On Wednesday, March 22, 2023 at 2:07:08=E2=80=AFAM UTC-5, Douglas Goncz

*Post by Douglas Goncz A.A.S. M.E.T. 1990*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

Einstein & Laub, as well as by Minkowski - both in 1908 and - as such -

comprise a set of relations known today as the Maxwell-Minkowski

relations.

Only the case alpha beta > 0 permits the constitutive laws to be written

as G-independent, i.e. for media that are not just isotropic, stationary

and homogeneous, but also boost-invariant (i.e. the effective definition

of what we call a "vacuum").

And that occurs precisely when epsilon mu = alpha/beta = 1/c^2.

The ability to pose the equations in Lagrangian form hinges on what

epsilon and mu are, as functions of the force fields and/or potentials.

The Lagrangian will serve as a generating function for the constitutive

laws:

D = @L/@E, H = -@L/@B

where "@" denotes the partial derivative operator.

In the most general case of a Lorentz-invariant Lagrangian L(E,B) of the

field strengths alone, the function will reduce to a function of the

invariants I = 1/2 (E^2 - B^2 c^2) and J = E.B, one may define the

derivatives:

epsilon = @L/@I, theta = @L/@J

and write the constitutive laws as

D = epsilon E + theta B, H = B/mu - theta E,

defining mu = 1/(c^2 epsilon). This requires that you be away from a

"Landau Pole" (which is equivalently described as epsilon = 0!)

The "theta" coefficient is 0 if the Lagrangian is parity symmetric.

In general, epsilon, mu and theta will be functions of I and J that

satisfy the pre-condition:

@(epsilon)/@J = @theta)/@I, epsilon mu = 1/c^2.

For free fields - if you characterize "free" as "null", i.e. fields

where I = 0 and J = 0 - one has a reduction to epsilon(I,J) = epsilon_0

:= epsilon(0,0), and theta(I,J) = theta_0 := theta(0,0)

and without loss of generality (by adding suitable residuals from E and

B to D and H) one can take theta_0 = 0, so a free field is one that is

approximated by a Maxwell-Lorentz Lagrangian (i.e. one where epsilon, mu

are both non-zero constants such that epsilon mu = 1/c^2).

This is true *irrespective of the actual Lagrangian*.

Stated equivalently, in the backwards direction: there is nothing that

empirically justifies the use of the Maxwell-Lorentz Lagrangian, in

place of other Lorentz-invariant Lagrangians where @L/@I != 0.

That's an important point to note, both classically (since the

Maxwell-Lorentz Lagrangian leads to self-force and self-energy

infinities) and in quantum field theory (which uses the Maxwell-Lorentz

Lagrangian as the classical limit to quantize off from).

Most of what's been described above applies equally well to gauge

theory, with the following generalizations:

(A, phi, E, B) are now Lie-vector valued, component-wise (J, rho, D, H)

are now Lie-covector valued, (epsilon, mu) generalize to a Lie metric

and Lie dual-metric (epsilon, theta, mu) are Lie tensors, in general.

The gauge analogue of the Maxwell-Lorentz Lagrangian is the Yang-Mills

Lagrangian.

The gauge analogue to the Maxwell-Minkowski relations has no name or

published citation that I know of.