1

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

The victorious truther
  • 286
  • 4
    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? – G. Smith May 27 '20 at 19:15
  • Some older questions on this topic: https://physics.stackexchange.com/q/5456/123208 & https://physics.stackexchange.com/q/27845/123208 & https://physics.stackexchange.com/q/458136/123208 – PM 2Ring May 27 '20 at 19:22
  • @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. – The victorious truther May 27 '20 at 20:12
  • @G.Smith, Of course it's also for my own amusement. – The victorious truther May 27 '20 at 20:13
  • @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? – The victorious truther May 28 '20 at 03:57
  • The person who posted it deleted it. The equation was Nordström’s 1912 equation mentioned here. The d’Alembertian or box operator is the four-divergence of the four-gradient, and is a generalization to 4D spacetime of the usual 3D Laplacian. A relativistic gravitational field could be defined as the negative four-gradient of the potential. – G. Smith May 28 '20 at 06:23
  • @G.Smith, would it be correct to rephrase then gauss law of electromagnetism as $\square^{2} \phi = \frac{\rho}{\varepsilon_{0}}$. To Take into account the speed of propagation of the electromagnetic field. – The victorious truther May 29 '20 at 22:25
  • This isn’t called Gauss Law. It’s the inhomogeneous wave equation for the EM scalar potential. It takes into account the finite speed limit of propagation. For the gravitational potential it is a failed theory with only historical interest. No more new questions as comments please. – G. Smith May 29 '20 at 22:43

1 Answers1

0

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.

G. Smith
  • 51,534