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2008

Does light have mass?

This is one of the most common questions that keeps cropping up when people are studying physics, and its actually a good question.

The answer is ... yes and no.

Light certainly carries energy (shine a spotlight at an object and that object gets hot), and it also carries momentum, that is, a lightbeam has a certain amount of "push" to it. A lightbeam aimed at you should push on you with a certain amount of force, and the object projecting the beam should feel a certain amount of recoil. For an object placed in the path of the beam, we talk about it experiencing a certain amount of light-pressure, and technically, when a body absorbs light, and is given a small "shove" by the beam , the momentum that originally belonged to the light is now owned by the (now-moving) object.

1: The energy and momentum of light

For everyday purposes, the amount of momentum carried by light is pretty small, but it's proportional to the beam's energy, which is in turn proportional to the quantity and frequency of the light. Higher frequencies carry more energy, and have a correspondingly stronger "shove".

If someone shines a torch at you, and you run straight at the beam, the pressure of the light hitting you in the face should be slightly be stronger, and the apparent frequency of the signal will be higher, too, because you're intercepting the light-signal at a faster rate.  If you try to run directly away from the lightbeam, the light-signal's apparent frequency will be lower ( thanks to its Doppler shift), it's apparent energy will be lower, and the amount of momentum that you feel the light to have when it hits you should be smaller, too, by exactly the same amount.

If the person then throws the torch at you, then the light coming at you from the approaching torch will again have a higher frequency, energy and momentum. But the hurled torch itself, as a moving body, will also have kinetic energy and momentum. So since we say that these things, for the torch, are standard properties of a  moving mass, why don't we say that light is made up of little moving particles that also carry mass?

2: What is mass?

When we look at our operational definitions for "mass", we find that they were designed around the sorts of properties that we associate with solid objects, and one of the most important properties that objects have but light doesn't is persistence. Objects tend to stay put, and if they move, they tend to move slowly enough for us to be able to see them by bouncing light off them. Objects have a location, even if that location isn't always fixed with time. We can grab a moving object and stop it, and then poke it about to measure its properties.

But with light, transience is part of its very nature. If you could stop light dead, it would cease to be light. It's frequency would be zero, it's energy would be zero, and its momentum would be zero. All the properties that we came across earlier that sounded rather like the properties of matter would disappear. We can suggest (as Einstein did in his E=mc² paper  ) that light can transfer momentum between an emitting object and a receiving one, but while the light is "in flight", it's difficult to say that it has "mass" itself, in any conventional way. A single pulse of light just isn't a complicated enough thing to be able to support the properties that we normally associate with "mass"

What we can do, though, is to trap light, and when we do that, the rules change.

3: Trapped light

Although an individual "stopped" light-beam wouldn't seem to have doesn't have energy or momentum, we can (in theory) trap light within a mirrored capsule or container. We can imagine a complex of light bouncing about within the container, exerting pressure equally on all of the container's walls. The light-complex still has energy, but the momentum of its different components, acting in different directions, cancels out. The container stays put. The energy of the light is now confined to its container, and we can now carry out thought-experiments with it.  

If we try to move the container, our lightcomplex doesn't like it. The container wall that's pushed against the light-complex feels a greater resistance due the fleeting increase in light-pressure, and the wall that moves away form the light-complex feels a reduced light-pressure. When we push against the container, it seems to resist, just as if it contained a piece of matter, rather than trapped light-energy.

Once the container is moving, and its internal light has quickly reached a state of equilibrium at the container's new velocity, the moving container's mass has momentum, and so does the light-complex inside it. The light traveling in the same direction as the container now has more energy and momentum that the light components acting in the opposite direction.  The trapped light-complex has a total momentum of zero when it is stationary, but a conventional-looking momentum when the complex moves.

This is much more like the behaviour of mass, and in fact, we find that the initial energy of the light-complex and its net momentum change with velocity in exactly the same way as the relationship between the rest mass of a body, and the momentum that a body has when it's moving. If we use Newtonian mechanics, we say that the momentum of a body, p, is equal to its mass times its velocity, p=mv ... and if we calculate the way that Doppler shifts change the forward and rearward energies and momenta of the different light-components, we find that the overall momentum of the complex is proportional to velocity and energy, p=E(something)v. This suggests a simple scaling factor between light-energy E and the equivalent amount of mass, m

If we then do the same exercise under special relativity, the Doppler shifts and the mass-momentum relationships are both a little bit more complicated than under Newtonian physics, but again, we get the result that the captive light-complex has properties equivalent to that of a body, with the amount of equivalent mass being proportional to the amount of trapped energy.

What's the proportionality between the energy and the apparent mass that it gives to its container?

Well, if we calculate the exact amount of momentum for a moving light-complex based on its initial "rest-frame" energy, and work out how much mass would have to be in the container to have the same momentum, we find that with either Newtonian physics or special relativity  , it's

m = E / c2

So, if the apparent total mass of a container and its trapped light reduces by an amount m when we open the container and we allow its energy to escape, the amount of escaping energy should be found to be

E=mc2

4: Generality of E=mc2

From here, we can show that the relationship should be general. Let's take our moving container of light, the one whose apparent mass and momentum is increased by the amount of light-energy that it contains. Does the energy have to be in the form of light?

Suppose that the container has an internal mechanism that suddenly exposes solar panels that soak up the light and use it to charge an internal battery. Should the container's speed or momentum change? What if that stored energy is then used to drive an internal lightsource that replaces the original lightcomplex? Should the container's speed or momentum change again?

For an external observer, we want to say say that the total energy and momentum (and apparent mass) that the sealed box presents to the outside world should remain the same, regardless of what might be going on inside. The energy and momentum of the box-system should be conserved.

So, the fact that this trapped energy is in the form of light shouldn't be relevant. If the light is absorbed by the walls of the box, heating it, and the energy is turned into the additional thermal energy of the jiggling molecules that make up the box structure, then this heat energy should again contribute the same amount of mass, according to the same E=mc² formula. If that heat is absorbed by chemical reactions inside the box, then the energy of those chemical bonds will again contribute mass, according to E=mc².

Mass and energy are interchangeable, just as Newton had suggested a few hundred years earlier, in  "Opticks"  ... what E=mc² gives us is the conversion factor (or "exchange rate") that applies when we want to convert one into the other. Although Einstein's famous paper on E=mc² was originally presented as a followup to his paper on special relativity, the relationship is a general one. 

References
1. John Michell tried (and failed) to measure the momentum of sunlight in the Eighteenth Century.
2. A. Einstein, " Does the Inertia of a Body depend upon its Energy-Content?" ("Ist die Trägheit eines Körpes von seinem Energiegehalt abhänging? ") Annalen der Physik 18 639-641 (1905)
3. Shortly before Einstein's paper, a number of other researchers had already been asking, "What effect does radiation-pressure have on a moving cavity containing trapped electromagnetic energy?". Some of these researchers came tantalisingly close to Einstein's result, but stopped short of actually saying, as Einstein did, that the resulting "mass-like" effects should be considered as real.
4. E. Baird, Relativity in Curved Spacetime (2007), chapter 2, " Gravity, Energy and Mass "
5. E. Baird, " Two exact derivations of the mass/energy relationship, E=mc²physics/0009062
6. E. Baird, Relativity in Curved Spacetime (2007),  " Calculations 2: E=mc² from Newtonian mechanics
7. I. Newton, Opticks, Query 30:
" Are not gross Bodies and Light convertible into one another, and may not Bodies receive much of their Activity from the Particles of Light which enter their Composition? ... "
8. A. Einstein, " An Elementary Derivation of the Equivalence of Mass and Energy " Bull. Am. Math. Soc. 41 223-230 www.ams.org/bull/2000-37-01/