Contravariant Gravity on Poisson Manifolds and Einstein Gravity
arXiv:1610.06554 · doi:10.1088/1361-6382/aa6e5b
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.
arXiv comment21 pages; v3 version accepted for publication in Classical and Quantum Gravity; v2 minor revisions and some improvements, including discussions on divergence theorem
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