What if the two Black Holes spiraling around each other are evaporating via their Hawking radiation?

Question

A non-technical explanation

In a Gravitational Wave event, the point is that in this event when Black Holes are spiraling around each other, they will cause the very fabric of Space-time to ripple and this event is an energetic event and those ripples travel at the speed of light and we can describe mathematically that how this could happen and how these Waves of Gravity travel. Now, my question stops here and my question is that as we all know Stephen Hawking came up with a Beautiful explanation that Black Holes aren't so much Black as we thought of, in time, they can evaporate and that would bear the name, ''Hawking Radiation''. Now, this implies to Black Holes, and still we haven't detected Hawking Radiation yet and Hawking didn't had a Quantum Theory of Gravity.

So, if we consider for example, that Black Holes can evaporate, so they must evaporate when they are also spiraling and what will happen in this case? If the spiraling Black Holes can evaporate via Hawking Radiation, how will we detect it and what will the mathematical equations describing this Event be?

Answer 1

Gravitational waves are efficiently emitted by massive black holes orbiting each other - the power emitted increases with mass. Hawking radiation on the other hand is a process that increases with decreasing mass. As a result only very tiny black hole binaries would emit more power in Hawking radiation than they do in gravitational waves; at least towards the end of the inspiralling phase.

Details:

The characteristic timescale on which a black hole binary system spirals to merger is $$\tau_{\rm GW} \simeq \frac{20c^5}{256 G^3}\left(\frac{a_0^4}{M^3}\right)\ ,$$ where $a_0$ is the separation, $M$ is the total system mass and I've assumed the binary components are of equal mass.

The characteristic evaporation timescale by Hawking radiation (again assuming each black hole is of mass $M/2$, is $$ \tau_{\rm Evap} \simeq 640\pi \frac{G^2 M^3}{\hbar c^4}$$

For $\tau_{\rm Evap}$ to be shorter or even comparable with $\tau_{\rm GW}$, then $$ 640\pi \frac{G^2 M^3}{\hbar c^4} \leq \frac{20c^5}{256 G^3}\left(\frac{a_0^4}{ M^3}\right),$$ which means $$ \frac{M^6}{a_0^{4}} \leq 3.9\times 10^{-5} \frac{\hbar c^9}{G^5}$$

For black hole binaries, a reasonable value for $a_0$ would be a few times the Schwarzschild radius of the final merged black hole since most of hte power in gravitational waves is radiated in the final few orbits before merger, i.e. $a_0 \sim 10 GM/c^2$, so we can say $$ M \leq 0.62 \sqrt{\frac{\hbar c}{G}} \sim 2\times 10^{-8}\ {\rm kg}\ ,$$ which is the Planck mass.

So the only merging black holes for which energy loss by Hawking radiation will play a significant role, close to the end of their inspiral, would have a mass of a few $\sim 10^{-8}$ kg or less. But the dynamics of such tiny black holes would not be controlled by gravity at all and they would evaporate in a fraction of a second.

Answer 2

Well, I may not know much, but the other answers on this question seem very well accurate. If you can find them, there are videos that give examples of what happens when black holes get nearer and nearer to each other. They create extreme gravitational waves and bend the space-time fabric itself. Correct me if I'm wrong, but I also think that they create sounds that can be heard. I can't describe it to well, but the sound is similar to a ripping. I know it's not the greatest example, but a black hole in the Perseus cluster actually produces the B-flat note- which is 57 times deeper than the human ear can hear. The note has been sounding for millions of years.