Poincaré Stresses

Poincaré Stresses

Using the fields E and B from a charge q moving at constant velocity v, as found at the end of Notes 6-1, you can compute the energy density and the momentum density of these fields. You can then integrate these over all space to find the total electromganetic momentum P_EM and energy U_EM of the moving charge. Show to lowest order in v (i.e. in the limit that v → 0), that P_EM and U_EM are related by,

P_EM = (4/3) ( U_EM/c²) v

Note, this calculation is only simple if you work to lowest order in v. You might wish to assume the charge is a spherical shell of radius R and constant surface charge σ = q/4πR², so that your integrals do not diverge.

If all the mass of the charge came from its electromagnetic fields (as in problem 2 on Problem Set 7), then one might think that the rest mass of the charge (i.e. its mass as v → 0) is

m = |P_EM|/|v| = (4/3) ( U_EM/c²)

However, from special relativity (Unit 7) we expect that the rest mass should be related to the rest energy by U = m c². The factor 4/3 is therefore a problem!

One can fix this 4/3 problem by use of Poincaré stresses. As we commented on at the end of Notes 4-2, the surface charge on the spherical shell representing the charge creates a force pushing outwards. If nothing was holding the charge together, the charge would therefore explode outwards. To keep this from happening, one invokes the Poincaré stresses, which one thinks of as mechanical forces holding the charge together. These should lead to a diagonal mechanical stress tensor T_mech = pI , where I is the identity tensor, and p can be thought of as the mechanical pressure pushing inward on the spherical shell to keep the total stress balanced and the charge stable.

In the rest frame of the charge, the Maxwell stress tensor of the spherical shell of charge is also diagonal, T_EM=T_xxI (you should show this), so for the Poincaré stresses to balance out the electromagnetic stresses, one should have p = - T_xx, so that the total stress T_EM + T_mech = 0.

This mechanical pressure p gives rise to a mechanical contribution to the charge's energy U_mech. If one thinks of the Poincaré stresses like rubber bands holding the charge together, then increasing the charge's radius will stretch the rubber bands and increase their energy. For an increase in the charge volume ΔV, the energy will increase ΔU_mech = p ΔV. From this you can compute U_mech for the charge.

If one now uses U_total = U_EM + U_mech, then you can show that

P_mech = (U_total/c²)v

and the 4/3 problem goes away!

This argument shows that all the mass of a charge cannot be electromagnetic in origin, but there must also be some additional mechanical part that comes from the Poincaré stresses.

To do the above calculation, you do not need to know anything from Unit 7 other than U = mc². However, if you do use the results of Unit 7, then there is a more sophisticated way of doing the calculation. You don't have to do it this way, but I'll mention it in case you are interested!

If one uses the relativistic field strength 4-tensor F_μν, one can define the relativistic Maxwell stress 4-tensor as

T_μν = (1/4π) [ F_μλF_λν - (1/4) δ_μν F_λσF_σλ]

where δ_μν is the Kronecker delta, and the summation convention on repeated indices is implied. One finds that the spatial parts of T_μν are just the usual Maxwell stress tensor, while the time parts give the electromagnetic momentum and energy densities. The relativistic way to write the laws of momentum and energy density then become,

(∂T_μν/∂x_ν) = F_μν j

Start in the rest frame K of the charge and compute T_μν. Then, using the proper Lorentz transformation for a 2nd rank 4-tensor, transform to a new frame K′ in which the charge is moving with velocity v in the x-direction. The T′_i4 components (i = 1, 2, 3) then give the electromagnetic momentum density in K′, which you can integrate to find the total electromagnetic momentum P′_EM in K′. You will see that this transformation gives P′_EM in terms of the energy U and stress T_xx of the charge in the rest frame, and if these are taken as the electromagnetic energy and stress, one gets the same result as above, with the 4/3 factor. However, if one takes T_μν to be the total EM + mechanical Poincaré stresses, then the 4/3 factor goes away! One also finds that it is only the total momemtum and energy of the charge (i.e. the EM + mechanical Poincaré parts) that transform like a 4-vector, and that the electromagnetic parts alone do not.

There are some subtleties in this calculation, so be careful if you attempt this!