From Spinor Geometry to Complex General Relativity

We present a comprehensive introduction to spacetime algebra that emphasizes its practicality and power as a tool for the study of electromagnetism. We carefully develop this natural (Clifford) algebra of the Minkowski spacetime geometry, with a particular focus on its intrinsic (and often overlooked) complex structure. Notably, the scalar imaginary that appears throughout the electromagnetic theory properly corresponds to the unit 4-volume of spacetime itself, and thus has p...

The possibility of a complex 4-dimensional space-time manifold is suggested. This may imply the existence of a matter wave.

We expect the final theory of gravity to have more symmetries than we suspect and our research points in this direction, to start with, standard general coordinate invariance can be extended to complex holomorphic general coordinate transformations This is possible by introducing a non Riemannian Measure of integration (NRMI) and where we avoid the non holomorphic standard $\sqrt{-g}$. measure of integration. Second, although globally signed transformations produce a change o...

In this paper, a complex daor field which can be regarded as the square root of space-time metric is proposed to represent gravity. The locally complexified geometry is set up, and the complex spin connection constructs a bridge between gravity and SU(1,3) gauge field. Daor field equations in empty space are acquired, which are one-order differential equations and not conflict with Einstein's gravity theory.

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...

In order to ask for future concepts of relativity, one has to build upon the original concepts instead of the nowadays common formalism only, and as such recall and reconsider some of its roots in geometry. So in order to discuss 3-space and dynamics, we recall briefly Minkowski's approach in 1910 implementing the nowadays commonly used 4-vector calculus and related tensorial representations as well as Klein's 1910 paper on the geometry of the Lorentz group. To include micros...

I withdraw the previous version of the paper since it contains conceptual and mathematical mistakes. I will soon replace it with a radically revised version.

The following are expanded lecture notes for the course of eight one hour lectures given by the second author at the 2014 summer school Asymptotic Analysis in General Relativity held in Grenoble by the Institut Fourier. The first four lectures deal with conformal geometry and the conformal tractor calculus, taking as primary motivation the search for conformally invariant tensors and diffrerential operators. The final four lectures apply the conformal tractor calculus to the ...

Coupling spinor fields to the gravitational field, in the setting of general relativity, is standardly done via the introduction of a vierbein field and the (associated minimal) spin connection field. This makes three types of indices feature in the formalism: world/coordinate indices, Lorentz vector indices, and Lorentz spinor indices, respectively. This article will show, though, that it is possible to dispense altogther with the Lorentz indices, both tensorial ones and spi...

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...