# I Invariance of elastic potential energy

1. Jun 21, 2017

### e2m2a

At non-relativistic speeds is the elastic potential energy of a compressed spring frame-invariant? That is, would all reference frames agree on how much elastic potential energy is stored in the spring?

2. Jun 21, 2017

### dextercioby

Rather than asking this question and waiting for an answer, can you state your opinion first? And what mathematical formulas would you use to support it?

Last edited: Jun 21, 2017
3. Jun 21, 2017

### e2m2a

It is my opinion that at non-relativistic speeds, all reference frames would agree on the elastic potential energy of the spring. I assert this because the elastic potential energy is given by: 1/2 k (delta l) squared, where k is the spring constant and delta I is the change in length of the spring. Since k is a constant, all reference frames will agree on this value, and all reference frames would agree on delta l or the change in the length of the spring. But am I missing something?

4. Jun 21, 2017

### dextercioby

No, you are not missing anything. Since time is absolute, the length between two points is an invariant when hypothetically measured, thus this purely potential energy k delta l^2 should be invariant under Galilean relativity.

5. Jun 21, 2017

### pervect

Staff Emeritus
I'd agree with this, though as it stands it's a statement about non-relativistic physics in the relativity forum. Are you interested in a discussion of the relativistic case at all? My thoughts on this would be that in the relativistic case, we represent the compressed spring by its stress energy tensor $T^{ab}$. We generally don't try to separate out "potential energy" from "kinetic energy", in the relativistic case we concentrate on how $T^{ab}$ transforms via the tensor transformation rules.

There is a lot more that could be said and maybe should be said, but it's not clear that you're interested in the relativistic case - or what your background knowledge of tensors might be. I will say that I don't think there is a good way to treat the issue of a relativistic compressed spring without tensors (in particular the stress-energy tensor).

[add]I'll add that I believe Rindler has a treatment of the relativistically moving compressed spring (I think the text uses a rod, which is makes no difference to the analysis) which uses the stress-energy tensor in "Essential Relativity".

6. Jun 21, 2017

### e2m2a

Thank you for your responses. Maybe I should have posted this in the general physics forum because I am not interested in the relativistic case.

7. Jun 21, 2017

### Staff: Mentor

Note: this thread has been moved to General Physics.