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I am studying special relativity and I can't figure out what is the difference between the matrix - index notation between:

$$ Λ_{α}{}^{β}, Λ^{α}{}_{β}, Λ^{αβ} ,Λ_{αβ} $$

Why do we introduce this kind of notation in SR ? In Linear Algebra, we denote matrices simply with $ Λ_{ij} $ where $i,j$ are indices that run in some common (or not) set. I know that the answer is somehow related with covariance and contravariance (which is important in SR mathematical formalism) but I don't know how exactly.

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For the sake of concreteness and without going into details of covariant and contravariant indices, let's just show the difference with the help of an example:

$\Lambda^\alpha_{\,\,\beta}$ is usually used to denote a Lorentz transformation. Let's take a transformation in $x$-direction:

$$\Lambda=\Lambda^\alpha_{\,\,\beta}=\left(\begin{array}{c c c c}\gamma & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &\gamma\end{array}\right)$$

When this transformation is applied to a four-vector $x^\beta=\left(\begin{array}{c}x\\y\\z\\t\end{array}\right)$, i.e. $\Lambda\cdot x=\Lambda^\alpha_{\,\,\beta} x^\beta $, one obtains the standard Lorentz transformation $x^\prime = \gamma(x-vt)$, $y^\prime = y$, $z^\prime = z$, and $t^\prime = \gamma(t-\frac{v}{c^2}x)$.

Now, it is easy to show the difference between the different index notations: $$ \Lambda_{\alpha\beta}= \eta_{\alpha\gamma}\Lambda^\gamma_{\,\,\beta}=\left(\begin{array}{c c c c}1 & 0 & 0 &0 \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\0& 0 & 0 &-c^2\end{array}\right)\cdot \left(\begin{array}{c c c c}\gamma & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &\gamma\end{array}\right)=\left(\begin{array}{c c c c}\gamma & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\\gamma v & 0 & 0 &-\gamma c^2\end{array}\right)$$ and similarly $$ \Lambda^{\alpha\beta}= \Lambda^\alpha_{\,\,\gamma}\eta^{\gamma\beta}=\left(\begin{array}{c c c c}\gamma & 0 & 0 &-\gamma v \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &\gamma\end{array}\right)\cdot \left(\begin{array}{c c c c}1 & 0 & 0 &0 \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\0& 0 & 0 &-\frac{1}{c^2}\end{array}\right)=\left(\begin{array}{c c c c}\gamma & 0 & 0 &\gamma \frac{v}{c^2} \\ 0 & 1 & 0 & 0\\0& 0 & 1 & 0\\-\gamma\frac{v}{c^2} & 0 & 0 &-\gamma\frac{1}{c^2}\end{array}\right)$$

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  • $\begingroup$ $Λ_{α}^{\ \ β} $ is just $ (Λ^{-1})^{α}_{\ \ β} $, right ? $\endgroup$ – Unstoppable Tachyon Apr 19 at 10:13
  • $\begingroup$ I think I get it. $ Λ_{αβ} $ and $ Λ^{αβ} $ are defined as the way you wrote them down, which is the metric of space times the transformation matrix (in the correct order). Also, what's the difference between those four matrices as tensors ? $\endgroup$ – Unstoppable Tachyon Apr 19 at 10:25
  • $\begingroup$ If you want to write it as equation, the indices on the left hand side and on the right hand sight have to agree with each other. But in principle you are right, raising the first and lowering the second index results in a change of sign for the velocity $v\rightarrow -v$ (which is the inverse transformation, maybe up to transposition). $\endgroup$ – p6majo Apr 19 at 10:33
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    $\begingroup$ $\Lambda^\alpha_{\,\,\beta}$ is used to transform components of vectors, i.e. $x^{\alpha\prime}= \Lambda^\alpha_{\,\,\beta}x^\beta$. $\Lambda_{\beta}^{\,\,\alpha}$ is used to transform components of forms, i.e. $e_{\beta}^\prime=\Lambda_{\beta}^{\,\,\alpha}e_{\alpha}$. $\Lambda_{\alpha\beta}$ is not used at all. Non of them are tensors, they are just coordinate transformations. They are used to transform tensors like $\eta_{\alpha\beta}$. $\endgroup$ – p6majo Apr 19 at 10:39
  • $\begingroup$ The OP asks a "why" question, which this answer hasn't addressed. $\endgroup$ – Ben Crowell Apr 19 at 12:11

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