# Mass Conversion from Potential Energy in Merging Black Holes

I am considering a hypothetical system of two black holes, each with a mass of 10 solar masses, uncharged and without spin (for the sake of simplification). Initially, they are separated by a distance of 1 light-year and are at rest relative to each other. Over time, they start moving towards each other due to their gravitational interaction. I'm curious about how the potential energy of this system at the moment of the black holes merging converts into mass and what would be the total mass of the resulting black hole, neglecting the radiation of gravitational waves?

I would appreciate any insights and clarifications on this topic!

• If you neglect radiation of gravitational waves, the Bondi mass of the system is constant. You thus get the trivial answer that the final mass is simply 20 solar masses. Sep 3, 2023 at 15:01
• There is no potential energy in General Relativity. The result is $M_1+M_2$ (less radiation) simply due to energy conservation. Sep 3, 2023 at 15:52

For two black holes of mass $$M_1$$ and $$M_2$$ at rest at separation $$d$$, the system's total rest-frame energy is

$$M_1+M_2-\frac{GM_1M_2}{d}$$

in units where $$c=1$$. This assumes the separation is far enough that a Newtonian approximation can be made. If gravitational radiation is neglected, energy conservation implies that will also be the mass of the post-merger black hole.

For $$M_1=M_2=10~\mathrm{M}_\odot$$ and $$d=1$$ light year, $$M_1+M_2-\frac{GM_1M_2}{d}=(20-1.56\times 10^{-11})~\mathrm{M}_\odot=19.999999999984~\mathrm{M}_\odot.$$

Note that there's no clear sense in which potential energy "converts into mass". The potential energy of the system simply contributes from the outset to the invariant (rest-frame) mass of the system. During the system's evolution, potential energy converts into kinetic energy, which also contributes to the invariant mass, such that the invariant mass remains conserved.

Let us first consider gold ions accelerated to opposite directions in two particle accelerators. The mass of each ion stays constant. The mass of the two-ion- system increases, because the formula of relativistic addition of masses says so.

When these ions get close to each other at some point, there is a large mass at that point. Nuclear reactions occur at that point. Said reactions may produce massive particles, because it does not violate conservation of mass.

Let us now consider two black holes, each with a mass of 10 solar masses, accelerating themselves to opposite directions in space, using their energy. (Not potential energy as there is no potential energy in general relativity)

The mass of each black hole decreases ("mass defect"). The mass of the two-black-hole-system stays constant, because the formula of relativistic addition of masses gives such result when the decreased masses and the increased speeds are given as parameters.

When these black holes get close to each other at some point, there is a 20 suns mass at that point. A fusion reaction will occur at that point producing a black hole with mass of 20 suns, because that reaction does not violate conservation of mass.