[Paper Review] Exact solutions to quadratic gravity generated by a conformal method

Vojtěch Pravda, A. Pravdová+2 more|arXiv (Cornell University)|Jun 8, 2016

Cosmology and Gravitation Theories20 references3 citations

TL;DR

This paper presents a conformal method to construct exact vacuum solutions in quadratic gravity by solving a single nonlinear PDE for the conformal factor on Einstein spacetimes or backgrounds with vanishing Bach tensor. The key result is that all spacetimes conformal to Kundt are either Kundt or Robinson–Trautmann, and explicit solutions are derived by solving the conformal equation on specific Kundt backgrounds.

ABSTRACT

We study the role of conformal transformations in constructing vacuum solutions to quadratic gravity. We find that such solutions can be obtained by solving one non-linear partial differential equation for the conformal factor on any Einstein spacetime or, more generally, on any background with vanishing Bach tensor. We show that all spacetimes conformal to Kundt are either Kundt or Robinson--Trautmann, and we provide explicit Kundt and Robinson--Trautman solutions to quadratic gravity by solving the above mentioned equation on certain Kundt backgrounds.

Motivation & Objective

• To investigate the role of conformal transformations in generating vacuum solutions for quadratic gravity.
• To identify conditions under which conformal transformations preserve or classify spacetime types such as Kundt or Robinson–Trautmann.
• To develop a systematic method for constructing exact solutions by solving a single nonlinear PDE for the conformal factor.
• To provide explicit examples of Kundt and Robinson–Trautmann solutions in quadratic gravity using this conformal approach.

Proposed method

• Utilize conformal transformations to map known Einstein spacetimes with vanishing Bach tensor into solutions of quadratic gravity.
• Derive a nonlinear partial differential equation for the conformal factor that ensures the vacuum condition in quadratic gravity.
• Apply the method to Kundt spacetimes, which are characterized by a geodesic, shear-free null congruence.
• Analyze the conformal class of Kundt spacetimes to determine whether they remain Kundt or become Robinson–Trautmann solutions.
• Solve the conformal factor equation explicitly on selected Kundt backgrounds to generate new exact solutions.
• Verify that the resulting spacetimes satisfy the field equations of quadratic gravity in vacuum.

Experimental results

Research questions

• RQ1 Under what conditions does a conformal transformation of an Einstein spacetime yield a solution to quadratic gravity?
• RQ2 Which spacetimes conformal to Kundt spacetimes remain in the Kundt or Robinson–Trautmann class?
• RQ3 Can the conformal factor equation be solved explicitly on specific Kundt backgrounds to yield new vacuum solutions?
• RQ4 What is the role of the Bach tensor in enabling the conformal method for quadratic gravity?
• RQ5 How do the geometric properties of the background spacetime influence the structure of the resulting solutions?

Key findings

• All spacetimes conformal to Kundt spacetimes are either Kundt or Robinson–Trautmann, establishing a classification under conformal transformations.
• Vacuum solutions to quadratic gravity can be systematically constructed by solving a single nonlinear PDE for the conformal factor on any background with vanishing Bach tensor.
• Explicit Kundt and Robinson–Trautmann solutions to quadratic gravity are derived by solving the conformal factor equation on specific Kundt backgrounds.
• The method applies to any Einstein spacetime or more generally to any background with vanishing Bach tensor, broadening the scope of solvable cases.
• The conformal factor equation ensures that the resulting spacetime satisfies the vacuum field equations of quadratic gravity.
• The approach provides a unified framework for generating exact solutions in quadratic gravity using conformal geometry.