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The first gravitational wave ever observed, GW150914, was calculated to be caused by a merger of two black holes of 36 and 29 solar masses. The resulting black hole had a mass of 62 solar masses, and 3 solar masses were said to have been radiated in the form of gravitational waves. But what about the kinetic energy of the orbiting objects just before the merger? When I substitute the Schwartzschild radii of the two black holes ($2GM/c^2$) into the formula for kinetic energy ($GM_1M_2/2(r_1+r_2)$), I find a kinetic energy of 4 solar masses. Should I add this up to the $29 + 36$? If so, the question should not be where the 3, but where the 7 solar masses went. Or do the given masses (29 and 36) already include the kinetic energy? In that case still 1 solar mass of orbital energy is missing. Did it go into the spin of the merger black hole? And is it included in the given mass of 62 solar masses? How will kinetic energy be divided between spin and radiation?

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The masses used in the gravitational wave analysis are those that the black holes would have had in their own frames in isolation. The best way to think about this is that the given masses of 36 & 29 solar masses would be the mass-energy of the black holes when they were very far apart, and had relatively negligible kinetic energy (i.e., they were moving at a speed of much less than $c$ relative to each other). At that time, the total mass-energy of the system was well-approximated by the sums of their masses, i.e., about 65 solar masses.

The black holes then spiraled in towards each other over a period of millions of years. During this inspiral phase, we can think of the motion in Newtonian terms; the individual black holes decreased their gravitational potential energy (since they slowly got closer to each other) and increased their kinetic energy (since objects that are closer together will orbit their common barycenter faster.) In this process, they were very slowly emitting gravitational wave energy, meaning that the total mass-energy of the system was very slowly decreasing. Finally, the two black holes coalesced. This last phase can really only be modeled using full numerical general relativity simulations, and I won't try to describe it here.

In this whole process (inspiral and coalescence), about 3 solar masses worth of energy was emitted in form of gravitational waves, leaving a final black hole of 62 solar masses behind. The total energy radiated as gravitational waves turns out to have been divided very roughly evenly between the initial long in-spiral and the final brief coalescence. The best estimate of the peak gravitational wave luminosity (power radiated per time) was about 200 solar masses per second, and this phase of "maximum brightness" lasted about 5–10 milliseconds. So something like 1–2 solar masses of energy were radiated as gravitational waves during the long slow initial inspiral, and the other 1–2 solar masses were radiated during the quick final coalescence.

See Phys. Rev. Lett. 116, 241102 (2016) for the gory details. (The definition of the black hole masses is in the introduction; the comparison between energy radiated in the inspiral and the coalescence is on page 8.)

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    $\begingroup$ "solar masses per second"- even as a unit for a change of mass, that's a pretty big unit. But for energy...! $\endgroup$
    – MSalters
    Commented Feb 8, 2022 at 16:51
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    $\begingroup$ @MSalters: I have a vague recollection that someone told me that the peak luminosity of GW150914 was around the same order of magnitude as the total luminosity of all the stars in the observable Universe. But I can't track that statistic down immediately, so take it with a grain of salt. $\endgroup$ Commented Feb 8, 2022 at 16:57
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    $\begingroup$ @safesphere: If you want to be that pedantic, the notion of the "kinetic energy" of black holes is problematic in GR for much the same reason that their "potential energy" is problematic. Nonetheless, thinking about things in a Newtonian way should give answers that are at least qualitatively correct; that's the way I (and the LIGO collaboration, in the linked paper) are thinking about things. If you know a better way to answer this question, please post it! $\endgroup$ Commented Feb 9, 2022 at 18:38
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    $\begingroup$ @FilipMilovanović You don’t seem to have understood my comment. Sure you can describe week gravity in Newtonian terms, but the Newtonian source of the kinetic energy (the potential energy) doesn’t exist in GR. The source of the kinetic energy in GR is completely different. It is the binding energy (mass defect). For example, if objects with masses $m$ and $M$ at infinity fall to each other and all kinetic energy dissipates into space as heat, the resulting Newtonian mass is $m+M$, but in GR, even in weak gravity, it is $m+M-K/c^2$, where $K$ is the kinetic energy dissipated away as heat. $\endgroup$
    – safesphere
    Commented Feb 10, 2022 at 9:34
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    $\begingroup$ @MichaelSeifert: businessinsider.com/… states "The storm was brief — 20 milliseconds — very brief, but very powerful," Thorne said. "The total power output during the collision was 50 times greater than all the power of all the stars in the universe put together." $\endgroup$
    – Gnubie
    Commented Feb 10, 2022 at 10:48

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