Premetric Electrodynamics

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 :smiling_face:

  • 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 :smiling_face:, 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:

  1. Are my arguments correct so far? (Without details of differentiability of the functions and so on.)
  2. 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”?
  3. 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. :smile:

I’m a complete beginner but trying to learn. SoI wonder if you can give a reference to the meaning of ‘premetric’. I believe that if you take the exterior products of vectors, the resultant multivector does not depend on the metric. But this would be unmetric, not premetric. Help please.

@wfmcgee thank you for your question. As far as I understood, the word ‘premetric’ is a concept of this community and means that they try to introduce the metric as “late” as possible within their axioms. E.g., they write the conservation of the charge and the conservation of the magnetic flux only with the exterior derivative and therefore without metric. Maybe this arxiv paper helps. Although I think, the reasoning is a bit short. (Maybe it is more elaborate in the cited book, which I don’t have.)

Thanks for your reply. My concern was that the definition of the vector derivative depends on the metric since it includes a reciprocal vector, but, once you have the vector derivative, the rest of the calculation seems metric free according to both definitions. My eyes were opened by a recent post of Enki on Discord explaining that solving equations using GA can be metric-free…

A few weeks ago, I read a paper which argued that the exterior derivative d and the exterior product with the vector derivative \nabla \wedge are both metric free.

Are the Discord logs accessible somewhere? It would be nice to dig through them. :smile: