Potential Energy in a Stretched String in SR

[LarryS]

TL;DR Summary

What is formula for the potential energy stored in a stretched string in SR?

In classical mechanics, the potential energy stored in an elastic stretched string is the work done by the tension during the stretching process. Is this concept and formula the same in Special Relativity?

Thanks in advance.

 

[kuruman]

What is the real question you want to ask? Does the definition of potential energy depend on whether some observer is moving relative to the spring?

 

[pervect]

Unfortunately, I don't have an intermediate level answer to this question, but I do have an A-level (graduate level) answer to some of it.

The density of energy and momentum per unit volume in a stretched string (or, anything else for that matter) in Special Relativity is given by the stress-energy tensor, $T_{ab}$. This is a rank two tensor.

A unit 3-volume in special relativity depends on the reference frame in which the volume element is at rest. The useful formulation of this concept in this particular concept turns out to be $u^a$, the four-velocity of the unit volume. There is another formulation which I won't get into which is a rank-3 completely anti-symmetric tensor which also defines a volume element.

Thus, $T_{ab} \, u^a$ gives the energy-momentum 4-vector of a unit volume of the stretched string, the total energy and momentum contained in the unit volume.

The conservation of energy, which is what I think you are asking about when you talk about the work done in stretching the string, is given by the fact that the divergence of the stress energy tensor, $\nabla_a T^{ab} = 0$.

A stretched string is not an isolated system. For a point particle, one can write the mass of the particle as the length of the energy-momentum 4-vector, and this quantity is invariant, but if you sum up the total energy $E_t$ and momentum $p_t$ of a stretched string via adding together the energy-momentum in all the volume elements comprising the string, the quantity given by $E_t^2 - (p_t c)^2$ , which is invariant for a point particle and equal to m^2 c^4 of said particle, is no longer a relativistic invariant.

This issue relates to the relativity of simultaneity, the mass of a distributed system requires a notion of "now", and the notion of "now" depends on the frame of reference.

The closest I can come to a intermediate level answer is that energy-momentum is still conserved in special relativity, but the details are different. I would say that the root cause of the differences is the relativity of simultaneity. When one is talking about the energy of a distributed system, one adds up the energy of the pieces of the system at some specific time. In special relativity, because of the relativity of simultaneity, the notion of "at the same time" depends on the reference frame one chooses. This leads to some extra complexity into the "bookeeping" of energy.

 

[Mister T]

TL;DR Summary: What is formula for the potential energy stored in a stretched string in SR?

$E_o=mc^2$. The energy stored is equivalent to the increase in mass of the string.

 

[PeterDonis]

Is this concept and formula the same in Special Relativity?

If you are asking whether the work-energy theorem holds in relativity, yes, it does, as long as you correctly evaluate the work and the energy.