Non-relativistic gravity and its coupling to matter
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Received: April 7, 2020 Accepted: May 26, 2020 Published: June 23, 2020
Non-relativistic gravity and its coupling to matter
a
Institut f¨ ur Theoretische Physik, Eidgen¨ ossische Technische Hochschule Z¨ urich, Wolfgang-Pauli-Strasse 27, 8093 Z¨ urich, Switzerland b School of Mathematics and Maxwell Institute for Mathematical Sciences, University of Edinburgh, Peter Guthrie Tait road, Edinburgh EH9 3FD, U.K. c Nordita, KTH Royal Institute of Technology and Stockholm University, Roslagstullsbacken 23, SE-106 91 Stockholm, Sweden d The Niels Bohr Institute, Copenhagen University, Blegdamsvej 17, DK-2100 Copenhagen Ø, Denmark
E-mail: [email protected], [email protected], [email protected] 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 Schr¨ odinger-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. Keywords: Classical Theories of Gravity, Space-Time Symmetries ArXiv ePrint: 2001.10277
c The Authors. Open Access, Article funded by SCOAP3 .
https://doi.org/10.1007/JHEP06(2020)145
JHEP06(2020)145
Dennis Hansen,a Jelle Hartongb and Niels A. Obersc,d
Contents 1 Introduction 1.1 Background and motivation 1.2 Outline and summary of the main results
1 1 2 6 6 10 11 13 15 17 18 19
3 Non-relativistic gravity 3.1 Theory from 1/c2 expansion 3.1.1 General structure 3.1.2 NNLO Lagrangian: non-relativistic gravity 3.1.3 Equations of motion 3.2 Theory from gauge invariances 3.2.1 Lagrangian 3.2.2 Equations of motion 3.3 Equality of LNRG and L0NRG 3.4 Comments on imposing τ ∧ dτ = 0 and dτ = 0 with a Lagrange multiplier
20 20 20 22 25 29 29 32 33 34
4 Coupling to matter 4.1 Expansion of the matter Lagrangian 4.2 Ward identities 4.2.1 Expansion of the Hilbert energy-momentum tensor 4.3 Boost invariant currents 4.4 Newtonian gravity
35 35 37 39 40 41
5 Examples of matter couplings 5.1 Point particles 5.1.1 Lagrangian 5.1
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