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!
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!