I have been reading about symmetries of systems' actions, e.g. the Polyakov action, and I have encountered Lorentz transformations of the form: $\Lambda^{\mu}_{\nu} X^{\nu}$. I am moderately familiar with $\Lambda$, the Lorentz matrix. If the indices are in superscript then it is the inverse of $\Lambda_{\mu \nu}$. However, what is $\Lambda^{\mu}_{\nu}$ in terms of the Lorentz matrix?

I have chosen mathematical physics as the tag, as I do not think any discipline of pure mathematics is appropriate based on the context. Please correct me if I am wrong.


1 Answer 1


Let's recap: upper indices are vectors ($x^\mu$), the inner product on Minkowski space is given by $g_{\mu \nu}$ so "dual vectors" have lower indices $x_\nu = g_{\nu \rho} x^\rho.$

Then you see that a matrix (in the sense of linear map between vectors) has one upper and one lower index, because it maps a vector to another vector:

$$x^\mu \mapsto A^\mu{}_\nu x^\nu.$$

So actually $\Lambda^\mu {}_\nu$ is a very natural object, it's a matrix that rotates/Lorentz boosts vectors.

If you prefer working with the tensor $\Lambda_{\mu \nu}$, then the link between both tensors is given by

$$\Lambda^\mu{}_\nu = g^{\mu \rho} \Lambda_{\rho \nu} \Leftrightarrow \Lambda_{\mu \nu} = g_{\mu \sigma} \Lambda^\sigma {}_\nu.$$


Your Answer

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

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