I decided to open a new topic on the discussion of premetric electrodynamics and therefore copied (and improved) the post from the ‘Dirac equation in GA (Proca)’ topic.
From the link there I found some links to papers about “premetric” electrodynamics. There, the different authors formulate Maxwell’s equations with the help of differential forms and start at the charge conservation (where the charge-current is modelled as a 3-form). Based on this, they formulate the equations with sources and the source-free equations with help of the exterior derivative and therefore metric-free. They argue that the only point, where the metric could enter the Maxwell equations is the constitutive equation where the field strengths (F = E + I B) and the excitations (G = D + I H) are coupled. Is there any possibility to show this metric-free formulation of Maxwell equations with the help of GA (maybe STA)?
In the meanwhile I did some calculations and therefore I can answer some of my own questions, but they lead to new questions
- In GA the “standard” formulation of Maxwell equations seem to be: \nabla\cdot G = J and \nabla\wedge F = 0. (In GA4P, an abbreviation which I learned recently , they write \nabla\cdot F = J and \nabla \wedge F = 0, since they use the microscopic field equations).
- The charge conservation can be calculated by \nabla \cdot J=0 since \nabla\cdot(\nabla\cdot G) = 0 (or the other way around). So G can be thought of as a “potential” for J.
- \nabla \cdot is NOT metric free, since it is an inner product. If I decompose, e.g., G and \nabla into a basis and components, at one point there occur inner products between the basis vectors which lead to a metric. Is it possible to formulate the inner product metric-free?
- If I follow the lines from the premetric literature and rewrite the current J into a trivector in STA (which also has 4 components), I could write the divergencelessness of J as \nabla \wedge J = 0. From there I could introduce G as a bivector potential \nabla \wedge G = J in analogy to A for F.
- With the last point the divergence condition of J is also metricfree, but I broke the possibility to rearrange the whole thing into one equation in the microscopic case, i.e. \nabla F = J.
My questions are:
- Are my arguments correct so far? (Without details of differentiability of the functions and so on.)
- It seems that the formulation of the Maxwell theory is not unique, neither in STA nor in the calculus of differential forms. How to decide, which formulation is “the right one”?
- The premetric guys argue that we may not start with a microscopic formulation of the Maxwell equations in the first place. Is it possible that the rearrangement of the microscopic equations into one equation \nabla F = J is only an accident? (Because, already in the macroscopic case, this does not hold, if I am not mistaken. I cannot rearrange \nabla\cdot G = J and \nabla \wedge F = 0 into one equation in general.)
I appreciate the opinions and corrections of the experts here.