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If we carry out a linear coördinate transformation, $$x'_i=\sum_{k=1}^3c_{ik}x_k+\overset\circ x'_i,\quad i=1,2,3,$$ (from Introduction to the Theory of Relativity by Peter Gabriel Bergmann)

I came across this in a book about relativity and I've never seen this before. Can somebody explain that notation? Is it commonly used and what exactly does it mean? Thanks in advance!

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    $\begingroup$ The meaning depends from the context. Isn't there a series of notations defined at the end of the book? $\endgroup$
    – Vincent
    Sep 4, 2016 at 11:23
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    $\begingroup$ It might represent a derivative with respect to time. It is hard to say without context. Also, the spelling "coördinate" seems a bit off. $\endgroup$
    – TZakrevskiy
    Sep 4, 2016 at 11:24
  • $\begingroup$ @TZakrevskiy I can tell that it is not the derivative with respect to time as the author uses a normal dot for that. $\endgroup$
    – Jannik Pitt
    Sep 4, 2016 at 11:25
  • $\begingroup$ @Vincent Unfortunately no there is nothing at the end of the book that explains the notation $\endgroup$
    – Jannik Pitt
    Sep 4, 2016 at 11:26
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    $\begingroup$ In my opinion it is a constant (one constan for each i). For context see books.google.es/… $\endgroup$
    – mfl
    Sep 4, 2016 at 11:47

1 Answer 1

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$$x_i'=\sum_{k=1}^3 c_{i,k}x_k + \overset{\circ}{x_i}'$$

The new coordinate equals a scale and skew transform of the old coordinate plus a shift vector; which is the new coordinates of the origin (zero vector) after the transformation.

$$\because\quad\begin{pmatrix}x_1'\\x_2'\\x_3'\end{pmatrix} = \begin{pmatrix}c_{1,1}&c_{1,2}&c_{1,3}\\c_{2,1}&c_{2,2}&c_{2,3}\\c_{3,1}&c_{3,2}&c_{3,3}\end{pmatrix}\begin{pmatrix}x_1\\x_2\\x_3\end{pmatrix}+\begin{pmatrix}\overset{\circ}{x_1}'\\\overset{\circ}{x_2}'\\\overset{\circ}{x_3}'\end{pmatrix}$$

$$\therefore\quad\begin{pmatrix}\overset{\circ}{x_1}'\\\overset{\circ}{x_2}'\\\overset{\circ}{x_3}'\end{pmatrix} = \begin{pmatrix}c_{1,1}&c_{1,2}&c_{1,3}\\c_{2,1}&c_{2,2}&c_{2,3}\\c_{3,1}&c_{3,2}&c_{3,3}\end{pmatrix}\begin{pmatrix}0\\0\\0\end{pmatrix}+\begin{pmatrix}\overset{\circ}{x_1}'\\\overset{\circ}{x_2}'\\\overset{\circ}{x_3}'\end{pmatrix}$$

That is all.

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  • $\begingroup$ Yeah I guessed so, I just wanted to know whether that's a common notation for an additive constant and if there's more behind it or if that's just something the writer came up with. Apparently it's just unique notation of the author or outdated notation. Thanks for your answer! $\endgroup$
    – Jannik Pitt
    Sep 5, 2016 at 14:17

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