  Wick rotation is usually performed by rotating the time coordinate to
imaginary values. In a general curved spacetime, the notion of a time
coordinate is ambiguous. We note here, that within the tetrad formalism of
general relativity, it is possible to perform a Wick rotation directly in the
tangent space using considerably less structure: a timelike, future pointing
vector field, which need not be Killing or hypersurface orthogonal. This method
has the advantage of yielding real Euclidean metrics, even in spacetimes which
are not static. When applied to a black hole exterior, the null generators of
the event horizon reduce to points in the Euclidean spacetime. Requiring that
the Wick rotated holonomy of the null generators be trivial ensures the absence
of a `conical singularity' in the Euclidean space. To illustrate the basic
idea, we use the tangent space Wick rotation to compute the Hawking temperature
by Euclidean methods in a few spacetimes including the Kerr black hole.
