nLab
Wick rotation

Idea

Wick rotation is a method for finding a solution to a problem in Minkowski space from the solution to a related problem in Euclidean space. It is motivated by the observation that the Minkowski metric (with the $-1,1,1,1$ convention) and the four-dimensional Euclidean metric are equivalent if the time components of either are allowed to have imaginary values.

Example

Consider the Minkowski metric with the $-1,1,1,1$ convention for the tensor:

$d{s}^2=-(dt{)}^2+(dx{)}^2+(dy{)}^2+(dz{)}^2$

and the four-dimensional Euclidean metric:

$d{s}^2=d{\tau }^2+(dx{)}^2+(dy{)}^2+(dz{)}^2$.

Notice that if $dt=i\cdot d\tau$, the two are equivalent.

Method

A typical method for employing Wick rotation would be to make the substitution $t=i\tau$ in a problem in Minkowski space. The resulting problem is in Euclidean space and is sometimes easier to solve, after which a reverse substitution can (sometimes) be performed, yielding a solution to the original problem.

Technically, this works for any four-vector comparison between Minkowski space and Euclidean space, not just for space-time intervals.