# What is the rest/invariant mass of an electron?

An electron moves around the nucleus with a high speed. According to Einstein's theory of relativity the relativistic mass of a body is increases as it's speed increases. m=9.1×10^-31 kg shouldn't be it's relativistic mass? How it can be equaled with electron's rest/invariant mass?

For light atoms even the inner electrons can be satisfactorily described by the - nonrelativistic - Schrödinger equation. For heavy atoms the Dirac equation is used. This the field of quantum chemistry. Even radiative corrections are sometimes applied. On condensed matter physics relativistic local density theory is used. So it is safe to say that relativity is adequately taken into account.

In all equations the mass of an electron is $~9.11 \cdot 10^{-31}$ kg. The concept of relativistic mass is no longer in use.

• This might be better if you included more details (e.g., a link to this post, for instance) Jun 25, 2018 at 12:27
• While this answer doesn't use relativistic mass, which is completely fine, I feel like relativistic effects are missing here. The question is more about how relativity affects electrons in bound states of atoms. Jun 26, 2018 at 6:14

According to special relativity, the momentum of a body increases with higher and higher velocities and not anymore in a linear fashion. The mass at velocity $v = 0$ is called the rest mass and for an electron has a value of about $9.11 \times 10^{-31}$ kg. By now relativistic mass is not used anymore, because momentum is the more general concept.

But a bound electron 'orbits' the atomic nucleus and has a certain velocity. Be careful, that we are actually talking about a probability distribution here, so the effect is connected to the mean velocity $<v>$. Because of that the electron's momentum differs from the usual newtonian momentum and the behavior of the system should change at high velocities.

And it does! If you derive the quantum mechanical solution for the hydrogen atom, you can see the energy levels get shifted, when the momentum dependance changes. The amount is in what is usually called the fine structure and can be measured.

The whole fine structure consists of a combination of effects, but one is generally attributed to the relativistic kinetic energy (connected to relatistic momentum): Wikipedia article for the fine structure the effect is on the order of $10^{-4}$ eV.

This also has been experimentally verified, the hydrogen fine structure is one of the best known atomic spectra and it fits the theory behind it extremely well.

• "According to special relativity, the mass of a body increases [...]" A significant fraction of physicists have abandoned this unnecessary way of framing relativity as too easily leading to errors in thinking. We've discussed it many times (usually in the comments), but try physics.stackexchange.com/questions/133376/… for a start. Jun 25, 2018 at 17:33
• I thought about including a disclaimer about this at the top of the question, but chose to focus on the relativistic effects for bound electrons. I feel, that this is more in line with the intention of the question and the discussion about whether to use or not to use relativistic mass is a different subject. Jun 26, 2018 at 6:13
• I now changed the explanation from relativistic mass to relativistic momentum. Jun 26, 2018 at 6:51

The electron (rest) mass is a constant entering the electron equations of motion, whatever they are. Non relativistic equations contain $m$ as a measure of the electron inertia in response to a weak external force. It can be measured in mass spectrometers.

In relativistic case the electron response to an external force is more complicated as it must include a possible radiation losses, but the constant $m$ is still the same, just the equations are different.