# Is it true that all particles that have a speed less than the speed of light must also have mass?

I've previously learned that massive particles cannot achieve the speed of light.

But recently I read that, concerning the gels that refract and bounce light within around enough that it can travel at worldly speeds, and by extension how electromagnetism propagates through matter, that pure photons are thought of as interacting with the massive objects, gaining mass and moving at a speed less than the speed of light as a result.

I'm not sure if this is just how the paradigm is set up to frame the phenomena, or if this paradigm carries over seamlessly into other accepted theories elsewhere in science, but this made me think of the possibility of the exclusivity of mass and velocity applies not only to particles with speeds at c being unable to have mass, but also the converse, particles without mass being unable to have speeds less than c.

My question about the properties of all particles, with syntactical brevity:

Mass exclusive-or speed of light is true?

• Conceptually it's not the photons that gain mass, but it's the coupled states (which are quasi-particles), that are massive (because the electronic/atomic part of their state is). – CuriousOne Apr 27 '15 at 0:36
• I foresee that very well might lead to an answer of no, depending on how not-a-particle a quasi-particle is. EDIT: Of course, that also means the photon still is required to be a part of this quasi-particle to move at a speed less than c. – 0xFFF1 Apr 27 '15 at 0:38

First, we will look at the energy of a free relativistic particle of (rest) mass $m$ moving with velocity $v$: $$E = \frac{mc^2}{\sqrt{1-\frac{v^2}{c^2}}}$$ where $E=mc^2$ when $v=0$. We now consider a few cases:

1. $m\ne0$: In this case, $E\rightarrow\infty$ as $v\rightarrow c$. Therefore, a massive particle that at any point of time is moving at less than the speed of light cannot practically be accelerated to achieve the speed of light. A massive particle that has always existed with $v$ = $c$ however, has infinite energy - this leads to trouble, like infinite transfers of energy to ordinary matter being possible in interactions all over its path - and is therefore considered unphysical.
2. $m\rightarrow0$: Here, we demand that the particle has a finite energy (for various reasons such as its being able to produce some measurable effects) in spite of its vanishing mass, and this can only be possible if $v=c$, so that $mc^2 = E\sqrt{1-\frac{v^2}{c^2}} = 0$. In these cases, the energy is often determined by other things. For example, a photon has an energy determined by its frequency $\nu$: $$E = h\nu$$

So, at least from an energy point of view, massless particles can only ever travel at the speed of light, and massive particles only at lesser speeds. This conclusion, of course, assumes that the above expression for energy also holds for massless particles. While a potentially suspect assumption, it is ok if we expect a smooth transition between the massless case and the limiting case of very low mass.

Every particle needs to have energy to be a particle (if it had none it wouldn't even exist). Since energy is equivalent to mass and therefore gravitates I would say YES, all particles that have a speed less than the speed of light must also have mass.

Because the speed of the particle is less than the speed of light an observer could travel with the same velocity as your particle and experience the particle's rest mass (in contrast to photons, which have energy and gravitate, but have no rest mass).

It depends on how you define mass. I like to think of it as mass is just rest mass. I mean the mass you weight on a scale when nothing is moving. On different media light moves slower not because it gains mass but because of its interaction with the atoms in the media. The photons get absorbed and reemited in such a manner that when you sum the waves for each of the photons the end up moving slower. But the photons themselves always move at the speed of light