Boundary Terms in Complex General Relativity

TL;DR

This paper explores boundary terms in complex general relativity, revealing conditions for Ricci-flat space-times, the role of holomorphic multimomenta, and connections to twistor theory, extending real GR boundary analyses to complex manifolds.

Contribution

It introduces a boundary term analysis in complex general relativity, linking boundary conditions to complex Ricci-flat spaces and twistor theory, with a novel geometric structure involving holomorphic multimomenta.

Findings

Boundary terms depend on imposed conditions on complex surfaces.

Holomorphic multimomenta should vanish on three-complex-dimensional surfaces.

A link between complex space-times and twistor theory is established.

Abstract

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-dimensional surfaces. One then finds that the holomorphic multimomenta should vanish on an arbitrary three-complex-dimensional surface, to avoid having restrictions at this surface on the spinor fields which express the invariance of the theory under holomorphic coordinate transformations. The Hamiltonian constraint of real general relativity is then replaced by a geometric structure linear in the holomorphic multimomenta, and a link with twistor theory is found. Moreover, a deep relation emerges between complex space-times which are not anti-self-dual and two-complex-dimensional surfaces which are not totally null.