What is meant by "non-relativistic"?

Question

I am reading a document on cosmology and particle physics (this s the first time I look into something like this). It frequently states the word "non-relativistic" but I do not understand what is meant by this. When I try to Google it, I don't get a straight answer. The passage I'm referring to is:

"This means that away from temperatures where particles become non-relativistic we find that the factor of proportionality, i.e. the slope of the decrease of the temperature, is constant and depends on the relativistic degrees of freedom. If one species drops out of equilibrium because it becomes non-relativistic, then its entropy density (like its energy density) decays exponentially. However, the net entropy has to stay constant so the particle that becomes non-relativistic has to transfer its entropy to the particle species that are still in thermal equilibrium. For example, when electrons and positions are in thermal equilibrium we have the reaction:..."

My question is: what does the author mean when he says the particles become non-relativistic? Why does that change the equations he provides as examples?

Answer 1

In special relativity, the total energy of a particle in free space (i.e. in the absence of external fields) is given by:

$$\mathrm{E^2 = p^2 c^2 + m^2 c^4}$$

The energy of the particle is therefore dependent on a sum of two quantities, the first being the kinetic energy (which contains the momentum $\mathrm{p}$), and the second being the "rest energy" (which contains the mass $\mathrm{m}$).

To simplify calculations, in some cases we can approximate the energy by just calculating the biggest term. When the kinetic energy of a particle is much smaller than its rest energy (specifically, when $\mathrm{p \ll mc}$), then $\mathrm{E^2 = p^2 c^2 + m^2 c^4 \approx m^2 c^4}$, from which $\mathrm{E \approx mc^2}$. This is the so-called "non-relativistic limit", where classical mechanics is a good description. This turns out to be the case for most of chemistry. Particles said to be "non-relativistic" obey this approximation with good accuracy.

For completeness, the kinetic energy of a particle can also be much greater than its rest energy ($\mathrm{p \gg mc}$), such that $\mathrm{E^2 = p^2 c^2 + m^2 c^4 \approx p^2 c^2}$, from which $\mathrm{E \approx pc}$. This is called the "ultra-relativistic limit", which for most particles is only applicable in extreme conditions (e.g. astrophysical events). Lastly, if a particle has kinetic energy similar to its rest energy ($\mathrm{p \approx mc}$), then no simplification can be made. The term "relativistic" is often used to indicate that special relativity cannot be ignored.