Re: Magnetic Monopole
From: Rob Woodside (rwmw_at_telus.net)
Date: 09/28/04
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Date: Tue, 28 Sep 2004 15:20:05 +0000 (UTC)
"robert j. kolker" <nowhere@nowhere.net> wrote in message news:<2rjo3nF1bfevoU1@uni-berlin.de>...
> How much disruption or inconvenience would mainline physics theories
> suffer if a magnetic monopole were found. GTR would not be impacted.
> What of the quantum type theories?
>
> Bob Kolker
Good Question. Classically the monopoles existence hinges on whether
or not the vector potential is global or only local. How this
topological difference works in Quantum feild theory where the vector
potential is of paramount importance is not clear to me.
Although the Einstein equations are not impacted, the Maxwell
equations (F = dA, d*F = *Je) certainly are. A duality rotation of a
quarter turn produces a field FD = *F, interchanging the roles of F
and *F. So the field equations, no longer Maxwell's, now become dFD =
*Je and *FD = -dA and the Je is now identified as the magnetic current
Jm.
The Maxwell stress-energy tensor for the electromagnetic field is
invariant under all duality rotations. Identifying the matter tensor
of the electric current as that of a magnetic current, then leaves the
total stress-energy tensor invariant under a quarter turn with duality
rotation. Thus any Einstien-Maxwell solution can be written as a
magnetic current solution to the Einstein-FD equations. This means
that the geometry is invariant under a quarter turn of the field, with
electric and magnetic current interchange;
and the caveate that the "electromagnetic field equations" are
different.
As an example, topological electric charges arise when *F is closed
(d*F = 0) and NOT exact (*F = dB, B is a global one form). So
electrically charged black holes obey the Maxwell equations F = dA,
d*F = 0 and magnetically charged black holes obey dFD = 0, FD = -dA
and both have the same geometry.
This has a nasty consequence for null electromagnetic fields where
duality rotations do give rise to NEW solutions of Maxwell's equations
which alter the polarization or phase of the Maxwell field. As the
Einstein-Maxwell equations are currently used this phase or
polarization information is not passed on to the geometry!!!
The conformal tensor describes gravitational waves along with their
phase and polarization. A Maxwell field with a Maxwell stress-energy
tensor has a piece of curvature quadratic in the field, i.e. (FF +
*F*F)/2, by Einstein's equations. So by the full second Bianchi
identities there should also be a piece of the Conformal tensor
quadratic in the field. It is straight forward to find the piece of FF
that has the same symmetries as the conformal tensor, but the overall
constant is undetermined. Checking examples, it has one value for
Maxwell fields, the opposite value for FD fields and of course zero
for conformally flat solutions. Picking this constant gives the global
coupling between the Maxwell field and the curvature. Specifying a
piece of the CURVATURE (having a trace as above and a traceless piece
with the right coupling) explit in F and specifying the divergence of
this piece of curvature in terms of current and derivatives of the
field, allows the derivation of Maxwell's equations from these
curvature assumptions alone. This is more restrictive than the usual
application of the Einstein-Maxwell equations as that makes no demand
on the conformal piece of the curvature. Currently the conformal
tensor is what ever it needs must be, including conformally flat. In
the curvature approach, the Einstein-Maxwell equations are required to
produce a conformal tensor that contains the right coupling between
field and curvature. If this restriction removes any physical
solutions then it must be altered or abandoned.
This is pushing Einstein's equations into Curvature. To handle
mechanics one specifies the traces of cuvature with the usual Einstein
equations. To handle Maxwell's electrodynamics one specifies the
traces of curvature as well as a traceless piece. To handle
gravitational radiation surely one must make demands on the traceless
bits of curvature.
In the quest for quantum gravity it may well be that both the
classical and quantum sides require radical changes.
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