Re: Conserving Q ? (long)

[David Bowman <dbowman@TIGER.GEORGETOWNCOLLEGE.EDU>]

Regarding Jerry Epstein's comment:
> I missed a lot of this thread, but I am quite baffled by what I now see.
> The displacement current is quirte obviously a source of B via the curl
> equation, just as is a current of charges. If it were not, there could
> be no electromagnetic waves in the vacuum. The energy osciallates back
> and forth between electgric and magnetic fields. The time varying
> magnetic field acts as generator of electric field (as is well known
> experimentally from Faraday), and conversely, to complete the chain, the
> time varying electric field (the "displacement current") must act as a
> source of the magnetic field. Otherwise the oscillation stops and there
> is no such thing as light.

This is *one* way to view the situation (as a view in terms of a 3-d
spatial geometry with time as an external parameter that is not a
coordinated part of the geometric manifold).  Another way to view the
situation is from the vantage point of 4-d Minkowski spacetime.  In
spacetime a field does not need any sources present for it to propagate as
a wave at speed c (the causal speed limit) in that propagating fields are
seen as described by *homogeneous* PDEs.  This is a consequence of the
indefinite nature of the signature of the metric of spacetime where *non*-
propagating 'elliptic' behavior of fields obeying homogeneous PDEs in space
become propagating and oscillatory 'hyperbolic' behavior for fields
obeying homogeneous PDE's in *spacetime*.

Since the situation for electromagnetism is, somewhat excessively,
complicated by the fact that the the electromagnetic field strength tensor
is a 2-form (antisymmetric 2nd rank) on spacetime it is easier to explain
what I mean using a lower rank tensor example where the resulting behavior
is qualitatively similar to the case of E&M.

Consider a 3-vector field e(r) obeying (source-free) homogeneous PDEs:
curl(e) = 0 and div(e) = 0.  By standard methods we can solve for the
behavior of this field by realizing that the curl-free nature of e requires
that e be a gradient and by setting e = -grad([phi]) for some 3-scalar
field [phi](r).  The transverse condition (div(e) = 0) on e when applied to
[phi] becomes Laplace's equation: div(grad([phi])) = 0.  The solution of
this elliptic PDE is unique once the conditions on the boundary of the
region of the space over which the solution is sought are appropriately
specified.  If the boundary conditions (BCs) are static, then so is the
solution.  If the BCs are varied in time then the [phi] field responds
everywhere instantaneously as a whole and the time-dependent configuration
solves the problem for the current instantaneous BCs as they are varied.
There is no propagation or wave-like oscillation in the solution.  Since
the e field is just -grad([phi]) we see that the e field also does not
propagate or oscillate without it being locally forced to do so by a
propagating/oscillatory inhomogeneous source term.

OTOH, this is *not* the situation in Minkowski spacetime.  Again consider a
4-vector field e with contravariant components e^j (j =0,1,2,3) where the
0-th component refers to the time-component and 1,2,3 components refers to
the spatial components of the field.  (I'll use ^ for contravariant
superscripts and _ for covariant subscripts.)  Since I can't describe
boldface type or vector arrows using plain ASCII text let me surround a
3-vector by asterisks so that the contravariant 4-vector field e^j is
noted by (e0,*e*) and the covariant version e_j is (e0,-*e*) where
e0=e^0=e_0, and *e* is the 3-vector spatial part of e.  Note I'm taking the
metric to have the signature: +---.  Now suppose that e obeys spacetime
versions of the previous conditions (curl(e) = 0 & div(e) = 0) which can be
stated as: d_i(e_j) - d_j(e_i) = 0  and d_i(e^i) = 0 where d_i means
differentiate w.r.t. the i-th contravariant coordinate and the
summation convention holds for a repeated lower/upper (_,^) index pair.
The first of these equations says the generalized curl (4-d version
involving 6 antisymmetric components) of e vanishs and the second equation
says that the 4-divergence of e is also taken to vanish as well.  Because
of the vanishing of this "curl" we know that e must be the 4-gradient of
some 4-scalar field, i.e. e_i = d_i([phi]), or using separate temporal and
3-vector notation: e0 = (1/c)*d[phi]/dt & *e* = -grad([phi]).  The
requirement that the 4-divergence of e vanish is equivalent to the
statement that the e field obeys the continuity equation and thus
represents a flux of a locally conserved quantity (however abstract that
quantity may be).  IOW, d_i(e^i) = 0 means d(e0/c)/dt + div(*e*) = 0 when
written in 3-vector notation.  Thus *e* is the flux current density of some
locally conserved 'stuff' and e0/c is the density in space of that stuff.
If we substitute our representation of e as a 4-gradient into the
4-divergence equation we get the result that the D'Lambertian of [phi]
identically vanishes which is equivalent to the wave equation for waves
propagating (dispersionlessly) at speed c.  IOW, d_i(d^i([phi])) = 0 means
that: (1/c^2)*d^2([phi])/dt^2 - div(grad([phi])) = 0.  This is a hyperbolic
PDE which supports oscillatory and propagating advanced and retarded
solutions.  Since [phi] obeys this wave equation its 4-gradient e does so
as well, and we see that e-waves and [phi]-waves propagate in empty
(source-free) space at speed c (and thus behave as a massless scalar
field).  There is no need for different parts of e to act alternately as
sources for other parts to keep the wave propagating.

For electromagnetism the situation is quite similar.  The main difference
is that here the field which obeys the 1st order homogeneous PDEs (in a
region devoid of charged sources) is not a 4-vector field e but the 2-form
(antisymmetric 2nd rank) field strength tensor F.  In this case the role of
the scalar potential field [phi] is played by the 4-vector (or 1-form)
potential A^j (or A_j) whose temporal component is the usual electric
potential and whose spatial components are the usual magnetic vector
potential.  In this case, as well, the potential field A and the field
strength field F both obey an appropriate version of the wave equation
describing wave propagation at speed c.  The 3-vector versions of the
1st-order homogeneous PDEs for F are the usual source-free Maxwell
equations: div(*B*) = 0 & curl(*E*) + d(*B*)/dt = 0 (from the vanishing of
the generalized curl, i.e. exterior derivative, of F, or equivalently, the
vanishing of the 4-divergence of the *dual* of the field strength tensor),
and div(*E*) = 0 & curl(*B*) - (1/c^2)*d(*E*)/dt = 0 (from the vanishing of
the generalized divergence of F).  These equations are homogeneous and
source-free just as much as was the case in our simpler example involving e
above.  There is no need to put the time derivatives on the RHS of these
equations and treat them as sources for the spatial derivative terms.

David Bowman
dbowman@georgetowncollege.edu