Why do we call the time dimension 'imaginary'?

[circa 1991]

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?
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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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