Notes on Emergent Gravity

TL;DR

This paper explores the inverse relationship in emergent gravity, using specific scalar-flat Kahler metrics to clarify foundational issues in constructing Riemannian geometry from gauge fields on noncommutative spaces.

Contribution

It demonstrates how to invert the emergent gravity construction to derive gauge fields from Riemannian metrics, focusing on the LeBrun metric and related geometries.

Findings

Clarifies the inverse construction in emergent gravity.

Analyzes the LeBrun metric and its special cases.

Provides insights into the geometric structures underlying emergent gravity.

Abstract

Emergent gravity is aimed at constructing a Riemannian geometry from U(1) gauge fields on a noncommutative spacetime. But this construction can be inverted to find corresponding U(1) gauge fields on a (generalized) Poisson manifold given a Riemannian metric (M, g). We examine this bottom-up approach with the LeBrun metric which is the most general scalar-flat Kahler metric with a U(1) isometry and contains the Gibbons-Hawking metric, the real heaven as well as the multi-blown up Burns metric which is a scalar-flat Kahler metric on C^2 with n points blown up. The bottom-up approach clarifies some important issues in emergent gravity.