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26.6 Proofs by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.

In section 26.5 I gave physical arguments to the effect that relativistic momentum should be greater than `mv` and that an energy `E` should be equivalent relativistically to some amount of mass `m`. In this section I'll prove that the relativistic equations are as claimed: `p=mgammav` and `E=mc^2`. The structure of the proofs is essentially the same as in two famous 1905 papers by Einstein, “On the electrodynamics of moving bodies” and “Does the inertia of a body depend upon its energy content?” If you're interested in reading these arguments as Einstein originally wrote them, you can find English translations at www.fourmilab.ch. We start off by proving two preliminary results relating to Doppler shifts.

On p. 694 I showed that when a light wave is observed in two different frames in different states of motion parallel to the wave's direction of motion, the frequency is observed to be Doppler-shifted by a factor , where `c=1` and `v` is the relative velocity of the two frames. But a change in frequency is not the only change we expect. We also expect the *intensity* of the wave to change, since a combination of electric and magnetic fields observed in one frame of reference becomes some other set of fields in a different frame (p. 668). There are equations that express this transformation from `E` and `B` to `E'` and `B'`, but they're a little complicated, so instead we'll just determine what happens in the special case of an electromagnetic wave.

Since the transformation of `E` and `B` to `E'` and `B'` is a universal thing, we're free to imagine that the wave was created in any way we wish. Suppose that it was created by a uniform sheet of charge in the `x-y` plane, oscillating in the `y` direction with amplitude `A` and frequency `f`. This will clearly produce electromagnetic waves propagating in the `+z` and `-z` directions, and by an argument similar to that of problem 7 on p. 645, we know that these waves' intensity will not fall off at all with distance from the sheet. Since magnetic fields are produced by currents, and the currents produced by the motion of the sheet are proportional to `Af`, the amplitude of the magnetic field in the wave is proportional to `Af`. The oscillating magnetic field induces an electric field, and since electromagnetic waves always have `E=Bc`, the oscillating part of the electric field is also proportional of `Af`.

An observer moving away from the sheet sees a sheet that is both oscillating more slowly (`f` is Doppler-shifted to `fD`) and receding. But the recession has no effect, because the fields don't fall off with distance. Also, `A` stays the same, because the Lorentz transformation has no effect on lengths perpendicular to the relative motion of the two frames. Since the fields are proportional to `Af`, the fields seen by the receding observer are attenuated by a factor of `D`.

Since the fields in an electromagnetic wave are changed by a factor of `D` when we change frames, we might expect that the wave's energy would change by a factor of `D^2`. But the square of the field only gives the energy per unit volume, and the volume changes as well. The following argument shows that the volume increases by a factor of `1"/"D`.

If an electromagnetic wave-train has duration `Deltat`, we already know that its duration changes by a factor of `1"/"D` when we change to a different frame of reference. But the speed of light is the same for all observers, so if the length of the wave-train is `Deltaz`, all observers must agree on the value of `Deltaz"/"Deltat`, and `1"/"D` must also be the factor by which `Deltaz` scales up.^{10} Since the Lorentz transformation doesn't change `Deltax` or `Deltay`, the volume of the wave-train is also increased by a factor of `1"/"D`.

Combining the two preceding results, we find that when we change frames of reference, the energy *density* (per unit volume) of a light wave changes by a factor of `D^2`, but the volume changes by `1"/"D`, so the result is that the wave's energy changes by a factor of `D`. In Einstein's words, “It is remarkable that the energy and the frequency of a [wave-train] vary with the state of motion of the observer in accordance with the same law,” i.e., that both scale by the same factor `D`. Einstein had a reason to be especially interested in this fact. In the same “miracle year” of 1905, he also published a paper in which he hypothesized that light had both particle and wave properties, with the energy `E` of a light-particle related to the frequency `f` of the corresponding light-wave by `E=hf`, where `h` was a constant. (More about this in ch. 34.) If `E` and `f` had not both scaled by the same factor, then the relation `E=hf` could not have held in all frames of reference.

Suppose that a material object O, initially at rest, emits two light rays, each with energy `E`, in the `+z` and `-z` directions. O could be a lantern with windows on opposite sides, or it could be an electron and an antielectron annihilating each other to produce a pair of gamma rays. In this frame, O loses energy `2E` and the light rays gain `2E`, so energy is conserved.

We now switch to a new frame of reference moving at a certain velocity `v` in the `z` direction relative to the original frame. We assume that O's energy is different in this frame, but that the change in its energy amounts to multiplication by some unitless factor `x`, which depends only on `v`, since there is nothing else it could depend on that could allow us to form a unitless quantity. In this frame the light rays have energies `ED(v)` and `ED(-v)`. If conservation of energy is to hold in the new frame as it did in the old, we must have `2xE=ED(v)+ED(-v)`. After some algebra, we find . In other words, an object with energy `E` in its rest frame has energy `gammaE` in a frame moving at velocity `v` relative to the first one. Since `gamma` is never zero, it follows that even an object at rest has some nonzero energy. We define this energy-at-rest as its mass, i.e., `E=m` in units where `c=1`.

Defining an object's energy-at-rest as its mass only works if this same mass is also a valid measure of inertia. More specifically, we should be able to use this mass to construct a self-consistent logical system in which (1) momentum is conserved, (2) conservation of momentum holds in all frames of reference, and (3) `papproxmv` for `v"<<"c`, satisfying the correspondence principle.

Let a material object P, at rest and having mass `2E`, be completely annihilated, creating two beams of light, each with energy `E`, flying off in opposite directions. A real-world example would be if P consisted of an electron and an antielectron. As shown on p. 693, light has momentum. Because beams of light can be split up or recombined without violating conservation of momentum, a light wave's momentum must be proportional to its energy, `|p|=yE`, where the constant of proportionality `y` is found in problem 12 on p. 789 but not needed here. Let the momentum of a material object be `mvx`, where our goal is to prove `x=gamma`. In this frame of reference, `v=0`, and conservation of momentum follows by symmetry.

We now change to a new frame of reference, moving at some speed `v` along the line of emission of the two light rays. In this frame, conservation of momentum requires `2Evx=yE"/"D-yED`. We therefore have `vx"/"y=(1"/"D-D)"/"2`, which can be shown with a little algebra to equal `vgamma`. Since only `x` can depend on `v`, not `y`, and the correspondence principle requires `xapprox1` for `v"<<"c"`, we find that `x=gamma`, as claimed.

Problem 15 on p. 791 checks that this result also works correctly for a system consisting of material particles.

26.6 Proofs by Benjamin Crowell, Light and Matter licensed under the Creative Commons Attribution-ShareAlike license.