pass through each other.

     The first problem is that conventional Black Hole solutions  (e.g. Schwarzschild  or  Kerr–Newman Solutions)  just won't continue to apply (they won't even hold as a rough approximation) when two black holes come into such close proximity.
     As such it's not obvious that there would be two separate event horizons, or that they would pass through each other.   As the Black Holes approach each other, a different solution to the Einstein Field Equations would be exhibited.  I can barely guess what that might look like but our best models are going to be those used for orbiting black holes that eventually merge  (e.g. the sort of thing that LIGO has observed).   
     To paraphrase this,  a black hole is not a physical object like a big particle of matter.   It is a much more intangible thing,  it's just an unusual curvature in spacetime.   When two black holes get close we can't assume anything like two Black Holes continues to exist.   Instead you get a different thing, a new spacetime curvature appearing.

Now, if we have a photon moving perpendicular to that action (you should visualise this as moving straight upwards) such that it passes through the edge of both discs exactly at the moment when they pass through each other,...

     The problem is that there wouldn't be two recognisable EH discs in this region of space.   There should be just one bigger joined up event horizon surrounding an unusual gravitational source (a source that was previously recognisable as two separate black holes when they were further away from each other).
     Keeping everything simple:  Any photon that enters the region bound by that (new, conjoined and bigger) event horizon should not be getting out again.   (To the best of my knowledge - although I'm no expert in this.   With things like Hawking radiation, a similar photon might appear in the region around the event horizon if the two original black holes do manage to pass each other instead of merging and become recognisable as separate black holes again travelling away from each other).

Best Wishes.

[Halc]

Can a photon escape from inside the event horizon of two black holes?

It wouldn't be an event horizon if it could, so no, by definition.

Picture a situation where two black holes are moving in opposite directions at just a fraction under the speed of light. Their event horizons are length contracted so much that they practically become discs rather than spheres.

You're using special relativity concepts to describe something gravitational, which isn't the way to go about it. A stress energy tensor might describe the situation. You're picturing a Schwarzschild black hole in a non-Schwarzschild situtation. That doesn't work.

The two black holes pass each other in such a way that at the point of closest approach the edges of the two discs pass through each other.

Translation, two black holes with high relative velocity and similar Schwarzschild radius r pass at perhaps less than 2r of each other (coordinate separation?). Event horizons can't 'pass through each other'. Either there could be events between them from which light could escape or the event horizons merge, and it must become one black hole.

Picture one moving towards you and the other moving away, and imagine them side by side as they pass each other - both should be deflected sideways by each other's pull, and the amount of that deflection will be related to how fast they're moving.

Yes, a plot of their mass centers will curve. There will be a sort of coordinate distance between them at closest approach, and we assume that they're far enough aprat that they don't merge. You're at that midpoint, and by symmetry, you go nowhere. You live to see the day and tell others about it, so light has escaped from you, and thus you've never been within either event horizon.

[David Cooper (OP)]

     The first problem is that conventional Black Hole solutions  (e.g. Schwarzschild  or  Kerr–Newman Solutions)  just won't continue to apply (they won't even hold as a rough approximation) when two black holes come into such close proximity.
     As such it's not obvious that there would be two separate event horizons, or that they would pass through each other.   As the Black Holes approach each other, a different solution to the Einstein Field Equations would be exhibited.  I can barely guess what that might look like but our best models are going to be those used for orbiting black holes that eventually merge  (e.g. the sort of thing that LIGO has observed).

Clearly if it's possible for the two black holes to pass each other at very high speed without merging if their event horizons touch, then they will for a moment have a shared event horizon with two singularities in it with the shape still being that of two discs. What I expect to hear though is that there is no way for them to pass each other if this happens and that they will always merge into a single black hole, so what I really need to know first is if it's possible for such a merger to be avoided in a case with ridiculously high speeds of travel.

In the case of a photon getting close to a black hole, it can escape even if it travels close to the event horizon, but a black hole cannot copy that as it is increasing the depth of the gravity well there with its own mass, leading to a radically different situation. That increase in depth should also lead to the event horizons warping out to touch each other and join up for a moment even in cases where it initially looks as if they'll miss, so it isn't sufficient just to consider cases where the singularities are clearly on a path to being be separated by the distance of the addition of their initial radii because paths leading to greater separations than that can lead to event horizons touching too, but again in every single case where they touch I expect the answer to be that the black holes have to merge.

[David Cooper (OP)]

You're using special relativity concepts to describe something gravitational, which isn't the way to go about it.

I'm not. I'm running a different model which matches the predictions of both STR and GTR for all predictions of externally observable events. (In that model, black holes are full of stuff all the way through rather than having singularites though, so they would physically collide.)

Translation, two black holes with high relative velocity and similar Schwarzschild radius r pass at perhaps less than 2r of each other (coordinate separation?)

No - they can pass at greater than 2r from each other with the event horizons warping out towards each other to join up.

Event horizons can't 'pass through each other'. Either there could be events between them from which light could escape or the event horizons merge, and it must become one black hole.

This is the whole point of the question. They join up, and what I want to know is whether in all such case that leads to the black holes merging completely or if the singularities can remain just far enough to avoid that. I'd be very surprised if cases up to 2r separation haven't all been checked already and found to result in merger, but I want to make sure that all cases greater than 2r separation where the event horizons meet by warping towards each other also result in total merger.

There will be a sort of coordinate distance between them at closest approach, and we assume that they're far enough apart that they don't merge. You're at that midpoint, and by symmetry, you go nowhere. You live to see the day and tell others about it, so light has escaped from you, and thus you've never been within either event horizon.

There may be room for an argument about whether the event horizons join up for a moment or not in a case where the black holes don't merge and go their separate ways afterwards, but symmetry enabling your escape would not keep the event horizons apart in any case where you're at sufficient depth in the combined gravity wells of the two black holes for your clock to stop.

[David Cooper (OP)]

There may be room for an argument about whether the event horizons join up for a moment or not in a case where the black holes don't merge and go their separate ways afterwards

No there isn't. An event horizon isn't a location in space. It's a null surface, and there's no way for two null surfaces to touch and then separate. For example, it's not possible for the future light cones of two spatially separated events can intersect and then later on not intersect.

In which case you're describing a case that fails to meet my conditions.

[Bored chemist]

the edges of the two discs pass through each other.

I really don't think that can happen.
The notable thing about the event horizon is that anything in it gets "trapped" in that black hole.
If you have overlapping event horizons, how would anything "know" which hole to fall into?

As far as I can tell (and I simply wouldn't know where to start with the maths), the only way to address that is to say that if the event horizons meet, then the holes must merge.
That way our unfortunate particle that is inside both, only has one hole to fall into.

[David Cooper (OP)]

If you have overlapping event horizons, how would anything "know" which hole to fall into?

It wouldn't fall towards either singularity, but it would be inside the event horizon of both. The question here is whether the two singularities have to merge or if they can move further apart and decouple their event horizons. Here's the thing - I've been imagining tiny manufactured black holes which can be guided by moving masses near them while those masses can be held apart such that their locations can be controlled at all times with precision. We may be able to use these extra masses to prevent the black holes from being diverted off straight paths as they pass each other or at least reduce the bending of their paths enough to prevent their merger. If they fail to merge after their event horizons have connected up, those event horizons can disconnect and the photon can escape.

(Clearly those masses would be much bigger than the black holes and couldn't be moved fast enough to provide the necessary control, but I'm also imagining long chains of black holes in between the end masses with the two chains of black holes meeting at close to the speed of light in opposite directions such that each black hole in the central region of each chain always has an approximately equal pull on it from either side. Have all possible scenarios of this kind been checked carefully enough to be sure that event horizons touching always lead to their singularities merging?)

The reason I want to check whether this might or might not be possible is that it would potentially enable a difference between theories to be tested by experiment. The fuzzballs of string theory (which are like the black holes in LET) are full of stuff all the way through from the event horizon at one side to the event horizon at the opposite side of the black hole, whereas in GTR there is nothing physically there because the material continues to fall on down to the singularity. If you smash the edges of two black holes together, you don't normally get the chance to see if they are made of lots of stuff or consist of nothing because no evidence ever escapes to tell you, but if the black holes fail to merge, you could then access that evidence from a safe distance.

[Bored chemist]

It wouldn't fall towards either singularity, but it would be inside the event horizon of both.

In which case it must remain in both because it can't leave the event horizon of either.
That means they can not separate.
And that means they must merge.

Once the EH overlap, there's no force that can possibly separate them. Your idea of "

tiny manufactured black holes which can be guided by moving masses near them while those masses can be held apart such that their locations can be controlled at all times with precision.

breaks down at that point.

[Eternal Student]

Hi.

Sorry, this is going several posts back....

Clearly if it's possible for the two black holes to pass each other at very high speed without merging if their event horizons touch.....

    I'm not sure what this sentence or group os sentences was saying.   There's an  "IF"   in it.    Are you asking if this is possible,  or is the IF an accident and you are telling us it is possible and then the rest of the stuff described in the sentence does happen? 

Best Wishes.

[Bored chemist]

Hi.

Sorry, this is going several posts back....

Clearly if it's possible for the two black holes to pass each other at very high speed without merging if their event horizons touch.....

    I'm not sure what this sentence or group os sentences was saying.   There's an  "IF"   in it.    Are you asking if this is possible,  or is the IF an accident and you are telling us it is possible and then the rest of the stuff described in the sentence does happen? 

Best Wishes.

The trouble is it's like saying "if circles are square then pi is 4".
I don't need to worry about how you define the diameter of a square, because the idea already failed at the first hurdle.
I don't really need to worry about the "If".

[Eternal Student]

Hi again,

I don't really need to worry about the "If".

   OK,  I was just wondering how best to start replying.   I don't want the replies to seem like outright disagreement throughout.   

Let's start as far back as possible:

Picture a situation where two black holes are moving in opposite directions at just a fraction under the speed of light. Their event horizons are length contracted so much that they practically become discs rather than spheres.

    No, I don't even agree on this.
    Firstly the black holes aren't intrinsically changed into discs.  At best that's only how it will be for a distant observer that is not moving with the black hole.   Moreover, the black hole becomes a "disc" only in some co-ordinate system which the distant observer KNOWS is almost completely useless in the region of space that is not local to the observer and yet they persist in assuming distances are determined by the Minkowski metric.   In particular the metric is very different around the black hole.     
   Special relativity only gets you so far,   let's do it and see:    Let's have a black hole moving along the x-axis and centred on it.  Let's have an observer fixed at x=0, they assign x-axis co-ordinates  x₁  and  x₂  to the two sides of the black hole at a fixed time.    The observer never learnt any General relativity so they assume the black hole has a width along the x-axis of  x₂ - x₁.    We'll have another observer but this one will be moving with the black hole along the x-axis but starting as usual at x=0 when t=0.   That observer assigns co-ordinates x₁'   and x₂'  to the two sides of the black hole and, exactly as you might expect from special relativity, x₂' - x₁' is going to be slightly bigger (slightly wider) than the other observer reported.     The second observer is only using special relativity so they will assume that the width of the black hole was  x₂' - x₁'.   We're done.... the two observers have different widths and the black hole looks like a disc to one of them and a perfectly round sphere to the other.
     Now let's add some general relativity:   The Minkowski metric doesn't apply when you get too far away from x=0 because the space between the observer and the black hole (and also across the black hole) wasn't flat Minkowski space.    So the distance between  x₁ and x₂ on the x-axis is NOT equal  to  x₂ - x₁    when x₂  and x₁  are far away from the observer and close to the black hole.
       So, just to be clear,  the two observers have a different value  of   (x₂ - x₁ )  and one observer has a smaller value but neither of these quantities was the physical width of the black hole anyway.

IS THIS IMPORTANT?   No, not on it's own.   It's just one example where not using general relativity can lead us off the tracks early on.     Just because a conventional solid object would show length contraction when it is moving relative to an observer (in flat Minkowski space) it doesn't follow that a black hole would show the same sort of length contraction.

Does a black hole become flattened and disc shaped when it is travelling relative to a distant observer?   I honestly don't know.   I know that it's not obvious.  Shapes aren't what you might have thought in curved and non-Euclidean space.   For example, even if a black hole is considered to be stationary relative to a distant observer, it's still not like a sphere as the distant observer would imagine one in 3-D Euclidean space.  A black hole has no spatial centre, there's no place in space where I can put the point of my compass and start to draw a perfect circle around the black hole.

Best Wishes.

[David Cooper (OP)]

It wouldn't fall towards either singularity, but it would be inside the event horizon of both.

In which case it must remain in both because it can't leave the event horizon of either.
That means they can not separate.
And that means they must merge.

That's an assertion which may be correct, but I just want to check that it's guaranteed to be correct.

If there's another black hole to the other side of each of the original two, that might make it less easy for the two in the middle to spiral round each other and merge. If we have two long lines of black holes running into each other (with each black hole aimed at the open spaces between black holes in the opposite line, all the event horizons would link up into one with lots of singularities within it, while each of those singularities is moving at a fraction less than the speed of light in the opposite direction to the singularities to either side of it with an equal pull on it from either side leading to it carrying on along a straight path, so can they really be halted quickly enough to stop them separating again? (It wouldn't be an equal pull to either side for the ones towards the ends of the line, but it would be very close to equal for the central ones.)

_____________________________________________________________________________________

Firstly the black holes aren't intrinsically changed into discs.  At best that's only how it will be for a distant observer that is not moving with the black hole.

If you are observing these and see the black holes approaching each other at these high relative speeds which you measure as a fraction less than c in opposite directions, you will also measure the event horizons to be contracted so strongly that they are almost turned into flat discs, exactly as a spherical planet would be when observed to be moving at such a speed. For that not to happen, absolute speeds could be measured by the failure of those objects to conform to the maths of relativity.

Next, the complications of spacetime bending do not alter that picture at all: LET and GTR map to each other perfectly when it comes to their predictions about observations (until you're inside a black hole), and LET achieves this while maintaining Euclidean geometry throughout - it has the speed of light reduce in gravity wells instead of cramming extra space in there to make light travel further without slowing down, so we get a fully valid picture of events when we imagine two lines of discs passing each other with their edges momentarily touching, and there is very little impact on the space ahead of each disc as it travels along as all of that influence has been contracted down to a tiny distance too - the entire gravity well is contracted. This is not disputed in physics - it is what would be observed under LET, STR and GTR. You can certainly calculate under both LET and GTR that the black holes look spherical to observers moving at the same speed and in the same direction as them, but that does not negate the fact that they would be measured to be almost completely flat discs by our original observer. The edges of those discs would meet with practically no response to each other until the very last moment.

[Bored chemist]

That's an assertion which may be correct,

It's not an assertion, it's a deduction.
Which bit do you disagree with?

[Eternal Student]

Hi.

If you are observing these and see the black holes approaching each other at these high relative speeds which you measure as a fraction less than c in opposite directions, you will also measure the event horizons to be contracted so strongly that they are almost turned into flat discs, exactly as a spherical planet would be when observed to be moving at such a speed.

   and also,

You can certainly calculate under both LET and GTR that the black holes look spherical to observers moving at the same speed and in the same direction as them

    There is a difference between how things "look" and how they are.  You might be mixing the two.
I agree that a Black hole looks like a sphere to a distant observer moving with the Black Hole.   However, it is not actually a sphere as you would imagine one in 3-D Euclidean space.   The distant observer knows this if they apply the right metric.   For example, the Schwarzschild metric for a black hole where I've set the Schwarzschild radius to 1 unit is given by:

ds² =     +  terms involving the other co-ordinates  t,θ, φ

    As I mentioned, the Schwarzschild radius was set to 1 unit and a distant observer might very well see the black hole as a sphere with a radius of 1 unit.   For large r  (a long way from the black hole) then the r co-ordinate does behave exactly like a measure of the radial distance from the centre of the black hole.  In the local region around the distant observer, these co-ordinates are describing space perfectly adequately and naturally:  To the distant observer (r, θ, φ) act like spherical co-ordinates describing flat space (the metric would be the ordinary Euclidean metric) with r as the radial distance from the black hole and θ, φ as the polar and azimuthal angle from the black hole.
    Now, if the distant observer plots the location of the black hole event horizon on a piece of paper using these as if they were spherical co-ordinates in ordinary flat space then they do indeed produce a plot where the black hole looks like a sphere with radius 1 unit centred at r =0.   
    However, the physical distance from  r = 1   to r=0    is not 1 unit of length  (i.e. the physical distance from the Event horizon to where the observer would naively consider the centre of the sphere to be located).   This is because these locations are NOT local to the distant observer and the metric has long since stopped approximating the Minkowski metric when r is small.

   The path length   from r=1 to r=0  (holding constant t, θ and φ) is determined (as usual) from the metric shown earlier:

     =   

   Since  r ≤ 1   throughout that integral we are taking roots of a negative quantity.   So the integrand is imaginary and hence the total path length is imaginary.    Most significantly, it is certainly not +1 unit, the Schwarzschild radius.

Best Wishes.

[David Cooper (OP)]

That's an assertion which may be correct,

It's not an assertion, it's a deduction.
Which bit do you disagree with?

If the two singularities move further apart by continuing in the direction they were moving in at the start, the depth of that photon in the gravity well will reduce and it can end up outside the event horizon.

I've been thinking a bit more about the case with a line of black holes, and I've now found a situation where you can play with the strength of dark energy to change the size of a universe such that you can create an infinite line of black holes separated by vast distances, all moving at a tiny fraction under the speed of light, then shrink the universe down while maintaining equal distances between them, and then stop the contraction when the separation distances are just right for the edges of their event horizons to collide when the two lines pass each other. You then have all these black holes moving along parallel paths throughout the rest of the experiment as none of them are pulled more strongly to one side than the other, so they will not spiral together. All the event horizons will connect up for a moment, but then it looks as if they should separate again.

[David Cooper (OP)]

    There is a difference between how things "look" and how they are.  You might be mixing the two.
I agree that a Black hole looks like a sphere to a distant observer moving with the Black Hole.   However, it is not actually a sphere as you would imagine one in 3-D Euclidean space.

You have to remember that the 3D Euclidean space analysis of this is always valid as it matches the predictions of GTR, so the observer moving along with the black hole will measure it as spherical. What varies depending on which model you're applying is the calculated radius that would be measured from the inside because GTR rams in extra space there. What's more important here though when visualising the action is that our observer who sees these black holes moving relative to him at a fraction less than c in opposite directions will see them flattened to very narrow discs. If we make the speeds sufficiently high, we can compress the gravity wells to very narrow discs too such that the discs can have no detectable influence on each other almost up to the moment of contact. There will be a strong effect though just before the impact as the black holes finally begin to interact, but we then reach a point where it would take a detailed simulation to show exactly what would result from it. I want to see such a simulation test this. This case may have been overlooked up to now. Something extraordinary would need to happen to prevent those singularities from continuing on on their merry way at a fraction under the speed of light and with the event horizons detaching after connecting up for a moment. A possibility like that clearly needs to be explored, so I just wanted to put the idea out there. If anyone has simulation software capable of checking it, they should be able to provide a definitive answer on the matter as it will necessarily agree with STR, GTR and LET on the outcome. Doubtless the case of interactions between just two black holes will have been tested independently by many people and shown that any contact between their event horizons leads to a merger, but how much further has this been taken to test for extreme cases like the one I've devised?

[Eternal Student]

Hi again.

I can't keep up with the posts and respond to the issues systematically.   I'm just going to skip to some of the latest posts.

....  I've now found a situation where you can play with the strength of dark energy to change the size of a universe such that you can create an infinite line of black holes separated by vast distances, ....... You then have all these black holes moving along parallel paths throughout the rest of the experiment....

    You're doing an experiment?   I once mixed some baking soda with lemon juice and was amazed with the results but your experiment does sound more impressive.
  - - - - -
I think the next post may have answered this, you are looking for simulations...

If anyone has simulation software capable of checking it, they should be able to provide a definitive answer

   OK.   Regrettably I don't have any such simulation software.

Best Wishes.

[Halc]

If we have two long lines of black holes running into each other (with each black hole aimed at the open spaces between black holes in the opposite line, all the event horizons would link up into one with lots of singularities within it

Doesn't work. The correct answer involves coming up with a metric describing this that is a solution to Einstein's field equations, but that is beyond either of us. But some naive reasoning may still apply.

You have a series of masses, say 1 cm radius black holes (a bit more massive than Earth each).
There's some threshold of (coordinate) separation where the line is either a series of distinct masses, or is one large mass (regardless of the number of them that you put in the line).  So we presume the separation is greater than that, so they're spaced over 2cm apart. Any less than that and the mass of any pair of adjacent ones is greater than their mutual Schwarzchild radius since the latter is directly proportional to mass (well, at least for the two of them in isolation). So any finite line of these masses will have a Schwarzchild radius greater than the length of the line, and thus it will just be one big black hole.
So they're further apart than 2 cm.  When the oncoming 2nd line of BHs comes on, for a moment they'll be one line with half the separation between them. Same story. If that new half-separation is under 2cm, both lines become one black hole and nothing gets out. If they're still further apart than that, then none of your event horizons (assuming naively that they don't distort) will overlap.
So they pass without incident. High speed of passing doesn't help. If anything, that just adds energy and makes it more likely to be that one big BH. If slow and steady doesn't work, doing it fast isn't going to help, at least not for the line scenario.

so can they really be halted quickly enough to stop them separating again?

If they're one big black hole, then there's no meaningful coordinate 'speed of halting'. It's just there. Under the presentism that you love to push, it takes nearly infinite time for the speedy objects to come to a complete halt, assuming equal speeds/masses in opposite directions. There is probably a brief but intense pulse of gravitational waves that you'd not want to be near.

If the two singularities move further apart by continuing in the direction they were moving in at the start, the depth of that photon in the gravity well will reduce and it can end up outside the event horizon.

Said photon was never inside any EH then, by definition. See my very first sentence of my first reply. You're positing this photon outrunning a null surface, which requires it to move faster than light, a self contradiction.

[Bored chemist]

it can end up outside the event horizon.

No.
It can't.