A Kahler manifold is essentially a complex manifold with a metric that may be written as,

$$g_{ij}=\frac{\partial^2}{\partial z^i \partial z^j}\mathcal{K}$$

where $\mathcal{K}$ is denoted the Kahler potential. The form of $g_{ij}$ ensures the Kahler form (which is a natural choice for a volume form) is closed. Kahler manifolds are a specific type of Riemannian manifold and hence are not extensions of the concept of a Riemannian manifold. As such, if we wanted to perform general relativity on a Kahler manifold, it would require no modification of the field equations,$^{\dagger}$

$$R_{ij}-\frac{1}{2}g_{ij}R = 8\pi T_{ij}$$

I would think it is not sensible to take a spacetime manifold to be a complex manifold, as in reality we are 'living' on real pseudo-Riemannian manifolds. However, sometimes it is convenient to use complex coordinates. For example, in string theory, we define new complex wordsheet coordinates,

$$z=\tau+i\sigma \quad \bar{z}=\tau-i\sigma$$

which are Euclidean analogues of lightcone coordinates. If we consider $z$ and $\bar{z}$ as separate independent variables, then we are going from $\mathbb{R}^2$ to $\mathbb{C}^2$. However, at the end, we need to keep in mind we are on the real slice which is a subset, i.e. $\mathbb{R^2}\subset \mathbb{C}^2$. I guess one could apply a similar procedure to a solution to the Einstein field equations, and in that sense we are, in a manner, temporarily extending ourselves to a complex manifold.

$\dagger$ In units where $c=G=1$.

This post imported from StackExchange Physics at 2016-01-07 09:54 (UTC), posted by SE-user JamalS