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Dear physics stack exchange,

I've been trying to consider exactly how to program a gravitational simulation with time-retarded gravitational potential fields. Its proven difficult given each time one of the source masses is instantaneously accelerated the original sources potential fields spherically vanishes at a speed "c" from the last place it had its previous velocity and a new potential field moving at the new velocity is created also at a speed "c". It would be, however, computationally extreme to separately keep track of every instantaneous new field and how it's expanding or decaying. Especially because I hope to have the field determine the movement of multiple bodies and their new velocities resulting from time delayed changes to the overall net potential field. Put simply, is there a way of calculating the time retarded potential without it being so computationally extensive? Such as translating this into maybe something like an elastic PDE that could be solved by multidimensional finite difference methods.

If anyone can assist in either minor or major ways I would be grateful.

Sincerely, a freshman college student going on sophomore year

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    $\begingroup$ Newtonian gravity doesn’t have a retarded potential, and simply adding retardation doesn’t give you Einsteinan gravity. Is your simulation supposed to have a physical basis or is it just for amusement? $\endgroup$
    – G. Smith
    Commented May 27, 2020 at 19:15
  • $\begingroup$ Some older questions on this topic: physics.stackexchange.com/q/5456/123208 & physics.stackexchange.com/q/27845/123208 & physics.stackexchange.com/q/458136/123208 $\endgroup$
    – PM 2Ring
    Commented May 27, 2020 at 19:22
  • $\begingroup$ @G.smith, i'm investigating what i've seen other papers claim is the reason for the sort of velocity distributions we get in galaxies. That being that the potential field changes at some finite speed rather than instantaneously. I'm aware that in General Relativity most of those higher retarding terms are extremely small or just cancel out. $\endgroup$
    – The victorious truther
    Commented May 27, 2020 at 20:12
  • $\begingroup$ @G.Smith, Of course it's also for my own amusement. $\endgroup$
    – The victorious truther
    Commented May 27, 2020 at 20:13
  • $\begingroup$ @G.Smith, it seems that one of the previous persons post was deleted either by me accidentally or themselves I do not know. Again, was his equation that the four gradient of the four gradient of the potential was equal to $4 \pi G p$ then you stipulated that the gravitational field be equal to the negative of the four gradient of the potential right? $\endgroup$
    – The victorious truther
    Commented May 28, 2020 at 3:57

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A completely different approach would be to integrate the Einstein-Infeld-Hoffman equations for stars in a galaxy. These equations of motion have all the right post-Newtonian corrections due to General Relativity at low velocities. They should require much less computation because you are not dealing with the gravitational field at all but rather only its effect on the point masses that are creating and feeling it.

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  • $\begingroup$ Amazing! Thank you for making this post as this is wonderfully intriguing. $\endgroup$
    – The victorious truther
    Commented May 27, 2020 at 20:38
  • $\begingroup$ However, this does not consider the time delay for gravity to travel from the source, which was the question. $\endgroup$
    – eshaya
    Commented Feb 1 at 19:54

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