[Paper Review] A note on Dirac spinors in a non-flat space-time of general relativity

Руслан Шарипов|ArXiv.org|Jan 11, 2006

Algebraic and Geometric Analysis5 references3 citations

TL;DR

This paper provides a geometric and algebraic framework for Dirac spinors in non-flat space-times by rigorously linking the SL(2,ℂ) group homomorphism to the Lorentz group SO⁺(1,3,ℝ). It establishes that Dirac γ-symbols transform as a spin-tensorial field of type (1,1|0,0|0,1), deriving their transformation laws under frame changes and clarifying the role of P and T symmetries as frame transformations rather than global operators in general relativity.

ABSTRACT

Some aspects of Dirac spinors are resumed and studied in order to interpret mathematically the P and T operations in a gravitational field.

Motivation & Objective

• To establish a geometric and algebraic foundation for two-component spinors in general relativity using the SL(2,ℂ) → SO⁺(1,3,ℝ) group homomorphism.
• To define and characterize the transformation properties of Dirac γ-symbols under changes of orthonormal frames in a space-time manifold.
• To clarify the interpretation of P and T symmetries in general relativity, treating them as frame transformations rather than global physical operations.
• To extend the scope of γ-symbol transformation laws to include chiral and orthonormal frame pairs via matrix homomorphisms and reflection operators.

Proposed method

• Utilizes the standard group homomorphism φ: SL(2,ℂ) → SO⁺(1,3,ℝ) to relate 2×2 complex matrices (spinors) to 4×4 Lorentz matrices (spacetime vectors).
• Defines moving frames on a 4-dimensional space-time manifold M with a Lorentzian metric and polarization, ensuring orthochronous and right-handed orientation.
• Introduces a complex vector bundle SM with a non-degenerate skew-symmetric spin-metric tensor d, enabling the definition of spinor structures.
• Derives transformation laws for γ-symbols under frame changes using transition matrices S and T, which are inverse Lorentz transformations from the homomorphism.
• Applies matrix representations of P and T operators (as -I matrix) to derive transformation rules for γ-symbols under parity and time reversal in the spinor formalism.
• Establishes that γ-symbols transform as a spin-tensorial field of type (1,1|0,0|0,1) via explicit component formulas (7.3–7.6) and verifies consistency through generalized transformation laws (7.7–7.9).

Experimental results

Research questions

• RQ1 How do Dirac γ-symbols transform under changes of orthonormal frames in a non-flat space-time manifold?
• RQ2 What is the precise geometric and algebraic structure of the γ-symbols when derived from the SL(2,ℂ) → SO⁺(1,3,ℝ) homomorphism?
• RQ3 How should parity (P) and time-reversal (T) symmetries be interpreted in general relativity, given the fixed metric and manifold?
• RQ4 What is the tensorial type and transformation behavior of the γ-symbols under spinor and frame transformations?
• RQ5 Can the transformation laws of γ-symbols be generalized beyond orthonormal frames using matrix homomorphisms and reflection operators?

Key findings

• The γ-symbols are shown to be components of a spin-tensorial field of type (1,1|0,0|0,1), as formalized in Theorem 7.1.
• Transformation laws for γ-symbols are derived via matrix homomorphisms, with equations (7.7) and (7.8) generalizing their behavior under frame changes.
• The matrix Q = -I is identified as the product of P and T reflections, leading to the antisymmetrized transformation law (7.9) for γ-symbols.
• The formulas (7.3)–(7.6) define γ-symbols in terms of orthonormal chiral frames and their associated spacetime frames, valid across the diagram 6.19.
• P and T symmetries are interpreted not as global physical operators but as frame transformations in general relativity, due to the fixed metric and manifold structure.
• The framework consistently extends the transformation rules of γ-symbols to include chiral and orthonormal frame pairs, ensuring geometric coherence in curved spacetime.