Can one explain the relativistic energy transformation formula:(adsbygoogle = window.adsbygoogle || []).push({});

[itex]E = \gamma\ E',[/itex]

where the primed frame has a velocity [itex]v[/itex] relative to the unprimed frame, in terms of relativistic time dilation and the quantum relation [itex]E=h\ f[/itex]?

I imagine a pair of observers, A and B, initially at rest, each with an identical quantum system with oscillation period [itex]T[/itex].

Now A stays at rest whereas B is boosted to velocity [itex]v[/itex].

Just as in the "twin paradox" the two observers are no longer identical: B has experienced a boost whereas A has not. Both observers should agree on the fact that B has more energy than A.

From A's perspective B has extra kinetic energy by virtue of his velocity [itex]v[/itex]. Relativistically A should use the energy transformation formula above.

But we should also be able to argue that B has more energy from B's perspective as well.

From B's perspective he is stationary and A has velocity [itex]-v[/itex]. Therefore, due to relativistic time dilation, B sees A's oscillation period [itex]T[/itex] increased to [itex]\gamma\ T[/itex].

Thus B finds that his quantum oscillator will perform a factor of [itex]\gamma\ T/T=\gamma[/itex] more oscillations in the same period as A's quantum system.

Thus B sees that the frequency of his quantum system has increased by a factor of [itex]\gamma[/itex] over the frequency of A's system.

As we have the quantum relation, [itex]E=h\ f[/itex], this implies that B observes that the energy of his quantum system is a factor of [itex]\gamma[/itex] larger than the energy of A's system.

Thus observer B too, using his frame of reference, can confirm that his system has more energy than observer A's system.

Is this reasoning correct?

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# Lorentz transformation of energy and E = h f

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