# Conservation of mass energy and kinetic energy in different reference frames

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!

• 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 '14 at 20:45

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.
• @Mr. Gentleman It is right that when we switch frames, the quantity $E^2 -p^2$ taking $c=1$, remains invariant as this is same as $m_0^2$ which is a Lorentz scalar. However in reactions where the rest mass of things change from LHS to RHS of the reaction, what remains conserved when we switch between multiple inertial frames? – Naman Agarwal Mar 1 at 6:12