In order to make the Minkowski metric, in special relativity, equivalent to the Euclidean metric, one idea is to allow time to take imaginary values. As far as I have learned about SR, it does make calculations and derivations better, but I'm not sure how far can it go. I've been told by some that it does not work at all in general relativity, for some coordinate-systems and for non-inertial frames. Is this correct? How far can the concept of imaginary time go?
1$\begingroup$ Related: physics.stackexchange.com/q/327318 $\endgroup$– G. SmithDec 5, 2020 at 4:06
This is an interesting historical blip in my opinion. You say that imaginary time makes derivations "better" but you don't state how. Perhaps better is in the eye of the beholder. Some older advanced quantum mechanics texts (the precurser to quantum field theory) use imaginary time instead of the Minkowski metric and few ever drone on about how ridiculous the notion of a metric that is not positive definite is and should never be taken seriously. However, I think this is no more ridiculous than imaginary time. Treating time and a purely imaginary number for the sake of preserving the metric and (1, 1, 1, 1) provides no real advantage as far as I can see, it sweeps the issues in dealing with space-time to a different spot under the rug so to speak. I suppose it make covariant and contravariant tensors equivalent so there is no need to track upper and lower indices but to call it easier or better is subjective. I have not seen a generalization of relativity or particle physics to curved space-times that use the imaginary time concept. That doesn't mean it is impossible but the Minkowski metric does generalize with little effort. SO the is the paradigm that has survived. In another direction altogether there are Finsler metric spaces, where the metric depends on the direction of travel within the space. Relative to Minkowski or its curved equivalent these are quite strange beasts. I think they were once looked at in GR, and are used in solid state physics and even ordinary acoustics.