Non-relativistic gravity and its coupling to matter

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Abstract

We study the non-relativistic expansion of general relativity coupled to matter. This is done by expanding the metric and matter fields analytically in powers of 1/c² where c is the speed of light. In order to perform this expansion it is shown to be very convenient to rewrite general relativity in terms of a timelike vielbein and a spatial metric. This expansion can be performed covariantly and off shell. We study the expansion of the Einstein-Hilbert action up to next-to-next-to-leading order. We couple this to different forms of matter: point particles, perfect fluids, scalar fields (including an off-shell derivation of the Schrödinger-Newton equation) and electrodynamics (both its electric and magnetic limits). We find that the role of matter is crucial in order to understand the properties of the Newton-Cartan geometry that emerges from the expansion of the metric. It turns out to be the matter that decides what type of clock form is allowed, i.e. whether we have absolute time or a global foliation of constant time hypersurfaces. We end by studying a variety of solutions of non-relativistic gravity coupled to perfect fluids. This includes the Schwarzschild geometry, the Tolman-Oppenheimer-Volkoff solution for a fluid star, the FLRW cosmological solutions and anti-de Sitter spacetimes.

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Palatini formulation of f(R, T) gravity theory, and its cosmological implications

Article Open access 29 May 2018

Aspects of hot Galilean field theory

Article Open access 22 April 2015

On a Generalization of the Equations of General Relativity Based on Weyl’s Principle of Scale Invariance

Article 01 July 2021

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Institut für Theoretische Physik, Eidgenössische Technische Hochschule Zürich, Wolfgang-Pauli-Strasse 27, 8093, Zürich, Switzerland
Dennis Hansen
School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Peter Guthrie Tait road, Edinburgh, EH9 3FD, UK
Jelle Hartong
Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91, Stockholm, Sweden
Niels A. Obers
The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100, Copenhagen Ø, Denmark
Niels A. Obers

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Hansen, D., Hartong, J. & Obers, N.A. Non-relativistic gravity and its coupling to matter. J. High Energ. Phys. 2020, 145 (2020). https://doi.org/10.1007/JHEP06(2020)145

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Received: 04 April 2020
Accepted: 26 May 2020
Published: 23 June 2020
DOI: https://doi.org/10.1007/JHEP06(2020)145

Non-relativistic gravity and its coupling to matter

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Palatini formulation of f(R, T) gravity theory, and its cosmological implications

Aspects of hot Galilean field theory

On a Generalization of the Equations of General Relativity Based on Weyl’s Principle of Scale Invariance

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Non-relativistic gravity and its coupling to matter

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Palatini formulation of f(R, T) gravity theory, and its cosmological implications

Aspects of hot Galilean field theory

On a Generalization of the Equations of General Relativity Based on Weyl’s Principle of Scale Invariance

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