# I Symmetry of the metric tensor

1. Oct 28, 2016

### kent davidge

I was thinking about the metric tensor. Given a metric gμν we know that it is symmetric on its two indices. If we have gμν,α (the derivative of the metric with respect to xα), is it also valid to consider symmetry on μ and α? i.e. is the identity gμν,α = gαν,μ valid?

2. Oct 28, 2016

### andrewkirk

It is not valid. A counterexample is the FLRW metric, in which $g_{11,0}$ is nonzero because the metric changes over time with the expansion of the cosmos, while $g_{01,1}$ is zero because $g_{01}$ is uniformly zero.

3. Oct 28, 2016

### kent davidge

How would it look if we symmetrize / antisymmetrize it on its first and third indices (μ and α)?

4. Oct 28, 2016

### andrewkirk

What do you mean by symmetrise/antisymmetrise? In my understanding those are operations one performs on a tensor, and $g_{ab,c}$ is not a tensor, because partial differentiation (what the 'comma' does) is not a valid tensor operation.

5. Oct 28, 2016

### kent davidge

I'm sorry. Actually I mean how could we obtain gμα,β + gμβ,α - gαβ,μ from gαβ,μ. (I'm trying to derive the Christoffel Symbol to put it in the geodesic equation.)

6. Oct 29, 2016

### Orodruin

Staff Emeritus
You are being unclear. These are not the same things.