Matrices and Tensors (Example in special relativity)

Question

I was studying special relativity and i found this derivation of the Lorentz transformations \begin{equation} \left( \begin{array}{cccc} x'^0 \\ x'^1 \\ x'^2\\ x'^3 \end{array} \right)= \left( \begin{array}{cccc} \gamma & -\gamma \beta & 0& 0 \\ -\gamma \beta &\gamma &0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right)\left( \begin{array}{cccc} x^0 \\ x^1 \\ x^2\\ x^3 \end{array} \right) \end{equation}

and then he denotes \begin{equation} Λ^μ{}_ν= \left( \begin{array}{cccc} \gamma & -\gamma \beta & 0& 0 \\ -\gamma \beta &\gamma &0 & 0 \\ 0 & 0 & 1 & 0\\ 0 & 0 & 0 & 1 \end{array} \right) \end{equation} as the lorentz transformation matrix. If that's the case which matrix is for example $Λ_{νμ}$ or $Λ^{νμ}$ or even $Λ_μ^ν$.I am confused about which matrix is which.

Anyone to clarify?

Also, how do i know which index is first meaning is it $Λ_{ν}{}^{\ μ}$ or $Λ^μ{}_{\ ν}$?

I will appreciate if you have any reference to check these.

Answer 1

Usually for a 2d tensor ${\Lambda_\mu}^\nu$ the first index ($\mu$) is refered to rows while the second one ($\nu$) to columns. A Lorentz transformations will be seen then as:

$$x'^\mu={\Lambda_\nu}^\mu x^\nu$$

When you transpose the matrix, rows and columns are interchanged $[{\Lambda_\mu}^\nu]^T={\Lambda^\nu}_\mu$

The metric tensor $\eta_{\mu\nu}$ provides a natural isomorphism between the tangent (space of vectors) and cotangent space (space of 1-forms), so it let us "lower" and "rise" indices. $$\Lambda_{\mu\nu} = \eta_{\nu\sigma}{\Lambda_\mu}^\sigma \Rightarrow \Lambda_{00} = \eta_{0\sigma}{\Lambda_0}^\sigma\qquad\mbox{and so on}\\ \Lambda^{\mu\nu} = \eta^{\mu\sigma}{\Lambda_\sigma}^\nu\Rightarrow\Lambda^{00} = \eta^{0\sigma}{\Lambda_\sigma}^0\qquad\mbox{and so on}$$ Using the same notation and $\eta_{\mu\nu}=diag(-1,1,1,1)=\eta^{\mu\nu}$:

$$ \Lambda_{\mu\nu} = \eta_{\nu\sigma}{\Lambda_\mu}^\sigma = \left(\begin{matrix} \gamma & \gamma\beta & 0 & 0 \\ \gamma\beta & -\gamma & 0 & 0 \\ 0 & 0 & -1 & 0 \\ 0 & 0 & 0 & -1 \end{matrix}\right) $$

It is possible to work solely with matrix notation, and there are good books in this topic (for example Einstein in Matrix Form by Günter Ludyk), however is usefull to consider a tensor as a list of components labeled by indices and use sums instead. A First Course in General Relativity by Bernard Schutz, specially chapters 1, 2, 5 and 6, is a good book to exercise with index notation and gives a simple but usefull overview of tensor calculus.