Undergrad Deriving e=mc^2, how is it possible?

[rupcha]

TL;DR

How does rest energy "magically" emerge in SR?

I was recently very surprised when I had a looked up relativistic kinetic energy.

All sources gave the kinetic energy as the difference between total energy and rest energy, in some or other variant of the formula $E_k=(\gamma−1)mc^2$.

I didn't really understand at first. It seemed overly "deep" and indirect to me, to start with total energy and introduce rest energy. Surely, it should be possible to just integrate the work done and come up with some relativistic but recognizable variant of $E_k=\frac 1 2 mv^2$.

So I did the integration and, not surprisingly (but surprising to me then), the result was the very formula $E_k=(\gamma−1)mc^2$.

But what really blew me away was that the rest energy $mc^2$ was being spat out "for free" as the integration constant.
I still don't quite understand how that's possible. There just seems to be too little information going into the integral for such a result to emerge.

I mean, the only ingredients going into the calculation are Newton's $F=m\cdot a$ and the Lorentz transformations. How the hell can math extract an equivalence of mass and energy from that? I would have expected that you had to add some deep insights into the nature of matter and possible conversions to come to a result like $E_0=mc^2$.

Still absolutely blown away.

Grateful for anyone who can help me understand.

Edit: formulas got broken, trying to reenter (looked fine in preview)

 

[PeterDonis]

I did the integration

How? Please show your work. And please use the PF LaTeX feature to make your equations readable.

 

[PeterDonis]

I am experimenting with a rather unconventional "reference frame"

Which is personal speculation and is off limits here. And of course explains why you're confusing yourself.

Hoping of course, that I didn't simply make several mistakes

Your mistake was trying to experiment with personal speculation instead of doing standard SR math.

Thread closed.