How Can Photons Contribute Mass to a Container?

[PhyCurious]

TL;DR

If photons are massless, how could trapping them in a container increase the mass of the container?

My question comes from reading the wikipedia page on mass-energy equivalence. The statement would seem to be contradictory:

In a similar manner, even photons (light quanta), if trapped in a container space (as a photon gas or thermal radiation), would contribute a mass associated with their energy to the container. Such an extra mass, in theory, could be weighed in the same way as any other type of rest mass. This is true in special relativity theory, even though individually photons have no rest mass. The property that trapped energy in any form adds weighable mass to systems that have no net momentum is one of the characteristic and notable consequences of relativity.

So photons contribute to the energy (and therefore mass) of the container; but photons are massless - that is, they have 0 rest mass. But doesn't this mean that we're basically adding 0s to a container of X rest mass and eventually getting X+1? This doesn't make sense so the gap in understanding must be on my end.

 

[PeterDonis]

photons contribute to the energy (and therefore mass) of the container

Yes.

photons are massless - that is, they have 0 rest mass. But doesn't this mean that we're basically adding 0s to a container of X rest mass and eventually getting X+1?

No. Rest mass is not additive. But energy is, and, as you have said, the photons add to the energy of the container. More precisely, they add to the rest energy of the container (the container's energy in its own rest frame), and therefore add to its rest mass.

 

[Ibix]

Mass isn't additive in relativity. The energy-momentum four-vector is additive, and mass is the modulus of that total vector.

Light pulses have null energy-momentum four vectors. Null vectors have modulus zero, so light has zero mass. But the sum of two null vectors isn't necessarily null - in particular, the energy-momentum four vectors of two light pulses traveling in opposite directions do not add to a null vector. So the combination of two light pulses can have non-zero mass.

 

[Nugatory]

Summary: If photons are massless, how could trapping them in a container increase the mass of the container?

So photons contribute to the energy (and therefore mass) of the container; but photons are massless - that is, they have 0 rest mass. But doesn't this mean that we're basically adding 0s to a container of X rest mass and eventually getting X+1?

No, because the rest mass of a system is not in general equal to the sum of the rest masses of its components.

Much of the confusion around these questions would go away if we would talk about the "rest energy" of the container and its contents instead of "rest mass". They're the same thing, just expressed in different units (related by $E=mc^2$) but there's nothing surprising about the energy of the electromagnetic fields contributing to the total rest energy.

For a sort of intuitive picture of what's going on, imagine that your container is a box, so perfectly sealed and insulated that nothing - not energy, not matter, not radiation, not heat - can enter or leave. No matter what happens inside the box, its mass cannot change because that would violate conservation of energy - the total energy of the box is $mc^2$ and that cannot change because nothing is entering or leaving. So if the box contains an electron and positron (non-zero rest mass) and they annihilate producing photons (zero rest mass, but the same energy as the rest energy of the two particles) the mass of the box doesn't change; we count the photon energy instead of the electron rest energy towards the total mass/energy of the box.

 

[vanhees71]

The mass of the container including the photons is larger by $\Delta m=E_{\text{photons}}/c^2$ than the box without the photons.

Note that today we only use invariant mass to describe mass, i.e., the photon mass is always and in any frame 0. Nevertheless photons have energy and momentum. Within a box, the photon energy contributes to the total energy of the entire system in the center-momentum frame, where the system is at rest as a whole, and the total energy of the system in this frame is by definition its mass (times $c^2$ in conventional units).

 

[PhyCurious]

So if the box contains an electron and positron (non-zero rest mass) and they annihilate producing photons (zero rest mass, but the same energy as the rest energy of the two particles) the mass of the box doesn't change; we count the photon energy instead of the electron rest energy towards the total mass/energy of the box.

OK, this is helpful. I am still struggling to understand how something with zero rest mass can have energy without violating mass-energy equivalence. Would it be correct to say that the photons have zero rest mass but have relativistic mass (I understand this is a somewhat problematic or confusing term, but essentially their relativistic energy / c²) because they possesses kinetic / some other kind of energy?

The energy of a photon is E = hf (Planck's constant * the frequency of the photon). But why wouldn't the mass be mc² = hf and therefore m = hf/^c2?

 

[PhyCurious]

Note that today we only use invariant mass to describe mass, i.e., the photon mass is always and in any frame 0. Nevertheless photons have energy and momentum.

This, in a nutshell, is what I'm struggling to understand. The mass is zero but the energy is non-zero. Just trying to wrap my head around this, I've been reading different explanations of this all day (here and on other websites). It seems to have something to do with the inertial frame of the photon and system in question.

 

[Ibix]

Mass and energy are different things. Mass is the modulus of the energy-momentum four vector, energy is one component of the vector. They are equal (give or take factors of $c$) under some circumstances (when momentum is zero, only the energy component of the energy-momentum four vector is non-zero), but not in general.

inertial frame of the photon

There is no such thing.

 

[Nugatory]

This, in a nutshell, is what I'm struggling to understand. The mass is zero but the energy is non-zero.

The relationship between mass and energy is $E^2=(mc^2)^2+(pc)^2$.

For a particle at rest, $p=0$ so this equation reduces to the familiar $E=mc^2$.

For a massless particle such as a photon, $m=0$ so this equation reduces to $E=pc$ and the energy is non-zero even when the mass is zero.

A massive particle not at rest will have both $m$ and $p$ non-zero so both te4ms will contribute to the total energy.

 

[PhyCurious]

For a massless particle such as a photon, $m=0$ so this equation reduces to $E=pc$ and the energy is non-zero even when the mass is zero.

I get what you are saying and could probably repeat it back to you, but this is a fairly non-intuitive idea, right? Basically a photon can never be at rest, so it has momentum of magnitude $p$? And as a result has energy but no mass (thereby violating the words of "mass-energy equivalence" but not violating the equations of relativity that produce mass-energy equivalence, just our English transliteration of it?)

 

[weirdoguy]

thereby violating the words of "mass-energy equivalence"

Because there is no mass-energy equivalence, and I don't know why some pop-sci sources (and even "serious" ones) insist on using this phrase. Mass can be seen as form of energy (rest energy $E_0=mc^2$), but not the other way around. Energy (in general) is a different property than mass and as you can see in equations posted by @Nugatory, also momentum comes into play.

 

[PeterDonis]

The mass is zero but the energy is non-zero.

That's because all of a photon's energy is kinetic energy; none of it is rest mass energy. A photon still has kinetic energy (or, to put it another way similar to what others have said, it has energy from its momentum even though it has no rest mass).

 

[PhyCurious]

That's because all of a photon's energy is kinetic energy; none of it is rest mass energy. A photon still has kinetic energy (or, to put it another way similar to what others have said, it has energy from its momentum even though it has no rest mass).

OK this is helpful and makes sense. Does this mean that photons are always moving (and always at the speed of light)?

 

[Orodruin]

thereby violating the words of "mass-energy equivalence"

Do not overinterpret the name given to something. The mass-energy equivalence is the realization that the rest energy of an object is (up to a factor c^2) the same thing as the inertia in its rest frame. This is the reason we call rest energy ”mass” in the first place.

 

[PeterDonis]

Does this mean that photons are always moving (and always at the speed of light)?

Of course.

 

[vanhees71]

OK, this is helpful. I am still struggling to understand how something with zero rest mass can have energy without violating mass-energy equivalence. Would it be correct to say that the photons have zero rest mass but have relativistic mass (I understand this is a somewhat problematic or confusing term, but essentially their relativistic energy / c²) because they possesses kinetic / some other kind of energy?

The energy of a photon is E = hf (Planck's constant * the frequency of the photon). But why wouldn't the mass be mc² = hf and therefore m = hf/^c2?

I think it's ironic that the most quoted of all formulas in physics is wrong and most misunderstood. Einstein was pretty strictly against using the idea of "relativistic mass" after he thought about it shortly after his paper of 1905, where this idea is used. The physicists soon realized that if you define something like "relativistic mass" it is not only dependent on the speed of the particle put also on the velocity vector, i.e., it becomes dependent on the angle between the momentary velocity and the force. That's too complicated for physicists and that's why today they only use well-defined covariant quantities, i.e., scalars, vectors, and tensors (the quantum field enthusiasts also need various kinds of spinors, but that's another story).

Then there are massive particle. Any classical particle, i.e., a lump of matter whose extension can be neglected for the physical situation analyzed, is always massive. It's mass is the same quantity in special relativity as an Newtonian mechanics, and it is an invariant quantity. Energy and momentum together give a four-vector, $(p^{\mu})=(E/c,\vec{p})$ with
$$E=\frac{m c^2}{\sqrt{1-\beta^2}}, \quad \vec{p}=\frac{m \vec{v}}{\sqrt{1-\beta^2}}$$
with $\vec{\beta}=\vec{v}/c$, $\beta=|\vec{\beta}|$.

You can easily show from this that energy is related to momentum by
$$E=c \sqrt{m^2 c^2+\vec{p}^2}.$$

Now sometimes you can think about photons as if they were particles with zero mass. That's almost always wrong, but concerning energy and momentum it's ok. Using the energy-momentum relation above you get by setting $m=0$
$$E=c|\vec{p}|.$$

 

[Herman Trivilino]

I get what you are saying and could probably repeat it back to you, but this is a fairly non-intuitive idea, right? Basically a photon can never be at rest, so it has momentum of magnitude ppp? And as a result has energy but no mass (thereby violating the words of "mass-energy equivalence" but not violating the equations of relativity that produce mass-energy equivalence, just our English transliteration of it?)

To understand it you have to realize it's an equivalence between mass and rest energy. The term rest energy just means total energy minus kinetic energy. For a photon, the total energy equals the kinetic energy, it's a purely relativistic particle.

To really understand it you need to look at composite bodies, that is, bodies that are composed of more than one particle. Consider, for example, a system consisting of two photons moving in opposite directions along the same line. There will be a frame of reference in which they have equal but opposite momenta, so in that frame the total momentum of the system is zero. This frame is called the rest frame, or the center of momentum frame. The total energy of this two photon system in this rest frame will be $2hf$. This is by definition the rest energy. Thus the mass of the two photon system is $2hf/c^2$. The total energy will be different in different frames, but the mass will be the same in all frames.