Relativistic Simulation of Charged Particles: How a Physicist Would Go About It

[Jeronimus]

Supposed i wanted to do a relativistic simulation of charged point particles moving at different velocities and interacting with each other.

My simulation would give me the x,y,z coordinates of each particle seen from an arbitrary observer's point of view, at a given t.
The t given however, is the simultaneity axis.
Basically a 3D slice of 4D spacetime.

The problem i would be facing, as it seems to me, is that when i compute the next slice, i would use charged particles located on that simultaneity axis.

But would that be the right way to go about it?

Creating a simulation for the twin paradox, it appeared to me, that the simultaneity axis, while mathematically and physically well defined, does not really represent the "present" or what we call as "now" but only serves as a point(axis) of reference.
Events on the simultaneity axis do happen, but only the even at x=0 is an even which happens in the "now".

When computing the next 3D slice of my simulation however, i would have to use events that are in the "present" or wouldn't I?

Maybe my approach is corrupted altogether, so maybe someone can shine some light on this.

How would a physicists go about creating the most accurate relativistic simulation of charged particles in empty space?

 

[Ibix]

The obvious solution is to build yourself a block universe. Pick a frame and run the whole simulation, recording x,y,z coordinates for each time step. Then pick the observer's frame and use the inverse Lorentz transforms to work out where on each particle's worldline is "now".

@m4r35n357 may have useful ideas.

 

[Jeronimus]

The obvious solution is to build yourself a block universe. Pick a frame and run the whole simulation, recording x,y,z coordinates for each time step. Then pick the observer's frame and use the inverse Lorentz transforms to work out where on each particle's worldline is "now".

@m4r35n357 may have useful ideas.

No, that is not the problem. I already did that with my simulation of the twin paradox.

The problem is computing the interaction between particles, attracting and repelling each other. Which 3D slice of the universe to use for computing those interactions.

 

[Ibix]

Any will do. The only reason to prefer one over another is if the maths is easier, perhaps because there are symmetries in that frame. It sounds like you are considering arbitrary charges, though, so you can't assume symmetry.

I presume you aren't just hoping to use Coulomb's law in some frame.

 

[m4r35n357]

The obvious solution is to build yourself a block universe. Pick a frame and run the whole simulation, recording x,y,z coordinates for each time step. Then pick the observer's frame and use the inverse Lorentz transforms to work out where on each particle's worldline is "now".

@m4r35n357 may have useful ideas.

Relativistic charged n-body simulations are beyond my reach, I'm afraid!

A little googling led me to this approach, but it's not really my cup of tea. Hope it helps.