Re: Field theory for continua

From: Igor Khavkine (k_igor_k_at_lycos.com)
Date: 10/27/04 

Date: Wed, 27 Oct 2004 16:00:41 +0000 (UTC)

I've mentioned in a previous post that I have trouble interpreting your
notation and conclusions. Let me try to interpret what I understand and
perhaps you can fill in some blanks.

On Mon, 25 Oct 2004 13:09:48 +0000, Van Jacques wrote:

> Now the point of the post. What about continua. What is the homogenous
> eqn. which gives the field for continua? It is the eqn. of continuity. Let
> j = *J = n*u, where J = nu = matter current density, n = (rest) mass
> density, and u = 4-velocity.

The first point is that I do not think it is a trivial fact that
current density is represented by a 3-form in 4D spacetime. Second, I am
confused by your use of the Hodge star. You seem to be claiming that J
is a 1-form which is proportional to u, where u is a 4-velocity vector.
I guess the identification can be made with the metric, but this step
cannot be omitted.

For my own benefit and that of anyone interested, let me sketch that a
current density in an (k+1)-dimensional spacetime must indeed be an
k-form. Start off in familiar territory and assume that space and time
are split. Let n' be the particle density (n' integrated over a k-volume
of space gives a number, must be a k-form on space). Let j' be the particle
current density. Given a (k-1)-dimensional hypersurface, the integral of
j' over it also gives a number, the flux through the hypersurface per
unit time. Hence j' must be a (k-1)-form on space. The continuity
equation takes the form (where V is a k-volume of space, and @V its
boundary)
  
  @/@t int_V n = int_(@V) j

or equilvalently in integral form
  
  int_(V at t2) n' - int_(V at t1) n' = int_(t1 to t2) int_(@V) j' dt
                                      = int_(@Vx[t1,t2]) j' /\ dt .

Now we have two k-forms on spacetime, n' and j' /\ dt, let me denote
their sum by j = n' + j' /\ dt. Looking at the above equation carefully,
we note that (V at t2) - (V at t1) - (@Vx[t1,t2]) is exactly the
boundary of the spacetime volume Vx[t1,t2]. Now the continuity equation
can be rewritten as

  int_@(Vx[t1,t2]) j = 0 which implies int_(Vx[t1,t2]) dj = 0.

And since I can build any spacetime volume out of little pieces that
look like Vx[t1,t2], the differential form of the continuity equation
is just dj = 0.

The above discussion justifies two things. First that the current
density is given by a k-form j in a (k+1)-dimensional space time.
Integrated over a space-like k-volume, j gives the number of particles
contained in it. While integrated over a timelike k-hypersurface, j
gives the total number of particles that have passed through it. Second
it justifies the simple dj = 0 form of the continuity equation in
space-time. I might have messed up with the sign of the second term in
j, perhaps it is dt /\ j, but I think the general idea is fine.

Now, about the relation between the velocity field u and the current
density j. They can't be simply related by a Hodge transformation, since
one is a form and the other is a vector field. And how does the
application of the Hodge star follow from first principles anyway?

> Then write the eqn. of continuity div(J) = 0
> as dj = 0. The exterior derivative of the 3-form j = 0, so that j is
> closed and therefore exact. Because of the isotropy of 3D space, we have j
> is the exterior product of the exterior derivative of 3 scalar fields z^1,
> z^2, z^3.
>
> j = dz^1 /\ dz^2 /\ dz^3 = *J

I don't understand your last comment. How does "isotropy of 3D space"
lead to this particular form of j? By analogy with E&M, and by Poicare's
lemma, since dj = 0, then j = dL for some (k-1)-form A on spacetime.
What would this (k-1)-form dL represent?

> If this is the way to do EM, then this is the way to do continuous matter.
> Its clear to me that this is how to do both theories. Doing anything else
> leads to a
> mess and to errors. For example I do waves in perfect fluids, MHD, and
> plasma in my paper, including the energy and momentum of the waves, and
> the eqns. obeyed by the waves as they propagate, as well as the usual
> dispersion and polarization relations. Everything falls out easily if one
> start from this, the correct framework.

As far as I know, people have been doing hydrodynamics before
differential geometry in its current for was formulated. So notation and
formalism are definitely not an obstacle for theoretical physics. It may
be true that using the notation you suggest it is easier to get all the
equations of motion and such, but it is certainly not the only way. The
most probably reason for this formalism not being used in hydrodynamics
is that people who are doing the latter are usually not familiar with
the latter.

> As I said, Soper published some of this in his book, but no one seems to
> have noticed.

Most probably so. And they probably never will unless someone points it
out along with definite advantages to introducing a new formalism in
their work.

Igor