Re: Vector gravitational field equations -- gen. covariance

Hello:

I would like to know if it is proper for me to claim my unified field 
equations are non-trivially generally covariant.

I've been reading a lot in this newsgroup and in "Subtle is the Lord..." 
how central this idea is.  Here is how Lubos defines it (and it sounded 
similar to many things I've read recently):

>A generally covariant (or diffeomorphism invariant) theory is a theory
>defined by some fundamental equations whose form does not change if we
>perform a diffeomorphism: only the *dynamical variables* are allowed to
>change, and the dynamical variables can't be totally auxiliary fields
>that can be eliminated easily (and without breaking other physical
>symmetries) because the general covariance in that case would be an
>artificial mathematical construct.

The Lagrange density I have been studying is this:

    L_GEM = -(Jq^u - Jm^u) A_u - A^u;v A_u;v

There is the coupling of two currents to the same 4-potential and the field 
strength tensor which is the complete (i.e. all possible derivatives) 
covariant derivative of the 4-potential.  A diffeomorphism would change all 
contraction (yes?) by altering the metric that applies to that contraction, 
but the form of the Lagrange density does not change.  If one performs a 
diffeomorphism, that would also change the outcome of the covariant 
derivative, but again not the form of this equation.

>From here, I treat the potential A_u and the field strength tensor A^u;v as 
my variables, and derive the field equations by varying these in the action 
(that is what happens by applying the Euler-Lagrange equations).  I have 
read similar claims about general covariance when varying the 
Einstein-Hilbert action, and want to know if people think I can also claim 
my field equations are generally covariant.  If my field equations are not 
generally covariant, it would be acceptable to dismiss them, but I don't 
think that is the case.


doug
quaternions.com