# 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). Apr 27, 2015 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. Apr 27, 2015 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 $\nu$: $$E = h\nu$$