Antisymmetrised 2p-forms generalising curvature 2-forms and a corresponding p-hierarchy of Schwarzschild type metrics in dimensions d>2p+1

TL;DR

This paper introduces a recursive construction of antisymmetrised 2p-forms generalising curvature forms, leading to new Schwarzschild-like metrics in higher dimensions and exploring their topological and instanton properties.

Contribution

It develops a novel recursive method for constructing p-Riemann tensors and extends Schwarzschild solutions to higher dimensions with new geometric and topological features.

Findings

Construction of p-Riemann tensors via antisymmetrised 2p-forms

Existence of Schwarzschild-like solutions in dimensions d>2p+1

Special features and topological aspects in specific dimensions

Abstract

Starting with the curvature 2-form a recursive construction of totally antisymmetrised 2p-forms is introduced, to which we refer as p-Riemann tensors. Contraction of indices permits a corresponding generalisation of the Ricci tensor. Static, spherically symmetric ``$p$-Ricci flat'' Schwarzschild like metrics are constructed in this context for d>2p+1, d being the spacetime dimension. The existence of de Sitter type solutions is pointed out. Our 2p-forms vanish for $d<2p$ and the limiting cases d=2p and d=2p+1 exhibit special features which are discussed briefly. It is shown that for d=4p our class of solutions correspond to double-selfdual Riemann 2p-form (or p-Riemann tensor). Topological aspects of such generalised gravitational instantons and those of associated (through spin connections) generalised Yang-Mills instantons are briefly mentioned. The possibility of a study of surface deformations at the horizons of our class of ``p-black holes'' leading to Virasoro algebras with a p-dependent hierarchy of central charges is commented on. Remarks in conclusion indicate directions for further study and situate our formalism in a broader context.