# Would the gravitational waves a binary black hole system make a feasible weapon?

Backstory: Okay, so we have decided to wage war on numahs for the macguffenite. We mostly are doing things like shooting lasers at each other ships, and occasionally shooting high velocity projectiles at each other. The humans than come up with a plan. The plan is to hit the numah bases with a bunch of micro black hole missiles (which have an explosive amount of hawking radiation at the end of their life) and then throw a couple planetary mass black holes to eat up their bases and suns and what not. Have way through gathering up all the resources, one of our space diplomats/explorer discovers that in space, they have a set of treaties governing space warfare (which applies whether you signed it or not.) One of the rules (which I shall conveniently translated to English for you is

No space civilization shall cause a black hole to enter a system of another civilization for the purposes of gravitationally disrupting or accreting the persons or properties of that civilization, nor shall they expose the persons or properties of any civilization to damaging levels of black hole particle emissions (hawking radiation).

So well, darn, are we were all ready to start making our black holes. It also turns out that the last civilization to violate the treaty got pelted by antimatter rail guns, so that is out of the question. What are going to do with our black holes now. Wait, is there a loophole?

Question: How feasible of a weapon would it be to set two rather massive black holes in rapid orbit around each other, for the purpose of generating gravitational waves.

• How large would the black holes have to be, and how fast would they have to spin?
• How would you prevent them from ramming into each other?
• What would be the effective range of the weapon?
• How do we not hit ourselves?
• Will it violate the space warfare treaty (the answer is yes, but the humans don't know that yet (the humans should study more carefully))
• I'm assuming that energy loss from a binary black hole system would be a decent gauge of its ability to function as a weapon. – Green Aug 3 '15 at 22:12
• See the chapter 4.2 of this article but it is dealing with 2 neutron stars instead. I'm intimidated by the mathematics, imagine what if... – user6760 Aug 4 '15 at 2:49

Green found an interesting way to calculate the energy emitted via gravitational waves. The calculations are all in the framework of general relativity and tensor calculus. A simpler way is given on Wikipedia, when calculating power: $$P=\frac{dE}{dt}=-\frac{32}{5}\frac{G^4}{c^5}\frac{(m_1m_2)^2(m_1+m_2)}{r^5}$$ We can integrate to find that the energy emitted between $t_1$ and $t_2$ is $$E_{t_1\to t_2}=\left(-\frac{32}{5}\frac{G^4}{c^5}\frac{(m_1m_2)^2(m_1+m_2)}{r^5}\right)(t_2-t_1)$$

This change in energy means that the bodies will eventually spiral together and collide.

Using the relativistic framework, we have $$P=\frac{dE}{dt}=-\int_{d\mathcal{V}}t^{0i}n_id^2x$$ Here, we are using Einstein summation notation, where $$v_a=\sum_{i=1}^{3}v_i$$ Here, $t^{\mu\nu}$ is the "effective" stress-energy tensor, instead of the typical $T^{\mu\nu}$: $$t^{\mu\nu}=-\frac{1}{8\pi}G^{\mu\nu}+\frac{1}{16\pi}H^{\mu\nu\alpha\beta}_{,\alpha\beta}$$ Fortunately, the second half of that is zero. After all the calculations, $$\langle \dot{E} \rangle=-\frac{1}{5}\langle \dddot{Q}_{ij}\dddot{Q}_{ij} \rangle$$ In an earlier lecture, $Q_{ij}$ is defined: $$Q_{ij}=I_{ij}-\frac{1}{3}I_{kk}\delta_{ij}$$ where $I_{ij}$ is the inertia tensor and $\delta_{ij}$ is the Kronecker delta, defined as $$\delta_{ij}=\begin{cases}0 &\text{if }i\neq j\\ 1 &\text{if }i=j\end{cases}$$ I suspect that you could produce substantial amounts of energy from a proper system, but I doubt you could control the waves at all. Actually, I doubt you could control the system at all, given the incredible gravity at work and the masses of the black holes (and the possible instability of the system).

• +1 I like it(mathematics), they looks intimidating! and please recommend a good translator I think google translate and bing are both not working :( – user6760 Aug 4 '15 at 2:54

I'm not sure it matters whether gravity waves make a great weapon given that the sources for those weapons will wreak havoc on anything even remotely close through gravitational disturbances of the solar systems normal orbits.

Gravitational waves are only weakly interacting compared to electromagnetic radiation. However, even weakly interacting particles such as neutrinos can be lethal in sufficient quantities.

I don't have the math to prove anything but they will be one of two scenarios.

1. Gravitational waves are insufficiently strong to do any damage at distances where gravity from the black holes doesn't kill (directly or indirectly). Thus they behave according to the same laws that govern how strong gravity is at a certain distance.
2. The waves do deliver lethal energy at distances outside the black hole's gravity kill zone.

I strongly suspect it is the first situation. If so, then an attack by gravitational waves is equivalent to an attack using black holes.

• When 2 massive objects particularly neutron stars orbiting each other rapidly as they draws closer to each other the gravitational waves will carry away some of its energy like a series of ripples in a pond. When the collision finally occurs it is thought that there will be a burst of gravitational waves carrying lots of energy away mostly from event such as supernova, it will be like a splash with a shockwave! – user6760 Aug 4 '15 at 5:14