Question

DERIVE E=MC^2 and explain it deeply​

Answer 1

Answer:

Einstein’s equation E=mc^2 describing the equivalence of mass and energy is arguably the most famous equation in physics. But his beautifully elegant derivation of this formula (here is the English translation) from previously understood laws of physics is considerably less famous. (There is an amusing Far Side cartoon in this regard, with the punchline “squared away”, which you can find on-line by searching hard enough, though I will not link to it directly.)

This topic had come up in recent discussion on this blog, so I thought I would present Einstein’s derivation here. Actually, to be precise, in the paper mentioned above, Einstein uses the postulates of special relativity and other known laws of physics to show the following:

Proposition. (Mass-energy equivalence) If a body at rest emits a total energy of E while remaining at rest, then the mass of that body decreases by E/c^2.

Assuming that bodies at rest with zero mass necessarily have zero energy, this implies the famous formula E = mc^2 – but only for bodies which are at rest. For moving bodies, there is a similar formula, but one has to first decide what the correct definition of mass is for moving bodies; I will not discuss this issue here, but see for instance the Wikipedia entry on this topic.

Broadly speaking, the derivation of the above proposition proceeds via the following five steps:

Using the postulates of special relativity, determine how space and time coordinates transform under changes of reference frame (i.e. derive the Lorentz transformations).

Using 1., determine how the temporal frequency \nu (and wave number k) of photons transform under changes of reference frame (i.e. derive the formulae for relativistic Doppler shift).

Using Planck’s relation E = h\nu (and de Broglie’s law p = \hbar k) and 2., determine how the energy E (and momentum p) of photons transform under changes of reference frame.

Using the law of conservation of energy (and momentum) and 3., determine how the energy (and momentum) of bodies transform under changes of reference frame.

Comparing the results of 4. with the classical Newtonian approximations KE \approx \frac{1}{2} m|v|^2 (and p \approx mv), deduce the relativistic relationship between mass and energy for bodies at rest (and more generally between mass, velocity, energy, and momentum for moving bodies).