Understanding the Relationship between Connection and Metric in Curved Spacetime

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Discussion Overview

The discussion revolves around the relationship between connection and metric in curved spacetime, specifically addressing how changes in the metric generally lead to changes in the connection. Participants explore the implications of this relationship and inquire about potential formulas connecting different connections derived from different metrics.

Discussion Character

  • Exploratory
  • Technical explanation
  • Conceptual clarification

Main Points Raised

  • One participant expresses difficulty in understanding why a change in metric implies a change in connection, questioning if there is a general relationship between different connections derived from different metrics.
  • Another participant notes that there is no singular connection, emphasizing that multiple connections exist and that in general relativity, the Levi-Civita connection is typically chosen based on the metric.
  • A further elaboration is provided using analogies of stretching surfaces, suggesting that changes in the induced metric affect the parallel transport of vectors and thus the connection, as geodesics are not preserved under such changes.
  • One participant suggests that to find the new connection, one must recalculate the Christoffel symbols using the new metric.

Areas of Agreement / Disagreement

Participants do not reach a consensus on the nature of the relationship between connections and metrics, with some emphasizing the variability of connections while others focus on the implications of changing metrics. The discussion remains unresolved regarding the existence of a general formula for converting between connections.

Contextual Notes

Limitations include the lack of clarity on specific conditions under which the relationships hold, as well as the dependence on the definitions of connections and metrics. The discussion does not resolve the mathematical steps involved in deriving connections from metrics.

gnieddu
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Hi,

I'm struggling to grasp the physical reason behind the fact that, in a curved spacetime, a change of metric implies, in general, a change of connection, i.e. if I have two metrics g_{ab} and \hat{g}_{ab}, in general \nabla_a \neq \hat{\nabla}_a.

Besides this, is there any relationship between the two connections? In other words, if I know \nabla_aT for a given tensor T, is there a general formula which converts it into \hat{\nabla}_aT?

Thanks
 
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There isn't such a thing as "the" connection. There are many possible connections. In general relativity, the connection is chosen to be the Levi-Civita connection, which is defined by the metric.
 
gnieddu said:
Hi,

I'm struggling to grasp the physical reason behind the fact that, in a curved spacetime, a change of metric implies, in general, a change of connection, i.e. if I have two metrics g_{ab} and \hat{g}_{ab}, in general \nabla_a \neq \hat{\nabla}_a.

Imagine stretching a surface embedded in Euclidean space in a non-uniform way. This is a change of the induced metric on the surface, and you can see that parallel transported vectors on the original surface are no longer parallel transported. This is why the connection changes when the metric is changed. Another way to see it is to imagine taking a flat surface with straight lines drawn on it, and then stretching it over a sphere. There are many ways to do this, and in generel, the straight lines need not become great circles on the sphere, i.e. they are no longer geodesics with the new induced metric. Since the set of geodesics determine the connection, and since geodesics are not preserved by changes of metric, the connection must change.

gnieddu said:
Besides this, is there any relationship between the two connections? In other words, if I know \nabla_aT for a given tensor T, is there a general formula which converts it into \hat{\nabla}_aT?

Thanks

You simply have to re-calculate the Christoffel symbols with the new metric.
 
Last edited:
Thanks!
 

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