🚀 go-pugleaf

On Apr 5, 2:00 pm, "Ian Parker" <ianpark...@gmail.com> wrote:
> The Earth is here. Maxwell's simple analogue cannot reflect
> experiment.

At the same time, an outright dismissal of the general concept must,
itself, be outrightly dismissed, simply because equations governing
GR, itself, can be written in a form analogous to Maxwell's equations.

The difference, of course is that duality is not defined as it would
be for a Maxwell field or, more generally, a Yang-Mills field; so that
no analogue of (D,H) comes out; one only has the analogue of (E,B).
The reason for that, simply, is that duality is an expression of the
metric, whereas the background to gravity is a bare manifold (the
metric itself is the dynamical variable).

But with that out of the way, one can write down the half of the
Maxwell theory governing the analogue (B,E,A,phi) set of fields;
equations immediately.
A = (e^a_{mu} p_a + omega^{ab}_{mu} s_{ab}) dx^{mu}
F = 1/2 F_{mu nu} dx^{mu} ^ dx^{nu} = dA + A^2

(where, for purposes of evaluating the square, the "gauge" generators
p_a, s_{ab} are assumed to lie in an enveloping algebra, where the Lie
bracket becomes a commutator). The generator p_a, s_{ab} are those for
the Poincare' group.

The curvature R^{ab} and torsion T^a 2-forms come out from the
decomposition
F = 1/2 R^{ab} s_{ab} + T^a p_a,
these are the field strengths corresponding to the generators s_{ab}
and p_a.

That, in itself, also serves to explain why there's no dual: Poincare'
is not a semi-simple group, whereas a Yang-Mills theory requires a
semi-simple symmetry group to define duals over (otherwise, there's no
gauge metric, since no non-degenerate adjoint-invariant metric exists
for a group unless it's semi-simple).

The other half of the equations, of course, are those expressing the
field law, and these come out of the Lagrangian

L = 1/(16 pi G) e_a ^ e_b ^ R^{ab}

where e_a is formed from the inverse e^{mu}_a of the matrix e^a_{mu}.
This yields the field equations with 0 torsion, when coupled to a
matter Lagrangian.

Somewhere underneath all this may be a closer analogue to the Maxwell
equations involving (D,H). An analogy surely exists when generalizing
the gauge group to the conformal group or aDS (since that's standard
fare in Supergravity), but the real problem is to try and find a
theory closer in spirit to Yang-Mills that does not requiring going
that route, but allows you to stay with Poincare'.

An obstacle to this is that a Yang-Mills theory has a stress tensor. A
gauge theory in a similar mould will also have a stress tensor. But,
as is well-known, there are fundamental problems with localizing
gravitational stress/energy/momentum (but, as is not as well-known:
the problems are not insurmountable in principle).