Re: Connection not a tensor..

"mike.james" <mike.james@xxxxxxxxxxxxxxxxxx> wrote in message
news:M_OdnVOoYL_tlwLeRVnyiQ@xxxxxxxxxxxxxxxxx
>
> 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.

[Hammond]
The "physical" quantity associated with the "connections"
(Christoffel symbols) is actually the gravitational force.
The metric as is well known is equivalent to the potential
of Newtonian theory. As we know the gradient of the
potential is the gravitational force, and since the connections
are formed from the first derivatives of the metric, it turns
out that "physically" the connections describe the gravitational
"force" field.

This can be easily seen in the linearized case where:

ds^2 = -(1 + 2phi)dt^2 + (1 - 2phi)dR^2

and:
__i
| jk = d(phi)/dx = "g" the acceleration of gravity.

hence, physically, the connections may be thought of as "g"
the acceleration, ("force" field) of gravity..

crudely speaking...
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