I read somewhere that part of Minkowski's inspiration for his formulation of Minkowski space was Poincare's observation that time could be understood as a fourth spatial dimension with an imaginary coefficient.

Clearly, taking the Euclidean norm of the vector $$(i \Delta t, \Delta x, \Delta y, \Delta z)$$ gives the correct spacetime interval (assuming appropriate units), but I don't really know where it goes from there (possibly something to do with Moebius transforms?)

I think this is mentioned in Taylor and Wheeler's book, but I may have read it elsewhere. After the historical note, the author (whoever it was) said it was "preferable" to use Minkowski geometry straight off, rather than mucking about with time as an imaginary space coordinate.

Could anyone elaborate on Poincare's formulation? Why is Minkowski's methodology better?

  • $\begingroup$ You might like to check out my answer here and other answers to the same question, but I think I'm making much the same point (messing with and beclouding signature) as in the big (teetering on graviational collapse) black book (Misner, Thorne and Wheeler). $\endgroup$ Jun 28, 2014 at 1:07
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    $\begingroup$ BTW sometimes, just sometimes, when the issues of signature aren't at the fore, the imaginary time is useful: see the idea of Wick Rotation. This is also used in "quantitative" finance to turn the Black Scholes stock option pricing equation into a Schrödinger equation and apply path integral methods, sometimes with disastrous results. Mathematics is a descriptive language, so sometimes certain descriptions are apposite, sometimes not, just as in "natural" language. $\endgroup$ Jun 28, 2014 at 1:14
  • $\begingroup$ BTW: great article on stochastic DEs. $\endgroup$ Jun 28, 2014 at 1:15
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    $\begingroup$ You will find answers to the question in [physics.stackexchange.com/q/107443/], in particular: "Of course, the two approaches are completely equivalent to each other." and "The primary reason to abandon the "ict" notation and use one of the last two is because we eventually want to go beyond special relativity and do general relativity. That is best done using (pseudo-) Riemannian geometry, and Riemannian geometry requires real-valued coordinates." $\endgroup$
    – Moonraker
    Jun 28, 2014 at 7:47

3 Answers 3


but why isn't SR taught with an imaginary time coordinate as standard?

From "Gravitation", page 51, via Google books.

enter image description here

I'll type up a paraphrase later.

  • 9
    $\begingroup$ it's been a couple of years - where's the paraphrase, you slacker :p $\endgroup$
    – Christoph
    Mar 27, 2017 at 15:46
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    $\begingroup$ I've been wondering the same thing... $\endgroup$ Mar 27, 2017 at 22:55
  • $\begingroup$ The Bible says, therefore, it is correct. BTW it is said in application to GR only. GR does not work otherwise (i.e. with $ict$). $\endgroup$
    – Eddward
    Dec 7, 2019 at 7:49
  • $\begingroup$ In The Universe in a Nutshell (Bantam, 2001), Hawking discusses imaginary time in several places. He points out (pp.61-63) that it can have "a much richer range of possibilities than the railroad track of ordinary real time." So I wonder if the cited critique may be a little antiquated? $\endgroup$ Feb 6, 2020 at 14:04

The reason comes down to having your system invariant under Lorentz transformations. With the imaginary time, such transformations are nothing more than rotations (of some sort).

For further detail:

  • $\begingroup$ I get that the Lorentz transforms are orthogonal transforms in Minkowski space - but why isn't SR taught with an imaginary time coordinate as standard? $\endgroup$ Jun 22, 2014 at 20:17
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    $\begingroup$ @SimonLyons because it doesn't generalize to GR, as outlined by Alfred $\endgroup$
    – Danu
    Jun 22, 2014 at 21:05

Special Relativity, and a proper understanding of time, can be confusing if you just look at the effect rather than both the cause and the effect. In other words, to start with, you have an absolute foundation or cause, that due to its structure, produces a relativistic outcome.

But today's school system in general only teaches you of the relativistic outcome. This is shocking in a way since an event is constructed of both cause and effect. So why inform the students of only the effect?

If you Google, with quotes included, "KSP SPECIAL RELATIVITY - YouTube", you will find a video coverage of the analysis of motion that leads to a full understanding of Special Relativity and its equations. ( Some of the folk who watch the videos are mystified as to why Special Relativity is not taught in this simplistic way within schools. )


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