Complex Geometry of Nature and General Relativity

A review of selected topics in mathematical general relativity

A recent analysis of real general relativity based on multisymplectic techniques has shown that boundary terms may occur in the constraint equations, unless some boundary conditions are imposed. This paper studies the corresponding form of such boundary terms in complex general relativity, where space-time is a four-complex-dimensional complex-Riemannian manifold. A complex Ricci-flat space-time is recovered providing some boundary conditions are imposed on two-complex-dimens...

We formulate holomorphic theory of gravity and study how the holomorphy symmetry alters the two most important singular solutions of general relativity: black holes and cosmology. We show that typical observers (freely) falling into a holomorphic black hole do not encounter a curvature singularity. Likewise, typical observers do not experience Big Bang singularity. Unlike Hermitian gravity \cite{MantzHermitianGravity}, Holomorphic gravity does not respect the reciprocity symm...

We provide an introduction to selected recent advances in the mathematical understanding of Einstein's theory of gravitation.

Complexifying space time has many interesting applications, from the construction of higher dimensional unification, to provide a useful framework for quantum gravity and to better define some local symmetries that suffer singularities in real space time. In this context here spacetime is extended to complex spacetime and standard general coordinate invariance is also extended to complex holomorphic general coordinate transformations. This is possible by introducing a non Rie...

These are the lecture notes from the 26th Winter School "Geometry and Physics", Czech Republic, Srni, January 14 - 21, 2006. These lectures are an introduction into the realm of generalized geometry based on the tangent plus the cotangent bundle. In particular we discuss the relation of this geometry to physics, namely to two-dimensional field theories. We explain in detail the relation between generalized complex geometry and supersymmetry. We briefly review the generalized ...

These notes represent approximately one semester's worth of lectures on introductory general relativity for beginning graduate students in physics. Topics include manifolds, Riemannian geometry, Einstein's equations, and three applications: gravitational radiation, black holes, and cosmology.

For various reasons, it seems necessary to include complex saddle points in the "Euclidean" path integral of General Relativity. But some sort of restriction on the allowed complex saddle points is needed to avoid various unphysical examples. In this article, a speculative proposal is made concerning a possible restriction on the allowed saddle points in the gravitational path integral. The proposal is motivated by recent work of Kontsevich and Segal on complex metrics in qua...

This paper summarises oral contributions to the parallel session {\it Complex and conformal methods in classical and quantum gravity} which took place during the 20th International Conference on General Relativity and Gravitation held in Warsaw in July 2013.

We present a brief overview of some key concepts in the theory of generalised complex manifolds. This new geometry interpolates, so to speak, between symplectic geometry and complex geometry. As such it provides an ideal framework to analyse thermodynamical fluctuation theory in the presence of gravitational fields. To illustrate the usefulness of generalised complex geometry, we examine a simplified version of the Unruh effect: the thermalising effect of gravitational fields...