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Way back in 2007, I wrote a blog post giving Einstein’s derivation of his famous equation ${E=mc^2}$ for the rest energy of a body with mass ${m}$. (Throughout this post, mass is used to refer to the invariant mass (also known as rest mass) of an object.) This derivation used a number of physical assumptions, including the following:

1. The two postulates of special relativity: firstly, that the laws of physics are the same in every inertial reference frame, and secondly that the speed of light in vacuum is equal ${c}$ in every such inertial frame.
2. Planck’s law and de Broglie’s law for photons, relating the frequency, energy, and momentum of such photons together.
3. The law of conservation of energy, and the law of conservation of momentum, as well as the additivity of these quantities (i.e. the energy of a system is the sum of the energy of its components, and similarly for momentum).
4. The Newtonian approximations ${E \approx E_0 + \frac{1}{2} m|v|^2}$, ${p \approx m v}$ to energy and momentum at low velocities.

The argument was one-dimensional in nature, in the sense that only one of the three spatial dimensions was actually used in the proof.

As was pointed out in comments in the previous post by Laurens Gunnarsen, this derivation has the curious feature of needing some laws from quantum mechanics (specifically, the Planck and de Broglie laws) in order to derive an equation in special relativity (which does not ostensibly require quantum mechanics). One can then ask whether one can give a derivation that does not require such laws. As pointed out in previous comments, one can use the representation theory of the Lorentz group ${SO(d,1)}$ to give a nice derivation that avoids any quantum mechanics, but it now needs at least two spatial dimensions instead of just one. I decided to work out this derivation in a way that does not explicitly use representation theory (although it is certainly lurking beneath the surface). The concept of momentum is only barely used in this derivation, and the main ingredients are now reduced to the following:

1. The two postulates of special relativity;
2. The law of conservation of energy (and the additivity of energy);
3. The Newtonian approximation ${E \approx E_0 + \frac{1}{2} m|v|^2}$ at low velocities.

The argument (which uses a little bit of calculus, but is otherwise elementary) is given below the fold. Whereas Einstein’s original argument considers a mass emitting two photons in several different reference frames, the argument here considers a large mass breaking up into two equal smaller masses. Viewing this situation in different reference frames gives a functional equation for the relationship between energy, mass, and velocity, which can then be solved using some calculus, using the Newtonian approximation as a boundary condition, to give the famous ${E=mc^2}$ formula.

Disclaimer: As with the previous post, the arguments here are physical arguments rather than purely mathematical ones, and thus do not really qualify as a rigorous mathematical argument, due to the implicit use of a number of physical and metaphysical hypotheses beyond the ones explicitly listed above. (But it would be difficult to say anything non-tautological at all about the physical world if one could rely solely on ${100\%}$ rigorous mathematical reasoning.)

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:

1. Using the postulates of special relativity, determine how space and time coordinates transform under changes of reference frame (i.e. derive the Lorentz transformations).
2. 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).
3. Using Planck’s law $E = h\nu$ (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.
4. 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.
5. 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).

Actually, as it turns out, Einstein’s analysis for bodies at rest only needs to understand changes of reference frame at infinitesimally low velocity, $|v| \ll c$. However, in order to see enough relativistic effects to deduce the mass-energy equivalence, one needs to obtain formulae which are accurate to second order in v (or more precisely, $v/c$), as opposed to those in Newtonian physics which are accurate to first order in v. Also, to understand the relationship between mass, velocity, energy, and momentum for moving bodies rather than bodies at rest, one needs to consider non-infinitesimal changes of reference frame.

Important note: Einstein’s argument is, of course, a physical argument rather than a mathematical one. While I will use the language and formalism of pure mathematics here, it should be emphasised that I am not exactly giving a formal proof of the above Proposition in the sense of modern mathematics; these arguments are instead more like the classical proofs of Euclid, in that numerous “self evident” assumptions about space, time, velocity, etc. will be made along the way. (Indeed, there is a very strong analogy between Euclidean geometry and the Minkowskian geometry of special relativity.) One can of course make these assumptions more explicit, and this has been done in many other places, but I will avoid doing so here in order not to overly obscure Einstein’s original argument. Read the rest of this entry »