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?
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3$\begingroup$ Energy and its conservation are relative as well. $\endgroup$– HiamphCommented Apr 4, 2023 at 17:28
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1$\begingroup$ A quantity being "conserved" is absolutely not the same as a quantity being "invariant". Kinetic energy is not invariant: it depends on the frame. But if two objects collide elastically, the final kinetic energy will be equal to the initial kinetic energy, and this will hold in all frames. $\endgroup$– printfCommented Apr 5, 2023 at 3:02
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2$\begingroup$ Possible duplicates: physics.stackexchange.com/q/51220/2451 , physics.stackexchange.com/q/1368/2451 and links therein. $\endgroup$– Qmechanic ♦Commented Apr 5, 2023 at 4:58
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2 Answers
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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.
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$\begingroup$ Ok, the frame is non-inertial. But how about this situation. Earth and Person are already born 1 m/s relative to each other, but are not accelerating. Relative to Person, Earth has 3 septillion Joules, and relative to Earth, person has 35 Joules. If they collide (ignoring the 11 km/s acceleration of Person due to the gravity of Earth, as it is insignificant compared to 3 septillion joules), the collision occurs and now Earth has 0 Joules relative to Person. But I guess since Person is now in the gravity of Earth, they are again, no longer in an inertial reference frame. $\endgroup$– CPlusCommented Apr 4, 2023 at 17:59
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$\begingroup$ 11 km/s is not an acceleration, but you can calculate the force felt by the person when earth comes to rest against them in their frame (this is a very large number of course). This already shows that it's a non-inertial frame of reference. An external observer sees an inelastic collision with the earth continuing to move on with only negligibly less energy as previously. $\endgroup$ Commented Apr 4, 2023 at 18:32
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1$\begingroup$ @user16217248 During the collision, the person and the reference frame they inhabit undergo acceleration. The person's measurement of the earth's KE before the collision, and their measurement after are not taken in a common inertial frame. This would happen even if there were no gravity at all, it's solely due to the fact that the person accelerates during the collision - an external observer in an inertial frame would see a relative velocity between the person before and after the collision, indicating acceleration. $\endgroup$ Commented Apr 4, 2023 at 18:50
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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.
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$\begingroup$ Would the relevant inertial reference frame be the common center of mass of person and the Earth? $\endgroup$– CPlusCommented Apr 4, 2023 at 17:33
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$\begingroup$ But if the reference frame is non-inertial, the real kinetic energy, i.e., that which would be measured by an inertial reference frame, would depend on the mass of the observer corresponding to the non-inertial reference frame, right? $\endgroup$– CPlusCommented Apr 4, 2023 at 17:37
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2$\begingroup$ @user16217248 any inertial frame will be fine, but yes the inertial frame of the center of mass is a good one to use. There is not always a specific object with a specific mass associated with a reference frame. So you cannot rely on the mass of a reference frame observer, there may not be one $\endgroup$– DaleCommented Apr 4, 2023 at 19:00
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1$\begingroup$ When it comes to non-inertial reference frames, someone mentioned that time-translation symmetry doesn't hold in non-inertial reference frames, so by Noether's theorem it is expected that energy is not conserved. I think that would be something important to point out here. $\endgroup$ Commented Apr 5, 2023 at 5:03