[Paper Review] Exact Solutions of General Relativity and Quadratic Gravity in Arbitrary Dimension

Tomáš Málek|arXiv (Cornell University)|Apr 2, 2012

Black Holes and Theoretical Physics99 references6 citations

TL;DR

This thesis presents exact solutions in higher-dimensional general relativity and quadratic gravity using generalized Kerr–Schild (xKS) metrics and the Brinkmann warp product. It establishes algebraic classifications of Weyl tensors in higher dimensions, derives conditions for vacuum solutions in quadratic gravity, and identifies that only specific Weyl types (N, III(B)) with an effective cosmological constant satisfy the field equations, while null radiation terms require Kundt-class geometry for consistency.

ABSTRACT

In the first part of this thesis, Kerr-Schild metrics and extended Kerr-Schild metrics are analyzed in the context of higher dimensional general relativity. Employing the higher dimensional generalizations of the Newman-Penrose formalism and the algebraic classification of spacetimes based on the existence and multiplicity of Weyl aligned null directions, we establish various geometrical properties of the Kerr-Schild congruences, determine compatible Weyl types and in the expanding case discuss the presence of curvature singularities. We also present known exact solutions admitting these Kerr-Schild forms and construct some new ones using the Brinkmann warp product. In the second part, the influence of quantum corrections consisting of quadratic curvature invariants on the Einstein-Hilbert action is considered and exact vacuum solutions of these quadratic gravities are studied in arbitrary dimension. We investigate classes of Einstein spacetimes and spacetimes with a null radiation term in the Ricci tensor satisfying the vacuum field equations of quadratic gravity and provide examples of these metrics.

Motivation & Objective

• To extend the Kerr–Schild formalism to higher-dimensional spacetimes and analyze its geometric and algebraic properties.
• To generalize the Newman–Penrose formalism and Weyl tensor algebraic classification to arbitrary dimensions for studying curvature structures.
• To investigate the impact of quadratic curvature invariants on the Einstein–Hilbert action and derive exact vacuum solutions in quadratic gravity.
• To identify which Einstein spacetimes and spacetimes with null radiation satisfy the field equations of quadratic gravity in arbitrary dimensions.
• To construct new exact solutions using the Brinkmann warp product and verify their compatibility with quadratic gravity constraints.

Proposed method

• Employing higher-dimensional generalizations of the Newman–Penrose formalism to analyze null congruences and Weyl tensor algebraic types in Kerr–Schild spacetimes.
• Using algebraic classification based on Weyl aligned null directions (WANDs) to determine compatible Weyl tensor types in higher dimensions.
• Applying the Brinkmann warp product construction to generate new exact solutions from known four-dimensional solutions.
• Deriving field equations for quadratic gravity by adding Gauss–Bonnet and other quadratic curvature invariants to the Einstein–Hilbert action.
• Imposing the optical constraint and analyzing r-dependence of the Weyl tensor to detect curvature singularities in expanding xKS spacetimes.
• Using the Ricci tensor with aligned null radiation to classify solutions and derive constraints for compatibility with quadratic gravity.

Experimental results

Research questions

• RQ1 Which higher-dimensional spacetimes admit a generalized Kerr–Schild form, and what are their geometric and algebraic properties?
• RQ2 How do the algebraic types of the Weyl tensor in higher dimensions constrain the structure of Kerr–Schild and extended Kerr–Schild spacetimes?
• RQ3 Under what conditions do Einstein spacetimes in higher dimensions satisfy the vacuum field equations of quadratic gravity?
• RQ4 What role does the null radiation term in the Ricci tensor play in determining the class of solutions in quadratic gravity?
• RQ5 Can the Brinkmann warp product construction generate new exact solutions that are consistent with quadratic gravity field equations?

Key findings

• Only Einstein spacetimes of Weyl type N or III(B) with an effective cosmological constant Λ determined by B=0 satisfy the vacuum field equations of quadratic gravity.
• Type III(A) Einstein spacetimes, including Ricci-flat pp-waves, do not solve the field equations of quadratic gravity unless a null radiation term is present.
• Spacetimes with aligned null radiation in the Ricci tensor must belong to the Kundt class to be solutions of quadratic gravity, with the constraint B=0 determining the effective cosmological constant.
• The charged rotating CCLP black hole in five-dimensional minimal gauged supergravity is of Weyl type I_i, while the uncharged Myers–Perry black hole reduces to a generalized Kerr–Schild form and is of type D.
• Explicit solutions of quadratic gravity with Λ=0 and Λ≠0 are constructed, showing that the equations for the null radiation term Φ and the vacuum Kerr–Schild function H_vac can be decoupled in non-critical cases.
• Solutions of type III(A) with null radiation in the Ricci tensor may exist in quadratic gravity, unlike their Einstein counterparts, indicating a non-trivial extension of the solution space.