# The physical significance of $\sqrt{-1}$ prior to Einstein

The spacetime metric of relativity

$$(ds)^2 = - (cdt)^2 + (dx)^2 + (dy)^2 + (dz)^2$$

attaches physical significance to $\sqrt{-1}$. (In order to achieve invariance the time differential used must be $-(cdt)^2 = (c\sqrt{-1}dt)^2$ rather than $(cdt)^2$.)

Prior to Einstein, which physical theories treated $\sqrt{-1}$ as physically significant?

• Anything having a phase component, i.e. using sine and cosine... – Jon Custer Jul 23 '17 at 18:16
• @JonCuster My understanding is that prior to the 20th century, physicists largely used complex numbers simply to make the analysis easier. They did not attach any direct physical significance to them. For example, I have read that introducing Euler's identity to a Fourier series results in negative frequencies (symmetric to positive frequencies). But negative frequencies have no physical meaning. – Nick Jul 23 '17 at 19:09
• The metric does not have a factor of $\sqrt{-1}$ in it: all of its coefficients are real, either 1 or $-1$. Your insertion of $\sqrt{-1}$ is inside the square but it is not visible in the actual formula. To stress my point, you could rewrite $(dx)^2$ as $2017(dx/\sqrt{2017})^2$ and ask for the physical significance of $\sqrt{2017}$, or in Newton's second law rewrite $F = ma$ as $F = \sqrt{-1}(ma/\sqrt{-1})$ and ask for the physical significance of $\sqrt{-1}$ in classical physics because of this "new" way of writing Newton's second law. – KCd Jul 24 '17 at 10:18
• @KCd The text I'm reading (Nahin's history of complex numbers) gives the analysis I have stated here. It seems reasonable to me, although I know next to nothing about physics. – Nick Jul 24 '17 at 14:35
• @Mozibur Ullah, I have heard of the mathematical device of Wick rotation in physics, but as far as I am aware there is not a genuinely physical meaning to replace $t$ with $it$ in relativity. – KCd Jul 27 '17 at 10:53

Complex numbers enter physics in the work of Fresnel on wave theory of light, namely in his derivation of total reflection. This was in the early 19 century.

https://en.wikipedia.org/wiki/Fresnel_equations

For more details, see Whittaker, E. T., A History of the Theories of Aether and Electricity.

• That's much earlier than I expected. – Nick Jul 24 '17 at 1:41

I must say that I don't understand in which way does the expression$$(ds)^2=-(c\,dt)^2+(dx)^2+(dy)^2+(dz)^2$$“attaches physical significance to $\sqrt{-1}$”. Nobody has to think in terms of $\sqrt{-1}$ in order to understand it.

Besides, what has Einstein to do with it? This expression was introduced by Minkowski in his 1907 lecture The fundamental equations for elecromagnetic processes in moving bodies. If I am wrong, please tell me where did Einstein mention it before.

• I am only just learning a bit about physics. The presence of $\sqrt{-1}$ in the metric is highlighted and explained in the text I am reading (Nahin's history of complex numbers). – Nick Jul 24 '17 at 14:38
• @NickR Paul Nahin is a former professor of electrical engineering. I think that the best books on the history of Mathematicas are those written by hisorians of Mathematics. By the way, I took a look at that book and I was not impressed, to say the least. – José Carlos Santos Jul 24 '17 at 14:41
• @Nick R: Regarding Nahin, I own and have read parts of several of his books, and I consider them very nice, especially for supplementary topics one might present in a class or for pleasure reading. However, I would NOT use Nahin for some mathematical or math-historical topic I wanted to investigate any more than I would use a newspaper article about catastrophe theory for information about singularity theory of $C^{\infty}$ manifolds, or someone's blog about infinity for what I investigated here. – Dave L Renfro Jul 24 '17 at 15:23
• @DaveLRenfro You are right. I should have added that I have no problem with the strictly mathematical parts of Nahim's book. – José Carlos Santos Jul 24 '17 at 15:31
• @DaveLRenfro Yes, I have also read Nahin's book on Euler's identity and I thought both books were well written and helpful for someone at my (undergraduate) level. But you are right when you say that they are not suitable for serious study since they very much "lean" towards a pop-math audience. – Nick Jul 24 '17 at 16:44