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Name: Unknown
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Date: Around 1993


Question:
With regard fourth dimension, I wonder what an imaginary dimension of time in the equations of relativity has to do with reality? Does this dimension actually manifest itself in observable ways, if it is imaginary? If so, why call it imaginary?



Replies:
Stephen Hawking uses the term "imaginary time" in his book "A Brief History of Time". First of all, this is NOT a reference to some physical quantity different from "real" time, but another way of handling "real" time mathematically. As Hawking says on page 135, "...we may regard our use of imaginary time and Euclidean space-time as merely a mathematical device (or trick) to calculate answers about real space-time." There is an important quantity given by ds^2 = dx^2 + dy^2 + dz^2 - (c^2)*dt^2 where "^2" means "squared"; c is the speed of light; dx, dy, and dz are tiny changes in the three spatial directions; and ds is the resultant change in a sort of 4- dimensional "length". Suppose we define a new variable w by w=i*c*t where i is the square root of -1 (i is an example of an "imaginary" number. Then we get ds^2 = dx^2 + dy^2 + dz^2 + dw^2. Now, all four terms have the same form, and w (like x,y, and z) has units of length. One may think of w as "imaginary time", but the only thing that has changed is the mathematical form, not the underlying physics.

R.C. Winther



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