Is there anything for which mass is undefined?

Question

I read that atoms do not have a temperature (neither zero nor other).

Is there anything that doesn't have a mass (neither zero nor other)?

Answer 1

We can roughly divide properties into two classes: fundamental and emergent. Temperature is an emergent property because it only exists for large numbers of particles. Pressure is another such example - a single particle does not have a pressure. By contrast mass is a fundamental property since even a single particle has a mass. Other examples of fundamental properties would be spin and charge.

Emergent properties can be undefined, but fundamental properties are always defined. So while a system may not have a temperature it always has a mass.

Answer 2

First of all, as many have already said, saying that something "doesn't have mass" will lead most physicists to think about things having zero mass, so it is better to talk about physical systems with undefined mass. (This is equivalent to saying that, if I tell you that I don't have any money, you will understand that I have 0 dollars in my bank, not that money is undefined.)

Anyway, I should disagree with @John Rennie's answer, saying that mass is always defined.

The first example that comes to mind is in quantum mechanics. In quantum mechanics, a particle can be in a superposition of states. A practical conclusion of this is that, in general, quantum states don't need to have, e.g., a well-defined energy. Thus, if you construct a quantum state that is a superposition of two mass eigenstates with different masses (states with well-defined unequal masses), then this new physical state won't have a well-defined mass. Now, there might be some rules that forbid physics to construct such a state (these are known as superselection rules). According to Weinberg (Section 2.4 of QFT Vol I book), if nature were invariant under the Galilei group, there would in fact be a mass superselection rule. However, we know our universe is not invariant under the Galilei group, and these undefined mass states are actually very relevant in particle physics. Probably the most known example are the neutrinos. The particles* known as "electron neutrino", "muon neutrino" and "tau neutrino" do not have a well-defined mass.

*Whether we can call a "particle" to a quantum state with no definite mass is a matter of terminology, which I will not discuss.

Another example of a physical system with undefined mass could be a macrocanonical thermodynamic ensemble. This is quite equivalent to what you mention about temperature. In the macrocanonical ensemble, the system doesn't have a well-defined number of particles and thus doesn't have a well-defined mass. (Of course, in this case, you can say that the combination of the macrocanical ensemble and particle reservoir does have a well-defined mass.)

Answer 3

If tachyons (particles that travel faster than light) would exist, some theories predict their mass would be imaginary. This is not quite the same as undefined, but it is also unlike the mass of any known particle.

But currently the scientific consensus is that tachyons probably do not exist.

Answer 4

Light spots, and shadows.

These are quite well defined, and I am going to twist special relativity and use it to prove they cannot have mass, since it is rather easy to obtain superluminal speeds with them. Therefore they cannot have m>0 (requires v<c), nor m=0 (requires v=c). (and m<0 would be quite a different level of "fun").

Of course, it depends on your definition of a "thing", if a shadow is a "thing" or not.

Answer 5

In general relativity, the mass of an extended system is generally undefined. One way to see this is that for an extended system, the different parts have energy-momentum four-vectors that can only be added by first parallel-transporting them to the same place, but parallel transport is path-dependent. An important example is that the mass of the entire universe is not defined.

Answer 6

Is there anything that doesn't have a mass (neither zero nor other)?

This is a fascinatingly difficult philosophical question. It actually has to dig at what does it mean to have a property. Philosophers have thousands of pages on that topic, so we will not succeed at answering it here.

The interesting detail is you mention that you're looking to exclude the cases where something has "zero mass." I'd like to explore what could that mean.

First off, there is the question of scientific realism vs. insturmentalism. Not getting too deep into a gigantic topic, this can be boiled down to whether the product of the scientific method (the concept of mass, in this case) is a true property of the world or not. Scientific realists argue that the scientific method does indeed find such truths. Instrumentalists argue that it merely models the behaviors of the real world but does not have to be founded on the true properties of the world.

That debate is unresolved, and like all good philosophical topics, is likely to remain unresolved. I think the only thing we can do on the Physics.SE forum (Philosophy.SE is that way!) is to focus on interpretations of the question which work in both viewpoints. We have to focus on how mass is used in our predictive equations.

In this sense, many corner cases get pruned early. Zero is the additive identity. $x+0=x$ for all values of $x$. In most of mathematics that we do in physics, mass gets rolled up into some term that is added, either directly or through calculus and integration. So in that sense, it's very hard to tell them apart.

But what about an equation like $a=\frac{F}{m}$? All I did there was reorganize $F=ma$ into a formula one might use to solve for acceleration. Obviously in such cases, $m=0$ yields an undefined result. If you are considering $m=0$ as "a defined mass," then you accept that there may be equations where mass is defined, but you still can't use the equation because the mass is outside the domain of the function that you are using it in. If you are considering light as a wave, it has either "no mass" or "0 mass." In both phrasings, $a=\frac{F}{m}$ is an equation that cannot be used.

The one case I can think of where we could really dig at a useful meaning of "mass is undefined" is a case where there is no one value of mass which can be used in all circumstances. As is mentioned in another answer mass has peculiar behaviors in general relativity. In GR, mass is a frame dependent value. Different observers will observe that an object has different masses, depending on the motion of the observer. If you calculate mass in one place (using equations like $F=ma$), and try to apply that mass in another frame, you will get the wrong answer. Instead, GR defines a slightly different concept, rest mass/rest energy. Whether that counts as "mass is undefined" may be subjective.

There's also the peculiar question of the inertial mass versus the gravitational mass. Inertial mass is the mass used in $F=ma$. It is the mass associated with how hard it is to accelerate an object when a force is applied to it. Gravitational mass is the mass used in $\frac{GMm}{r^2}$. It is the mass associated with how much force gravity applies to bodies. To date, all experiments have shown that these are the same. The Equivalence Principle is the hypothesis that assumes they are the same (so that I can write the above equations as is, rather than having to use subscripts like $m_i$ and $m_g$). It remains a hypothesis. While all experimental evidence points to it being true, we do not have any underlying reason for why it is true. It just seems to be so.

Were this hypothesis to prove false, we could no longer define the "mass" of any object. We would be obliged to specify which of these two measurable constants we are using. From a practical perspective, I find "now we need two numbers, because one is not good enough" is a pretty reasonable translation of "there is no one defined number."

Answer 7

In the field of thermodynamics, you can look at the grand canonical ensemble, which can exchange particles with a reservoir at some chemical potential. Consequently, the mass is not fixed, though you can of course derive the expectation value. (Note that this is usually formulated in terms of the particle number, but the effect is the same.) Hence, you may say that the mass doesn't, strictly speaking, exist.

Answer 8

I actually disagree with the premise, in that a single atom can have a well-defined temperature if it is in thermal equilibrium with its surroundings. Indeed, researchers have made a single atom heat engine.

That aside, fundamental particles all have a well-defined mass. If you're willing to accept quasiparticles, which are emergent excitations that appear at low energies in condensed matter systems, the possibilities are richer. I would say that one possible answer is a type of quasiparticle known as a fracton. Roughly speaking, these quasiparticles are immobile in isolation, but can move in correlated pairs. As such, at least colloquially, they can't really be said either to have a finite mass (and be independently movable) or to have an infinite mass (and be immobile).

Answer 9

Our notion of mass is tied to physical effects we can observe at least in principle. For example, we expect a massful physical something to move slower than c. So for mass to be undefined, we might look for things whose expected effects are themselves undefined because they are not observable.

Commenters have mentioned dreams, thoughts as well as holes. Are dreams and thoughts slower or as fast as light? Is this a meaningful question? If not, they don't have defined mass. What about holes?

I would also like to mention a purely mathematical analogy to this question. Mathematics has a concept of volume that coincides very well with our physical notion of it. This concept is able to assign volume to any realistic physical shape, as well as to shapes that have no physical correspondence. Now there is a mathematical result which states that one can split a threedimensional solid sphere into finitely many pieces, move those pieces around and obtain two solid spheres each of which has a volume of the original sphere. The insight here is, that those pieces do not have a volume even in mathematical sense - neither zero nor otherwise.

However, unlike in mathematics, if we were able to produce such pieces in reality, we would try to measure them in various ways. How much water would such a piece displace? I expect the answer to be either none or some. And this is what counts in physics - observations.

Answer 10

About the closest we get to real physical objects with no mass are photons and neutrinos, which have no rest mass, because they are never at rest.