Imaginary Time: Stephen Hawking's Favorite Physics Concept Relativity by Parth G

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When asked about the one thing Hawking wished everyone knew about his work, he replied "Imaginary Time". #imaginary time #stephenhawking #physics

Hi everyone, in this video I wanted to discuss the basics of the concepts behind Imaginary Time. As some of you will have guessed, it has nothing to do with imagination, and everything to do with imaginary numbers.

In this video, we briefly discuss the definition of the imaginary number i, sometimes called j by engineers. It represents the square root of -1, and imaginary numbers can be represented on an axis perpendicular to the real number line. Together, the real and imaginary axes form what is known as an Argand diagram.

Imaginary time consists of a simple substitution where any mention of time in our mathematics is replaced with the imaginary number i, multiplied by a new quantity being defined, which we will now call imaginary time (tau).

Firstly, it's worth mentioning that this kind of substitution, where a real quantity is replaced with an imaginary one, is known as a Wick rotation after Gian Carlo Wick. The reason it's known as a rotation is because in going from a real value to an imaginary one or vice versa, we rotate from one axis to another on the argand diagram.

Wick rotations, especially imaginary time, are useful in relativity as a mathematical tool. In this video we see how using imaginary time can take us from a complex Minkowski spacetime (time has the opposite sign to spatial components) and bring us to more familiar mathematics where all components have the same sign. This makes many problems easier to solve mathematically, after which we can revert the imaginary time substitution.

But Stephen Hawking took the idea of imaginary time further than just a mathematical convenience. He hypothesized that in the early universe, time could take both real and imaginary values, and that at some point the imaginary time dimension was lost (which is why we only experience time running in one direction now). The reason he hypothesized this was because having an extra time dimension would allow us to deal with infinities in our mathematics much more easily due to the extra degree of freedom.

Specifically, most of our current theories suggest that the universe was initially densely packed into an infinitely small volume, meaning its density was infinite at the Big Bang. This is problematic for mathematics, and using imaginary time solved this issue.

Unfortunately, this was not testable according to Hawking, since we have no way of knowing if an extra time dimension existed. Additionally, other scientists have found it difficult to align the idea of imaginary time with other observations we can make of our universe today.

Here are some resources you may like:
Stephen Hawking Interview -    • Stephen Hawking Extended Interview: L...  
My Relativity Playlist:    • Relativity by Parth G  
Relativistic Interval: https://en.wikipedia.org/wiki/Special...

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