# Rest Mass, Where is it Stored?

Where is rest mass stored in a system? Reading Tipler, it says that a system of two 4 kg objects have a rest mass of 10kg, and atoms have rest masses smaller than their combined masses... Originally I thought the bonds in the nuclei would have given rise to rest mass, but even electrons which have no bonds have a rest mass. What is rest mass, and where is it stored? Furthermore, the atomic bomb is cited at taking advantage of E=mc^2, but isn't it really just breaking the bonds in the nuclei? I feel like it's just taking advantage of potential energy from bonds, and while I know these bonds store mass, it seems very different from the quality of the rest mass of an electron.

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An equivalent question is "Energy/momentum: Where is it Stored ?" Quantum mechanically, even for a single particle, there are restrictions due to the Heinsenberg uncertainty principle. –  Trimok Jul 23 '13 at 10:11
I am puzzled by this statement. The rest mass of a system is always smaller than its relativistic mass. Can you give a link or a direct quote? –  anna v Aug 2 '13 at 4:29
By Tipler, I hope you don't mean Omega Point nonsense, it scores 341 cranks (including a -5 starting credit) on this index. –  Dimensio1n0 Aug 2 '13 at 15:09
I don't understand this question , . What do you mean by "What is rest mass?"? Yup, it's the binding energy and the energy from the Higgs field (in the SM Lagrangian Density, you've probably seen a $\bar\psi\phi\psi$. . . –  Dimensio1n0 Aug 2 '13 at 15:13
I am very confused by this question too, and to me it is not so clear what it is asking ... At least it would need some more and better descriptive tags, no? Rest mass is a term (that shoul no longer be applied as I have heard people say) from special relativity. What do you for example mean by the bonds in the nuclei, are you rather asking about nuclear physics and mass defects of something? –  Dilaton Aug 2 '13 at 22:10

The answer to, "where is the rest mass stored" is easy: it's stored in the particle. I think what you are really asking is, "is there a way to explain the origin of the rest mass in terms of something more fundamental?" That is a great question. It depends on the level of answer you want, and the deepest answer available right now is, 'we can explain some rest mass, but some rest mass is unexplained.'

If you want a classical special relativity answer, then rest mass is just a brute fact: an object is allowed to have a rest mass, and some objects do. Possibly there is a deeper explanation, if you look into the inner workings of the particle, you might find that there is some binding energy that you were not aware of that accounts for some or all of the rest mass. For example, you might think a hydrogen atom's mass was all rest mass, but if you looked at its structure you would find some of what you thought was rest mass was really binding energy. It would be great if all rest mass could be explained in this way, so that there was no mass at a fundamental level, only potential energy. However, there might also not be an explanation. It might just be that "rest mass" is one of the things you have to take for granted about the physical world.

If you want a quantum field theory answer, well, again, the mass is just a bare fact. Particles are allowed to have a nonzero frequency even at zero wavelength, and that's the mass. ($\omega^2 = \vec{k}^2 + m^2$ with $\vec{k}=0$). Again, we always observe the world at some scale. It's possible if you probe deeper into the structure of the particle you'll find that the mass is due to some interaction energy, but it is allowed to simply be a number that is not explained.

In the standard model, the rest mass of most particles comes from one of two sources (or a combination). The rest mass of the electron comes (almost) entirely from it's coupling to the Higgs VEV. The mass of the proton, meanwhile, comes mostly from binding energy due to the quarks interacting with the gluons (confinement) and has relatively little to do with the Higgs. Most of the particles we know have have their mass explained in one of these two ways or a combination (the exceptions are the neutrinos and the Higgs particle itself).

However, even given the Higgs field and confinement, you can still imagine that there are particles that just have rest masses. There might be a right handed Majorana neurino with a bare mass that has no 'interaction energy' interpretation (it certainly couldn't get its mass from the standard model Higgs field). The Higgs particle itself has a mass that is really just a bare fact. The Higgs mass does not come from an interaction energy per se, it is just allowed to have a rest mass and it does have one. Furthermore, even if the mass of the electron can be said to be an interaction energy, you still have to ask, what sets the electron's coupling strength with the Higgs field, which is what sets the scale of the interaction energy? In some sense our current explanations of the origin of mass just push the same underlying problem to a new place, and we are still left with brute facts that are unexplained.

If you want a string theory answer, then you can attribute some masses to the vibrational energies of the string (plus quantum mechanical zero point energy). However you should be aware that all of the particles we know and love are really massless compared to the vibration energy of the string. So the masses of the particles we know about, even in string theory, arise from other effects--the geometry of the extra dimensions and the interaction energies with various fields. In that case the rest masses of particles all are ultimately explained in terms of interaction energies, in a very nontrivial way. Now, again you have to ask, to what extent are you really explaining the origin of mass, and to what extent are you simply pushing that issue into other things that are unexplained? You can see how this gets very tricky.

So... make of that what you will. It's a good question, and explaining the origin of masses and the coupling constants is a major driver of HEP research. But it's also a slippery, thorny issue, and the current best answer is that at least some of the rest mass in the world cannot be explained in terms of anything more fundamental, but maybe we will be able to in the future.

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The mass of an object, whether it be inertial (its resistance to being accelerated) or gravitational (its ability to bend spacetime), does not have to reside anywhere at all. That is, for compound objects, mass is a property of the system, not just a sum of properties of the components.

Your connection to potential energy is more apt than perhaps you realize - it is much the same story with that. When you raise an object off the ground, you might say you increase its potential energy, but this is just speaking colloquially. It is more proper to say the potential energy of the Earth+object system increases, rather than that one object holds the potential energy. After all, it is the potential for the system to undergo a change.

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For systems of elementary particles, the bonding energy in the fields between them corresponding to the electroweak and strong nucluear interactions is counted as part of the system's rest mass. As far as we can tell, both the inertial and gravitational mass does indeed correspond to these energies. That is, if we count these energies in the mass, it corresponds to both our ability to move the system of particles with a force and to the ability for it to bend space-time. So your question really: is what is the nature of rest mass of elementary particles? This is an open question, and is part of what the search for the Higgs boson is all about. The theories of quantum gravity also generally try to address this. Some theories tend towards ideas that rest mass is indeed localized energy of some form (energy associated with vibrations on strings or membranes or discrete meshes that make up space), but understanding this is in fact the cutting edge of modern physics.

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What is rest mass, and where is it stored?

A system of objects will have a total energy and a total momentum. The rest mass (most physicists would say: the mass) is the total energy of the system as observed from a frame of reference in which the total momentum is zero.

Your example of two masses of 4 kg each having a total mass of 10 kg probably refers to the following scenario:

Two identical objects move away from each other. An observer moving with either of these objects measures the object to have vanishing momentum and an amount of energy corresponding to ($E=c^2m$) a mass of 4 kg.

An observer in a central position with respect to both objects sees both objects moving away from her at equal speed. This observer will not conclude that the combined system (comprising both objects) will have a total mass of 4 kg + 4 kg = 8 kg. For instance, if both objects speed away from the observer at $\frac{3}{5}$ times the speed of light $c$, the observer will judges the combined system to have zero momentum and total energy corresponding to a mass of 10 kg.

That 10 kg is the rest mass of the total system comprising of two objects which in isolation each have a rest mass of 4 kg. There is no contradiction there, as both objects are in relative motion that represents an amount of kinetic energy. Furthermore, if the two objects flying apart would have been the sole result of the disintegration of a larger object, that larger object would have had a mass of 10 kg.

Where is the kinetic energy equivalent of 2 kg stored? I can give you an answer, but only if you tell me first where the relative motion is stored...

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