Comparing potential energy in non-relativistic classical physics and in special relativity

Question

Preface

In classic physics the potential energy of a compressed (or stretched) spring is

$E_{\rm sp} = (\frac{1}{2})k_{\rm sp}(\Delta l)^2$

where $\Delta l$ is the length that the spring is compressed by, $k_{\rm sp}$ is spring constant.

Also in classic physics the potential energy of two charged particles is

$E_{\rm el} = k_{\rm el}\frac{q_1q_2}{r}$

where $q_1$ is charge of particle 1, $q_2$ is charge of particle 2, $r$ – distance between the particles, $k_{\rm el}$ – Coulomb constant.

The main part

Now, suppose we have 2 observers: one is at rest with the spring and the two charged particles (lets call this observer Ralf); and the other is moving relative to these objects with a speed close to speed of light (lets call this observer Maria).

For Maria the formulas above will be different due to length contraction:

$l’=\frac{l}{\gamma} $

where $l$ – proper length, $\gamma$ – Lorentz factor.

As I understand, for Maria the potential energies of the spring and the two particles will be $\gamma$ times the potential energies measured by Ralf, that is

$E_{\rm sp}' = \gamma E_{\rm sp}$

$E_{\rm el}' =\gamma E_{\rm el}$

where $E_{\rm el}'$, $E_{\rm sp}'$ - energies measured by Maria, $E_{\rm el}$, $E_{\rm sp}$ – by Ralf.

Here I have two questions.

1. If we try to calculate potential energy by lengths or distances, then potential energy turns out to be frame dependent. I can barely believe that, because I always thought that any type of internal energy (including potential one) is invariant both in classic physic and in special relativity. Is this true? If it is, then Ralf and Maria should get the same values for potential energies.

2. If potential energy is frame dependent, then how can energy conservation law hold in special relativity? Examine a situation of a cart at the base of a hill and it is being pushed up to the hill by a compressed spring. Let cart start its motion with a smallest possible speed just sufficient to pass the peak of the hill. All the observers (let them move horizontally) must agree on that the cart succeeds in passing the peak of the hill. This event requires that spring gives some energy ($E=mgh$, where $h$ – height of the hill) to the cart, otherwise the cart can’t get to the peak.

Answer 1

The value of any energy measured by different observers (relativistic or non-relativistic) might be different. The conservation of energy states that whatever the energy measured by one observer remains constant for him/her. For example, if you have an object of mass $1kg$ moving at speed $v=2m/s$, its kinetic energy in your frame is $2J$. If there are no forces it will remain $2J$ forever. For someone moving with the object, the speed is 0, hence kinetic energy is $0J$. Similarly, the value of potential energy will depend on the observer. Even a given observer can measure different values of potential energy depending on where they put their reference point. (center of earth/surface of the earth)

Answer 2

Examine a situation of a cart at the base of a hill and it is being pushed up to the hill by a compressed spring. Let cart start its motion with a smallest possible speed just sufficient to pass the peak of the hill.

Moving observer sees this:

The extra small internal potential energy of the spring becomes the extra small internal kinetic energy of the cart-planet system, and that kinetic energy becomes the extra small internal gravitational potential energy of the cart-planet system.

You have a correct idea: If some of the internal energies is not extra small, then there is a problem.

So we can conclude that all types of internal energies of a moving object are equally small.

Now, if we consider a moving arrow hitting a tree, we now know that there occurs an increase of all internal energies of the arrow, which we probably have not thought about until now.

So there is extra energy involved in the collision. So the arrow had extra kinetic energy.

Se we know what becomes of the internal energies of an object whose internal energies decrease as the object's speed is increased. Kinetic energy.