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
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.
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