Two posts that I made in the Science Advisor subforum are relevant here. I'll just repost the text here. See also #15 in
this thread.
Post 1
Regarding the "why c?" question, here's an outline of an answer: We define (four-)momentum as mass times the (four-)velocity, so the square of momentum is (in units such that c=1): p^2=m^2u^2=-m^2, but also p^2=-(p^0)^2+\vec p^2, so E^2=\vec p^2+m^2. When we restore factors of c, we get E^2=\vec p^2c^2+m^2c^4. There are two ways to interpret E=mc
2. One is to set \vec p=0 in the previous equation, and take the square root. The other is to write the mass as m
0 instead, and
define the "relativistic mass" by E=mc
2. I'm one of those who feel that the latter is a really pointless thing to do.
So where did the c "come from"? If we include explicit factors of c in every step, one must be included in the definition of the four-velocity. See
this post for more about four-velocity. The place where c enters is where we note that the slope of the world line is 1/v=u^0/(c|\vec u|).
Post 2
Yes, what I said above doesn't really explain why the 0th component of the four-momentum must be interpreted as energy, or why four-momentum is defined as mass times four-velocity. The post I linked to has a partial answer to the latter. (But I think a full answer should explain why this definition makes four-momentum a conserved quantity in particle interactions).
How are you going to derive E=mc
2? The only other way I know to obtain that result as a consequence of something else is to calculate the work performed when accelerating a massive particle from 0 to v. (The result is \gamma m-m). But to obtain that result, we "just assumed" that the relativistic expression for work is obtained from the non-relativistic
W=\int F\ dx=\int m\ddot x\dot x\ dt=\int m\frac{d\dot x}{dt}\dot x\ dt
by substituting the
first of the two velocities in the last step (but not the second) for the spatial components of the four-velocity. I know that we can make this look like a "natural" thing to do by expressing the non-relativistic work as
W=\int\dot\vec p\cdot d\vec x
but that doesn't change the fact that we have made an
assumption about the relationship between energy and the spatial components of four-momentum, and got E=mc
2 from that. This justification of E=mc
2 isn't better just because the assumption is a few algebraic steps away from the conclusion.
I think that to really answer the question, we'd have to show that the components of four-momentum behave like energy and momentum in non-relativistic mechanics. In particular, we should show that they are all conserved quantities in particle interactions.
