Connections of degenerate metrics

[haushofer]

I have a question about connections derived from degenerate metrics, like in Newton-Cartan.

The question is simply: can the transformation properties of the connection change if one considers connections derived from metrics which are degenerate?

One one hand, I would say that one can follow the usual GR-analysis, check how covariant derivatives of general vectors/covectors must transform and conclude how the connection must transform (with a inhomogeneous term).

On the other hand, in Newton-Cartan one has the metric conditions

$$\nabla_{\mu}\tau_{\nu} = 0, \ \ \ \ \nabla_{\rho}h^{\mu\nu}=0$$

where $h^{\mu\nu}\tau_{\nu}=0$; h plays the role of spatial metric, and tau the role of temporal metric.

So, do the transformation properties of the connection depend on the metrical structure?

 

[arkajad]

I have a question about connections derived from degenerate metrics, like in Newton-Cartan.

So, do the transformation properties of the connection depend on the metrical structure?

I would rather say: connection is a connection and it can be considered independently of the metric. Then, if you have a metric, degenerate or not, you may impose conditions on your general connection and solve these conditions as much as you can - given the circumstances. If your metric is non-degenerate and if you impose zero-torsion, then your metric determines the connection uniquely. Otherwise you can have some freedom. Sometimes this freedom can be eliminated by requiring that your connection is smooth across some lower dimensional submanifold where your metric degenerates. If you would quote some particular page in some particular paper, then I could be more specific.

In Newton-Cartan you will have a certain freedom in your connection. With the right setting you can interpret this freedom as some kind of a (electromagnetic?) potential.