Abstract
We consider the non-relativistic limit of gravity in four dimensions in the first order formalism. First, we revisit the case of the Einstein-Hilbert action and formally discuss some geometrical configurations in vacuum and in the presence of matter at leading order. Second, we consider the more general Mardones-Zanelli action and its non-relativistic limit. The field equations and some interesting geometries, in vacuum and in the presence of matter, are formally obtained. Remarkably, in contrast to the Einstein-Hilbert limit, the set of field equations is fully determined because the boost connection appears in the action and field equations. It is found that the cosmological constant must disappear in the non-relativistic Mardones-Zanelli action at leading order. The conditions for Newtonian absolute time be acceptable are also discussed. It turns out that Newtonian absolute time can be safely implemented with reasonable conditions.
Highlights
In four-dimensional MZ action, besides the EH term, gravity gains topological terms, a cosmological constant term and two extra terms associated with torsion
To distinguish from the Galilei gravity obtained from the EH action, we call the non-relativistic theory obtained from the MZ action by Galilei-Cartan (GC) gravity
The torsionless vacuum solution can be adjusted in order to account for the non-relativistic limit of the maximally symmetric solution of the MZ relativistic theory. It is shown (In appendix A) that if we keep the cosmological constant in the GC theory, the field equations are inconsistent with the Bianchi identities
Summary
We review the construction of the LC theory of gravity in four dimensions [7, 8] which generalizes Lovelock gravity theories [34] within Einstein-Cartan first order formalism by including torsional terms in the gravity action. The Poincaré group is not a semi-simple Lie group, a property that makes the task of constructing a gauge invariant action quite difficult.. A solution for this problem is to consider only the Lorentz sector for gauging while keeping the translational sector of the group to. Aiming the construction of a general gravity action, we consider the fundamental independent 1-form fields: the vierbein E and the Lorentz connection Y , defined respectively through. The MZ theorem [7, 8] states that the most general four-dimensional action which is gauge invariant, polynomially local, explicitly metric independent, and that depends only on first order derivatives is given by SMZ = κ +. Let us take a look at the Inönü-Wigner contraction [9] from the Poincaré group to the Galilei group
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