# Two black holes in rotation - Energy conservation

I was looking at this problem:

Let consider two black holes of mass $$M$$ rotating around a point $$O$$ (circular montion). The first black hole center is $$M_1$$, the second $$M_2$$. Each of them has a distance $$R$$ from $$O$$ (i.e. $$|OM_1| = |OM_2| = R$$). And they are both at oposite points on the trajectory (i.e. $$|M_1M_2| = 2R$$).

They both radiate, so they lose some energy.

Do the radius R increases or decreases? What is the total energy loss?

This problem is meant for student that doesn't have any knowledge about relativity.

First, what type of radiation could they emit?

Then, I want to answer to theses questions.

What I did:

Let $$E$$ be the energy of the system "The two black holes". We have: $$\boxed{E = E_m + U} (1)$$ where $$E_m$$ denotes the mechanical energy and $$U$$ the intern energy. The first law of thermodynamics gives: $$\dfrac{dU}{dt}= -\mathcal P_{radiation}$$ with $$\mathcal P_{radiation}>0$$.

Then, according to the conservation of angular momentum of this system, I've got: $$R^2 = \dfrac K {\omega(t)}$$ where $$K$$ is a constant and $$\omega$$ is the angular speed.

Then, after derivating $$(1)$$, $$\dfrac{dE}{dt} = -\mathcal P_{radiation} + \alpha\times\dfrac{\frac{dR}{dt}}{R} + \beta \frac{dR}{dt}\frac{d^2R}{dt^2}$$

with $$\alpha,\beta$$ positive constants.

What I don't understand:

• What is $$\dfrac{dE}{dt}$$?
• I guess the hypothesis "circular motion" means to neglect some terms, like $$\beta \frac{dR}{dt}\frac{d^2R}{dt^2}$$. But what would justify it rigorously?

What would be the answer to this problem?

EDIT 1: I think we can suppose that all the mechanical energy loss is due to the radiation. So we should have $$\dfrac{dE_m}{dt} = -\mathcal P_{radiation}$$. But I'm curious of any justification for this asumption.

• where did you get this homework, this requires advanced knowledge in general relativity, I wonder why you tagged that with Newton. dE/dt is the power (Watt) that is radiated away via gravitational waves, if you integrate that over time you get the loss in orbital energy. – Yukterez Jun 23 at 11:18
• This is an entrance exam for a french school. And that's sure they want us to solve it withoyt general relativity. But, if dE/dt is the power radiated, what is dU/dt? Where is my mistake in my energy conservation equation? – MiKiDe Jun 23 at 11:22
• dU/dt seems to be the power that is radiated away, that should be the same as dE/dt in this case – Yukterez Jun 23 at 11:25
• if it was the case, that would mean that there is no loss of mechanical energy... I'm sure I'm wrong somewhere... – MiKiDe Jun 23 at 11:27
• if the radiation is not spherical but in direction of motion the black hole would lose the same amount in kinetic energy. Are there any other hints in that examn or is this the whole text? – Yukterez Jun 23 at 11:37

I think is about mass loss due to radiation. It is loosing mass only, not mechanical energy (or indirectly due to the mass lose). That's mean your $$K$$ is not constant (because you have your mass in) but the angular momentum is still because radiation is assume to be isotropic. The gravitational potential drop. From there the solution is not very far.