Relativistic equivalent of a spring-force?

Question

Usually what helps me understand a concept better in physics is to write a simulation of it. I've got to the point where I'm competent in the basics of special relativity, but, I can't figure out how to write a relativistic simulation! My normal approach in Newtonian mechanics is just to attach two objects together with a hookean spring force, maybe add a damping effect, and then just to use the calculated force to get acceleration, and numerically integrate over that.

The first issue is length contraction. The issue isn't as easy as a contraction of $1/\gamma$. If I have two point events simultaneous in some reference frame, corresponding to locations of masses connected by a spring, I do know their spacelike separation, which is an invariant. However, even though the proper length is invariant, if I change reference frames, the events are no longer simultaneous, so I can't really get a consistent definition from this... I figured that I could apply the spring force instantaneously, "faster than light", and show how when I do that that leads to violation of causality, but I can't even define force consistently!

I understand that if I use a wave equation or some sort of electromagnetic force, then I can have a force field that transforms correctly, but I really don't want to do this, because I'm not great at electromagnetism, and this is really just for me to better my understanding of special relativity. Plus it would be difficult computationally.

I haven't been able to make any headway on assuming that the masses are connected by a chain of springs, whose velocities relative to each other are $<<c$, but I think the solution may lie there.

Answer 1

If you know about Lagrangian and Hamiltonian formalisms yo might try to find first the equations of motion. This is done in the paper Relativistic harmonic oscillator. In a nutshell, what is done is the following, a "relativistic" hamiltonian (for slow particles) is set up (we set c=1):

$$ H = \sqrt{p^2+m_0^2} + \frac{1}{2}k x^2 $$

Then the evolution of a particle will be given by the Poisson Brackets:

$$ \{\cdot,H\} = \frac{p}{\sqrt{p^2+m_0^2}}\partial_x - kx\partial_p $$

then $$ \dot{x} =\{x,H\} =\frac{p}{\sqrt{p^2+m_0^2}} $$ and $$ \dot{p}= \{p,H\} = -kx $$ and then use a integration algorithm (I personally like the velocity verlet algorithm) to get your evolution.

An alternative aproach can be seen in Relativistic (an)harmonic oscillator

Answer 2

Your question was interesting to me, but you hadn't shown any effort to solve the problem yourself. Which might be because you couldn't even begin.
I don't have enough karma to comment on the question directly. Which is why I am posting this as an answer. Please don't vote on this at all, thus!

1. For a short explanation of how the force relation works in STR, look at page 41 of this document.
2. Also as Eric suggests, the form of the spring force does not generalize uniquely to A relativistic force field, which is perhaps why it does not have an unique satisfying answer. You could try to classify the set of potentials which reduce to the spring force potential as $c>>1$. This would be why i was intrigued by the question. Do expect an edit from me in the future with the results of my feeble attempts at a solution to this.
3. Although the spring force might not be a good potential to look at to gain intuition in STR, it might be an interesting problem in its own right. However if you still want you can try to simulate two charged particles in the relativistic EM potential of each other. It would be rather fun to observe and you might test out the Lienard-Wiechert potential. Or, sigh, you can play this free game. My suggestion - do both.
4. And do post a link to your simulation here when it is done. Would love to see your work on it!

EDIT1: This paper takes on solving a relativistic Harmonic Potential for those who are interested.

Answer 3

My suggestion is that if your trying to model Special Relativity using anything but the equations provided that you are asking for trouble. Special Relativity should be handled with equations first, so that you don't confuse yourself trying to wrap your head around the implications.

Also, modeling a force transfer as faster then light is breaking the laws of what your trying to model.