# Example of a metric incompatible connection

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#1
Hi,

I'm doing a Relativity course, and I'm a little confused by the concept of metric compatibility.Given (I assume these are correct):
Code:
`gab  =  ea.ebea  =  gab eaΓabc  =  ea ∂c eb`
Code:
```∂c gab  =  eb ∂c ea  +  ea ∂c eb  (product rule)
=  gbd Γdac  +  gad Γdbc```
But this is supposedly only the case if the connection is metric compatible.So my question is, how could the connection not be metric compatible? (what incorrect assumptions have I just made?) An example of this would be helpful, but I can't seem to find one (since people generally assume metric compatibility)

Thanks,
0
4 years ago
#2
(Original post by sunarcs)
Hi,

I'm doing a Relativity course, and I'm a little confused by the concept of metric compatibility.Given (I assume these are correct):
Code:
`gab  =  ea.ebea  =  gab eaΓabc  =  ea ∂c eb`
Code:
```∂c gab  =  eb ∂c ea  +  ea ∂c eb  (product rule)
=  gbd Γdac  +  gad Γdbc```
But this is supposedly only the case if the connection is metric compatible.So my question is, how could the connection not be metric compatible? (what incorrect assumptions have I just made?) An example of this would be helpful, but I can't seem to find one (since people generally assume metric compatibility)

Thanks,
The basic issue here is that a connection and a metric are two different geometric concepts - and a-priori there is no reason for any compatibility between them. So the first thing to say is that a metric is compatible with a connection if the parallel transport of the connection preserves the inner product defined by the metric. So take any two vectors, take their inner product w.r.t the metric; you get the same answer if you parallel transport the vectors to another tangent space and then take their inner product. Equivalent to this is that the covariant derivative of the metric tensor is zero. Indeed the covariant derivative of a metric tensor (if it is non-zero) is sometime called the non-metricity of the connection.

What is sometimes confusing is that the connection that is often wheeled out first in relativity courses is the Levi-Civita connection. This is defined to be the unique torsion free connection that is compatible with the metric. If you trundle through the calculations, the Christoffel symbols of the connection can be expressed in terms of the dual of the metric tensor and of derivatives of the metric tensor. The Levi-Civita connection is then by definition compatible withe metric - but what you also get is that any other torsion free connection will not be compatible with the metric.
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