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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?

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    $\begingroup$ 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? $\endgroup$
    – Brick
    Nov 5, 2020 at 1:18
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    $\begingroup$ 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. $\endgroup$
    – knzhou
    Nov 5, 2020 at 1:48
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    $\begingroup$ I'm not sure what you mean. In electromagnetism, you can have non-terminating field lines too. There's nothing forbidden about them. $\endgroup$
    – knzhou
    Nov 5, 2020 at 1:56
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    $\begingroup$ 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. $\endgroup$ Nov 5, 2020 at 2:50
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    $\begingroup$ 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. $\endgroup$ Nov 5, 2020 at 3:05

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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.

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