Abstract
We consider two distinct limits of General Relativity that in contrast to the standard non-relativistic limit can be taken at the level of the Einstein-Hilbert action instead of the equations of motion. One is a non-relativistic limit and leads to a so-called Galilei gravity theory, the other is an ultra-relativistic limit yielding a so-called Carroll gravity theory. We present both gravity theories in a first-order formalism and show that in both cases the equations of motion (i) lead to constraints on the geometry and (ii) are not sufficient to solve for all of the components of the connection fields in terms of the other fields. Using a second-order formalism we show that these independent components serve as Lagrange multipliers for the geometric constraints we found earlier. We point out a few noteworthy differences between Carroll and Galilei gravity and give some examples of matter couplings.
Highlights
Alternative approach one ends up with a non-relativistic vibrating string in the bulk [10]
Using a second-order formalism we show that these independent components serve as Lagrange multipliers for the geometric constraints we found earlier
Note that the Galilei-Maxwell action above does not correspond to the action of what is known in the literature as Galilean electrodynamics [31], coupled to an arbitrary nonrelativistic background
Summary
Before taking limits we first summarize some relevant formulae of General Relativity including matter couplings which will be of use . ΔEμA = ∂μηA + ΛABEμB − ΩAμ BηB , δΩAμ B = ∂μΛAB + ΩAμ C ΛBC − ΩBμ C ΛAC These gauge fields transform as covariant vectors under general coordinate transformations with parameters ξμ. We consider the following action which is invariant under general coordinate transformations and local Lorentz transformations:. EE[μAEBν ECρ EDσ ]Rμν AB(J )RρσC (P )ηD + δP Smat This shows that only for D = 3 the gravity kinetic term in the action (2.7) is invariant under both Lorentz and P -transformations. This is related to the fact that for D = 3 this kinetic term can be rewritten as a Chern-Simons gauge theory. For the equations of motion to be consistent, these identities require the following on-shell relations among the currents: T[AB]
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