When does a gravitational wave beam become dangerous?

Question

A follow-up on this old question.

If you've got ridiculous amounts of energy to play with, and not much reaction mass, a photon rocket is the way to go for moving your spaceship--it gives you the best possible power-to-thrust ratio.

Based on the comments section, this apparently needs some proof, so:

The relativistic energy equation is $E^2 = E_K^2 + E_M^2 = (pc)^2 + (m_0c^2)^2$ Note that, for a fixed fungible energy budget, the more energy we lock up in mass, the less there is to allocate to the kinetic energy/momentum term. Thus, the maximum amount of momentum per unit energy, and therefore the maximum thrust per unit power, is obtained when mass goes to zero, as we are left with the luxon energy-momentum relation, $E = pc$, or $p = E/c$, and the corresponding power-thrust relation, $P = Tc$. We only use rockets that throw mass out the back because we don't have a fungible energy budget--the mass-energy of, e.g., liquid hydrogen fuel isn't something that we can easily turn into photons. If it were, it would produce more thrust! So if you have a ton of energy and not much reaction mass, it is always a losing proposition to try turning that energy back into mass.

But, there's a problem: when all of the energy in your exhaust is kinetic energy rather than mass energy, it becomes much more noticeable! To merely suspend 1 kg of mass against gravity at the surface of the Earth, you have to provide 9.8 newtons of thrust--which corresponds to 9.8 newtons * 299,792,458 m/s = ~2.938 gigawatts of radiative energy. That's one frickin' powerful flashlight, just to suspend one kilogram. Actually accelerating upward, or suspending a ship weight many thousands of kilograms, requires proportionately more power, and exawatt flashlights tend to turn anything behind them into rapidly expanding plasma. If you want to launch from a planet... or just be in a solar system where you might at some point accidentally point your engine towards a planet... that's inconvenient, to say the least. Fortunately, it turns out photons aren't uniquely special--any luxon will do, as they all have the same energy-momentum relation! Including, for example, gravitons. So, if we have a magical means of producing high-power, collimated gravitational waves, they'll produce exactly the same amount of thrust, and interact with matter much less strongly, so we don't vaporize the spaceport when we take off!

Or... do we? Gravitational waves still do interact with matter a bit, so some energy will be transferred into the ground, and the rest of the Earth, if we fire a gravitational wave beam into it. Which raises the question: just how powerful can we make our graviton rocket before it becomes dangerous?

Answer 1

I found this relevant Physics StackExchange answer from a few years ago, which references the paper Seismic Response of the Earth to a Gravitational Wave in the 1-Hz Band. Now, you can dump a lot of energy into the entire Earth without ill effects, but it turns out that gravitational wave absorption only occurs at density discontinuities--so, the surface of the Earth, and the core-mantle boundary. The Earth is supposed to absorb about $10^-21$ of passing gravitational waves at 1Hz, with absorption varying as the inverse square of frequency--higher frequencies are absorbed less strongly. So, as a first approximation, we can guess that a specific boundary will absorb one quarter of that, but this is all super-approximate, order-of-magnitude stuff anyway, and absorption will be higher for more elastic materials (like, say, squishy human flesh), so for safety margin calculations I think we can just go with "$10^-21$ of a 1Hz wave will be absorbed at each surface of a body", and it's fine if that's an overestimate.

We need about 600 degrees to start melting rocks, and the heat capacity of silica is about 733 J/kg, which works out to around 440kJ to melt a kilogram of rock as a rough estimate. If we could melt a kilogram of rock in less than a minute, that seems like a pretty decent estimate for "power level that's dangerous to the spaceport"--we don't want to melt, or significantly weaken, the launchpad underneath the rocket, for example! So, we're looking at an absorption rate of around 7.3 kilowatts. That means that, if we used 1Hz gravitational waves, we could safely push the engine power up to 7.33 yottawatts, applying a thrust of 24.45 petanewtons.

That's, uh... quite a lot. Now, there is reason to believe that, if gravitons can be artificially produced at all, they can be produced at fairly high frequencies, in the gigahertz range, which would cut down on the absorption coefficient by an additional factor of 10^18. Which pushes us well into the range of "you could pour the entire power output of the Sun into this drive and the exhaust would still not be dangerous to squishy humans", so I think we can reasonably say:

A graviton rocket will not be dangerous to anyone or anything caught in its wake, until you are at the level of needing to apply multiple gs of thrust to entire planets.