So, if two black holes, each with mass M, were moving at nearly the speed of light and collided head-on (resulting in a final velocity of zero), what would happen to all that momentum? Would the resulting black hole have a mass greater than 2M? If so, how and why would this occur?
I think I'm going to answer my own question by saying both momentum and energy are conserved. The momentum of the entire system was zero before and after the collision. Energy must also be conserved, and since the final object is at rest, all the kinetic energy gets converted into rest mass energy, minus what is radiated away as gravitational waves.
I'm not a physicist, but I took a class on special relativity in college, and I still remember some of it ... If I'm remembering it right, we still have conservation of momentum and energy in special relativity, with the caveat that these are defined differently than in classical mechanics. Specifically, E = γmc^2 and p = mvγ, where γ = 1 / sqrt(1 - v^2/c^2) and m is the invariant mass (aka the "rest mass"). [1] Note that when v=0 (so γ=1), this equation for momentum is the same as the classical p=mv, which is generally a good approximation when v << c.
So, using those relativistic definitions for energy and momentum, I think you're exactly right, at least up to the part about "since the final object is at rest". However:
- As I understand it, invariant mass, aka "rest mass" (which is equivalent to "rest energy", aka "rest mass energy"), is invariant, and it's the same before and after the collision, so the kinetic energy doesn't get "converted into rest mass energy". Rather, if the final object is at rest, then all of its kinetic energy has been radiated away; kinetic energy (E_K) is is total energy (E) minus rest energy (E_0 = mc^2, where m is invariant mass)
- I have no idea whether gravitational waves are the only way for the kinetic energy to be radiated away. I imagine other forms of energy could also be emitted.
- In order to know that the final object is at rest/has no kinetic energy (in an inertial frame), I worry that we might need to have specified more in the original question. In particular, I don't know how to handle spin. (I know that black holes have some concept of "spin", but I don't know if this is like rotational spin, or more like quantum mechanical spin, or something else, and I don't know how it figures into the black holes' total energy.) If we change the original question to say that the black holes are not spinning, then I think we can ignore this (since the collision is head-on).
Energy and momentum are always conserved in EVERY physical process. We can distinguish three types of collisions: “sticky” ones, in which the kinetic energy decreases (typically, it is converted into heat); “explosive” ones,
where the kinetic energy increases; and elastic ones, in which the kinetic energy is conserved. Since the total energy (rest plus kinetic) is always conserved, it follows that rest energy (and hence also mass) increases in a sticky collision, decreases in an explosive collision, and is unchanged in an elastic collision. The resulting black hole in other words would have way more of a mass than 2M since you're talking about a 'sticky' collision in the above instance. You can see an example of why this is in Griffiths' text (Introduction to Elementary Particles (which I highly recommend)) -- page 101 contains a great example of what happens to the mass of particles in 'sticky' collisions: https://www.hlevkin.com/hlevkin/90MathPhysBioBooks/Physics/Q...
That is about something entirely different. It more or less just says that energy might be lost if you have a flux towards infinity. It does not in any way claim e.g. that the divergence of the stress energy tensor is non-zero (which would be how I think most people would interpret energy/momentum conservation).
My hunch is they would briefly pancake and much of the mass/energy contribution from their initial velocities would dissipate as incredibly high amplitude gravitational waves from the ring-down.
It would create a universe, obviously. First all the mass would attempt being squished at a singularity. WHILE the squishing continues, the first-in-line stuff would've already started to explode back-out inside the event horizon. From the inside viewpoint, this looks like the big bang. Once all the mass from the two black holes collide and loose momentum, the inside-universe no longer expands as fast. Things wobble a bit as all this happens, creating tangles and non-homogeneity. Could be caused by initial Planck-scale uncertainties even when having a perfect head-on collision.
