Protons have mass? I didn’t even know they were Catholic. –Woody Allen

As is often the case, this story began with a misquote.

Someone on Twitter quoted the above, but wrote “photons” instead of “protons”. When another person (not me) challenged him on it, he doubled down and said that everything has mass because of *E* = *mc*^{2} : anything possessing energy could be considered to have an effective mass equal to *m* = *E*/*c*^{2}. Of course, you *could* do that—write a mass value for anything, including stuff like energy released in a chemical reaction—but that’s not what mass is. It may feel intuitively right, but it’s not *meaningful* from a physics point of view. Photons are indeed massless in a fundamental sense, but to see why that’s important, it’s good to understand mass, energy, and their role in the most famous equation in physics.

(A quick aside on what I’m *not* going to talk about: mass has another purpose in gravity, where it is sometimes called gravitational mass. In both Newtonian physics and general relativity, gravitational mass is equivalent to the more ordinary form of mass, something known as the [weak] equivalence principle. The equivalence principle is an important subject in its own right, but it’s beside the point for the moment.)

### Mass and energy

Mass is a measure of inertia: how difficult it is to change an object’s state of motion. To phrase it another way, if an object is moving at a steady speed (its state of motion is constant), mass is what tells you how hard it is to slow it down or speed it up. Large masses are more difficult to accelerate, which is the same thing as saying they have greater inertia. From a physics point of view, energy is a measure of the ability of a system to do things: punch through barriers, break chemical bonds, escape gravity, etc. Moving objects have kinetic energy, which larger for faster motion. Greater energy is needed to change the state of motion of a larger mass, and a larger mass carries greater kinetic energy if it is moving at the same speed as a smaller mass.

### Relatively speaking

Things get fun when we add relativity to the mix. All measurements depend on the relative motion of the observer and the observed, and no frame of reference is more valid than another. To put it another way, there’s no such thing as an object being absolutely at rest: from another reference point, that object will appear to be moving. (On Earth, we use the ground as a convenient reference frame, but in no real sense is it stationary. Earth rotates on its axis and revolves around the Sun, which in turn orbits the galactic center, and so forth.) In other words, kinetic energy is relative to a reference frame: if you’re moving at the same speed in the same direction as an object, you won’t measure any kinetic energy from it. Already something interesting is going on, since energy is *conserved* in physics: it’s not created or destroyed, only transformed.

To see what’s going on, consider the related concept of momentum: a quantity that depends on the mass and the speed of the object. (Light also carries momentum, but hold off on that for a moment.) Momentum is also conserved, and is intimately related to kinetic energy. However, unlike kinetic energy, momentum tells which direction an object is traveling, as well as how fast. In relativity, that associates momentum with space, while energy is associated with time. Just as space and time are united as spacetime in relativity, energy and momentum are unified as energy-momentum. The total energy-momentum content is conserved, rather than energy and momentum separately.

When you’re moving at the same speed in the same direction as an object, it has no momentum or kinetic energy, from your perspective. However, conservation of energy-momentum tells us that there must still be *some* energy left over, which we call the “rest energy” of the object. That energy can be written as (wait for it):

where *E*_{0} is the rest energy, *m* is the object’s mass, and *c* is the speed of light. In other words, the most famous equation in physics is actually a statement of the *minimum* energy an object can have, what’s left over when it is at rest with respect to the observer. When that same object is in motion, the equation changes towhere *v* is the speed the object is moving relative to the observer. When *v* is 0, the equation reduces to the famous form*, *but as* *the speed approaches the speed of light, energy approaches infinity: you would need an infinite amount of energy to bring a massive particle to light-speed. [**Update**: see the note at the end of the post for a comment about "relativistic mass".]

As I pointed out in an earlier piece, Albert Einstein didn’t often use the famous form of his equation because it’s inaccurate. The version he preferred to use makes a point:As you can see, it’s exactly the same *meaning* as the famous equation, but with the items shuffled (as in a sentence spoken by Yoda). The point Einstein was trying to make by writing it this way comes out in the title of his second 1905 paper on relativity, “Does the inertia of a body depend upon its energy-content?” We already established that mass is a measure of an object’s inertia — resistance to forces — so the purpose is to say that mass is equivalent to the rest energy of the object. The speed of light is a fundamental constant of nature, so its presence in the equation isn’t particularly relevant. Physicists, including Einstein, often set *c* equal to 1 for that reason, allowing everything to be written in terms of energy.

### Back to photons

Light is different. As we saw, you can’t bring an object with mass up to light-speed, so you can never observe or perform measurements on a photon while moving at the same speed. In other words, its momentum will never be zero — and it has no rest energy, because it is never at rest. A photon has no minimum energy, so the equation *E*_{0} = *mc*^{2} is zero on both sides. Another way to think of it: a photon will always travel at light-speed, so you can’t accelerate it. It has no inertia, and hence no mass.

For a photon, the energy is where *h* is a number called Planck’s constant and λ is the wavelength of the light. If you observe the photon while moving relative to the source that emitted it, it will exhibit the Doppler effect and the beaming effect (though both effects are going to be small unless the relative speed is pretty big). In the Doppler effect, the wavelength will appear shorter (bluer) if the source is moving toward you, or longer (redder) if the source is moving away — the measured energy is relative to the motion.

So, you could technically write a “mass” for a photon using Einstein’s formula, but it would have a different value depending on how fast the light source was traveling relative to the observer. With such a large potential variation in “mass” value, it’s obvious it has nothing to do with inertia and therefore has no correspondence to what mass means for matter.

### Addendum: “Relativistic mass”

A commenter below referred to the notion of mass appearing to increase near the speed of light. That’s based on the energy equation again,but saying that it’s the mass itself increasing, relative to an observer. From that perspective, there are two kinds of mass in relativity: rest mass and relativistic mass. That’s the way Einstein originally formulated it, and a lot of physics books (including textbooks) talk about it that way.

The problem is that “relativistic mass” isn’t a useful concept. An object’s inertia doesn’t increase in a meaningful way, because from that object’s perspective, any force acting on it acts within its own reference frame. To put it another way, we can’t measure “relativistic mass”: any mass measurement will turn up the inertial mass. It’s much better to talk about the increased energy near the speed of light, even if Einstein didn’t think so back in 1905.

Excellent explanation on why a photon does not have a rest mass! Well done, and thank you for posting this article, Matthew!

I agree with Christian Luca — well done, Matthew!

-Don

I now have two questions:

Firstly, since measurements are relative and we can treat any reference frame as being ‘still’ (So a train passing me can consider itself still while I am moving by rapidly.) What happens when I take two objects and accelerate them to say, 0.75c from my reference frame but in opposite directions. (So that if one were considered still the other object would be approaching it at 1.5c)?

Secondly I understand how energy, momentum and the doppler effect works for waves in a medium, but how does it work for photons? A star emits photons of a certain momentum and thus energy. If I am moving away from it at speed I will see lower energy ‘redshifted’ photons; energy seems to have vanished somewhere. I have always assumed that it has something to do with p = mv, since v and p are constant the ‘mass’ and thus the energy drops but this sounds wrong to me somehow. I just can’t wrap my head around it.

Photons do have momentum, but it’s equal to h/λ (since they have no mass and they all move at the speed of light). If you’re moving relative to their source, some of the energy gets shuffled into momentum or back, just like it does for massive particles.

There’s a special formula for adding velocities together in relativity, instead of just adding them up like you can for smaller speeds.

Ok, so what is conserved? The sum of energy and momentum? And this works for massive particles too? I know rapidly moving particles gain mass at the expense of velocity, is that related?

It’s not useful to think of mass increasing (though it’s common to think of it that way). I added an extra piece to the end of the post for that.

Both energy and momentum are conserved, but remember that this ideal light source as we’re seeing it moving relative to us is emitting in all directions. The energy increase we measure in one direction is offset by the energy decrease in the opposite direction.

(That’s not the whole story, though, since if the source beamed light preferentially in one direction, my explanation would break down. There’s more concepts we need before I can give you a full answer, but in brief: in special relativity we take a measurement relative to our own frame of reference. If we measure light from a stationary source, then take a second measurement while moving relative to that source, we changed the state of motion, either ours or the source’s. To do that requires changing the energy: using rocket thrusters or similar.)

New favourite physics / Woody Allen quote!