I get mail: Does a box of photons have mass?

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 E=mc^2, 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:

  a=\frac{F}{m}=\frac{10N}{10kg}=1 m/s^2

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:

  E=mc^2=(10kg)(3\times 10^8m/s)^2=9\times 10^{17}J

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:

  1. The laws of physics are the same in all inertial frames of reference.
  2. 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 E=mc^2 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:

  p=\frac{h \nu}{c}

where h is the Planck constant, and nu 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:

  \frac{\nu_f}{\nu_0}=\sqrt{\frac{1+v}{1-v}}\simeq 1+v/c

and similarly

  \frac{\nu_b}{\nu_0}\simeq 1-v/c

So the total momentum of the forward going photons are:

  P_f=\frac{N}{2}\frac{h\nu_f}{c}=\frac{N}{2}\frac{h\nu_0}{c}(1+v/c)

and backward gets a minus sign in two places:

  P_b=-\frac{N}{2}\frac{h\nu_b}{c}=-\frac{N}{2}\frac{h\nu_0}{c}(1-v/c)

Adding them together yields:

  P_{box}=\left(\frac{Nh\nu_0}{c^2}\right)v

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:

  P_{box}=Mv=\left(\frac{Nh\nu_0}{c^2}\right)v

The bit in the parentheses is the mass. It’s also worth noting that:

  E_{tot}=Nh\nu_0=E

so

  P_{box}=\left(\frac{E}{c^2}\right)v

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.

  1. 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.
  2. 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.

-Dave

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7 Responses to I get mail: Does a box of photons have mass?

  1. Joe Bedard says:

    Dave, you said a photons in a box have mass even though each individual photon is massless, so if we let the photons out of this box, do released photons go back to being massless?

    If answer is yes then suppose we don’t let photons escape completely, we just let them escape to a bigger box which contains the smaller box, do photons still have same mass they had in smaller box? What if we let size of box get a lot bigger, do photons ( whose confinement is going down, but is not zero ) still have same mass? What if size of box is size of a black hole?

    You might say, mass goes down as confinement goes down, so would you then say inertia is a property of confinement or a function of applied force which would imply inertia is not a property of the particles ( at least photons) by themselves in absence of force?

    If photon inertia is a function of confinement, then could I change the center of mass of a closed system box by confining and releasing sub boxes of photons in various sub regions of this main box thus changing a system’s center of mass without applying an eternal force?

    Or if you say photons don’t go back to being massless when freed, how is that?

  2. dave says:

    Here’s the deal:
    Inside the box, or outside the box, the individual photons never have mass. They do have energy, and one of the strange quirks of relativity is that energy — not mass — is the source for gravity.

    But as for inertial mass, you can think of it as a sort of ensemble property, in the same way that a single atom flying around doesn’t have a temperature, but a bunch of atoms flying around in random directions does. By the same token, a box full of photons has inertial mass even though the individual photons don’t.

    The point is that it’s not the size of the box, but rather the fact that you are considering the photons collectively rather than individually.

  3. Joe says:

    Ok, so if photons (with mass) are released from box will photons go back to individual state and become massless? And if they do become massless, then I assume they can never be collected again because that would imply they were still never released from the criteria of “photons collectively” Ok. So if you take a bunch of individual independend massless photons and put them in a box, to the photons acquire mass? Mass (inertia) may not be a fundemental properity (like temperature), but what physics says that inertia of a system can not change when particles are converted from fermions to bosons? after fermions and bosons are different in so many ways.

  4. Joe Bedard says:

    Follow up of previous comment.
    What I meant to ask in previous comment is how do photons interact when they are in a state of confinement to manifest inertia to external force? My limited understanding of QED tells me photons don’t interact with each other because no loops in Feynman diagrams for photons..right? And I am also asking if I have a box of particles and I can convert the particles from fermions to bosons and back at will, and nothing escapes the box, does the box have any external difference in properties? Can i tell what kind of particle I have in the box by only measuring motion of the box?

  5. Joe Bedard says:

    When the photons are in the box, they are bosons right? Ie they must obey Bose Einstein statistics right? Although the energy density is the same regardless of if box contains matter/antimatter or photons , the quantum states contained within the box wll be very different right? So that means the interaction of the stuff contained in the box must interact different ly with sides of box depending if contents photons or m/anti right? Is there any relationship between the quantum state something is composed of and how gravity interacts with it?

  6. John Macken says:

    Dave,
    You may be interesting in reading a different treatment of the question about whether confined photons have inertia. The first chapter of the book: The Universe Is Only Spacetime deals with this subject and goes much further. (available at http://onlyspacetime.com/ ). For example, the appendix at the end of chapter 1 shows that the inertia of confined photons matches the inertia of an equal energy in the form of matter even at relativistic velocity. Furthermore, confined photons have a total of 8 particle-like properties including the equivalent of de Broglie waves and time dilation. This similarity between confined light and particles is developed into a model of the universe made of the quantum mechanical properties of 4 dimensional spacetime. The development of this idea includes the derivation of gravity and electric fields.

  7. bossini Antonietta says:

    Non sono la persona più adatta per fare un commento. Ma per quanto possa valere eccolo! Ammesso che si possano ottenere 5 kg di materia oscura, io avrei paura a manipolarla! Dico che é giunto il momento di dire BASTA! Non sarebbe la prima volta che l’umanità deve ricominciare a causa di disastri inimmaginabili! ( vedere le tracce di rocce fuse).
    Certamente i vostri calcoli ( nei quali non ho capito nulla) sono esatti ed é quello che mi fa paura!!!!! Scusate lo sfogo.

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