Abstract
We derive an action whose equations of motion contain the Poisson equation of Newtonian gravity. The construction requires a new notion of Newton-Cartan geometry based on an underlying symmetry algebra that differs from the usual Bargmann algebra. This geometry naturally arises in a covariant 1/c expansion of general relativity, with c being the speed of light. By truncating this expansion at subleading order, we obtain the field content and transformation rules of the fields that appear in the action of Newtonian gravity. The equations of motion generalize Newtonian gravity by allowing for the effect of gravitational time dilation due to strong gravitational fields.
Highlights
The idea that gravity is geometry was pioneered by Einstein in his celebrated theory of general relativity (GR)
Spacetime covariance is a property of any physical theory, which led Cartan [1,2] to geometrize Newtonian gravity using what is known as Newton-Cartan (NC) geometry
We present in this Letter a novel type of NC geometry, dubbed type II torsional Newton-Cartan (TNC) geometry, which for zero torsion includes the standard NC geometry used to geometrize Newtonian gravity, and which allows us to formulate an action, in any spacetime dimension D 1⁄4 d þ 1
Summary
We derive an action whose equations of motion contain the Poisson equation of Newtonian gravity. The construction requires a new notion of Newton-Cartan geometry based on an underlying symmetry algebra that differs from the usual Bargmann algebra This geometry naturally arises in a covariant 1=c expansion of general relativity, with c being the speed of light. We present in this Letter a novel type of NC geometry, dubbed type II TNC geometry, which for zero torsion includes the standard (type I) NC geometry used to geometrize Newtonian gravity, and which allows us to formulate an action, in any spacetime dimension D 1⁄4 d þ 1 To this end, it is crucial to allow for more general time (lapse) functions than the absolute time of Newtonian gravity. While type I TNC geometry follows from gauging the Bargmann algebra [7,14] (see [15,16]), type II TNC geometry follows from a novel nonrelativistic symmetry, which turns out to be a nontrivial contraction of the direct sum of the Poincareand Euclidean algebras in D 1⁄4 d þ 1 dimensions
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