Index notation Lorentz transfromation matrix

Question

A Lorentz transformation between two different coordinate systems is given by:

$$(x')^\mu = {\Lambda^\mu}_\nu~ x^\nu$$

where $\Lambda$ is the Lorentz transformation matrix. I am a bit confused by the ordering of the indices using this notation, that is why do we write ${\Lambda^\mu}_\nu$ instead of $\Lambda^\mu_\nu$ or ${\Lambda_\nu}^\mu$. Is there an explicit convention of which index comes first in the left-to-right ordering? Is ${\Lambda_\nu}^\mu$ different from ${\Lambda^\mu}_\nu$ and if yes, how do they relate to each other.

What about the ordering of the indices from left to right of the general tensor? Does the order here just specify which position in the "N-dimensional matrix" we are considering, i. e. first index gives the row, second the column, third the "depth", etc.?

Any help in clarifying these points will be greatly appreciated.

Answer 1

You may have noticed this is a matrix equation, which might be more succinctly written as $x'=\Lambda x$. However, when you write such equations with explicit indices, and you sum over repeated indices, you need one upstairs and one downstairs, or equivalently you connect them with a metric tensor, viz. $x'^\mu=\Lambda^{\mu\rho}\eta_{\rho\nu}x^\nu=\Lambda^{\mu\rho}x_\rho$.

Before you study relativity you're familiar with an analogous calculation for which the metric is Euclidean, so the metric tensor is just the identity matrix. But outside of that context, you need to think very carefully about which indices are upstairs.

You also need to think very carefully about how to denote index-raising/lowering on non-symmetric matrices. Starting from $\Lambda_{\alpha\beta}$, if I raise one index but not the other I can get $\Lambda^\gamma_{\,\,\,\,\beta}=\eta^{\gamma\alpha}\Lambda_{\alpha\beta}$ or $\Lambda_\beta^{\,\,\,\,\gamma}=\eta^{\gamma\alpha}\Lambda_{\beta\alpha}$. Don't confuse the two!