d3cd4fc2   
   From: rockbrentwood@gmail.com   
      
   On Wednesday, March 22, 2023 at 2:07:08=E2=80=AFAM UTC-5, Douglas Goncz   
   A.A.S. M.E.T. 1990 wrote:   
   > I didn't study electrodynamics so my understanding of the Maxwell equations   
   > is limited but I believe there are four of them and they do not apply when   
   > db/dt equals 0, zero.   
      
   The fundamental equations of electromagnetic theory (and also: of gauge   
   theory) can be broken down into two sets that respect the stratification   
   of the geometric framework they are posed on top of:   
      
   (1) A non-metrical, non-causal part, which resides on bare differential   
   manifolds.  (2) A metrical part, which is minimal and confined to a   
   couple relations.   
      
   This is an issue that Hehl has made a big deal out of (as did Einstein,   
   later in the game, in the 1920's). It's also something Baez used to   
   point out, from time to time, as have I.   
      
   Geometry can be stratified in layers, much like in the way that type   
   hierarchies are built up in languages like C++, as well as in   
   object-oriented languages.   
      
   (1) The "base class" is the "topological layer".  That's topological   
   space and (on top of it), the Manifold.   
      
   (2) On this, a "derived type" adds in infrastructure suitable for   
   differentiation.  That's the Differential Manifold. Natural objects and   
   natural operations exist at this level.   
      
   The base type to derived type connection can also be treated, in the   
   context of category theory as an adjunction relation between categories.   
   So, adjunction hierarchies play a role analogous to type inheritance   
   hierarchies.   
      
   (3) On this, a further layering may be added on as an affine structure,   
   with the inclusion of a connection.   
      
   (4) On top of this, one may add further structure in the form of a   
   metric.   
      
   Between levels (3) and (4), different in-between levels may be   
   entertained (e.g. a causal structure, a conformal structure, etc.)   
      
   The distinction between space-like and time-like directions resides   
   entirely on level (4).  Level (3) is tone-deaf to any such distinction   
   ... which also means that there is no concept of "laws of motion",   
   "causality" or even "dynamics" at this level. In place of "dynamics" one   
   only has something like "unfolding". Equations unfold the structure of a   
   system from its boundaries, at level (3), they don't govern dynamics in   
   time.   
      
   The central hypothesis of Relativity resides at level (4) (and at the   
   in-between level on the "causal layer", if you go in between layers).   
   The very distinction between relativistic versus non-relativistic   
   physics exists only at level (4). Level (3) is blind to all such   
   distinctions.   
      
   Maxwell's equations can be formulated in such a way that almost all of   
   it resides at level (2).  A small residual core resides at level (4),   
   and is the *only* part that distinguishes between a set of equations   
   suitable for a relativistic universe versus a set suitable for a   
   non-relativistic universe (e.g. the equations that Maxwell & Thomson had   
   *actually* written, or later Lorentz, in contrast to those which   
   Einstein and Laub, or later Minkowski, wrote).   
      
   The equations at level (2) can be written entirely in the language of   
   natural objects and natural operations - as Maxwell (in fact) had   
   essentially done both before and in his treatise - i.e.  differential   
   forms. In equivalent 3-vector form, they consist of two fundamental   
   sets:   
      
   The equations for the "force" fields: E = -grad phi - d_t A, B = curl A   
   and corresponding "Bianchi identities": div B = 0, curl E + d_t B = 0.   
      
   The equations for the "response" fields: div D = rho, curl H - d_t D = J   
   and corresponding continuity equation: div J + d_t rho = 0.   
      
   The Bianchi identities and continuity equations are derived, not   
   fundamental.  Maxwell didn't even bother to write down the (curl E + d_t   
   B = 0) equation.   
      
   As differential forms: F = (Ex dx + Ey dy + Ez dz) dt + Bx dy dz + By dz   
   dx + Bz dx dy A = (Ax dx + Ay dy + Az dz) - phi dt (pardon the reuse of   
   A, I'd use boldface A for the vector here, if I could), one has: dA = F,   
   dF = 0.   
      
   For the response fields, one has G = (Dx dy dz + Dy dz dx + Dz dx dy) -   
   (Hx dx + Hy dy + Hz dz) dt Q = rho dx dy dz - (Jx dy dz + Jy dx dz + Jz   
   dx dy) dt with dG = Q, dQ = Q.   
      
   Maxwell never mixed the parts with "dt" with the other parts, though   
   there was no reason for him not to have. It actually complicated and   
   cluttered his analysis to not do so.   
      
   It bears to point out something here:   
      
   The apparent 4-dimensionality of these equations has *nothing* to do   
   with relativity.   
      
   These equations live at level (2) on which level there is no such thing   
   as any "Relativity" versus "Non-Relativistic" distinction at all. They   
   would even apply in a universe where light speed is 0 (i.e. a Carrollian   
   Universe) or even in a timeless space (i.e. a 4D spacelike geometry).   
      
   The equations that reside at level (4) are those that connect the force   
   fields and response fields.  That's where the distinction between   
   relativistic and non-relativistic resides. They can be written in a   
   common form that simultaneously encapsulates the equations posed by   
   Maxwell and Thomas, later by Hertz and by Lorentz (which are all   
   non-relativistic), as well as those posed at the same time by Einstein &   
   Laub, and by Minkowski in separate papers.   
      
   The Constitutive Laws: D + alpha G x H = epsilon (E + beta G x B) B -   
   alpha G x E = mu (H - beta G x D) where (alpha, beta) != (0, 0).   
      
   For these equations, there is a (generally unique) frame of reference in   
   which G = 0 and the constitutive laws reduce to isotropic form: D =   
   epsilon E, B = mu H.  In the 19th century literature, the dichotomy was   
   referred to as the "stationary form" (for G = 0) versus the "moving   
   form" (G != 0). In the 20th century literature, after Einstein & Laub   
   and Minkowski the "moving form" is referred to as the form for "moving   
   media".   
      
   These are the equations naturally associated with a geometry that has   
   the following as its invariants:   
      
   beta dt^2 - alpha (dx^2 + dy^2 + dz^2) beta del^2 - alpha d_t^2 (dx, dy,   
   dz) . del + dt d_t   
      
   If alpha beta < 0, the geometry is that for a 4-dimensional timeless   
   space (i.e. the "Euclidean" version).   
      
   If alpha beta = 0, with beta = 0, but alpha != 0, the geometry is one   
   which has 0 as an invariant speed (the Carrollean universe)   
      
   If alpha beta = 0, with alpha = 0, but beta != 0, the geometry is one   
   which has the speed of "at the same time" (i.e. simultaneity) as the   
   invariant speed.  All finite speeds are relative (the Galilean   
   universe).   
      
   The static universe (alpha = 0, beta = 0) isn't included in this list.   
      
   The case alpha beta > 0 includes the relativistic universe, with the   
   speed c = sqrt(beta/alpha) being a finite, non-zero invariant speed.   
      
   The above constitutive laws, for the Galilean case, are equivalent to   
   those posed by Maxwell, after the correction made by Thomson (the   
   inclusion of the -G x D term). They are also equivalent to the forms   
   posed by Lorentz, as well as by Hertz.   
      
   For the Relativistic case, they are equivalent to the forms posed by   
      
   [continued in next message]   
      
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