I recently read in the footnotes of 'The Elegant Universe' by Brian Greene about the formula for proper time, defined as
$d\tau^2=dt^2-c^{-2}(dx_1^2+dx_2^2+dx_3^2)$.
I am new to the subject of Special Relativity, and I am not well-versed in the math behind it. From what I've understood, this is similar to the Pythagorean theorem. But where did the negative sign come from?
The minus sign stems from the fact that 4-dimensional space time is not a Euclidean space, thus the analog of the Pythagorean theorem does not have the property of positive definiteness, rather the Minkowski metric, as the distance function in space time is known, has a minus sign to distinguish intervals between points as time-like or space-like. It is, however, possible to Euclideanize the Minkowski interval by continuing to the complex numbers. For example, since the pure imaginary number $i$ is such that $i^2=-1$, one can replace the minus sign accordingly: $$d\tau^2=dt^2+i^2c^{-2}(dx^2+dy^2+dz^2).$$ Now it is a matter of convention as to whether one writes the metric as you have seen it, or as below: $$-cd\tau^2=-dt^2+c^{-2}(dx^2+dy^2+dz^2).$$ In the example from your post, an interval is space-like if $d\tau^2<0$, and time-like if $d\tau^2>0$. In the case above, the conditions are the opposite, thus one must take care to be consistent in one's adoption of convention when working problems in special relativity. Geometrically, the indefiniteness of the Minkowski metric is illustrated by the "light cone", which divides the space time continuum into distinct regions: future, past, time-like and space-like.
