Lately, my my inbox hath overflowed. Yesterday, I got a thought-provoking email from a fellow physics instructor. It’s a good, non-intuitive question about special relativitiy. It’s also got the “technical” tag, so if you’re afraid of a few equations and some truly terrible MS Paint figures, this may not be the right blog for you.
Since you might not make it to the end, let me remind you now to like my new facebook page!. Our correspondent asks:
A box has a mass m. Push on the Box and it has an inertia proportionate to m. If by various processes, some of the matter is converted to energy .. Say by burning fuel or mechanical to electromagnetic or radioactive decay… But the energy is still contained in the box. Does the box have the Same inertia? If answer is simple yes by , how does Energy have inertia?
I’m going to rephrase this a little bit for concreteness. Suppose you had an essentially massless box, and inside there was 5 kg of matter and 5 kg of antimatter, separated by a magnetic field or some other such contrivance.
The total device, of course, would have a mass of 10 kg by any measure you wanted to consider. Pushing on it with a force of 10 N, for instance, would cause it to accelerate at:
Likewise, were you to measure the gravitational pull of the box (which would be tough but, in principle, doable), you’d find it has a gravitational mass of 10 kg.
No problem so far, but what happens when you remove the membrane, and that 10kg of mass turns into:
worth of photons. Photons are individually massless particles, so the question is, does your box still have inertial mass?
Yes. And it has gravitational mass, too.
To understand why we need to delve a little into special relativity, and in particular, into the postulates of special relativity:
- The laws of physics are the same in all inertial frames of reference.
- The speed of light in free space has the same value c in all inertial frames of reference.
This setup is not that dissimilar to how Einstein derived in the first place. So imagine (for simple mathematical convenience) that the light in your box were monochromatic, and the box is stationary, with half of the light traveling to the right, and half to the left.
Light does carry momentum, as we known since Maxwell, and can easily be seen in a radiometer:
where is the Planck constant, and is the frequency of an individual photon. In this case, the momentum cancels.
But now look at the box from a different inertial perspective, one where the box is moving to the right at v. This speed can be much less than the speed of light, and will still produce an interesting answer.
The 2nd postulate of special relativity tells us that all photons travel at the same speed. The only thing that changes if you look at them in a moving frame is their frequency (or equivalently, their wavelength). The frequency of the photons in the forward-going direction are higher than they would be if the box were at rest (blueshifted), and the backward-going direction are _lower_ than they would be if the box were at rest (redshifted). The relation is:
So the total momentum of the forward going photons are:
and backward gets a minus sign in two places:
Adding them together yields:
Feel free to check my algebra, but the upshot is that there are two terms each in the forward-going and backward-going momenta, and one of them cancels, and one of them adds.
This means that the impulse required to push the box is the value above, and since the box is moving at non-relativistic speeds, we can re-write this as:
The bit in the parentheses is the mass. It’s also worth noting that:
So yes, a collection of photons has inertial mass because it requires an impulse to increase their net momentum.
As a final bonus: does a box of photons have gravitational energy? Absolutely yes! I’m not going to prove this in detail, but I’ll simply give you a flavor for why.
- The equivalence principle of general relativity says that there is no distinguishing between being in free-fall and a true gravitational field. As a result, all massive bodies fall with the same acceleration in the same field. After all, the curvature describes the acceleration, not some inverse square law.
- But Newton’s 3rd law really does hold. It gives rise to conservation of momentum, which means that if my box of photons is accelerated toward the earth, Newton #3 says that the earth must be accelerated toward the box with the same force.
Tada! A box of photons has mass even though each individual photon is massless!
Of course, this shouldn’t be such a big surprise. After all, what is the Higgs but a way of turning interaction energy into mass? For that matter, would it surprise you to learn that protons are about 50 times more massive than the quarks that make them? The rest is all interaction energy.