How does c being the constant unifying energy and mass derive from c being constant in all frames of reference

Question

E=MC^2 is the bit in special relativity that laymen like me had heard of. After reading a bit on SR I understand (at a primitive level) the postulates and consequences re spacetime and EM radiation. The descriptions then jump to E=MC^2 without any explanation as to where it fits in and why c is again the critical factor. I am interested to kow what and where the link is connecting relativity as in c and frames of reference and relativity as in c and energy.

Answer 1

It is important to note that the result more correctly stated is as follows
E^2 = (mc^2)^2 + (pc)^2
from which the relation you quote results if the momentum p, is zero ie you are not moving, and you take the square root of both sides and keep the positive solution. So in no sense should any treatment jump straight to E = mc^2, it is a special case. You can find derivations online or in any college level physics textbook.
The key is that relativity joins space and time into one, as a result it also joins energy and momentum into one object, let me explain.
Energy conservation can be understood as due to the fact that an experiment performed today and an experiment performed tomorrow will yield the same result unless you change the experiment - called time invariance. Momentum conservation can be understood as due to the fact that an experiment performed at one end of a room will give the same results as at the other unless again you change the conditions. This is called spatial invariance. Well in our new spacetime prescription, we can't very well say that Energy conservation and momentum conservation are different, instead they are tied together in the above equation.

Answer 2

Assuming that c is constant in all reference frames, and assuming that space is homogeneous. isotropic, and translationally invariant lead directly to the necessity of the Lorentz transformations between frames. The LT's, in turn, required that the concept of kinetic energy be generalized from mv^2/2. Einstein did this and got an extra term that did not even depend on velocity v. It was, of course, mc^2. Despite his best efforts, it would not go away.

Answer 3

basically, because we don't know any better yet