General Relativity via differential forms -- explorations in Plebanski's Formalism for GR

Abstract: This thesis studies general relativity (GR) using chiral formulations, which take advantage of the decomposition of the four-dimensional Lorentz group into self-dual and anti-self-dual sectors. Within this framework, GR can be expressed using Plebanski's formulation, where the basic variables are triples of 2-forms rather than a metric, or alternatively through pure connection approaches. These viewpoints expose additional structure in Einstein's equations (EEs) and offer new analytical and numerical tools. Part I develops the geometric foundations using fibre bundles, where the 2-forms arise as soldering forms on an SO(3,C) bundle. Part II investigates the linearised form of EEs in the chiral setting, with particular attention to their gauge fixings. Part III extends this analysis to the nonlinear regime, and also examines the complex-geometric structure underlying black hole spacetimes. The final part turns to numerical relativity, exploring evolution schemes built from the chiral formulations and their associated gauge choices.

• The paper presents a chiral reformulation of GR that replaces the metric with self-dual 2-forms, enabling novel analytic and numerical techniques.
• It develops an elliptic complex structure and hyperbolic gauge fixings that ensure well-posed evolution and effective constraint control.
• The work bridges Yang-Mills theory and GR, offering practical schemes for simulating high-symmetry spacetimes like Type D metrics.

General Relativity via Differential Forms: Explorations in Plebanski's Formalism

Introduction and Context

This thesis presents a comprehensive study of general relativity (GR) using the language of differential forms, with specific focus on Plebanski's chiral formalism. The motivation arises from the algebraic and geometric structures manifest in four-dimensional spacetime, particularly the decomposition of the Lorentz group and the resulting split of the space of 2-forms into self-dual and anti-self-dual components. Encoding GR in one of these chiral halves—where the fundamental variables are not the metric, but a triple of 2-forms—enables a reformulation of the Einstein equations with distinct analytical, algebraic, and computational advantages.

The work contextualizes this reformulation within the broader setting of principal bundles and gauge theories, drawing explicit parallels between Yang-Mills (YM) fields and gravity to illuminate the unifying geometric framework underlying both. By implementing both linear and nonlinear perturbative analyses, as well as exploring numerical evolution schemes, the thesis contributes both to the foundational understanding and practical computation of gravitational physics in this formalism.

Mathematical Construction and Chiral Formulations

A central theme is the recasting of 4d GR into a framework where the basic fields are chiral (self-dual) 2-forms, $\Sigma^i$, on an $SO(3, \mathbb{C})$ principal bundle. Metrics are derived, not fundamental, objects—constructed algebraically from these 2-forms via Urbantke's formula. The soldering forms and associated torsion-free connections are developed systematically, making clear the analogy with principal bundle connections in YM theory.

The thesis scrutinizes both the linear and nonlinear structure of the Einstein equations when expressed in terms of these chiral fields. Theoretical insights include:

• The identification of an elliptic complex structure in the linearized equations about self-dual backgrounds, paving the way for a cohomological analysis analogous to that of instantons in Yang-Mills theory.
• Twisting operators and adjoint structure, which facilitate the decomposition of the linearized system into two truncated de Rham complexes, with associated Dirac operators that square to Laplacians.
• Elucidation of a two-parameter family of hyperbolic gauge fixings—with explicit partial differential equations (PDEs) whose principal part is the flat-space wave operator, assuring well-posedness for time evolution.

In addition to the self-contained construction of the chiral formalism, the work explores its relation to pure connection (Krasnov’s) formulations and to the Ashtekar variables, enhancing the bridge between classical and quantum representations of gravity.

Nonlinear Structure and Sector Decomposition

In the fully nonlinear regime, the thesis develops a nonlinear generalization of the linear two-parameter gauge-fixing scheme. The pivotal results include:

• Nonlinear analogs of the twisting operators that separate the field equations into two interacting sectors, of size 4 and 12. At linear order, these sectors are independent; nonlinearly, the smaller sector depends on the larger, a structure that enables partial decoupling and iterative solution strategies.
• Detailed analysis of solutions with high symmetry, especially Type D metrics (including the full Plebanski-Demiański family), exploiting the fact that in four dimensions, these spacetimes are conformal to pairs of Kähler geometries. The thesis provides an analytic method to reconstruct these solutions purely in the chiral language, bypassing the need to refer back to metric or spinor calculi.

Numerical Implications and Evolution Systems

Recognizing the computational import of robust, well-posed formulations, the thesis applies chiral methods to the construction of evolution systems for numerical relativity. Highlights include:

• The exploitation of the chiral hyperbolic gauge to produce evolution systems for the field variables with good numerical properties—most notably, systems in which constraints propagate naturally or are actively damped via the structure of the evolution equations.
• Comparative development of three evolutionary schemes: one in (complexified) Ashtekar variables, one using conformally separated variables (reflecting the conformal/numeric decoupling of sectors), and one “partially conformal” system blending both approaches.
• Basic numerical tests demonstrating consistency and stability, with emphasis placed on the propagation and control of constraints.
• Extension to pure connection formalisms, including the development of new hyperbolic gauge conditions to guide numerical evolution in the absence of an explicit metric.

Implications and Future Directions

Theoretical Implications: The thesis strengthens the case for chiral and self-dual variables as not only elegant but also pragmatically effective for both analytic and numerical work in 4d gravity. The elliptic complex structure and associated Dirac/spectral theory open avenues for rigorous index-theoretic analysis of gravitational moduli spaces, akin to the successes in Yang-Mills and Seiberg–Witten theory (2604.20772). These results suggest possible new invariants and localization principles for GR in four dimensions, especially in the context of Euclidean, self-dual, or conformally Kähler geometries.

Practical and Numerical Impact: The hyperbolic gauge fixings and sectoral separations provide a blueprint for the construction of stable, constraint-preserving numerical schemes for gravitational evolution. The explicit representation of constraints and their evolution, combined with the decomposition into sub-sectors, has potential to enhance efficiency and stability, especially in simulations of highly symmetric or near-Kähler gravitational systems.

Speculative Directions:

• The precise control over decompositions (self-dual/anti-self-dual, conformal/tracefree) may enable new approaches to perturbative or non-perturbative quantization, especially for theories based on metric-affine or connection variables.
• Modifications of Plebanski’s constraints yield a space of “chiral deformations” of GR, whose study may reveal alternative gravitational dynamics with the same local degrees of freedom but possibly different global or quantum properties.

Conclusion

This thesis offers an in-depth, technically rigorous development of four-dimensional general relativity via chiral differential forms, grounded in Plebanski’s formalism. By fully exploiting the algebraic and geometric structures inherent to self-dual 2-forms and their associated principal connections, it advances both the analytic and computational toolkit available for both classical and quantum gravitational studies. The identification of elliptic complexes, hyperbolic evolution systems, and sector decompositions, as well as the concrete analytic and numerical procedures detailed herein, establish Plebanski’s chiral framework not only as an alternative language for GR, but as a formalism with direct calculational and conceptual benefits.

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Reference: "General Relativity via differential forms -- explorations in Plebanski's Formalism for GR" (2604.20772)