A Difference of Christoffel Symbols Transforms as Tensor

[binbagsss]

My notes seem to imply this should be obvious :

If i consider the covariant deriviative then i get something like

christoffel= nabla ( cov derivative ) - partial

So difference of two of them will stil have the partial derivatuves present ,assuming these are labelled by a different index ? Whereas a difference of cov derivatives would ofc transform as a tensor i don't see how here with the partials

 

[Orodruin]

assuming these are labelled by a different index ?

Why would there be a different index? This would violate the index laws. The point is that if you have two connections, $\nabla$ and $\bar \nabla$, then
$$
\nabla_\mu V - \bar \nabla_\mu V = [\partial_\mu V^\alpha + \Gamma_{\mu \nu}^\alpha V^\nu]\partial_\alpha - [\partial_\mu V^\alpha + \bar\Gamma_{\mu \nu}^\alpha V^\nu]\partial_\alpha
= [\Gamma_{\mu \nu}^\alpha - \bar\Gamma_{\mu \nu}^\alpha] V^\nu \partial_\alpha.
$$

 

[dsaun777]

Why would there be a different index? This would violate the index laws. The point is that if you have two connections, $\nabla$ and $\bar \nabla$, then
$$
\nabla_\mu V - \bar \nabla_\mu V = [\partial_\mu V^\alpha + \Gamma_{\mu \nu}^\alpha V^\nu]\partial_\alpha - [\partial_\mu V^\alpha + \bar\Gamma_{\mu \nu}^\alpha V^\nu]\partial_\alpha
= [\Gamma_{\mu \nu}^\alpha - \bar\Gamma_{\mu \nu}^\alpha] V^\nu \partial_\alpha.
$$

would two connections imply two different metrics?

 

[Orodruin]

would two connections imply two different metrics?

In general, a connection does not necessarily have anything to do with a metric (you do not need a metric to define a connection).

However, in the context of GR, where your connection is the Levi-Civita connection, then yes. This is why the connection is varied when you vary the metric components.

 

[dsaun777]

In general, a connection does not necessarily have anything to do with a metric (you do not need a metric to define a connection).

However, in the context of GR, where your connection is the Levi-Civita connection, then yes. This is why the connection is varied when you vary the metric components.

Do you mean You could define the connection only in terms of covariant derivative of basis and not need the metric of levi civita connection. Not every connection needs a metric but every metric can be used to define a connection?

 

[Orodruin]

Do you mean You could define the connection only in terms of covariant derivative of basis and not need the metric of levi civita connection. Not every connection needs a metric but every metric can be used to define a connection?

Yes. A connection is only a concept of what it means for a field to "change" between points on a manifold. This does not necessarily have anything to do with a metric. There are generally many possible connections on any given manifold. However, once you introduce a metric, there is only one connection that is both metric compatible ($\nabla g = 0$) and torsion free - that is the Levi-Civita connection. (Note that the requirement of being torsion free is also needed, there can generally be several metric compatible connections, but there is only one torsion free connection.)

 

[binbagsss]

Why would there be a different index? This would violate the index laws. The point is that if you have two connections, $\nabla$ and $\bar \nabla$, then
$$
\nabla_\mu V - \bar \nabla_\mu V = [\partial_\mu V^\alpha + \Gamma_{\mu \nu}^\alpha V^\nu]\partial_\alpha - [\partial_\mu V^\alpha + \bar\Gamma_{\mu \nu}^\alpha V^\nu]\partial_\alpha
= [\Gamma_{\mu \nu}^\alpha - \bar\Gamma_{\mu \nu}^\alpha] V^\nu \partial_\alpha.
$$

And how does that final expression transform like a tensor ?

 

[Orodruin]

And how does that final expression transform like a tensor ?

It should be quite clear that it does from the fact that it is derived from the difference of two covariant derivatives.