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I am a bit confused about this situation: according to general relativity, when two masses orbit one another, they emit graviational waves, which carry away certain energy. For example, check out this lecture notes. However, each mass (assuming it is a point mass) follows a geodesic, and so it is not accelerating in their local frame. So each mass individually should not produce radiation, at least when seen from their local frames. So what is going on here? Do they emit gravitational radiation or not? I would like to understand this from a more matehmatical point of view (what equations to use), and also conceptually (what exaclty is going on wrong with the reasoning).

I know that a similar question was asked here but I do not really understand the answer given there and that thread seems to be dead.

Interestingly, I found that Rovelli has a similar question in his new introductory book to GR but he does not answer it.

Any advise would be very helpful!

Glorfindel
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curiousphilosopher
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  • related https://physics.stackexchange.com/questions/480249/proving-that-test-particles-in-gr-follow-spacetime-geodesics . I agree with the answers statement "A core aspect of the problem is that point particles do not make sense in general relativity (or similarly non-linear theories)." one needs quantization of gravity to avoid singularities – anna v May 09 '22 at 13:55
  • But what if we use small particles then? You could reformulate my question using small (not point) particles. What would be the answer in that case? – curiousphilosopher May 09 '22 at 14:09
  • compllicated . see this https://hal.archives-ouvertes.fr/hal-01571766/document – anna v May 09 '22 at 14:18
  • I am not sure if it will help you but for gravitation matter is merely a distortion of spacetime that is mathematically solution of Einstein field equations. Geodesics are actually not particle trajectories but "straight lines" in it. However, a geodesic is often interpreted as a "world line" of free-falling particle under the silent assumption that its mass (energy) does not affect the spacetime geometry. It is a good approximation for a small planet orbiting a big star (no gravitational radiation) but not for two comparable stellar objects emitting gravitation waves. – JanG May 09 '22 at 18:19
  • @annav thanks for the document, I will check it out. Jan, I was assuming that a body in GR will follow geodesics to an approximate degree (for example, usually textbooks say that the Earth is following geodesics). Would you say this is not really true? Even if it is not true, I could imagine that in the frame of the object, the object has no acceleration, right? So how does radiation emerges then? Thanks! – curiousphilosopher May 09 '22 at 19:45
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    "according to general relativity, when two masses orbit one another, they emit graviational waves" -- I believe this is only according to linearized general relativity, which is not General Relativity. I think nobody has ever convincingly analyzed two body problem in full GR and shown emission of gravity waves, and loss of energy. It is mathematically amazingly difficult.

     – Ján Lalinský May 09 '22 at 23:49
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    @JánLalinský You find numerical relativity simulations unconvincing? – TimRias May 10 '22 at 11:48
  • @mmeent numerical simulations of differential equations on discrete computers can be interesting and important effort, but they are almost never a proof. One would need to prove that total error of numerical integration of the discretized model is under control. I do not know if the breakthrough in numerical relativity in 2005 supplied such a proof. Maybe it did and then it would be convincing. – Ján Lalinský May 10 '22 at 13:25
  • Thanks for your comments @JánLalinský. I do notice that when discussing this problem, most people cite Landau & Lifshitz treatment (in chapter 11), or similar treatments that appeal to certain approximations and make use of pseudo-tensors. I also know some work that appeals to full GR to explain gravitaitonal waves in the case of asymptotically Minkowski spacetimes (see https://arxiv.org/abs/2201.11634). However, they do not treat the two body problem! I was wondering if you know any resource ellaborating your point further (that nobody has shown gravitational radiation in two bodies in GR). – curiousphilosopher May 11 '22 at 11:20
  • @curiousphilosopher Michal Krizek's https://users.math.cas.cz/~krizek/cosmol/pdf/B111.pdf , from 2019. In section 1 he claimed that there are "no suitable guaranteed error estimates for numerical solutions". Of course, it is risky to claim something does not exist. Maybe there were such estimates, or are now. – Ján Lalinský May 11 '22 at 15:03

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(...) according to general relativity, when two masses orbit one another, they emit gravitational waves, which carry away certain energy. However, each mass (assuming it is a point mass) follows a geodesic, and so it is not accelerating in their local frame. So each mass individually should not produce radiation, at least when seen from their local frames. So what is going on here? Do they emit gravitational radiation or not? I would like to understand this from a more mathematical point of view (what equations to use), and also conceptually (what exactly is going on wrong with the reasoning). (...)

It is the system of orbiting masses that emits gravitational waves and not the point masses themselves. They follow geodesics of the system spacetime which is solution of Einstein field equations for the two-body problem. Contrary to Kepler’s problem (Newton’s gravity) there is no stationary solution there. The spacetime of a multi-body system is always transient and includes radiation (see https://en.wikipedia.org/wiki/Two-body_problem_in_general_relativity). The geodesics in such a system are by the definition transient, too. Therefore, after some time, the gravitating bodies will crash together, due to energy loss caused by the emission of gravitational waves. By the way, this effect was the first indirect evidence for gravitational waves discovered by Russell Alan Hulse and Joseph Hooton Taylor, Jr. in 1974 for that they earned 1993 Nobel Prize in Physics. In the case of the Sun Earth system these gravitational waves are very weak. The energy loss is about 200 watts and thus not really detectable (see https://www.quora.com/Does-a-freely-falling-point-particle-emit-gravitational-waves).

JanG
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  • Thanks for your answer and for the very useful references! So, to make sure I got your point right, you are saying that if one has a spacetime with just two masses orbiting one another, then the geodisics of that spacetime are changing in time (you said they are transient)? I checked the Wikipedia article you sent, but I am not sure what equation shows that the geodisics are transient (or that the spacetime is transient). Thanks again – curiousphilosopher May 11 '22 at 11:12
  • You are welcome. Your question was indeed interesting. The equation I have meant is of course the famous $G_{\mu\nu}=\kappa~T{\mu\nu}$ applied to a (toy) spacetime generated by two masses. Such a case can be solved only numerically and fully transient (not static). Transient spacetime means transient geodesics. In a typical planetary system that effect is negligible but in case of orbiting black holes or neutron stars will be quite pronounced. I am not really expert on this field but it is what I think is a correct answer to your question. – JanG May 11 '22 at 14:36