The Newman-Penrose formalism in quadratic gravity

TL;DR

This paper reformulates quadratic gravity constraints using the Newman-Penrose formalism, turning the field equations into a linear algebraic system for Ricci components, aiding analysis of solutions.

Contribution

It introduces a Newman-Penrose-like framework for quadratic gravity, enabling algebraic manipulation of field equations and geometric restrictions based on curvature types.

Findings

Reformulation of quadratic gravity constraints in Newman-Penrose formalism.

Derivation of algebraic systems for Ricci tensor components.

Proof of geometric restrictions under specific curvature assumptions.

Abstract

The quadratic gravity constraints are reformulated in terms of the Newman-Penrose-like quantities. In such a frame language, the field equations represent a linear algebraic system for the Ricci tensor components. In principle, a procedure for the combination of the Ricci components with standard geometric identities can be applied in a similar way as in the case of general relativity. These results could serve in various subsequent analyses and physical interpretations of admitted solutions to quadratic gravity. Here, we demonstrate the utility of such an approach by explicitly proving general propositions restricting the spacetime geometry under assumptions on a specific algebraic type of curvature tensors.