Re: Connection not a tensor..


Robert Low wrote:
> mike.james wrote:
> > I've been thinking about what it means that the connection coefficients
> > don't transform like a tensor. Given that we start off with the idea
> > that thing that have a co-ordinate independent reality are tensors i.e.
> > transform like tensors does the fact that the connection isn't a tensor
> > mean that it isn't a "physical thing" or does it mean that other
> > transformation laws define physical things.
> > Given that a connection does define things that are independent of
> > co-ordinates such as curvature, parallel transport etc. presumably the
> > connection is as real as a tensor and hence the transformation law that
> > it obeys is "as good as" the tensor transformation law.
> > Or is there a bigger picture in which the connection is part of a tensor?
>
> You can think of the connection as providing a kind of correction
> term: you add this to a coordinate derivative (which also isn't
> a tensor) and the result is then a tensor---i.e. a nice geometric
> notion of derivative.

See pg. 11, "correction term" related to the Christoffel in,

http://arxiv.org/abs/gr-qc/0511073

I think Wald agrees with the sentiment of the OP.

The connection becomes single indiced in the tetrad
formalism, which removes some of the math clutter,
to reveal a "bare boned" physics, that Weinberg in
"Grav & Cosmo" pg 365-373 demonstrates using
action to derive the Einstein Field Equations.
Regards
Ken S. Tucker

.

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