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.
Highlights
Nature is relativistic at a fundamental level but it often effectively appears to us as non-relativistic (NR)
It is relevant to mention that for fixed backgrounds, non-relativistic geometry has proven to be useful for understanding aspects such as energy-momentum tensors, Ward identities, hydrodynamics and anomalies in the context of non-relativistic field theories, which are ubiquitous in condensed matter and biological systems
The main purpose of this paper has been the development of non-relativistic gravity (NRG) as it appears from a large speed of light expansion of general relativity (GR)
Summary
Nature is relativistic at a fundamental level but it often effectively appears to us as non-relativistic (NR). This typically happens in many-body or condensed matter type systems but it can be true for gravitational phenomena. General relativity (GR) can often be well-approximated by a theory of non-relativistic gravity such as the post-Newtonian (PN) approximation. Uncovering the mathematical description of non-relativistic geometries, their dynamics and interaction with matter (classical and quantum) is a very relevant subject. We will show that the Friedmann equations, the Tolman-OppenheimerVolkoff (TOV) fluid star and the usual effects due to the Schwarzschild geometry can all be captured by the theory of non-relativistic gravity described here. More generally we expect this approach to be relevant whenever the gravitational interaction can effectively be treated as instantaneous
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