# Energy carried by a gravitational wave

In reading about the recent gravitational wave detection results, I came across a statement that the two merging black holes converted about three solar masses worth of… uh,mass… into gravitational waves at the time of their merger. That’s a colossal amount of energy, equivalent to 15 trillion years of Solar output. This happened about 1.5 billion light years away, and the gravitational waves that reached Earth managed to warp space here by much less than the diameter of a proton. Needless to say I didn’t notice the chirp passing through last September.

What effect would these waves have had if we were closer to the merger? Would space be warped by macroscopic distances? Would molecules or atomic nuclei be disrupted? Or would the effect be more pronounced on large-scale structures held together only by gravity, like a star or planet?

I think the effect would be sort of like an earthquake in each solid object. The distances would grow and shrink between pairs of points in separate locations on solid objects, so the objects would have to do some combination of stresses and strains to accommodate this.

If you had two spheres some distance apart, and nothing to accelerate the spheres (nothing to push on them), as the wave passed by the sphere center distances would fluctuate because the space between them – the space itself, mind you – would shiver in and out.

If you had a mechanical link between the spheres, either the link would supply the forces to accelerate the spheres back and forth, or the spheres would succeed in making the link get longer and shorter, or some combination.

Whatever happened, it would be mechanical vibration in the link and in all solid objects.

According to this (equation 2) the characteristic amplitude is inversely proportional to D and the strain (dL/L) is proportional to the amplitude. I guess that means if we are 1.3 billion light years away and the strain is of order 10^{-21}, then one light year away the strain would be of order 10^{-12}.

Assuming a strain of 10^{-2} is “noticeable”, you can get that about 1000km out if my math is correct.

The relative strain is roughly equal to the reciprocal of the distance from the source, measured in terms of the radius of the black holes. That is, if you were only a few times the radius away, then your body would be stretched or compressed by a large fraction of your height. If you were 100 radii away, then a two-meter body would be stretched and compressed by about 2 cm, which would still be quite damaging. If you were 1000 radii away, then your body would be stretched and compressed by a couple of millimeters, which you might feel, but which wouldn’t be a big deal. At 10,000 times the radius, you’d be talking a stretch of a fifth of a millimeter, and good luck noticing that.

Lets say we imagine an object as being made up of a collection of particles in free-fall, where there is no gravity or other forces between the particles. The amount the body is deformed by a gravitational wave will scale with a body. So imagining a very, very strong gravitational wave that will deform a circle to an ellipse with a semi-major axis twice the radius of the circle, this is independent of the radius of the circle.

The size of the object enters into it as obviously the larger the object, the easier the deformation is to measure. Realistically in a body there will actually be forces between the particles in it and the scale will also affect the size of forces between the particles in comparison to the tidal forces due to gravitational waves and hence the actual deformation of the body.

Doing some extremely rough and ready calculations, even as close as 1 light year from the source of these two merging black holes the deformation to a circle of particles 1 metre radius would be on the order of mircometres (give or take several orders of magnitude) and that is without considering any possible forces between the particles.

Would a squishy body like a human even notice that? This isn’t like tidal forces where you are literally pulled in two directions. All your molecules just get a little farther apart for a short time. It doesn’t seem like enough to break any molecular bonds, and the forces would be distributed evenly enough that they wouldn’t snap any hard objects like bones.

A gravitational wave is essentially a wave of tidal forces. That said I have to agree with you, I don’t think in this situation, especially considering the wavelength is much larger than the human body that the effects would be true great. Perhaps it would cause motion sickness, but I wouldn’t expect too much more. In fact in the region of 10-100 radii, the “static tidal forces” would not be that much less than the tidal forces due to gravitational waves.

Fair enough; I should have said “static tidal forces”. If I am thinking of things correctly, it seems that it’s the peak derivative of the wave that more closely relates to normal tidal forces. You could experience a very high amplitude as long as the wavelength was long enough, since your body would have time to adjust and maintain the normal equilibrium. A short wavelength would be a problem since you would have a ripple of stretching bonds passing through you, and the momentum of the rest of the body wouldn’t allow a natural redistribution.

Hm, true, I was thinking in terms of putting the body on a rack and pulling it out two centimeters, which would definitely hurt… but we’re not clamping the ends of the body into a rack, here. I guess I’m not too used to thinking of gravitational waves as acting on solid objects (it’s so much easier when the ends are connected only by laser beams). Then again, that “long wavelength” would correspond to a period of only a few milliseconds, which isn’t all that long for a body to adapt.

I appreciate the responses so far. It looks like any sort of macroscopic stretching would require you to be within a relatively short distance of the merging black holes.

I think that even microscopic amounts of distortion would have major effects on a living organism, though. I’m thinking of Feynman’s ‘sticky bead’ analogy.

Even if an individual protein is only being stretched/compressed by, say, 1% by a passing gravitational wave, that’s going to result in some amount of heating, isn’t it? And as the black holes merge, the waves come faster and faster as that final ‘chirp’ approaches. Even a 1% stretch/compress cycle is going to dump a lot of heat into a body when it’s happening a thousand times a second over the entire volume of the body.

Also, the article that I read said something along the lines that a 30 solar mass black hole merged with a 20 solar mass black hole and produced a 47 solar mass black hole, with the missing mass being accounted for by the gravitational radiation that caused their orbits to decay. I guess I could understand if the kinetic energy or angular momentum of the two-black-hole system were turned into gravitational radiation, but I don’t see how the mass inside the event horizon can be ‘lost’ as gravitational radiation. Maybe calling it a black hole of 30 solar masses merging with a black hole of 20 solar masses really means something more like 28.5 solar masses and 18.5 solar masses orbiting a COM with an incomprehensible amount of orbital energy? Like, the E=mc^2 equivalent of 3 solar masses of orbital energy?

I find it satisfying and appropriate that LIGO is essentially a Michelson interferometer–the instrument that famously failed to detect any difference in the speed of light parallel vs. perpendicular to the direction of Earth’s movement. The negative result led to Special Relativity; the positive result reinforces General Relativity.

It’s wrong to imagine the stretching and compressing as something that necessarily happens to solid objects. If the stiffness of the object is sufficient, it doesn’t stretch and compress, its ends accelerate in and out instead. That is, the space goes in and out, but what parts of a solid body do are dictated by the interaction of stress and strain.

Space has no stiffness at all, but objects do.

No, the total mass within the event horizons does actually decrease. There isn’t actually any law against this. There is a law that the total area of the event horizons cannot decrease (this appears to actually be a special case of the Second Law of Thermodynamics), but that’s easily satisfied in a merger even with some mass loss, because the event horizon area is proportional to the square of the mass: In this particular case, you’re going from 900+400 to 2209, which is a very comfortable increase.

How does the squishing or stretching or whatever effect compare to G forces, like in astronaut training? Or diving underwater?

For example, a 2-centimer compression would be the equivalent of being x feet underwater or accelerating at y G forces?

I read the Times article about it yesterday. How do they know the sizes of the 2 black holes prior to their merger? Wouldn’t a variety of sizes produce the same effect?

From the precise shape of the waveform, especially right at the point where they’re merging. It’s a very complicated process which depends, to some degree, on all of the parameters of the system, and which can thus, in principle, tell you all of the parameters.

Which is not to say that it tells you all parameters equally well. IIRC, you get much smaller error bars on the total mass than you do on the two individual masses separately.

Design a theoretical device that extracts energy from the GW’s of this merger. We can locate it (for a while !) a few radii away. What’s its operating principle? Does it have springs or walls collide with moving objects in a way that adds momentum due to the gw, or what?

I am having difficulty reconciling the conversion of several solar masses into energy with the seemingly small effect very close by. A star going supernova that converted this amount of mass to light is detectable millions of light-years away. Now if all the mass was converted into neutrinos, I understand this would not be detectable. I would take this to be due to the weirdness of the Weak Force. Gravitation seems more tangible. Am I confusing detecting something massive with detecting gravitational waves?

I was also going to ask if there are any Weber resonant detectors in operation anywhere in the world. Did they see any signal? Was the amplitude or frequency of the 14 September 2015 event such that you might think a bar detector could see it?

According to the Wiki page, Weber bars have a sensitivity of 10[sup]-16[/sup]. Advanced LIGO has a sensitivity of 10[sup]-22[/sup] (and needed 10[sup]-21[/sup] to detect this event). Even if any still existed, they wouldn’t have detected the signal.

Believe it or not, there are times and places in supernovae where neutrino pressure becomes a significant part of the dynamics. Just because the weak force is named after how weak it is doesn’t mean it can’t kick gravity’s ass five ways to Sunday.