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With a little work it's easy to show that kinetic energy by itself is not necessarily preserved when switching between frames of reference. And it is my understanding that energy should be preserved in any reference frame; after all, isn't that the point of this energy construct? So, please help me with the following example, because I think there is a huge, obvious hole in my knowledge!

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In reference frame A, a baseball is moving at nonzero speed. In another reference frame B that baseball is still, not moving. Reference frame A appears to have a larger sum of mass + kinetic energy than reference frame B, because B has no kinetic energy and I believe the mass energy is the same in both (because mass in the equations refers to invariant mass, but I am not sure on that point.) I also believe that we can ignore other energies (electrical, nuclear, etc.) but again I am not completely convinced that is safe.

Now imagine that in reference frame A the kinetic and mass energies are manipulated in some reaction to create a new, heavier, static baseball. Energy is conserved because the new baseball has more mass energy but less kinetic energy. The problem is that reference B has seen the transformation of a light, static baseball, into a heavy, moving baseball! Certainly that breaks conservation of energy.

Where did I go wrong? Thank you!

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I don't quite follow your argument. It sounds like you raised the mass of baseball while also slowing it down? What do you mean by "Energy is conserved becuase ..."? At any rate, energy is conserved only if all measurements are done in the same inertial frame. Energy is not "conserved" if one measurement is done in one frame, and the next measurement in a different frame. But I may be completely missing your argument. –  garyp Mar 2 at 20:45
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Conservation of energy refers to systems looked from the same reference frame, it does not make sense to require that energy of the same system to be the same in different reference frames. As a consequence of time translational symmetry, energy conservation is usually true unless we drive the system externally which may break this symmetry. Similarly, momentum conservation is a consequence of space translational symmetry.

The (invariant) mass $m$ is the same in all inertial reference frames, on the other hand, energy $E$ and momentum $p$ are connected through the famous equation

\begin{equation} E^2=(pc)^2+(mc^2)^2 \end{equation} where $c$ is the speed of light. This equation is valid in any inertial reference frame, to go from one frame to another, one has to do Lorentz transformation of both energy and momentum, and it turns out the final result is that the changes in energy and momentum compensate each other and validate this equation in every frame.

For the example you gave, if there is only that ball in the universe, in reference frame A, it cannot stop by momentum conservation. If it stops, you have to exert an external force, which may explicitly change the energy of this ball even in reference frame A. Then from frame B, roughly speaking, you exert a force (you may want to work out the transformation of the force between these two frames) to the left direction and the ball gains energy, so nothing is wrong.

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This is more or less what I assumed -- that there are other rules in place other than conservation of energy (like conservation of momentum) and so my example doesn't capture a meaningful hypothetical situation. Thanks for the great explanation! –  user2555938 Apr 20 at 3:20
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You are quite right. Energy is not conserved between the reference frames. That is the biggest mystery. It is certainly going to change one's concept of understanding of energy. I m here giving a simple example where there is total failure of 'law of conservation of energy' Take an example of spaceship in space. suppose you start the spaceship and accelerate for one hour and spaceship traveled a distance say 200 km. now shutdown the propulsion engine. At this point the ship get some speed say 300 km/h and chemical energy of fuel of spaceship is converted into kinetic energy of spaceship. The reference frame of the ship changes. Now lets spaceship turns back and again accelerated (decelerated for initial reference frame ) for one hour more and traveled more 200 km and then shutdown the engine. At this point the speed of the ship will be zero hence its total kinetic energy is also zero. now think again, you have burnt fuel in acceleration and deceleration but finally total kinetic energy remains zero. Chemical energy of the ship wasted (destructed) but this energy have not converted in any forms. Hence its a burning example of "failure of Energy conservation law"

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