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The Gravitational waves detected by LIGO on 14th September 2015 are attributed to a collision of two black holes, which had been rotating near the speed of light around each other just before the actual collision. The collision happened 1.3 billion years ago, approximately 1.3 billion light-years away and apparently, the energy of the collision exceeded the combined energy of all stars in the observable universe.

LIGO was able to detect the deformation in the fabric of space induced by the gravitational wave caused by the collision (or by the high-speed rotation of the black-holes just before the collision), even though the deformation was the magnitude of a fraction of an atomic radius. The reason why the deformation was so "small" by the time the wave had reached planet Earth was due to the relatively large distance of the collision from Earth.

Question: could someone qualified give an estimate of the magnitude of the space-time deformation we would experience on planet Earth, had the collision happened closer to us? I.e. how close would the collision have to happen for the space-time deformation to be millimeters? What about meters? At what magnitude of the deformation would the gravitational wave be dangerous or lethal for humans on planet Earth?

Bryan Greene described the gravitational wave deformation of space-time as a temporary "shrinkage" or "compression" of Earth (and everything on it). Am I right to assume that being compressed by even 1 centimeter could potentially be lethal for all live on Earth?

Jan Stuller
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    The energy of the collision could not exceed all the stars in the universe. The power might, because power is energy divided by time, so if you release enough energy in a short enough time you can get enormous power. – Ross Millikan Oct 06 '20 at 02:58
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    It isn't the collision that generates the alleged gravitational waves. The waves are apparently emitted before the collision, as the two bodies swirl around each other. – White Prime Oct 06 '20 at 06:40
  • Thank you, @WhitePrime, pls feel free to edit the question accordingly. – Jan Stuller Oct 06 '20 at 06:53
  • Thank you, @RossMillikan, also please feel free to edit the question to make it more accurate. – Jan Stuller Oct 06 '20 at 06:54
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    "Am I right to assume that being compressed by even 1 millimeter would probably be lethal for all live on Earth?" I don't think so. What does it mean for something to be "compressed" in that sense? LIGO measures a distance through empty space, but anything solid would not feel much more than not-irresistible slight tidal force that wouldn't affect it much. The electromagnetic forces holding "the thing" together are much stronger. – David Tonhofer Oct 06 '20 at 20:42

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Part of the answer is easy. The strain measured in that event was about $0.25\times 10^{-21}$. That is an object $1m$ long would be squeezed by $0.25\times 10^{-21} m$ in one direction and stretched by the same amount in the orthogonal direction.

The strain drops off linearly with distance from the black hole, so to achieve a distortion of 1mm in something the size of the Earth (ie about $8\times 10^{-11}$ it would need to be about $5\times 10^{10}$ times closer, so about $0.03$ light years away, or about 2000 AU. Two 30 solar mass plus black holes at that distance would have disrupted the solar system quite a bit before they collided.

To achieve a distortion of 1mm in something the size of a human they would need to be another factor of $6\times 10^6$ closer, so about $\mathrm{30000 km}$, closer than geostationary orbit. In this case we would certainly have bigger problems than the gravity waves.

What I don't know how to calculate is the energy absorbed by the Earth, or a human, in one of these scenarios. I suspect it would not be all that much, at least from the wave.

Added later: this answer gives the total energy passing through the Earth (about 34GJ with the black holes at their current distance) but offers no ideas for how much is absorbed. This would increase according to the inverse square law if the black holes were closer.

Steve Linton
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    The absorbed energy is probably the real hard part. The gravitational waves will have to couple somehow to acoustic waves. – fraxinus Oct 06 '20 at 08:28
  • When you say "the strain drops off linearly with distance from the black hole", why would the black holes need to be closer to affect humans by 1mm than to affect the Earth by 1mm? Humans live on Earth: therefore humans and Earth are always the "same" distance away from the black holes: shouldn't they both be equally strained? – Jan Stuller Oct 06 '20 at 08:37
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    @JanStulter The Earth is bigger than humans. The distortion of an object by a given strain is proportional to the size of the object. – Steve Linton Oct 06 '20 at 08:46
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    @SteveLinton: I somehow assumed that the distortion induced by gravity-waves is independent of the size of the object. Thinking of ocean waves, they will lift an oil-tanker the same way as a human body, when they pass underneath these "objects". I intuitively thought that gravity-waves passing through space time would "ripple" through objects the same way. – Jan Stuller Oct 06 '20 at 09:29
  • @JanStuller pretty original (and somehow reasonable) idea, but wrong in regard to gravitational waves. – fraxinus Oct 06 '20 at 10:24
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    Nice! For reference: How would a passing gravitational wave look or feel? and How close would you have to be to the merger of two black holes, for the effects of gravitational waves to be detected without instruments? and Transfer of energy from gravity back to other “more familiar” forms of energy? and (barely) Does shortening the path length of an excited etalon do work? What about LIGO? – uhoh Oct 06 '20 at 10:43
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    @JanStuller think instead of sound waves in water. The water is alternately compressed and stretched. A soft jellyfish floating in the water will be compressed or stretched proportionately to its size. Another consideration is that the wavelength is pretty large, at least comparable to the Earth. – Steve Linton Oct 06 '20 at 15:27
  • @JanStuller It's a reasonable intuitive assumption, but remember that there are actually many waves being generated and not just one. Each time the two black holes orbited each other it created another wave, so it is really a wave with a frequency (slowly increasing until the merge) rather than a single ripple – Kevin Oct 06 '20 at 15:53
  • Thank you for that remark, @KevinWells: may I then ask an additional question: if the orbiting black holes generated waves with frequency, then how come LIGO just registered one single perturbation? Perhaps the actual collision generated one large wave (ripple), whilst the orbiting generated waves with frequency, that however are not detectable by the time they reach Earth? – Jan Stuller Oct 07 '20 at 14:30
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    @JanStuller LIGO detected a brief burst fo waves. See https://www.ligo.caltech.edu/video/ligo20160211v2 for example. The intensity and frequency increase as the black holes spiral inwards, and then die down rapidly after they merge, – Steve Linton Oct 07 '20 at 15:26
  • @SteveLinton: I suppose the black holes must have been orbiting each other for many months (if not years), whilst generating gravity waves. Why did LIGO then detect only a brief burst of waves? Would that be the big wave coming from the actual collision? – Jan Stuller Oct 07 '20 at 16:43
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    @JanStuller The closer they orbit (and therefore the faster) the more intense the gravity waves they emit (by quite a lot) and also the higher frequency, which makes them easier to detect. If our detectors were more sensitive and to lower frequencies, we could detect them earlier. – Steve Linton Oct 07 '20 at 20:46