Einstein's Greatest Equation

Deriving Einstein’s Greatest Equation

E = mc^2

Stationary box A is floating in Space.

A Photon is then emitted from the left and moves towards the right.

E=Energy of the photon

Momentum(p) of a Photon=E/c

The box experiences a recoil force and moves towards the left with a velocity v.

p of box=Mv

Where M = mass of the box

By Conservation of momentum,

M∆x/∆t=E/c - 1

Where ∆x is the distance the box moves in time ∆t when the photon makes it to the other side.

∆t=L/c

Where L = the length of the box

Substituting this in 1,

M∆x=EL/c^2

Assuming the photon has mass m, if the box has a position x1 and the photon has a position x2, the centre of mass for the whole system is,

=(Mx1+mx2)/(M+m)

Since no external force was applied, the centre of mass of the whole system should not change.

Therefore,

(Mx1+mx2)/(M+m)=(M(x1-∆x)+mL)/(M+m)

The photon starts at the left of the box.

Thus x2 + 0

Hence,

mL=M∆x

mL=EL/(c^2)

m=E/c^2

Or,

E=mc^2