# Fields = mass?

## Main Question or Discussion Point

Hi,

I was just wondering about the nature of fields. What are they? Since fields carry energy, and energy is mass, does that mean that fields are equivalent to mass?? I think this is probably incorrect, but I just have no clue why.

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Very interesting question, I'm not sure that fields carry energy though, it's more about the interactions with the field, the field itself isnt energy (at least I think so). And from the interactions, energy can convert into mass, thanks to the field but I don't think there is a field energy that would need to be "consumed" by the creation of mass.
But I think I know nothing more than you on this subject, it's just what I think without proofs.

I would take the example of an electron and a proton, they each have an electromagnetic field, and when they are separated their fields interact to get them closer, the energy being used is the electromagnetical potential energy, which is transfered to the kinetic energy of these particles to give them speed, thanks to the work of the electromagnetical force. So energy has been used, but when you look the state of the system afterwards (let's say after the creation of a hydrogen atom), each particle still has the same field than before, so I don't think field is energy, it would just be some kind of way of interaction I think.

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Yes, that was my first reaction too. But then when I learnt more about Maxwell's equations, the whole thing became a lot more blurry to me. Basically, light is an EM field, and light is energy and thus possesses relativistic mass. Doesn't that mean that fields (at least EM fields) have energy and thus relativistic mass?

CompuChip
Homework Helper
Basically, light is an EM field
Actually, photons are the particles that mediate the EM interactions. In field theory, one would say that the field itself just exists everywhere in space. Photons are then excitations of the EM field, and as such, have a certain energy (above the ground state of the field) which one could indeed view as relativistic mass. This is a general principle in modern physics (field theory, Standard Model, string theory), where particles are seen as excitations of some field, and the energy of the excitation is related to the mass of the particle; for example, the relation p2 = - m2, with p the four-momentum holds for stable ("on-shell") particles.

Ah CompuChip, I would like to know more about what you just said. I'm, as you can probably tell, quite ignorant of physics more modern than that of relativity. Does field theory assume that fields are "all pervading" in the universe (it doesn't expand or anything - but is "just there") and that excitations, or changes to these fields are what we see as energy (somewhat similar to the theory of gravity being curvatures in space-time)?

Thanks.

mathematically speaking the energy stored at a point in the field is proportional to the square of the field strength at that point. integrated over infinite space the total energy is finite and well below rest mass for electrons and protons.

think of an electron and positron that come together. in the end their fields cancel out completely and all potential field energy in converted to light.

Hi,

I was just wondering about the nature of fields. What are they? Since fields carry energy, and energy is mass, does that mean that fields are equivalent to mass?? I think this is probably incorrect, but I just have no clue why.
In some situations there is a direct equivalence between mass and energy given by E = mc2. This holds in all cases where the system being described is a closed system. In other cases like the mass density if the field the mass density is not generally proportional to the energy density. However it can definitely be said, at least with something like the EM field, that the field has energy. To each point in the field there is a corresponding energy density. A physical reality of the field is seen when the field detaches from its source and moves such as when a charge is wiggeled and it generates an EM wave. If one had a finite EM distrubance such as a finite EM wave then one can integrate over the entire wave to get a finite mass and energy.

Pete

integrated over infinite space the total energy is finite and well below rest mass for electrons and protons.

meaning, the energy stored in the field of an electron or proton integrated over infinite space is finite and well below the rest mass of the electron or proton.

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integrated over infinite space the total energy is finite and well below rest mass for electrons and protons.
What is it that your speaking of when you are integrating over infinite space? The total energy of what? And why do you think this is below the rest mass of both the proton and the electron? When I was speaking about integrating over all space I was refering to integrating the energy of the field over space.
meaning, the energy stored in the field of an electron or proton integrated over infinite space is finite and well below the rest mass of the electron or proton.
You seem to be speaking about electromagnetic mass. The eletromagnetic mass of a classical particle only makes sense of the "particle" has a finite size. If the particle is a true mathematcal point then the self-energy would be infinite.

If the particle has a finite size and one assumes that it is a sphere (either uniformly charged or has a uniform surface charge distribution where the charge is spread out over the surface of the sphere) then the total inertial mass of the proton, i.e. its rest mass, is given by

mrest = mbare + mem

[when (v -> 0)] where the mbare is the mass the particle would have if there was no field at all and mem is the mass contribution from the field itself. In the case of such a distribution there are other forces at play here. Those forces hold the particle together. The stress from such forces contribute the inertial mass. The stress in cases like this is referred to as Poincare stress. These stresses are required if one wishes to calculate the inertial mass when one uses the relationship between mass and momentum, i.e. p = mv, to define the mass.

(The term "particle," as used here, refers to the idea that the size of the body can be neglected in the problem one is considering. As such there is no objective way to determine whether something is a point particle or not.)

Pete