# Raising and Lowering Indices

1. Jun 18, 2015

### Physicist97

Hello! New here, so please bare with me :). I am not entirely sure if this was the best forum to ask this question, so if it is not within the topic of GR, please say so. In Einstein's theory of General Relativity the metric tensor is symmetric and has the property of lowering and raising the indices of a tensor. If we were to construct a metric that was not symmetric, would it still have this property?
Assuming $g_{\mu\nu}\neq g_{\nu\mu}$ , than $g_{\mu\nu}A^{\mu}_{\alpha}\neq g_{\nu\mu}A^{\mu}_{\alpha}$ . This would mean that you could not simply lower the indice of this tensor. So, can a non-symmetric metric tensor hold the property that it raises and lowers indices, and if it does, how would you go about raising and lowering indices.
Thank you, and also, I have a poor understanding of mathematical symbols. My knowledge of the mathematics of differential geometry are the bare minimum for self-studying GR. To give you an idea of my limits, I have watched Leonard Susskind's free online lectures on GR (by Stanford), and have read Sean Carroll's lecture notes on GR. I have not been able to comprehend General Relativity by Wald.

2. Jun 18, 2015

### Simon Bridge

Sounds like you just need some practice with tensors.

The metric is a tensor ... so what happens when you use a non-symmeric tensor on another tensor?

The metric is special: is it possible to construct a non-symmetric metric.

3. Jun 19, 2015

### Ibix

Hint: what does the metric define, particularly for objects in the tangent space?