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We know an object with positive mass cannot be accelerated to the speed of light because this would require an infinite amount of energy. My question is:

Is there anything in the universe that can travel less than the speed of light in a vacuum and yet has no mass?

Is it even possible within our universe's physical laws?

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    $\begingroup$ Note $\frac{v}{c}=\frac{pc}{E}=\frac{pc}{\sqrt{(pc)^2+(mc^2)^2}}$. $\endgroup$
    – J.G.
    Commented Jul 17, 2021 at 7:21

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In vacuum, no. Massless particles travel at the speed of light. In relativity, the definition of the (rest) mass of a particle is the energy (divided by $c^2$) of the particle in its rest frame. If something has zero energy in its rest frame, does it really exist? (No.) Massless particles exploit a tricky loophole in this argument because by moving at the speed of light, it is impossible to boost into their rest frame.

In other words: massless particles are only allowed to exist (ie, have a finite amount of energy), despite not having a mass (energy in their rest frame), because they don't have a rest frame. If a particle was massless and traveling less than the speed of light, we could go into its rest frame, find it had no energy at all, and be led to a philosophical conundrum because such an "object" cannot have any effect on the physical world.

However in a medium, particles that are massless in vacuum can travel at a speed different from the speed of light in vacuum. For example, light traveling through glass travels at a speed that is about 1.5 times slower than the speed of light in vacuum; we describe this phenomenon by saying glass has a refractive index of about 1.5. Having said that, one explanation of the refractive index is that the photon acquires an effective mass due to interactions with phonons in the glass, so in a way the refractive index is "an exception that proves the rule" and actually confirms the idea that if something is traveling at less than the speed of light in vacuum, then it has some form of mass.

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  • $\begingroup$ Maybe not phonons, but plasmons? After all, visible light influences more the electronic part of the glass than the nuclei. $\endgroup$
    – Ruslan
    Commented Jul 18, 2021 at 23:42
  • $\begingroup$ Why do you write that something doesn’t exist if it has zero energy in it’s rest frame? $\endgroup$ Commented Nov 29, 2021 at 7:36
  • $\begingroup$ @HelloGoodbye Keeping in mind $E=mc^2$, something with no energy has no mass. Can you think of any examples of physical objects with no mass? (Besides light, which has energy and can't be brought to its rest frame) $\endgroup$
    – Andrew
    Commented Nov 29, 2021 at 13:37
  • $\begingroup$ What do you mean by "physical object"? I'm not aware of any object with no energy, or equivalently (I suppose), a massless object with speed less than $c$, but unless we resort to QFT-based explanations (see the part about the Klein–Gordon equation in Nullius in Verba's answer), I'm not convinced that such an object cannot exist. $\endgroup$ Commented Nov 29, 2021 at 22:39
  • $\begingroup$ @HelloGoodbye I mean, let's just restrict ourselves to classical physics (but you could make an analogous argument for other domains of physics). The equations of motion of The Universe in classical physics are $\{\dot{\vec{p}_i} = \sum_k \vec{F}_{k,i}, m_i \dot{\vec{x}_i} = \vec{p}_i\}$ where $i$ is a label running over every particle, $F_{k, i}$ is the $k$-th force acting on particle $i$, $p_i$ is the momentum, $m_i$ is the mass, and $x_i$ is the position. How would you describe a particle (label $j$) using those equations that had $m_j=0$? $\endgroup$
    – Andrew
    Commented Nov 30, 2021 at 0:09
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It depends what you mean by "anything".

If you drill a hole in a solid object that is moving slower than light, then the hole has no mass, and is also moving slower than light.

If you have two large sheets of metal with slits cut in them, the slits at an angle to one another, and one sheet sliding across the other, then there is a hole through the combination where both slits intersect. Again, a hole has zero mass, and in this case can move at any speed.

Or the intersection point of two beams of light. The light has zero rest mass, and the intersection can move with any speed.

Or the shadow of an object circling a light source cast against a distant wall. The shadow has zero mass, and can move with any speed.

If 'things' can be considered to include geometrically defined features like intersections, edges, boundaries, shadows, wave peaks, and so forth, then the answer is 'yes'. A wavepacket has a group velocity and a phase velocity, the group velocity is slower than light, the phase velocity is faster than light, but are the phase peaks of the wave a 'thing' in the sense intended? They obviously can't have 'mass', or they would not be able to travel faster than light. But is it that their mass is zero, or is mass a meaningless concept in this case? 'No mass' could mean either.

But if you're talking about matter specifically, and its mass and velocity, then matter obeys the Klein-Gordon equation, which is a wave equation in which the speed of propagation of the wave becomes c when the mass is set to zero. So if your definition of 'anything' means only material stuff that obeys a Klein-Gordon equation, then the answer is 'no'. Zero rest mass results in a wave that can only propagate at the speed of light.

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  • $\begingroup$ Thanks for being precise. I wanted to mean “any matter” by the term “anything” in this context. $\endgroup$ Commented Jul 17, 2021 at 16:19
  • $\begingroup$ Can you provide a reference for your Klein-Gordon reasoning? Also, what specifically do you mean when you say the matter follows the Klein-Gordon equation? The Klein-Gordon equation simply provides the mass-shell condition for the quantum fields, their propagation is not governed by the Klein-Gordon equation, it is still governed by the Schrodinger equation with the relevant Hamiltonian. $\endgroup$
    – user87745
    Commented Jul 18, 2021 at 21:19
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Yes, all massless particles must travel at c.

If a massless particle traveled at less than c, there would be some frame in which the particle was completely at rest. But a particle at rest and without mass as its 4-momentum (or more specifically the lorentz invariant contraction of its 4-momentum $p^{\mu}p_{\mu}$) could not be conserved (all components would be 0).

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  • $\begingroup$ What do you mean by that it couldn’t be conserved? It seems to me like it would just be 0 constantly, which means that it would be conserved. $\endgroup$ Commented Nov 29, 2021 at 8:59
  • $\begingroup$ @HelloGoodbye that's probably a better way of saying it - a particle without mass traveling below c couldn't conserve any non-zero momentum. (and if its 0 then it doesn't exist) $\endgroup$
    – Señor O
    Commented Nov 29, 2021 at 17:40
  • $\begingroup$ Why do you say that the particle doesn't exist if its momentum is zero? This is certainly not something that holds in general, since any particle that has a speed less than $c$ can be given a momentum that is zero by observing it in its rest frame. So why would it be the case that a massless particle with zero momentum doesn't exist? What does the "existence" of something even mean? $\endgroup$ Commented Nov 29, 2021 at 22:44
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    $\begingroup$ Something that exists can affect other things. A massless particle that always has 0 momentum cannot affect other things under any circumstances, so it doesn't exist. $\endgroup$
    – Señor O
    Commented Nov 30, 2021 at 3:02
  • $\begingroup$ How do you know that it cannot affect the probabilities of the different outcomes of a scattering event? $\endgroup$ Commented Nov 30, 2021 at 3:42
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It depends on what you mean by vacuum.

If you accept a solid body in a vacuum, then there is a situation with velocity smaller than c. The photon is deflected at the edge of the body. This interaction with the edge takes time and changes the direction of the photon's path. Obviously, such a photon, which is in a vacuum, propagates with less than c.

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