1
$\begingroup$

Tensors, as I understand, are a sort of functions that contain information on how to transform a set of vectors and dual vectors, represented by a matrix. However, what I don't understand is the differing notation used to represent them. For example, I've seen both

$g_{\mu \nu} = \begin{pmatrix}-1 && 0 &&0&&0\\0&&1&&0&&0\\0&&0&&1&&0\\0&&0&&0&&1 \end{pmatrix} \tag*{}$

i.e., the metric tensor, denoted with the indices below, and other tensors denoted, say, $T^{\mu \nu}$ or $T^{\mu}{}_{\nu}$. These would transform different types, as in, two vectors ($g_{\mu \nu}$), two dual vectors, or a vector and a dual vector. I'm unable to understand how to, given a tensor, rearrange the components to form other tensors, say take $g_{\mu \nu}$ and find $g^{\mu}{}_{\nu}$, $g^{\mu \nu}$, or even $g_{\mu}{}^{\nu}$ for example. How would one do this?

$\endgroup$
1

1 Answer 1

1
$\begingroup$

You raise tensors using your metric tensor. For flat spacetime, this is the Minkowski metric $\eta_{\mu\nu}$. You must contract the Minkowski metric with one of the indices of your tensor in order to raise it:

${T^\mu}_\nu= \eta^{\mu\rho}T_{\rho\nu}$ and $T^{\mu\nu} = \eta^{\nu\rho} {T^\mu}_\rho$. Note that by the Einstein summation convention, contracted indices (in this case $\rho$) mean that you sum over each index. So $$\eta^{\mu\rho}T_{\rho\nu} =\eta^{\mu0}T_{0\nu}+\eta^{\mu1}T_{1\nu}+\eta^{\mu2}T_{2\nu}+\eta^{\mu3}T_{3\nu}.$$

$\endgroup$

Your Answer

By clicking “Post Your Answer”, you agree to our terms of service and acknowledge that you have read and understand our privacy policy and code of conduct.

Not the answer you're looking for? Browse other questions tagged or ask your own question.