Invariant stress tensor = Invariant force?

[Pengwuino]

So we know that

$$\frac{d}{dt}(P_{mech} + P_{field}) = \oint_S {T_{\alpha \beta } n_\beta da}$$

that is, the time rate of change of the momentum of a system plus the momentum of the electromagnetic fields is equal to the surface integral of the term with the Maxwell Stress Tensor where there is an implied sum over the index beta.

There's a problem in Jackson that asks about whether or not a duality transformation that involves introducing the concept of a magnetic monopole would change the force on the particle. The stress tensor was derived using the mechanical force law (Lorentz) and the momentum of the system we derived back when.

The new Lorentz force would be given by

$$F_{monopole} = q_e (E + v \times B) + q_m (\frac{B}{{\mu _0 }} - v \times E\varepsilon _0 )$$

Now, if we know the field momentum is invariant under this transformation, and we know the stress tensor is invariant, are we allowed to say that then the mechanical momentum is invariant? and thus, the force is invariant?

 

[NateD]

Yes, if the stress tensor and the field momentum are both invariant under the duality transformation, then the mechanical momentum must also be invariant, and thus the force is also invariant. This is because the time rate of change of the momentum of a system plus the momentum of the electromagnetic fields is equal to the surface integral of the term with the Maxwell Stress Tensor where there is an implied sum over the index beta, and if the stress tensor and the field momentum are both invariant, then this implies that the mechanical momentum must also be invariant.