- #1

George Keeling

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- TL;DR Summary
- I cannot find a decent proof that E=mc². Can someone help?

Sean Carroll says that in SR the time component of the 4-momentum of a particle is its energy. It is of course also ##mc^2dt/d\tau##. He uses that to prove that ##E=mc^2##. Which begs the question why does ##E=p^0##?

Misner, Thorne, Wheeler do roughly the same thing.

I find these 'proofs' unsatisfying but neither of the above books are on special relativity so I look elsewhere. I have found nothing useful on the web!

I checked two videos that claim to prove it (here and here). The first immediately picks a definition of relativistic doppler shift: "Transverse Doppler effect, geometric closest approach" is the second of eight different equations for special relativistic doppler shift given in this Wikipedia article. Why should I choose that one?

The second is long and proves after much calculus (about 23 minutes in) that a particle's kinetic energy is$$KE=mc^2\gamma-mc^2$$##\gamma## is the usual thing related to the particle's velocity. It then turns that round to say$$mc^2\gamma=KE+mc^2$$and that ##mc^2\gamma## is the total energy of the particle (

I'm sure Einstein did better than this. Can somebody point me to a decent proof of ##E=mc^2##? Or even quickly put the proof here?

Misner, Thorne, Wheeler do roughly the same thing.

I find these 'proofs' unsatisfying but neither of the above books are on special relativity so I look elsewhere. I have found nothing useful on the web!

I checked two videos that claim to prove it (here and here). The first immediately picks a definition of relativistic doppler shift: "Transverse Doppler effect, geometric closest approach" is the second of eight different equations for special relativistic doppler shift given in this Wikipedia article. Why should I choose that one?

The second is long and proves after much calculus (about 23 minutes in) that a particle's kinetic energy is$$KE=mc^2\gamma-mc^2$$##\gamma## is the usual thing related to the particle's velocity. It then turns that round to say$$mc^2\gamma=KE+mc^2$$and that ##mc^2\gamma## is the total energy of the particle (

**WHY?**) and therefore when the particle is at rest ##KE=0## and ##E=mc^2##. I am in despair!I'm sure Einstein did better than this. Can somebody point me to a decent proof of ##E=mc^2##? Or even quickly put the proof here?