[Paper Review] The gravitational content of lorentzian complex structures

Ch. Ragiadakos|arXiv (Cornell University)|Oct 6, 2011

Cosmology and Gravitation Theories7 references3 citations

TL;DR

This paper proposes a renormalizable 4D generally covariant model where gravity is encoded in Lorentzian complex structures rather than the spacetime metric. By identifying spacetime as a codimension-4 CR manifold within the $G_{2,2}$ Grassmannian and using Chern-Moser and Cartan methods, the model derives a Fefferman-like metric and links open CR manifolds at the $U(2)$ boundary to Poincaré group representations, suggesting a geometric origin for the particle spectrum and a path toward defining positive energy in a metric-independent quantum gravity framework.

ABSTRACT

The definition of a positive energy is investigated in a renormalizable 4-dimensional generally covariant model, which depends on the lorentzian complex structure and not the metric of spacetime. The gravitational content of the lorentzian complex structures is revealed by identifying the spacetime with special 4-dimensional surfaces of the G{2,2} Grassmannian manifold. The lorentzian complex structure is found to be a codimension-4 CR structure and its classification is studied using the Chern-Moser and Cartan methods. The spacetime metric is found to be a Fefferman-like metric of this codimension-4 CR structure. The open CR manifolds "hanging" from the points of the U(2) characteristic boundary of the SU(2,2) classical domain belong into representations of the Poincaré group and are related to the particle spectrum of the model.

Motivation & Objective

• To define a positive energy in a renormalizable 4D generally covariant model that depends on Lorentzian complex structures rather than the spacetime metric.
• To reveal the gravitational content of Lorentzian complex structures by embedding spacetime as a 4-dimensional surface in the $G_{2,2}$ Grassmannian manifold.
• To classify Lorentzian complex structures using Chern-Moser normal forms and Cartan connection methods.
• To establish a Fefferman-like metric on codimension-4 CR structures arising from these complex structures.
• To connect open CR manifolds at the $U(2)$ boundary of the $SU(2,2)$ classical domain to representations of the Poincaré group, linking geometry to particle physics.

Proposed method

• The model uses a Yang-Mills-like action (1.1) depending on the Lorentzian complex structure $J_{\mu}^\nu$ and gauge fields $A_{j\mu}$, with a metric-independent Lagrangian enforced via Lagrange multipliers (1.6).
• Spacetime is embedded as a 4-dimensional surface in the $G_{2,2}$ Grassmannian manifold, identified as a codimension-4 CR structure via the null tetrad formalism (1.2) and integrability conditions (1.3).
• The Chern-Moser method is applied to classify Lorentzian complex structures by reducing the structure equations to normal forms under holomorphic transformations.
• The Cartan connection method is used to analyze the geometry of the CR structure, providing a framework for curvature and symmetry analysis.
• A Fefferman-like metric (5.11) is derived on the CR manifold, with the Minkowski metric recovered on the real slice $y^a = 0$, linking the geometry to physical spacetime.
• The $SU(2,2)$ classical domain is used to realize the $U(2)$ boundary, where open surfaces 'hanging' from a point correspond to Poincaré group representations via isotropy subgroups.

Experimental results

Research questions

• RQ1 How can a positive energy be defined in a metric-independent, renormalizable 4D quantum gravity model based on Lorentzian complex structures?
• RQ2 What is the geometric and algebraic classification of Lorentzian complex structures using differential-geometric tools like Chern-Moser and Cartan methods?
• RQ3 How does the spacetime metric emerge as a Fefferman-like metric on a codimension-4 CR structure derived from the complex structure?
• RQ4 What is the role of the $U(2)$ boundary of the $SU(2,2)$ classical domain in realizing Poincaré symmetry and particle representations?
• RQ5 How do open CR manifolds at the $U(2)$ boundary relate to the physical particle spectrum in the model?

Key findings

• The Lorentzian complex structure is identified as a codimension-4 CR structure embedded in the $G_{2,2}$ Grassmannian, with integrability conditions equivalent to vanishing spin coefficients $\kappa, \sigma, \lambda, \nu$.
• The spacetime metric is realized as a Fefferman-like metric on the CR structure, with the Minkowski metric recovered on the real slice $y^a = 0$ via the metric (5.11).
• The $U(2)$ characteristic boundary of the $SU(2,2)$ classical domain supports open CR manifolds that are isomorphic to Poincaré group representations, with the isotropy subgroup of the $z=I$ point being the $Poincar\acute{\text{e}}\times$ Dilation group.
• The $SU(2,2)$ symmetry is spontaneously broken to Poincaré symmetry when the vacuum is taken as the Minkowski part of the Shilov boundary with a fixed $i^0$ point.
• The $U(2)$ group acts as a global symmetry that maps Poincaré representations from one boundary point to another, preserving isomorphism of isotropy subgroups.
• The model suggests that vector and scalar (Higgs) fields arise from $U(2)$-broken modes of Minkowski-like spacetimes, while Kerr-Newman-type solitonic sectors may correspond to the electronic multiplet of the Standard Model.