What is the Mass Density of a Magnetic Field in Relativity?

[pmb_phy]

I have placed several journal articles on the concept of mass online for anyone to read. Its probably not kosher to do so so shhhh. :-)

See bottom of http://www.geocities.com/physics_world/sr/sr.htm

where it reads References to Journal Articles on the Concept of Mass in Relativity

Pete

 

[Garth]

Okay Pete, as nobody else has responded I'll put in my two pennyworth!

Having read some of the articles you referred to I take it the issue is whether an observer defines the mass of a particle moving relative to them as relativistic mass or as invariant 'rest' mass.

Where in SR the relativistic mass increases with velocity
m_r = m₀dt/dtau and the invariant mass is simply and always m₀.

I believe it is a matter of choice of perspective; that is it depends on whether you define mass to be fundamental and invariant, or whether you see energy to be fundamental.

Nucleons have more total mass when free than when bound in an atom, the difference being released as nuclear energy, but have the nucleons themselves lost mass or has it been the energy fields holding the particles together? The standard answer is that the energy has been lost by the (negative) energy fields of the atom while the individual mass of the particles has remained constant. However as we continue ‘downwards’ seeking more fundamental particles we eventually arrive at string theory where the fundamental constituents of matter, strings, are “fibres of vibrating energy”. So, what is fundamental, the inertial mass of a particle or the energy stored in its constituent strings? If it is indeed its energy, then may not that particle mass-energy vary to accommodate the work done against various potentials?

In SR the geometric four-momentum equation is P = m.U where P is a particle's geometric four-momentum and U its four-velocity with components U^a = dx^a/dtau and in which m is an invariant. The ‘increase in mass’ normally associated with motion, i.e. kinetic energy, is accounted for by defining motion as a rate of movement wrt proper time tau. This equation is simple, powerful, elegant and appropriate for a 4D geometric approach to space-time.

However, no matter how elegant that 4D world may be, we can only observe the universe and do experiments from our particular frame of reference with a 3+1D foliation of space-time. We can only measure our clock time and not the moving object’s proper time that is unless we happen to be in its rest frame. Thus in our frame of reference we measure its relativistic mass and not its ‘proper’ or ‘rest’ mass, and have to do a calculation to obtain the latter.

Relativity theory behoves us to adopt the 4D block view of space-time complete with geometric objects in which mass is invariant and energy-momentum conserved. As we have corresponded on these Forums elsewhere I would also add that if we now introduce curvature of space-time, i.e. gravitation, into this geometric perspective then, although the energy-momentum of a particle continues to be conserved, its energy is in general not.

Nevertheless, we continue to exist in our 3+1D foliated space-time world in which energy is conserved, and classical dynamics works very well thank you, so should there not be a choice in perspective? That is, should there not be a choice in convention as to the definition of mass as ‘relativistic’ or ‘invariant’ in nature? In my approach in Self Creation Cosmology there are two perspectives that are two conformal frames of measurement. In SCC’s Einstein conformal frame energy-momentum is conserved and particle masses are constant as normal, in this frame the theory reduces to canonical GR in vacuo. However, in the Jordan conformal frame it is energy that is conserved while particle masses vary in order to absorb gravitational potential energy. This latter frame of measurement might be seen to extend your definition of relativistic mass so that it not only includes ‘kinetic’ energy but also ‘potential’.

Garth

 

[pmb_phy]

I actually just posted that for information purposes only. I've mentioned those articles a bunch of times. If there was someone here who wanted to read one of them it is now available.

However there is one item that I'm thinking about that I've never discussed before. It has got me wondering - How does one answer the question "What is the mass density of a magnetic field?" if one defines "mass" as proper mass?

The properties of a field are not the same as the properties of normal matter. Therefore one can't use the expression $E^2 - (pc)^2 = m_0^2c^4$ and get a meaningful answer. For example, let u_m be the energy density of an electric field as measured in a frame where there is no magnetic field. If one let's $\rho_0 = u_m/c^2$ and then assumes this is a mass density then they will run into trouble because if one transforms to another frame moving relative to this one the relationship $E^2 - (pc)^2 = m_0^2c^4$ will not hold. This is because in the original frame there is stress due to the field which becomes part of the momentum in the "moving" frame.

I have a new SR text which asks a question like this. It asks what the mass density is of a magnetic field. I guess the author wants the reader to assume that the energy density is measured in a special frame. I think it was a vauge question for the position on mass that the author takes, i.e. mass = rest mass.

Pete