Problem with special relativity?

Question

I am trying to re-create a complete *proof* that E(sub_0)=mc^2, or that E=(gamma)mc^2 (I am at the stage where proving one will lead on to the other).
I have the expression for kinetic energy:
K=(gamma)*mc^2 - mc^2
And I need a proof from that of the equations above. All of the books I have looked in simply seem to assert it. Where do I go from here?
NB-all proofs based on Einstein's postulates of course.

I can't start with E = mc² / √(1-(v/c)²).
I'm trying to prove that.
I only have that K=gmc²-mc²
And i want to get to an expression for Total energy or rest energy, without making any unjustified assertions.

NB-for 'g' read 'gamma'

Answer 1

If we start with the following:

E = mc² / √(1-(v/c)²)
p = mv / √(1-(v/c)²)

then it's not hard to show that, by squaring E and p and subtracting:

E² - (pc)² = (mc²)²

For mass at rest, p = 0, so we're left with:

E = mc²

There are many ways to derive the first 2 equations, but one way starts with the requirement that momentum be conserved even when spacetime is relativistically affected, which gives you the expression for relativistic momentum, and then the fact force is change in momentum over time gets you the one for relativistic energy. Check out the link for the complete derivation by this means.

Addendum: Another way to derive the momentum expression from E = gmc²
is from the fact that dE/dp = v, which is true in both Newtonian and Special Relativity physics. From this, we have:

∫ (1/v) dE = ∫dp = p so, begin with:

E = mc² / √(1-(v/c)²) and solve for v, and invert it, so that we have:

∫ E / c√(E²-m^4c²) dE integrating this gets us:

(1/c)√(E²-(mc²)²) = p or

E² - (mc²)² = (pc)² from this, we can derive the relativistic momentum

It should be noted that gmc² is the total energy, while mc² is the rest mass energy, and the difference is the relativistic kinetic energy. Quite obviously, if the kinetic energy K is

K = gmc² - mc², we can rearrange and get

K + mc² = gmc², and then we note that when v = 0, we have

K + mc² = mc², or

K = 0, as it should be. It has no kinetic energy, and thus mc² is the rest mass energy, and gmc² is the total energy, being the sum of rest mass energy and relativistic kinetic energy.