Review on non-relativistic gravity

Abstract

This study reviews the history of Newton–Cartan (NC) gravity with an emphasis on recent developments, including the covariant, off-shell large speed of light expansion of general relativity. Depending on the matter content, this expansion leads to either NC geometry with absolute time or NC geometry with non-relativistic gravitational time dilation effects. The latter shows that non-relativistic gravity (NRG) includes a strong field regime and goes beyond Newtonian gravity. We start by reviewing early developments in NC geometry, including the covariant description of Newtonian gravity, mainly through the works of Trautman, Dautcourt, Künzle, and Ehlers. We then turn to more modern developments, such as the gauging of the Bargmann algebra and describe why the latter cannot be used to find an off-shell covariant description of Newtonian gravity. We review recent work on the 1/c expansion of general relativity and show that this leads to an alternative “type II” notion of NC geometry. Finally, we discuss matter couplings, solutions, and odd powers in 1/c and conclude with a brief summary of related topics.

Dynamical de Sitter black holes in a quasi-stationary expansion

“…which agrees with (48) in [12]. Lastly, our (22) corresponds to (49) in [12]. 7 In the special case where Φ is constant a larger group of diffeomorphisms remains unfixed.…”

“…As we already briefly mentioned, the solution we will present is also of relevance to the nonrelativistic 1/c expansion of general relativity, which has recently been quite actively researched, see [22] for a review. We'll show how, upon identification of the slow time parameter ϵ with 1/c, the quasi-stationary expansion as introduced in this paper is indeed equivalent to the 1/c expansion of [21].…”

“…Indeed this is the case; as we show in detail in section 4.3, upon the identification ϵ ∼ 1 c , the expansion in ϵ becomes equivalent to the nonrelativistic 1/c expansion. We refer to [22] for an introduction.…”

“…By the construction in section 4.1, for small ϵ, all the time derivatives are suppressed such that the quasi-stationary setup can alternatively be viewed as a small velocity and hence nonrelativistic 1/c expansion, with c the speed of light. The 1/c setup, see [22,25,29] for a review, is a generalization of the more commonly used PN expansion. Indeed, where the PN setup additionally assumes weak gravitational fields (both small 1/c and G N limit), this is not the case in the 1/c expansion, which is a purely non-relativistic expansion where, at LO, the metric can have arbitrarily high curvature.…”

Geometric post-Newtonian description of massive spin-half particles in curved spacetime

“…In order to perform the post-Newtonian expansion systematically, we are going to implement it as a formal power series expansion 6 in the parameter c −1 , where c is the velocity of light 7 . Such formal expansions are a well-established device to implement Newtonian limits and post-Newtonian expansions of (locally) Poincaré-relativistic physics in a mathematically controlled manner: they appear, of course, in the İnönü-Wigner contraction from the Poincaré to the Galilei group [28], and have been applied, e.g., to systematically develop the post-Newtonian expansion of the Klein-Gordon equation [11,[18][19][20], or to discuss the rigorous post-Newtonian expansion of General Relativity and its modifications in the context of Newton-Cartan gravity (geometrised Newtonian gravity) [29][30][31][32][33][34]. In order to obtain a consistent post-Newtonian expansion 8 , we need to treat the orthonormal-basis components of the curvature tensor and its covariant derivative as being of order c −2 , i.e.…”