From Relativistic Gravity to the Poisson Equation

TL;DR

This paper investigates the conditions under which non-relativistic limits of higher-dimensional gravity theories yield a Poisson equation for Newtonian gravity, focusing on the role of scalar couplings and symmetries.

Contribution

It demonstrates how a Poisson equation arises from non-relativistic limits of relativistic theories, highlighting the importance of scalar coupling fine-tuning and dilatation symmetry.

Findings

Poisson equation can be derived from Lagrangian theories with scalar coupling tuning.

Poisson equation can also emerge from non-Lagrangian equations of motion without dilatation invariance.

The approach can be extended to include more p-branes for richer gravitational models.

Abstract

We consider the non-relativistic limit of general relativity coupled to a $(p+1)$-form gauge field and a scalar field in arbitrary dimensions and investigate under which conditions this gives rise to a Poisson equation for a Newton potential describing Newton-Cartan gravity outside a massive $p$-dimensional extended object, a so-called $p$-brane. Given our Ansatz, we show that not all the $p$-branes satisfy the required conditions. We study theories whose dynamics is defined by a Lagrangian as well as systems that are defined by a set of equations of motion not related to a Lagrangian. We show that, within the Lagrangian approach, a Poisson equation can be obtained provided that the coupling of the scalar field is fine-tuned such that the non-relativistic Lagrangian is invariant under an emerging local dilatation symmetry. On the other hand, we demonstrate that in the absence of a Lagrangian a Poisson equation can be obtained from a set of equations of motion that is not dilatation invariant. We discuss how our Ansatz could be generalized such as to include more $p$-branes giving rise to a Poisson equation.