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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!

Jannik Pitt
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    The meaning depends from the context. Isn't there a series of notations defined at the end of the book? – Vincent Sep 04 '16 at 11:23
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    It might represent a derivative with respect to time. It is hard to say without context. Also, the spelling "coördinate" seems a bit off. – TZakrevskiy Sep 04 '16 at 11:24
  • @TZakrevskiy I can tell that it is not the derivative with respect to time as the author uses a normal dot for that. – Jannik Pitt Sep 04 '16 at 11:25
  • @Vincent Unfortunately no there is nothing at the end of the book that explains the notation – Jannik Pitt Sep 04 '16 at 11:26
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    In my opinion it is a constant (one constan for each i). For context see https://books.google.es/books?id=3cE9jXr_QhwC&pg=PA48&lpg=PA48&dq=%22if+we+carry+out+a+linear+coordinate+transformation%22&source=bl&ots=eoJqqyt-z7&sig=Zntq31gq-wraC0wkoG6Hlcu_SWU&hl=gl&sa=X&ved=0ahUKEwj_uoLu0PXOAhWLcRQKHWMZBOkQ6AEIITAB#v=onepage&q=%22if%20we%20carry%20out%20a%20linear%20coordinate%20transformation%22&f=false – mfl Sep 04 '16 at 11:47
  • @TZakrevskiy: The "ö" in "coördinate" indicates that the two "o"s are separate vowels, not a diphthong. As Wikipedia mentions, this use of dieresis in English is rare these days, but it's common in older works. – PM 2Ring Sep 04 '16 at 12:15
  • @mfl Yes I think so too. Earlier in the book the author uses x,y,z (with circles above them) to align two coordinate systems so it's just an additive constant. I was just wondering whether there was more behind that notation and if it's actually in use outside of the book. – Jannik Pitt Sep 04 '16 at 12:16
  • @TZakrevskiy This book is a bit older (from the time of Einstein) and back in those days coordinate was spelled "coördinates" to emphasise that both "o" are pronounced separatly. It's basically the same as writing "co-ordinates". – Jannik Pitt Sep 04 '16 at 12:17
  • @PM2Ring @ Jannik ok, I see, thanks for the info. – TZakrevskiy Sep 04 '16 at 12:31

1 Answers1

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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.

Graham Kemp
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  • 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! – Jannik Pitt Sep 05 '16 at 14:17