special relativity - Is the change in kinetic energy of a particle frame independent?

Intuitively, I would expect the change in kinetic energy of a particle to be frame independent. It just doesn't "feel" right that between two points in time-space, one frame should measure a change in kinetic energy of a particle different to another frame.

Is my intuition right? Is the change in kinetic energy of a particle frame independent?

Answer

Your intuition is incorrect, but you should get new intuition. The reason energy isn't conserved the same way in every frame is because the energy is not separate from the momentum. This is clearest in relativity, where the energy is the time-component of the energy-momentum vector. Then if you change frames, what you called energy partly becomes momentum, and it is the conservation law of the total energy-momentum vector which is frame independent.

When you have a vector conservation law, it is not true that the change in magnitude of every component is the same in every frame. For example, the change in x-momentum turns into the change in y-momentum if you rotate the x and y axes into each other.

Even without relativity, just with Galilean invariance, energy is mixed up with the momentum. The difference is that without relativity, the momentum is not mixed back, it is the same in all boosted frames.

So consider a change in velocity from 10 to 20 meters per second in a 2kg object. This means that you have put in 300 Joules of energy. This doesn't mean your muscles did 300 Joule's of work though, you can do this with an elastic collision where you do no net work. If you consider the frame where the object starts out moving at -5 meters per second, it ends up moving at 5 meters per second, and there is no change in the kinetic energy. If you just rebounded the object off a very massive wall, this is what would happen, and you would do no work in the process.

But in the collision with the wall, the very massive wall will end up with a tiny recoil, which will change the energy of its motion so that the overall motion is conserved.

Formally

The law of conservation of energy in nonrelativistic mechanics is that the sum over all the N particles in the system, indexed by the integer variable i, is conserved in any frame. The velcity in the new frame is altered by the boost velocity v, so that the momentum $p'_i = p_i - m v$.

$$ \sum_i {{p'}_i^2\over 2m_i} = \sum_i {(p_i - m_i v)^2 \over 2m_i} = \sum_i {p_i^2\over 2m} + v \cdot \sum_i {p_i} + {v^2\over 2} \sum_i m_i $$

note that the separation gives three terms, the first of which is the energy in the new frame, the second term is the momentum, and the third is the total mass times one half the boost velocity.

The transformation law shows that if energy is conserved in any one frame for a system which also conserves momentum, then the energy is conserved in another frame. This means that the total energy change is only meaningful for a system where no momentum flows in or out.