# Emergent Gravity and Weyl's Volume Formula

## TL;DR

This paper generalizes Weyl's volume formula to derive a universal multipole expansion for localized energy in geometric theories, connecting it to emergent gravity and higher-dimensional quantum models.

## Contribution

It introduces a finite multipole expansion for energy near submanifolds, linking geometric invariants to emergent gravity theories and extending Weyl's volume formula.

## Key findings

- Derived a universal multipole expansion for localized energy.
- Connected geometric invariants to gravity-like theories.
- Discussed conditions for exactness and corrections to the formula.

## Abstract

In physical theories where the energy (action) is localized near a submanifold of Euclidean (Minkowski) 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 universal expression is a generalization of an exact formula of Hermann Weyl for the volume of a tube. We describe when our result is applicable, when our generalization gives an exact result, and when there are corrections (often exponentially small) to our formula. In special situations, dictated by spherical symmetry, the expression is a generalized Lovelock lagrangian for gravity, a class of theories that are interesting because they have no negative metric states. We discuss whether these results represent a true theory of emergent gravity by discussing simple models where a higher dimensional quantum field theory without a fundamental graviton leads to a gravity-like theory on a submanifold where all or some of the dynamical degrees of freedom are fluctuations of the metric on the submanifold.