Spacetime interval

In three-dimensions, the distance between two points can be defined using the Pythagorean theorem:

${\left(\Delta{d}\right)}^2 = {\left(\Delta{x}\right)}^2 + {\left(\Delta{y}\right)}^2 + {\left(\Delta{z}\right)}^2$

In special relativity, however, the distance between two points is no longer the same if it measured by two different observers when one of the observers is moving, because of the Lorentz contraction.

But special relativity provides a new invariant, called the spacetime interval, which combines distances in space and in time. All observers who measure time and distance carefully will find the same spacetime interval between any two events.

$(\Delta s)^2 = (\Delta x)^2 + (\Delta y)^2 + (\Delta z)^2 + (\Delta cti)^2$

$\Delta cti$ has units of imaginary distance and its square is therefore a negative number.

References

This page incorporates text from Wikipedia:Spacetime