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?
[Physics] Where does the kinetic energy of the orbiting black holes go after the merger
black-holesenergygeneral-relativitygravitational-waves
Related Solutions
Yes. The amount of energy radiated as gravitational waves will depend on the details of the two black holes before the merger. The answers to questions like:
- Was the orbit circular or elliptical?
- Were they spinning?
- Were the spins of the black holes aligned with the orbital plane?
will affect the energy of the gravitational waves. The most import detail in terms of energy radiated is the mass ratio of the two precursor black holes.
The final mass of the system $M_\mathrm{fin}$ is always less than the original total mass ($m_1 + m_2$), since some of the mass energy gets converted to gravitational waves, $M_\mathrm{rad}$.
$$M_\mathrm{fin} + M_\mathrm{rad} = m_1 + m_2 $$
Because of something akin to the second law of thermodynamics the final black hole must be bigger than the biggest original black hole. Basically, you can't radiate so much energy in gravitational waves that a black hole shrinks.
$$M_\mathrm{fin} > m_1 \quad \mathrm{and} \quad M_\mathrm{fin} > m_2$$
We can define the fraction of mass radiated aways as:
$$ e = \frac{M_\mathrm{rad}}{m_1 + m_2} $$
This is sometimes called the efficiency of radiation.
If the black holes have about the same mass (as they did in the LIGO detection), about 5% of the total mass will be radiated away. This is the most efficient possibility.
On the other extreme imagine the case where one black hole is way more massive than the other: maybe 1 million solar masses and 1 solar mass. In order to follow the two rules stated above, $M_\mathrm{fin}$ is less than 1 million and one solar masses and greater than 1 million solar masses. In this case the efficiency would be about $e=10^{-6}$ or 0.0001%. Extreme mass ratios produce the weakest gravitational waves.
You've forgotten an important player in the system: the gravitational field.
Here's a pretty argument that gravitational fields are physically meaningful objects that carry energy: imagine two masses accelerating towards each other from rest, from a great distance away. The rest energy of the system is $E_\text{rest} = (m_1+m_2)c^2$; the kinetic energy is $K\approx\frac12m_1v_1^2 + \frac12m_2v_2^2$, at least while things are nonrelativistic, and only increases as a function of time. We introduce an internal energy $U=-Gm_1m_2/r$ so that we can make statements like "the total energy of the system is constant in time."
Now let's make partitions of our system to see whether we can account for everything. Looking only at the first particle, we see a total energy $E_\text{1} \approx m_1c^2 + \frac12m_1v_1^2$ which starts off positive and grows larger in time. Looking only at our second particle we also see a total energy which starts off positive and grows larger in time. So apparently if we only consider the particles in our system, we can't duplicate our statement that the total energy of the system is a constant in time. We need also to account for the energy tied up in the interaction between the two particles: the gravitational field. In electrodynamics and in general relativity you learn to actually compute how much of this interaction energy $U=-Gm_1m_2/r$ is found in any particular volume of the space around your interacting objects.
When objects emit gravitational radiation without colliding, that radiated energy comes from the gravitational field. Perhaps better, gravitational radiation is a redistribution of the energy stored in the gravitational field: energy is removed from the field near the interacting particles, leaving them more tightly bound to one another, and appears at large distances from them, where it can do things like move interferometer mirrors.
When you have nonrelativistic objects collide, you have conversion of gravitational energy into other forms of internal energy, like heat; this is why asteroid impacts can melt things. Eventually the heat gets radiated away, too.
A black hole is an object whose total energy is stored in the gravitational field --- we talk about a black hole's mass as a shorthand for how much of this gravitational energy there is.
Best Answer
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.)