# What happens to the angular momentum of two merging black holes?

Suppose that two black holes of roughly equal mass in a binary system, formed from say a large mass stellar binary system, are in orbits around their center of mass. Further, suppose that we are looking at their equations of motion well in advance of their merger.

As a very crude approximation, I am using the Schwarzchild metric even though I know that this is not a spherically symmetric system. The effective potential in this metric would be of the form:

$$V_{eff}=\frac{\mu c^2}{2}[-\frac{r_s}{r}+\frac{a^2}{r^2}-\frac{r_s a^2}{r^3}]\ ,$$ where $$a=\frac{L}{mc}$$, $$r_s$$ is the Schwarzchild radius, and $$\mu$$ is the reduced mass.

So, as this system loses energy via gravitational waves, they would eventually reach a circular orbit with a radius $$r_{min}=\frac{a^2}{r_s}(1+\sqrt{1-\frac{3r_s^2}{a^2}})\ ,$$ which is a local minimum in this effective potential.

However, to lose more energy and merge, there is a centripetal barrier to overcome. The only way I see around this is for the system to lose orbital angular momentum to a point where $$a=\sqrt{3}r_s$$, when this barrier disappears.

I know this is fairly crude, and in reality requires computer simulations using Einstein's field equations, but it seems to me that for black holes to merge they would need to lose orbital angular momentum as well as energy.

1. Does the system transfer orbital angular momentum to the spin angular momentum of the black holes; basically spinning them up as they approach one another? I am assuming that overall angular momentum still has to be conserved in the center of mass frame. What would be the mechanism for this transfer? Would this involve an ergosphere around each black hole?

2. Can one infer the spins of the merging black holes from the gravitational wave signal detected here on Earth?

Yes. Orbital angular momentum can and does get transferred to spin, or rotation, of the merged black hole. It probably happened on the merger seen in 2015; they were able to estimate the final spin as 0.7 of extremal, but didn't see enough of the incoming black hole waves to estimate their initial spin. I forget the exact waveform parameters they need more of, I believe also a higher SNR.

As for a limit, yes all rotating black holes have a maximum angular momentum usually denoted as a fraction of the mass, called a, where the solution breaks. In merging they can get rid of a lot of angular momentum and spins through the gravitational waves. A black hole won't form if a is greater than 1, but the horizon areas and entropy cannot be reduced, so they could also just scatter each other as whole black holes. Look for the LIGO papers for the 2015 and 16 observations and estimates of spin. Also, they expect that if eLISA ever is launched in the 20's it Will be able to estimate the incoming spins much better.

Angular momentum loss

Most of the orbital angular momentum (and energy) would be lost in the form of gravitational waves in the final stages of the merger. When the black holes are separated by of order $$\leq 10^5 r_s$$ (for stellar mass black holes), the timescale for doing so becomes less than billions of years. The angular momenta of the individual or final black holes are quite negligible compared to this until the very final stages of merger.

For example, a system of two maximally spinning black holes of mass $$M$$ have a total spin angular momentum of $$J_{\rm spin} = 2\frac{GM^2}{c}\ .$$ The orbital angular momentum of a binary system containing two such black holes separated by $$nr_s$$ is (just using a circular Keplerian orbit). $$J_{\rm orbit} \simeq 2\sqrt{n} J_{\rm spin}\ .$$ Thus even at a separation of $$3r_s$$, which is where the merger occurs, the orbital angular momentum is still $$\geq 3$$ times the spin angular momentum, even if the merging black holes had maximal spin. Hence, the vast majority of orbital angular momentum is not transferred to the spin of the merging black holes or the remnant - it is lost as gravitational waves - see Burko (2017) - $$\dot{L} = \left(\frac{32G^{7/2}}{5c^5}\right)\frac{(m_1m_2)^2(m_1+m_2)}{a^{7/2}}\ ,$$ where $$a$$ is the separation.

Inferring spin

The gravitational wave signature is to some extent sensitive to the black hole spins through something called the effective spin parameter (see Roulet et al. 2021) $$\chi_{\rm eff} = \frac{ {\bf \chi_1} + q{\bf \chi_2}}{1 + q}\cdot {\bf \hat{L}}\ ,$$ where $${\bf \chi_1}$$ and $${\bf \chi_2}$$ are the (dimensionless) spin angular momenta of the individual black holes, $$q=M_2/M_1$$ is the binary mass ratio and $${\bf \hat{L}}$$ is a unit vector in the direction of the orbital angular momentum. It is notable that theoretical models to try and account for the observed distribution of $$\chi_{\rm eff}$$ (e.g. Bavera et al. 2020) do not allow for any modification of the spin parameters after both black holes have formed and prior to these objects being detected as merging black holes - presumably because the spin angular momentum is not modified prior to the merger.