Since light has inertia and experiences gravity, what does it mean for photons to be massless?

Question

I've been trying for a long time to figure out what the heck mass even IS. In introductory physics and chemistry, students are told that massive objects are those that are made of matter and take up space. But then matter is defined as anything that takes up space and has mass, which is circular. Later on, we learn that mass is related to inertia, or the ability to resist changes in motion and that mass is proportional to gravity and I've read multiple times about Einstein unifying those definitions. OK, that works well enough in classical physics, but then we learn that photons are massless -- logically, that must mean they don't have inertia and/or aren't affected by gravity. Except, that's not true -- light DOES have inertia and gravity. Plus, it turns out that mass isn't even required for gravity anyway -- plain old energy warps spacetime just fine, which implies that we shouldn't use gravity to define mass anyway.

At this point I'm tempted to just throw up my hands and decide "mass" is simply an ill-defined term and none of this matters. But that can't be right, because the idea of photons being massless is apparently very important to QM. OK, so if I look deeper I find that, in particle physics, mass is supposedly just the confinement of energy -- the Higgs field somehow "confines" massive fundamental particles and composite particles, like protons, gain most of their mass from the confinement of the fundamental particles that make them up. On a larger level, even atoms and molecules gain some additional mass from the confinement of their constituent parts. At first, that made sense to me because it harked back to the idea that massive objects take up space -- confining the particles must be what makes that happen, I thought. And it made sense that mass ultimately was an emergent property of a certain type of energy, since, you know, $E=mc^2$ and the more general, $E^2=(mc^2)^2+(pc)^2$. But then someone pointed out that the idea of "taking up space" doesn't really make sense on the level of particles because the uncertainty principle means they don't even have well-defined positions most of the time, plus they seem to behave as point-like objects.
So at this point the only thing I can think of is that photons don't interact with the Higgs field and they're fundamental particles and so that's why they don't have mass. Except that doesn't really help me understand anything -- we've known photons were massless since before we even knew the Higgs field was a thing and most of the mass of macroscopic objects isn't due to Higgs anyway but the confinement of quarks in protons and neutrons, so Higgs can't be what DEFINES mass. So what the heck IS it? Because it seems like the confinement definition has nothing to do with the classical physics definition, at which point, why are we even calling it "mass" anymore?

I know I have to be missing something here, but I can't figure out what and I'm pretty darn frustrated and confused. Can someone please help me understand?

Edit: I have no idea why this was closed. The linked question it's supposedly a duplicate of is only tangentially related. My question is NOT, "how does gravity affect light?", it's "what does it MEAN for photons to be massless?"

Answer 1

Mass is defined by $$ m^2 c^2=E^2/c^2-p^2$$

In your question you mention this formula, and follow it with

But then someone pointed out that the idea of "taking up space" doesn't really make sense on the level of particles

Taking up space has nothing to do with mass. Sometimes taking up space is used in a definition of matter. With that definition of matter not all things with mass are matter. But that fact is no objection to this definition of mass.

In relativity this definition of mass is particularly elegant and useful. In the same way that relativity unites space, $\vec x$, and time, $t$, into spacetime $x^\mu=(ct,\vec x)$. Relativity also unites energy, $E$, and momentum, $\vec p$, into the four-momentum $P^\mu=(E/c,\vec p)$.

The mass formula given above is simply the magnitude of the four-momentum vector. That is important because magnitudes of four-vectors are invariant, meaning all frames agree on their value. Since the four-momentum is conserved, an invariant of a conserved quantity is very useful.

Answer 2

One additional note: photons have zero rest mass but because they possess energy, they exhibit momentum while traveling.

Answer 3

The problem is a change of language, which causes a lot of confusion.

When I studied physics, long ago, one distinguished two different meanings for the word "mass".

One was the "rest mass", better called "invariant mass", which was a characteristic of the particle. Photons had a zero "rest mass" because they cannot be at rest. And it was more correct to say that their "invariant mass" was zero.

The other one was the "total mass", or "relativistic mass", which was the "total energy" divided by $c^2$. Total energy was "rest energy", "rest mass" multiplied by $c^2$, plus the kinetic energy.

For photons, whose rest mass was zero, the "total mass" was just their energy divided by $c^2$.

But now the language has changed. "Mass" is now only accepted to mean invariant mass. So one cannot say anymore that a photon has zero "rest mass" but nonzero "total mass".

Now one can only say : "Photons have zero mass, period."

Einstein, when writing $E=mc^2$, meant $m$ to meant the total mass.

By changing the language, one makes this equality false.

Now even though the "total mass" (I am here using deliberately an obsolete expression) of a photon, or of any particle the kinetic energy of which is not negligible compared to its "rest mass" times $c^2$ does not act exactly like the rest mass of a particle at rest, in a gravitational field it remains that the "gravitational mass" and the "inertial mass" of a photon (or of any particle the kinetic energy of which is not negligible compared to its "rest mass" times $c^2$) are essentially their "total mass".

So when people tell you that a photon, near a large mass, "follows geodesics even though they have no mass", they do tell you a truth, but not the whole truth.

Sure, photons follow geodesics.

But what these people do not tell you is that a photon also modifies the metric of space-time, because having energy it has "gravitational mass", and thus influences the motion of the massive object. Not by much, maybe. But it does. Claiming the "massless" photons only follow geodesics modified by the presence of another mass without adding that the photon does modify the geodesics and alters the motion of the other mass is, in my opinion, not quite honest.

When one used to distinguish "rest mass" (= "invariant mass") and "total mass" (= "relativistic mass") there was much less confusion.

Answer 4

Kinetic energy has no mass. Light has no mass. Fat has mass.

Maybe we can make an enlightening thought experiment out of those pieces of information.

Let's say Bob starts accelerating his body, using the fat of his body as energy source. Bob's mass decreases.

Let's say Bob is 100% fat, and that fat is 100% energy. Now then when all the fat is burned, speed of Bob is c, and mass of Bob is zero.

Light has energy and no mass. Bob has energy and no mass.

Energy of Bob is kinetic energy, that is massless. Energy of Light is some form of energy, that is massless.

(I am not saying anything about what energy light is made of)