- #1
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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
[tex]
\nabla_{\mu}\tau_{\nu} = 0, \ \ \ \ \nabla_{\rho}h^{\mu\nu}=0
[/tex]
where [itex]h^{\mu\nu}\tau_{\nu}=0[/itex]; 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?
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
[tex]
\nabla_{\mu}\tau_{\nu} = 0, \ \ \ \ \nabla_{\rho}h^{\mu\nu}=0
[/tex]
where [itex]h^{\mu\nu}\tau_{\nu}=0[/itex]; 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?