Natural Science Forum / Physics / Relativity / April 2006

In an SPR post dated Sunday, April 16, 2006 1:55 PM, Igor Khavkine said "the
best you can hope for is a solution for the equations of motion of some kind
of charged fluid (macroscopic), together with the EM field, together with
the metric.  There is no way to directly identify this scenario with the
description of a single electron (especially given that hydrodynamic
equations are not expected to hold exactly at the microscopic level).  On
the other hand, there are models of a classical electron as an extended
object. They usually aim at removing some of the pathologies associated with
the assumptions of its point particle nature.  The literature on this
subject is fairly large and goes back all the way to Abraham and Lorentz.
AFAIK, none of these models have a claim to being "fundamental". To the best
of our knowledge, this title goes to the QED treatment of the electron."

Can those of you who know the literature in this field point me to the best
efforts that have been made to date to specify exact solutions to Einstein's
Field Equations for a so-called "charged perfect fluid"?

I have in mind exact solutions for energy tensors of the general form T^uv =
T^uv_Maxwell + T^uv_Euler = G^uv, where T^uv_Maxwell = -(1/4pi)[F^usF^v_s -
(1/4)g^uvF^stF_st] is the "usual" electrometric energy tensor and T^uv_Euler
= (mu+p)U^uU^v - p g^uv is the Euler energy tensor for a perfect adiabatic
fluid with four-velocity U^u, energy density mu, and pressure p, and where
G^uv is the Einstein tensor, g^uv is the metric tensor, and F^uv is the
electromagnetic field strength tensor.  Above, I am using a metric tensor
with signature +---.

Stephani does not present such composite solutions.  I did some arxiv and
Google searching and did find a few things, but want to be sure I am using
references which are recognized as containing reliable and unique solutions.

I am interested in "off-the-shelf" solutions for this energy tensor which
are static and spherically symmetric, static and axially-symmetric,
stationary and axially-symmetric with rigid rotation, and most preferably,
stationary and axially-symmetric with differential rotation.  Or, in being
advised that nobody to date has succeeded in obtaining one or more of these
types of solution.

I am also especially interested in the best efforts that have been made to
date to use such charged perfect fluids to formulate what Dr. Khavkine
refers to as "models of a classical electron as an extended object."

I fully understand and recognize that the QED solutions for the electron, at
present, are the only models which have a claim to be "fundamental,"
particularly, which are in accord to great accuracy with experimental
observation (not to mention that the computer I am using at the moment would
not work if QED was wrong).  It is also clear to me that any other approach
must in the end, at minimum, replicate the same results as QED, and I am not
among those who would ever consider suggesting otherwise.

I do plan, using these solutions, together with a good study of quantum
field theory (right now am considering Zee as a starting point but am still
open to suggestions as to the best course of study for this research
project), to see if there is some way in which the charged perfect fluid
solutions to GTR, which like all solutions, will greatly constrain the
permissible forms for the metric tensor, can, from some viewpoint, with a
suitable QFT construct, reproduce the basic quantum-number-characterized
constraints which QED imposes on charge density distributions for the
"observed" electron.  See, e.g., http://www.orbitals.com/orb/orbtable.htm,
for an example of what I would hope to eventually cast in the form of a
solution to the GTR field equations.

I would also be curious if anyone who has looked into these questions in the
past can lay out a roadmap of the walls you expect I will run into, or why,
as is probable, you may think this is futile.

Perhaps in the end I will be "Dorothy" and you all will be the "Good Witch
of the North" telling me I had to learn this for myself.  But I do want to
try this, because I firmly believe that GTR, with suitable QFT development,
is the fundamental theory of nature even on the smallest scale, and that
some day, in some way, we will find a way to reproduce QED, with the added
benefit of non-linearity, from GTR.

If all I do is learn how daunting or perhaps impossible this is, so be it.
There is nothing ignoble about learning, even if it turns out that what is
learned has been learned before, or that what one is seeking to find cannot
be found.

Best regards,

Jay.
_____________________________
Jay R. Yablon
Email: jyablon@nycap.rr.com