General Relativity/Raising and Lowering Indices

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<General Relativity

Given a tensor \mathbf{T}, the components T^{\alpha \ \mu \nu}_{\ \beta} are given by T^{\alpha \ \mu \nu}_{\ \beta}=\mathbf{T}(\mathbf{d}x^\alpha, \mathbf{e}_\beta, \mathbf{d}x^\mu, \mathbf{d}x^\nu) (just insert appropriate basis vectors and basis one-forms into the slots to get the components).

So, given a metric tensor \mathbf{g}(\mathbf{u}, \mathbf{v})=<\mathbf{u} \ |\ \mathbf{v}>, we get components g_{\mu \nu}=<\mathbf{e}_\mu \ |\ \mathbf{e}_\nu> and g^{\mu \nu}=<\mathbf{d}x^\mu \ |\ \mathbf{d}x^\nu>. Note that g^\mu_{\ \nu}=g_\mu^{\ \nu}=\delta^\mu_\nu since <\mathbf{e}_\mu \ |\ \mathbf{d}x^\nu>=<\mathbf{d}x^\mu \ |\ \mathbf{e}_\nu>=\delta^\mu_\nu.

Now, given a metric, we can convert from contravariant indices to covariant indices. The components of the metric tensor act as "raising and lowering operators" according to the rules w^\alpha=g^{\alpha \mu}w_{\mu} and w_\alpha=g_{\alpha \mu}w^\mu. Here are some examples:

1. T^{\alpha \ \gamma}_{\ \beta} = g_{\beta \mu}T^{\alpha \mu \gamma}

Finally, here is a useful trick: thinking of the components of the metric as a matrix, it is true that \left( g^{\mu \nu} \right) = \left( g_{\mu \nu} \right) ^{-1} since g^{\mu \sigma}g_{\sigma \nu}=g^\mu_{\ \nu}=\delta^\mu_\nu.