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Answers to Where do gravitational field lines go exactly? We know where they start, but explain that for Newtonian gravity they go to infinity.

In a simple Newtonian gravity exercise there is no "speed of gravity" consideration. At any moment we can calculate the potential field from integrating $-G \rho(r) / r$ where $r$ is the distance to each element. The force field comes from the gradient of that potential and we can draw gravitational field lines in order to look at them by starting at some point and integrating the normal of the force.

There is this comment which informs us

We don't use field lines for gravity outside of Newtonian gravity. But (this) answer is also correct.

I was hoping that that comment would have evolved into a second answer which would have provided enough space in which to expand on it. Instead I'll ask here:

Question: Are gravitational field lines ever used in situations beyond classical Newtonian gravity? If so, must they still "go to infinity" or might they do more interesting things?

Qmechanic
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uhoh
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    I'm not clear that they are useful even in Newtonian gravity. It's somewhat of an arbitrary question as you have it written now. What do you really mean? – Brick Nov 05 '20 at 01:18
  • @Brick I really mean exactly what's written. The question asks if a tool that is often used (therefore some find it useful) in Newtonian physics have uses outside of it. Any introductory text in physics will show field lines, I don't see how their utility as a concept is in question. – uhoh Nov 05 '20 at 01:42
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    I mean, you can construct field lines for any vector field, including a velocity field, an electric field, a gravitational field, a polarization field, and so on. They're a generic mathematical tool for visualizing vector fields. Whether they're useful depends on what problems you're trying to solve. – knzhou Nov 05 '20 at 01:48
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    I'm not sure what you mean. In electromagnetism, you can have non-terminating field lines too. There's nothing forbidden about them. – knzhou Nov 05 '20 at 01:56
  • @knzhou See https://physics.stackexchange.com/q/278664/83380 and https://physics.stackexchange.com/a/363868/83380 for more on why I use a boundary in electrostatics – uhoh Nov 05 '20 at 02:03
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    I have to agree that this question is basically opinion based. Someone may find them useful, someone may not, someone else may sometimes find them useful and sometimes not. – StephenG - Help Ukraine Nov 05 '20 at 02:50
  • @StephenG a single example of them being used in this way, in a peer-reviewed text or publication would definitively answer this question, as would a thoughtfully constructed example. I've worded the question carefully. No opinion-based answers are sought here. – uhoh Nov 05 '20 at 02:54
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    No. An example of someone using them somewhere does not mean that doing so is useful (or "un-useful") to any particular person - that would be an opinion. – StephenG - Help Ukraine Nov 05 '20 at 03:05
  • I think a more interesting (and not opinion-based) question is whether field lines can be defined in general relativity, and if so, how. – Puk Nov 05 '20 at 03:07
  • Somewhat related: https://en.wikipedia.org/wiki/Speed_of_gravity – PM 2Ring Nov 05 '20 at 03:28
  • This community is certainly able to answer "useful" type questions: https://physics.stackexchange.com/search?q=is+useful%3F without using opinions. – uhoh Nov 05 '20 at 03:36
  • @PM2Ring Thanks. that's also related to Besides retarded gravitation, anything else to worry about when calculating MU69's orbit from scratch? whose answer led me to ask and then answer How to calculate the planets and moons beyond Newtons's gravitational force? But here my focus is only on the utility of field lines outside of classical Newtonian gravity. – uhoh Nov 05 '20 at 03:39
  • @Puk but in this heavily gamified environment the first comment under that question will be "What makes you think they could be?" followed by closure for lack of research + unclear what you're asking. Maybe you will have better luck than I; please do consider posting your interesting question! I don't think it's been answered anywhere here yet, and I think it will facilitate useful answers. – uhoh Nov 05 '20 at 04:05

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Gravitomagnetism is totally a thing. It is a weak field approximation to the Einstein equations where you get something very much like electric and magnetic field lines.

In turn, when you have suitable mass flow, such that you might have a changing gravitomagnetic field, you can get gravitational field lines which do not escape to infinity nor start on a charge but rather loop around on themselves.

So I wouldn't say that they're never used, but certainly most people you run into likely have never used this approximation.

CR Drost
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