From relativistic gravity to the Poisson equation

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.