From Spinor Geometry to Complex General Relativity

An attempt is made of giving a self-contained introduction to holomorphic ideas in general relativity, following work over the last thirty years by several authors. The main topics are complex manifolds, spinor and twistor methods, heaven spaces.

In complex general relativity, Lorentzian space-time is replaced by a four-complex-dimensional complex-Riemannian manifold, with holomorphic connection and holomorphic curvature tensor. A multisymplectic analysis shows that the Hamiltonian constraint is replaced by a geometric structure linear in the holomorphic multimomenta, providing some boundary conditions are imposed on two-complex-dimensional surfaces. On studying such boundary conditions, a link with the Penrose twisto...

A generalized theory unifying gravity with electromagnetism was proposed by Einstein in 1945. He considered a Hermitian metric on a real space-time. In this work we review Einstein's idea and generalize it further to consider gravity in a complex Hermitian space-time.

This document contains a description of physics entirely based on a geometric presentation: all of the theory is described giving only a pseudo-riemannian manifold (M, g) of dimension n > 5 for which the g tensor is, in studied domains, almost everywhere of signature (-, -, +, ..., +). No object is added to this space-time, no general principle is supposed. The properties we impose to some domains of (M, g) are only simple geometric constraints, essentially based on the conce...

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...

The proposal of this work is to provide an answer to the following question: is it possible to treat the metric of space-time - that in General Relativity (GR) describes the gravitational interaction - as an effective geometry? In other words, to obtain the dynamics of the metric tensor as a consequence of the dynamics of other fields. In this work we will use a slight modfication of the non-linear equation of motion of a spinor field proposed some years ago by Heisenberg, al...

In this paper we discuss the twistor equation in Lorentzian spin geometry. In particular, we explain the local conformal structure of Lorentzian manifolds, which admit twistor spinors inducing lightlike Dirac currents. Furthermore, we derive all local geometries with singularity free twistor spinors that occur up to dimension 7.

An alternative to the representation of complex relativity by self-dual complex 2-forms on the spacetime manifold is presented by assuming that that the bundle of real 2-forms is given an almost-complex structure. From this, one can define a complex orthogonal structure on the bundle of 2-forms, which results in a more direct representation of the complex orthogonal group in three complex dimensions. The geometrical foundations of general relativity are then presented in term...

The "Spinors" software is a "Mathematica" package which implements 2-component spinor calculus as devised by Penrose for General Relativity in dimension 3+1. The "Spinors" software is part of the "xAct" system, which is a collection of "Mathematica" packages to do tensor analysis by computer. In this paper we give a thorough description of "Spinors" and present practical examples of use.

We give a spinorial set of Hamiltonian variables for General Relativity in any dimension greater than 2. This approach involves a study of the algebraic properties of spinors in higher dimension, and of the elimination of second-class constraints from the Hamiltonian theory. In four dimensions, when restricted to the positive spin-bundle, these variables reduce to the standard Ashtekar variables. In higher dimensions, the theory can either be reduced to a spinorial version of...