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Inspired by If two black hole event horizons overlap (touch) can they ever separate again?, I was wondering what would happen if two black holes collided into each other in a head-on collision.

In this model there's two black holes on a 3d Cartesian plane. Each black hole has a mass of 1 billion solar masses, and they're both going 0.9c. They're traveling on the z axis in opposite directions.

If both black holes are not spinning, one traveling positive z, the other in negative z, and collide at the origin in a perfect head-on collision what would be the result?

How long would a full merger take, and what would the resulting spin be? Would there be a large wave of gravitational energy, or some other emission?

I don't believe this to be a duplicate of Would LIGO Detect Head-On Collision? as suggested by @John Rennie or @uhoh beucase I'm more asking what's the effect on the BH themselves and what type of emissions are involved. Not necessarily ones detectable by LIGO.

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  • $\begingroup$ There won't be spin - impulse moment conserves also in the GR. There would be gravitational wave emission, although much smaller and a very different characteristic than the currently observed ones. The time of the full merge will be calculable by a GR free-fall. $\endgroup$
    – peterh
    Commented Jun 27, 2019 at 14:54
  • $\begingroup$ Why would it be much smaller @peterh? If the two BH were moving in polar opposite directions in a head-on collision, wouldn't there be a large wave since all that momentum just... stops? There'd have to be a release of energy somewhere. $\endgroup$
    – KGlasier
    Commented Jun 27, 2019 at 14:55
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    $\begingroup$ No. The black holes are modelled as point-like masses, with an event horizon around them. The gravitational waves are coming from the event horizons, more exactly from the space around the event horizons. No one knows, what will happen if the point-like masses collapse, because the current science stops far before that. What is known, are the known conservation laws of the GR. These don't predict very big things. However, I am just an educated layman. Hopefully a professional will come up with more convincing arguments as well. $\endgroup$
    – peterh
    Commented Jun 27, 2019 at 15:00
  • $\begingroup$ In Newtonian physics, linear momentum is a vector, it has magnitude an direction, so the combined momentum of two bodies with equal & opposite momenta is zero. In General Relativity, momentum is incorporated into the stress-energy tensor, and the mathematics is more complicated, but it's still possible for momenta to cancel. $\endgroup$
    – PM 2Ring
    Commented Jun 28, 2019 at 5:56
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    $\begingroup$ Hi @KGlasier can you take a look at all of the answers to Would LIGO Detect Head-On Collision? and mention if there are any aspects of your question not answered there? Thanks! $\endgroup$
    – uhoh
    Commented Jun 28, 2019 at 10:12

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Well the spin is easy. Your system has zero angular momentum, so the spin will be zero.

I think there will also be no gravitational radiation. for the slightly technical reason that gravitational waves only come from changes to the quadrupole moment of a system and in your system the quadrupole moment is always zero.

So everything ends up in the final black hole which will have mass about 4.6 billion solar masses, coming from the mass of the two colliding holes, and their kinetic energy.

I'm not sure exactly how long the merger would last, but it will be comparable to the Schwarzschild radii involved, divided by $c$, so around three hours (as seen by a distant observer).

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    $\begingroup$ So the kinetic energy could end up adding more than double their combined mass, rather than radiate anything out? Does this mean that in theory we'd never be able to detect if this happened outside of maybe seeing the effect of a larger black hole to its surroundings? $\endgroup$
    – KGlasier
    Commented Jun 27, 2019 at 15:08
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    $\begingroup$ @KGlasier I think that's right. It is worth noting that this scenario is unrealistic on at least three levels: more or less all black holes are spinning; it is unlikely they would collide exactly head on, rather than a bit offset; and supermassive black holes are not normally seen moving at 0.9c relative to anything much. $\endgroup$
    – Steve Linton
    Commented Jun 27, 2019 at 15:36
  • $\begingroup$ I'm not totally certain that there won't be any gravitational waves, but I don't have the skills to do the necessary calculation. True, it's a perfectly straight-line collision, but in the final moments their relative speed will be large, the BHs are traveling in curved spacetime, and they change that curvature as they approach one another. And the interaction is not spherically symmetric. $\endgroup$
    – PM 2Ring
    Commented Jun 28, 2019 at 6:01
  • $\begingroup$ @PM2Ring I'm not certain either, but I think cylindrical symmetry is enough to suppress gravitational radiation. $\endgroup$
    – Steve Linton
    Commented Jun 28, 2019 at 8:00
  • $\begingroup$ Ok. The linked dupe target says that the gravitational wave production would be small, and a fairly simple pulse, although in that question the issue of spin (or lack thereof) is not addressed. $\endgroup$
    – PM 2Ring
    Commented Jun 28, 2019 at 8:30

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