# Emergent Gravity in Spaces of Constant Curvature

## TL;DR

This paper develops a multipole expansion for energy in constant curvature spaces, linking geometric invariants of submanifolds to emergent gravity phenomena, especially in anti de Sitter space.

## Contribution

It generalizes a Weyl-based formula to spaces of constant curvature, providing a framework for emergent gravity from embedded submanifolds in quantum field theories.

## Key findings

- Derived a finite multipole expansion for energy in curved spaces
- Connected geometric invariants to emergent gravitational constants
- Discussed implications for cosmological constant and Planck mass

## Abstract

In physical theories where the energy (action) is localized near a submanifold of a constant curvature space, there is a universal expression for the energy (or the action). We derive a multipole expansion for the energy that has a finite number of terms, and depends on intrinsic geometric invariants of the submanifold and extrinsic invariants of the embedding of the submanifold. This is the second of a pair of articles in which we try to develop a theory of emergent gravity arising from the embedding of a submanifold into an ambient space equipped with a quantum field theory. Our theoretical method requires a generalization of a formula due to by Hermann Weyl. While the first paper discussed the framework in Euclidean (Minkowski) space, here we discuss how this framework generalizes to spaces of constant sectional curvature. We focus primarily on anti de Sitter space. We then discuss how such a theory can give rise to a cosmological constant and Planck mass that are within reasonable bounds of the experimental values.