Sheaf Cohomology as the Mathematical Tool Necessary to Describe a Conformally Invariant Isomorphism. Twistors and Spinors Theoreticals. Note Quote.


The geometry of complex space-time in spinor form calculus is described in terms of spin-space formalism, i.e. a complex vector space endowed with a symplectic form and some fundamental isomorphisms. These mathematical properties enable one to raise and lower indices, define the conjugation of spinor fields in Lorentzian or Riemannian four-geometries, translate tensor fields into spinor fields (or the other way around). The standard two-spinor form of the Riemann curvature tensor is then obtained by relying on the (more) familiar tensor properties of the curvature.

Since the whole of twistor theory may be viewed as a holomorphic description of space-time geometry in a conformally invariant framework, the key results of conformal gravity, i.e. C-spaces, Einstein spaces and complex Einstein spaces necessitates a sufficient condition for a space-time to be conformal to a complex Einstein space.

On to twistor spaces, from the point of view of mathematical physics and relativity theory,  this is defined by twistors as α-planes in complexified compactified Minkowski space-time, and as α-surfaces in curved space-time. In the former case, one deals with totally null two-surfaces, in that the complexified Minkowski metric vanishes on any pair of null tangent vectors to the surface. Hence such null tangent vectors have the form λAπA′ , where λA is varying and πA′ is covariantly constant. This definition can be generalized to complex or real Riemannian four-manifolds, provided that the Weyl curvature is anti-self-dual. An alternative definition of twistors in Minkowski space-time is instead based on the vector space of solutions of a differential equation, which involves the symmetrized covariant derivative of an unprimed spinor field. Interestingly, a deep correspondence exists between flat space-time and twistor space. Hence complex space-time points correspond to spheres in the so-called projective twistor space, and this concept is carefully formulated. Sheaf cohomology can then be used as the mathematical tool necessary to describe a conformally invariant isomorphism between the complex vector space of holomorphic solutions of the wave equation on the forward tube of flat space-time, and the complex vector space of complex-analytic functions of three variables. These are arbitrary, in that they are not subject to any differential equation.

The generalization of Penrose’s non-linear graviton combines two-spinor techniques and twistor theory in a way, where it appears necessary to go beyond anti-self-dual space-times, since they are only a particular class of (complex) space-times, and they do not enable one to recover the full physical content of (complex) general relativity. This implies going beyond the original twistor theory, since the three-complex-dimensional space of α-surfaces only exists in anti-self-dual space-times. Roger Penrose defines twistors as charges for massless spin-3/2 fields. Such an approach has been considered since a vanishing Ricci tensor provides the consistency condition for the existence and propagation of massless helicity-3 fields in curved 2 space-time. Moreover, in Minkowski space-time the space of charges for such fields is naturally identified with the corresponding twistor space. The resulting geometric scheme in the presence of curvature is as follows. First, define a twistor for Ricci-flat space-time. Second, characterize the resulting twistor space. Third, reconstruct the original Ricci-flat space-time from such a twistor space. One of the main technical difficulties of the program proposed by Penrose is to obtain a global description of the space of potentials for massless spin-3/2 fields.

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