Does kinetic energy rely on the observer mass too since velocity is relative?

Question

There is no 'correct' inertial reference frame according to relativity. Objects are only 'in motion' relative to an arbitrary inertial reference frame. So let us take the following example. A person on Earth jumps. They are now moving at 1 m/s up relative to Earth. As per $K =\frac{1}{2}mv^{2}$, assuming they weigh 70 kg, they did 35 Joules of work. But relative to them, Earth is suddenly moving away from them at 1 m/s. Earth now has 3 septillion extra Joules of kinetic energy relative to them, but they did not do anywhere near that much work. In order to conserve energy, how can this be explained? Does kinetic energy also rely on the mass of the observer? Otherwise, what is the 'velocity' used to calculate kinetic energy relative to?

Answer 1

Kinetic energy is indeed relative, because it depends on velocity which is inherently relative. The kinetic energy of an object is entirely dependent on the observer. An observer on the ground may see a bullet fly past at high speed with lots of kinetic energy, but an observer moving at the same velocity as the bullet sees no relative velocity at all, and therefore sees that the bullet has zero kinetic energy.

There's no reason why energy should be conserved between reference frames. Each different observer may assign a different numerical value of kinetic energy to a system. Each is correct in their own frame, but they'll all disagree about how many joules of KE are present in the system.

What's also happening here is that when the person jumps, they are in a non-inertial, accelerating reference frame. In non-inertial frames, fictitious pseudoforces often appear as a means of balancing the force equations - these forces do not actually have a direct physical cause like gravity or electromagnetism, but are more like bookkeeping forces that account for the non-inertial frame. In the jumping person's non-inertial frame, there is indeed an enormous pseudoforce that causes the massive earth to accelerate away from the person's feet. But the magnitude of this pseudoforce is merely a result of the choice of the non-inertial reference frame, and is unrelated to the biomechanics of the person's legs.

Answer 2

In order to conserve energy, how can this be explained?

The reference frame in question is non-inertial. Energy is not always conserved in non-inertial frames, and particularly not in this one. The 3 septillion extra joules appear out of nowhere in this non-inertial frame.

It can be seen by Noether’s theorem that energy is not conserved in the non inertial frame since it lacks the time translation symmetry.