# How could 'rest mass' and 'invariant mass' be the same?

The terms rest mass and invariant mass are often interchanged, however i cannot reconcile these concepts:
Consider a photon inside a mirror-box, measuring the mass of the box in rest we must arrive at $$m=E_p/c^2$$ (disregarding the mass of the box itself), however we know that this measured mass is in fact relativistic mass of the photon and is not invariant as it is dependent on our velocity and apparent frequency of the photon.

Therefore the rest mass is not equal to invariant mass in the case of the box.

Now someone may object that putting a box around a photon doesnt change anything, regarding mass of the box or a photon, however that is basically what elementary particles are. According to wikipedia the mass of proton consists of 99% relativistic mass of otherwise massless gluons. One can imagine that due to the small size of proton we will never be able to tell the difference between its measured rest mass (which is actualy relativistic mass of the gluons) and its invariant mass. However that doesnt change the facts that they are conceptually completely different and probably hold very different values.

• Comments are not for extended discussion; this conversation has been moved to chat. – tpg2114 Sep 10 '19 at 16:57

Photons don't have rest mass or relativistic mass at all, so your example is flawed. For photons $$E=pc$$ where $$p$$ is the photon's momentum.

Rest mass is invariant because rest mass is defined as the mass of something when it is at rest relative to you. You can't really work your way around this.

I think I understand your confusion. We say that the rest mass is an invariant quantity because it does not depend on the frame in which it is determined. The rest mass is always the mass in the rest frame of the system you are looking at, so it is invariant.

If you want to define a quantity that is the "invariant mass" that depends on other things such as kinetic energy, potential energy, etc. then you can do that. But it doesn't change that your argument about the photon is flawed because we cannot define a rest mass for a photon.

From Wikipedia:

Because the invariant mass includes the mass of any kinetic and potential energies which remain in the center of momentum frame, the invariant mass of a system can be greater than sum of rest masses of its separate constituents. For example, rest mass and invariant mass are zero for individual photons even though they may add mass to the invariant mass of systems. For this reason, invariant mass is in general not an additive quantity (although there are a few rare situations where it may be, as is the case when massive particles in a system without potential or kinetic energy can be added to a total mass).

I think this is where the confusion is. You are right that for a system of particles the invariant mass is not the sum of the rest masses of the particles. I think we are all on the same page now.

• Comments are not for extended discussion; this conversation has been moved to chat. – tpg2114 Sep 10 '19 at 16:57

So i found the real answer:
Invariant mass is equal to Rest mass only in case of single particles. Otherwise the Invariant mass is computed differently and is said to be Effective rest mass.

Mentions can be found here: https://atlas.cern/glossary/mass-invariant-mass
Its also noted on wiki in some articles: https://en.wikipedia.org/wiki/Mass_in_special_relativity

Or in this answer form Quora:

Mark Barton, PhD physicist with University of Glasgow Answered Apr 2, 2016 Originally Answered: What is the difference between invariant mass and rest mass ? Invariant mass is the effective rest mass of a complex system. As well as the rest mass of the components, it also includes contributions from the kinetic energy of any moving components, plus any potential energy from conservative forces between the components.

It is to some extent a way of tiptoeing past the verbal mine field created by the modern fashion of saying that only rest mass is mass and wash your mouth out with soap if you ever mention "relativistic mass" (the ratio of momentum to velocity or total energy, rest+kinetic over c^2 m=p/v=E/c2), despite the fact that it's the full relativistic mass that's additive and contributes to "rest" mass for complex systems.

• I was referring to your part about the single photon – BioPhysicist Sep 10 '19 at 14:44
• See the edit to my question. I think there has just been a misunderstanding of terms. – BioPhysicist Sep 10 '19 at 14:49
• No, i will keep it in and you can keep your downvote here. I think its a fair example of that science is not a democracy and the most upvoted answers are not always the right ones. Anyway.. Have a nice day. – user1316208 Sep 10 '19 at 14:59
• With all due respect, it seems that you misunderstand what Dr Barton is trying to say in that quote. He's saying that the rest mass of a composite system is generally not simply the sum of the rest masses of its components. That's because the kinetic energies of the components, as well as any potential energies between them, also contribute to the composite system's rest energy. – PM 2Ring Sep 10 '19 at 15:10
• The (invariant) mass of a system is--in general--neither the sum of the invariant masses of the constituents nor the sum of the "relativistic masses" of the constituents (though you can find examples of both of those cases). Instead the (invariant) mass of a system is found from the systems four-momentum in exactly the same way that a particle's (invariant) mass is found from its four-momentum. There is only one definition of (invariant) mass and it applies to individual particle and to systems alike. – dmckee --- ex-moderator kitten Sep 10 '19 at 17:20