# Is imaginary time a fifth dimension? [duplicate]

This question already has an answer here:

I've read that by introducing the concept of imaginary time, the dimension of time can be treated like a spatial dimension mathematically. Assuming, without imaginary time, one considers the universe 4-dimensional (3 spatial dimensions and 1 time), does imaginary time make for a 5th dimension? I.e. is imaginary time an additional axis orthogonal to all 4 dimensions of spacetime?

I can't find any resources relating imaginary time to the total number of dimensions in the universe, only vague summaries stating that imaginary time makes time 2-dimensional. But I can't help but wonder if they mean 2-dimensional in the same way that space is 3-dimensional, or something more subtle than that.

I'm also aware that simply adding an axis to a Euclidean space doesn't necessarily increase the number of dimensions, e.g. drawing 3 non-parallel lines on a plane doesn't change the fact that it's a 2-dimensional plane. I can't help but wonder if this is a way to describe imaginary time: a useful number that makes calculations easier, but isn't actually orthogonal to the 4 other dimensions we're intuitively aware of.

Edit:

The linked QnA this question is marked as a duplicate of does not specifically answer whether or not imaginary time is an additional dimension beyond spacetime. Instead, it simply defines imaginary time.

If you understand the definition, you know that it's not, in-fact, an additional dimension, but just another way of looking at the regular time dimension we're used to working with.

## marked as duplicate by David Z♦Dec 19 '14 at 22:41

This question has been asked before and already has an answer. If those answers do not fully address your question, please ask a new question.

• "Imaginary" time is an archaic way to denote the 4th dimension, it is no longer an accepted practice and only found in older books. – Kyle Kanos Dec 19 '14 at 20:01
• @KyleKanos en.wikipedia.org/wiki/Euclidean_quantum_gravity en.wikipedia.org/wiki/Imaginary_time All i know, solely from Hawking's "My Brief History", is that imaginary time (in a sense beyond special relativity) is related to Euclidean quantum gravity. – user12029 Dec 19 '14 at 20:08
• @NeuroFuzzy: I recall people used to use $x^\mu=(it,\,x,\,y,\,z)$ because it lead to $x_\mu x^\mu =-t^2+x^2+y^2+z^2$, this caused people to view it as "imaginary time." Perhaps, though, I am incorrect and am conflating two different things. – Kyle Kanos Dec 19 '14 at 20:10
• Yes, I'm referring to the imaginary time Hawking mentions in A Brief History of Time. – Jo Bates Dec 19 '14 at 20:15
• @JoBates: You may be interested in reading More than one time dimension – Kyle Kanos Dec 19 '14 at 20:25

## 1 Answer

It's a trick to get the negative sign in the Pythagorean measure of distance. "Stick an $i$ here" vs "use a negative sign here" in the rules of how to figure things.

It's not another different dimension nor is time a complex value. It's an x i t in the function of the interval, where $x$ is a real number.

Of the several posts nominated as duplicates of this question, this one is indeed the same question, with good responses. It appears that two very different questions get conflated in the duplicate marking of both of them, because Hawking introduces Minkowsky spacetime in his popular book, and he is the author of a paper on the big bang singularity using complex numbers to turn the point into a smooth cup. I propose that further duplicate marking point to a disambiguation page, when it's not clear which question is being asked.

• After some more reading inspired by your answer and comments on my question, I think I'm starting to understand this. However, this answer needs a lot more explanation before I accept it. – Jo Bates Dec 19 '14 at 21:36
• The comment by @kylekanos shows the math I alude to in the second paragraph. I invite him to edit this answer to paste that here. (TeX on tablet is not easy) – JDługosz Dec 20 '14 at 5:14
• @JoBates you could still upvote it :) since it has been done before I don't want to repeat it here, but point to the existing post (edited) that is specifially what you were looking for. – JDługosz Dec 20 '14 at 5:27
• Thanks. "It's not another different dimension nor is time a complex value" is the real answer to my question, and the duplicate question/answer give the details. Maybe in your answer here, it'd be best if you just included the first two paragraphs and said "See the meaning of imaginary time for details." – Jo Bates Dec 21 '14 at 23:15
• @JoBates I brought up the confusion on the meta site. – JDługosz Dec 21 '14 at 23:25