[Paper Review] Intersecting hypersurfaces and Lovelock Gravity

Steven Willison|ArXiv.org|Feb 22, 2005

Black Holes and Theoretical Physics86 references3 citations

TL;DR

This paper investigates intersecting hypersurfaces in Lovelock gravity, a higher-curvature extension of General Relativity. It demonstrates that thin shells with discontinuous first derivatives of the metric—mild curvature singularities—are classically admissible, enabling matter localization at intersections. This provides a classical analogue to intersecting brane-worlds in string phenomenology, a feature absent in pure Einstein-Hilbert gravity.

ABSTRACT

A theory of gravity in higher dimensions is considered. The usual Einstein-Hilbert action is supplemented with Lovelock terms, of higher order in the curvature tensor. These terms are important for the low energy action of string theories. The intersection of hypersurfaces is studied in the Lovelock theory. The study is restricted to hypersurfaces of co-dimension 1, $(d-1)$-dimensional submanifolds in a $d$-dimensional space-time. It is found that exact thin shells of matter are admissible, with a mild form of curvature singularity: the first derivative of the metric is discontinuous across the surface. Also, with only this mild kind of curvature singularity, there is a possibility of matter localised on the intersections. This gives a classical analogue of the intersecting brane-worlds in high energy String phenomenology. Such a possibility does not arise in the pure Einstein-Hilbert case.

Motivation & Objective

• To investigate the dynamics of intersecting hypersurfaces in higher-dimensional Lovelock gravity, a generalization of Einstein-Hilbert gravity with higher-order curvature terms.
• To determine whether thin shells with discontinuous metric derivatives can be consistently embedded in Lovelock theories, particularly at intersections.
• To explore the possibility of matter localization on the intersection of multiple hypersurfaces, motivated by brane-world scenarios in string theory.
• To develop a geometric and variational framework using differential forms, homotopy parameters, and dimensionally continued Euler densities to describe junctions and intersections.
• To establish that such intersections support consistent gravitational sources without strong singularities, unlike in standard General Relativity.

Proposed method

• Formulates the Lovelock action using orthonormal frames and differential forms, enabling a geometric treatment of curvature and connections.
• Applies the variational principle to Lovelock gravity, deriving junction conditions across hypersurfaces via boundary terms and induced connections.
• Introduces homotopy parameters and interpolating forms to describe the transition between bulk regions and to compute intersection actions.
• Uses dimensionally continued Euler densities and invariant polynomials to define topological invariants and curvature invariants at intersections.
• Constructs a non-simplicial intersection framework via dual lattices and chain integration, generalizing the standard simplex-based approach.
• Employs the exterior derivative and curvature 2-forms to derive the Bianchi identity and prove the closure of the homotopy form, ensuring consistency of the action.

Experimental results

Research questions

• RQ1 Can thin shells with discontinuous first derivatives of the metric be consistently embedded in Lovelock gravity?
• RQ2 Do intersections of hypersurfaces in Lovelock gravity allow for the localization of matter, even with mild curvature singularities?
• RQ3 How do junction conditions in Lovelock gravity differ from those in Einstein-Hilbert gravity, particularly at non-simplicial intersections?
• RQ4 What role do dimensionally continued Euler densities and homotopy parameters play in constructing consistent action terms at intersections?
• RQ5 Can the formalism describe colliding branes or three-way intersections in anti-de Sitter space with finite, well-defined deficit angles?

Key findings

• Exact thin shells with discontinuous first derivatives of the metric are admissible in Lovelock gravity, representing a mild form of curvature singularity.
• The absence of second normal derivatives of the metric in the intersection action implies that the theory remains well-defined despite the discontinuity.
• Matter can be localized at the intersection of multiple hypersurfaces due to the presence of curvature singularities that are consistent with Lovelock dynamics.
• The junction conditions derived from the Lovelock action allow for non-simplicial intersections through the use of homotopy parameters and dual lattices.
• In a three-way intersection in AdS space, the solution exhibits finite deficit angles and consistent energy conservation, confirming the stability of such configurations.
• The dimensionally continued Euler density formalism ensures topological invariance and consistency of the action across different regions and intersections.