Understanding the Relationship between Connection and Metric in Curved Spacetime

[gnieddu]

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

 

[atyy]

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.

 

[dx]

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.

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.

 

[gnieddu]

Thanks!