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Here's a simple scenario: Let's say, that there are 2 stationary planetoids of equal mass on the Earth's orbital motion path, placed at equal distances from Earth just like on the image below:

scenario

Will both planetoids start to accelerate towards Earth at the same rate or will the rate of acceleration differ for each of them? To simplify it, let's say that Earth motion path is linear.

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  • $\begingroup$ Related, but not really the same: What is a Lagrange Point? $\endgroup$
    – mmesser314
    Commented Mar 16, 2021 at 1:54
  • $\begingroup$ I would say, that Lagrange points are created due to gravitational interactions between Earth and Sun, so it's rather not directly related to Earth's orbital motion. What I'm interested in, is the influence of source's motion on the geometry of it's gravitational field, as it is observed in a stationary frame. $\endgroup$
    – Bartłomiej Staszewski
    Commented Mar 16, 2021 at 1:58
  • $\begingroup$ Taged as special-relativity but arguably ought to be general-relativity as it deals with gravity. However in terms of Earth-sized bodies there is really little point in considering relativity at all as the effects are very, very small in this case. $\endgroup$
    – StephenG - Help Ukraine
    Commented Mar 16, 2021 at 2:17
  • $\begingroup$ Yeah, it's conneted partially with SRT and partially with GRT. I've used Earth just as an example, to demonstrate the problem but there is no problem with using a more massive body, as the moving source of a gravitational field. What I'm interested in, is the influence of relative motion of a source on the geometry/distribution of it's gravitational field. I've searched the internet for anything regarding this probem, but there's literally nothing... $\endgroup$
    – Bartłomiej Staszewski
    Commented Mar 16, 2021 at 2:52

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What I'm interested in, is the influence of source's motion on the geometry of it's gravitational field, as it is observed in a stationary frame.

If Earth is considered the object the planetoids are stationary relative to at the start (your question is somewhat amniguous), then the natural "stationary frame" would be Earth which would, in that frame of reference, have no motion at all. From then on the two planetoids will simply be attracted towards Earth (and each other) equally (but in opposite directions) and their motion would be symmetric about Earth if their initial motion was stationary relative to Earth.

If the Earth is considered to be moving towards one body and away from the other, the natural stationary reference frame is the mid point between the two planetoids at the start (where Earth is at the start).

Will both planetoids start to accelerate towards Earth at the same rate or will the rate of acceleration differ for each of them?

In this case relativistic effects are completely negligible and it's simply a Newtonian gravitational problem with the two bodies in effect starting out with different velocities relative to Earth and the result is qualitatively simple : the body starting out approaching Earth hits Earth sooner that the other body which is (from Earth's point of view) moving away from it at the start. Earth's motion relative to the "stationary frame" as defined in this case would be slightly perturbed (from simple linear motion at a constant velocity) by what would be different forces from the two bodies (as they would occupy different positions at different time and hence the gravitational forces between them and Earth would be different).

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  • $\begingroup$ Thanks! That's exactly, what I'm interested in. If we'll stick to the Earth example, we can then use the frame of Sun, as a stationary one (excluding the rotational motion). However I have here 2 issuess: 1. you said, that "it's simply a Newtonian gravitational problem with the two bodies in effect starting out with different velocities relative to Earth" - if we exclude the gravity, then in the very beginning (t=0) both planetoids have the same velocity but with opposite directions of their motion in relation to Earth. $\endgroup$
    – Bartłomiej Staszewski
    Commented Mar 16, 2021 at 3:13
  • $\begingroup$ 2 You said " the body starting out approaching Earth hits Earth sooner that the other body which is (from Earth's point of view) moving away from it at the start" - problem is, that at some distance from the source object, the test object placed "behind" the source (moving away from the source) will be accelerating faster, than the source object will be moving away from it, so from the rest frame of source it will appear that both objects are accelerating towards it at different rates (with the object "in front" accelerating faster, than the one "behind" it). $\endgroup$
    – Bartłomiej Staszewski
    Commented Mar 16, 2021 at 3:13
  • $\begingroup$ And one more question: how the initial velocity of a massive object is incorporated in the Newton's formula for gravity: GM/r^2 ? $\endgroup$
    – Bartłomiej Staszewski
    Commented Mar 16, 2021 at 3:17
  • $\begingroup$ @BartłomiejStaszewski Either you're using the Sun (orbital motion is important) or it's linear motion with no Sun. Those are two completely different problems. $\endgroup$
    – StephenG - Help Ukraine
    Commented Mar 16, 2021 at 3:45
  • $\begingroup$ I'm just trying to simplify the problem as much as I can - so maybe the best solution, is to reduce the scenario to linear motion of the source object and some undefined stationary bystander. $\endgroup$
    – Bartłomiej Staszewski
    Commented Mar 16, 2021 at 3:49

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