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The effect of the EM field upon a charge $q$ is given by the relativistic Lorentz force equation:

$$\frac{dP}{d\tau}= qF^{\alpha\beta}U_{\beta}$$

The expression on the RHS is then substituted via Maxwell's equations to define EM field conservation laws and their corresponding definitions of field energy and momentum within them. Is there a physical reason for defining the energy and momentum of the electromagnetic field this way?

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  • $\begingroup$ The energy/momentum of the EM field are not normally defined that way. For context, can you cite a source that defined them that way? They're normally defined either using Noether's theorem, or (better) by writing them in terms of the stress-energy tensor that is defined by varying the action with respect to the metric tensor. $\endgroup$ Sep 2, 2021 at 1:59

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There are two ways to arrive at energy and momentum in electromagnetic fields.

  1. First approach: start from force on charged particles, and insist on energy and momentum conservation over all.

  2. Second approach: start from a field Lagrangian density and use Noether's theorem.

The second approach is often preferred by theoreticians who are used to this approach to field theory in general, but it involves somewhat more sophisticated ideas, so the first approach has much to recommend it.

In the first approach one notes that as the force causes particles, or more generally a continuous charged medium, to accelerate, energy and momentum is being delivered to the medium. One then makes the hypothesis that there is overall conservation of energy and momentum. Consequently any energy and momentum acquired by the charged medium has been given up by the fields. So one writes down the differential equation expressing the energy and momentum flow to or from the charged medium, and then one interprets the terms (Poynting's beautiful argument and its generalization). Formulating this in complete generality does not fully determine the equations for field energy and momentum, but it constraints them sufficiently such that the standard results emerge as the simplest form that satisfies the constraints.

Someone might want to say they are still unconvinced that the fields really carry energy and momentum. Such a person can be invited to use all the same equations and wherever everyone else says "$u$=energy density of the field" they will have to say "$u$=combination of field properties that correctly accounts for energy lost by or gained by charged matter as it interacts with the field". That way of thinking is quite unnecessary and obscure however, so the standard practice, where we do talk about field energy, has strong credentials.

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The relativistic Lorentz force equation ensures that the four force $\frac{dP^\alpha}{d\tau}$ is a contravariant four vector. In particular, the quantity $$ \frac{dP^\alpha}{d\tau}\frac{dP_\alpha}{d\tau}=\left(\frac{dP^0}{d\tau}\right)^2-\left(\frac{dP^1}{d\tau}\right)^2-\left(\frac{dP^2}{d\tau}\right)^2-\left(\frac{dP^3}{d\tau}\right)^2 $$ is the same in every Lorentz frame. Physically this means that an electron moving in a magnetic field experiences the same force when we observe it in the frame where it is at rest. In that frame the force comes from an electric field. This was in fact the starting point for Einstein to develop SR.

The discussion here is closely related.

Note that the electron experiences only the spacial components of the force which in isolation are not Lorentz invariant. However this is approximately true at low speeds.

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  • $\begingroup$ Thanks, but this doesn't answer my question. $\endgroup$ Sep 2, 2021 at 21:41

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