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Can one just change the notation of four vectors so as instead of having $$ X^{\mu} =(X^0, \vec{X})$$we define $$ X^{\mu}=(X^0,i\vec{X})?$$ This way we could use the Euclidean metric instead of $$g^{\mu\nu}=\text{diag}(1,-1,-1,-1).$$

Qmechanic
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user728261
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    Older books tend to do this I think (they put the $i$ in the time component though). As for why people swapped - another user might have opinions on this. – jacob1729 Jan 04 '20 at 14:47
  • Why do you want to pretend that the metric is Euclidean, and spacetime is complex, when neither is true? (Yes, I know there are some reasons in QFT for doing this.) – G. Smith Jan 04 '20 at 14:52
  • @G.Smith why do you say that neither are true? If they are mathematically equivalent, they should both be ""true"", whatever true means – user728261 Jan 04 '20 at 14:56
  • I suppose that is one way to think about it. It isn’t my way of thinking about it. I see no reason to think that meter sticks or clocks are measuring imaginary distances and times rather than real ones. – G. Smith Jan 04 '20 at 15:00
  • I consider all measurable quantities to be real. In any case, this was a former way of doing relativity that has fallen out of fashion, so if you do it you will look 50 years out of date. – G. Smith Jan 04 '20 at 15:05
  • So you allow measures to be negative? Because, couldnt you say with that reasoning, that measures are always positive, and depending on the direction of the measure you put a negative sign in front or not, therefore negative numbers dont exist neither? – user728261 Jan 04 '20 at 15:09
  • Pretty much everyone today allows spacetime intervals to be either positive or negative. That is the whole point of time-like vs. space-like intervals, and light-cones. – G. Smith Jan 04 '20 at 15:13
  • @G.Smith yeah, I understand it is pretty old-fashioned. Now this raises to me the question of why did they change the notation, just because its easier to see that space-time is hyperbolic or are there other reasons? – user728261 Jan 04 '20 at 15:26
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    Here is my paraphrase of Misner Thorne Wheeler’s “Farewell to ict”: https://physics.stackexchange.com/a/327516/148184 – robphy Jan 04 '20 at 15:28
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    Does this answer your question? Minkowski's complex Euclidean space vs. the real pseudo-Euclidean version – Kyle Kanos Jan 04 '20 at 16:22
  • Going back a 100 years or more this is how it was done. It does not easily generalize to curved space models impo. –  Jan 04 '20 at 17:50
  • @KyleKanos yeah, mostly. Thanks – user728261 Jan 04 '20 at 17:52

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