From where does the energy for gravitational waves come from?

Question

As far as I understand, in the events detected by LIGO, about 4% of the total mass of merging binary black holes was converted to gravitational waves.

Where does this energy come from, i.e. what exactly gets converted into gravitational waves?

Is it simply the kinetic energy of the merging objects (velocities of these objects before merger are huge, up to 60% of c if I recall correctly), so does it mean emitting gravitational waves makes them orbit slower, but retain their original masses? Or do the compact objects really lose "real" mass, meaning they become lighter and in case of BHs their radius changes accordingly?

As an example, let's assume two BHs, both with 50 solar masses, orbiting each other far enough (say 1 light year) so that GWs nor kinetic energy has no significance to these initial mass measurements. During the merge, they should radiate about 5 solar masses in GWs. Would the resulting black hole have mass of 95 or 100 solar masses?

Answer 1

Radiating gravitational waves makes an inspralling binary orbit closer and faster. (Rob Jefferies)

The source of the energy for both increased kinetic energy, and the gravitational radiation is the same: gravitational potential energy. (PM 2Ring)

Two black holes at a distance of 1 light year have a huge amount of potential energy, about 10^48 Joules of potential energy. As they spiral, a significant amount of that energy is radiated as gravitational waves

This is real mass lost. The mass of resulting black hole is smaller than the sum of the two merging black holes, though at no point does any black hole itself become smaller.

Answer 2

As Rob correctly pointed out, the emission of gravitational waves reduces the orbital energy and result in an inspiral. This reduction in total energy also reduces the mass of the final BH, since $E=mc^2$. The bulk of the gravitational wave energy is emitted (and energy=mass lost) in the final chirp, when the separation approaches the Schwarzschild radius.

To quantify this, let's just make a simple energy budget calculation, starting from two equal-mass BHs of mass $M_\bullet$ orbiting each other at distance $d$ on a circular orbit. Then the orbital energy is $$ E_{\mathrm{orbit}} = -\frac{GM^2_\bullet}{2d} = -M_\bullet c^2 \frac{R_s}{4d} $$ where $R_s=2GM/c^2$ the Schwarzschild radius of each BH and we have assumed that $d\gg R_s$ such that the orbit is Keplerian. The total initial energy is then given by the rest mass energies plus the orbital energy as $$ E_{\mathrm{total}} = M_\bullet c^2 \left[2-\frac{R_s}{4d}\right]. $$ After coalescence, a remnant of mass $M_{r}$ emerges. The energy deficit is the difference between the initial and final energies \begin{equation} \delta E = M_\bullet c^2 \left[2-\frac{R_s}{4d}\right] - \frac{M_rc^2}{\sqrt{1-v^2/c^2}}, \end{equation} where $v$ is the speed of the remnant w.r.t. to the centre of mass of the progenitors. This energy has been lost by gravitational wave radiation. If this corresponds to a certain amount $\mu$ of rest mass, then from $ \delta E = \mu c^2 $ we find $$ M_r = \sqrt{1-v^2/c^2}\left[2M_\bullet -\mu - M_\bullet\frac{R_s}{4d}\right]. $$ Now for $v=0$ and $R_s\ll d$, the mass deficit $\delta m\equiv 2M_\bullet-M_r$ is identical to $\mu$: the radiated energy corresponds to the mass deficit; the final hole has 95$M_\odot$ if $M_\bullet=50M_\odot$ and $\mu=5M_\odot$. In particular, the gravitational wave energy cannot be taken merely from the orbital energy as suggested by another answer.

The mass deficit is even larger than the radiated energy if the remnant has undergone a considerable velocity kick, such that $v\neq0$ (caused by asymmetric gravitational wave radiation).

Answer 3

Two black holes starting far apart have gravitational masses $M_1$ and $M_2$ (i.e. the masses you would put into the formulae for their Schwarzschild radii).

When they merge they form a black hole with $M < M_1 + M_2$. The difference is radiated away in the form of gravitational waves.

$M_1$ and $M_2$ are "rest masses" and don't change during the inspiral, though it may not make sense to talk about the "mass of each black hole" once they get close together, since they are no longer governed by the simple Kerr metric. The system mass $M$ will get smaller as the black holes get closer together. See the discussion here.

Gravitational potential energy is turned into the kinetic energy of the black holes. The sum of their potential energy and kinetic energy is negative, as it is for any bound system, and since the orbit started with the black holes far apart with essentially zero potential + kinetic energy again, the difference is accounted for by the radiation of gravitational waves. The total mass/energy of the system has decreased.

For black holes there is no distinction between different kinds of mass and energy, they have only mass, spin and charge. In particular they do not have a baryon number. No particles are emitted and no matter is lost from the merging system, only gravitational waves.