# Contents

## 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:

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

and the four-dimensional Euclidean metric:

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

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.

Revised on April 2, 2010 20:05:16 by Toby Bartels (98.19.61.80)