# Relativistic equivalent of a spring-force?

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.

• – user4552
Apr 9, 2013 at 16:17
• The review mentioned is a very good exploration of the limitations that a rigid body faces in STR. The OP has to set up his simulation such that at any point the tension doesn't exceed the gradient of energy density in the material. Also, the results seem to depend on the material parameters from this. Any progress till now @Neurofuzzy ? Apr 29, 2013 at 12:44
• @DebanjanBasu I didn't make any progress for a while, so I put it on the back-burner. I'm studying tensors, simultaneously with other physics things, and I'm going to attempt it again once I finish the special relativity chapter in my mechanics book (Goldstein). But it's definitely not a concept I'll be able to forget about :)
– user12029
Apr 29, 2013 at 20:43

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

• This only works for a fixed base point though. Is there any spring force which works for a pair of point masses? May 12, 2014 at 19:35
• If one adds relativistic corrections to the kinetic term then one would add relativistic corrections to the potential term as well. Mar 22, 2019 at 8:16

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.

• The OP tried to write a computer simulation, and gives a clearly worded explanation of the issues that arose in attempting to do so. How you get from there to "you hadn't shown any effort to solve the problem yourself" is an absolute mystery to me. Feb 8, 2013 at 15:01
• @Nathaniel: Yeah .. I notice that now! It was my mistake to gloss over the material, not the OP's. Feb 8, 2013 at 16:34
• Thank you! This was very helpful. In trying some of the practice problems on the first link, I realized I don't have as firm a grasp on force as I thought I did... And the last section of that document is telling! Maybe I won't be able to write a computationally simple sim after all! Anyways, I'll definitely post a link, as an answer maybe, once I figure out what I'm doing. It sounds like I'll have to do particles in a field, I've never numerically done both at the same time though.
– user12029
Feb 9, 2013 at 10:15

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.

• I'm doing self-study, so "what equations are provided" isn't fixed. If it's more advanced (such as, if really the only way to simulate it is to use a wave equation) then I'll work towards modeling that.
– user12029
Feb 7, 2013 at 23:48
• Start with special relativity definitions for momentum, time, and length and go from there. The important part is to interpret after calculating. Feb 7, 2013 at 23:51