Contravariant Gravity on Poisson Manifolds and Einstein Gravity

TL;DR

This paper explores a novel form of gravity on Poisson manifolds, called contravariant gravity, which relates to Einstein gravity and involves unique connections and matter couplings.

Contribution

It introduces the contravariant Levi-Civita connection and formulates an Einstein-Hilbert-type action on Poisson manifolds, linking it to Einstein gravity with matter interactions.

Findings

Contravariant gravity includes couplings between metric and Poisson tensor.

The theory can be equivalently described as Einstein gravity coupled to matter.

On 2D manifolds, it reduces to a scalar field coupled to the metric.

Abstract

A relation between gravity on Poisson manifolds proposed in arXiv:1508.05706 and Einstein gravity is investigated. The compatibility of the Poisson and Riemann structures defines a unique connection, the contravariant Levi-Civita connection, and leads to the idea of the contravariant gravity. The Einstein-Hilbert-type action yields an equation of motion which is written in terms of the analog of the Einstein tensor, and it includes couplings between the metric and the Poisson tensor. The study of the Weyl transformation reveals properties of those interactions. It is argued that this theory can have an equivalent description as a system of Einstein gravity coupled to matter. As an example, it is shown that the contravariant gravity on a two-dimensional Poisson manifold can be described by a real scalar field coupled to the metric in a specific manner.