Abstract
The three-dimensional non-relativistic isometry algebras, namely Galilei and Newton-Hooke algebras, are known to admit double central extensions, which allows for non-degenerate bilinear forms hence for action principles through Chern-Simons formulation. In three-dimensional colored gravity, the same central extension helps the theory evade the multi-graviton no-go theorems by enlarging the color-decorated isometry algebra. We investigate the non-relativistic limits of three-dimensional colored gravity in terms of generalized \.In\"on\"u-Wigner contractions.
Highlights
In the nonrelativistic limit, Einstein’s general relativity reduces to Newtonian gravity
It turns out that one instead needs to consider the Bargmann algebra gþ0 [6], which is an extension of the Galilei algebra with a central generator M called “mass.” This mass generator is related to an additional Uð1Þ gauge field in Newton-Cartan gravity
We have studied the nonrelativistic limits of three-dimensional colored gravity in terms of generalized İnönü-Wigner contractions
Summary
Einstein’s general relativity reduces to Newtonian gravity. In the cosmological case, the three-dimensional Newton-Hooke algebra admits double extensions We refer to this as the doubly extended Newton-Hooke (NH) algebra gþΛþ. The corresponding theories admit action principles through Chern-Simons formulation These nonrelativistic gravity theories are sometimes called extended Bargmann gravity and extended NH gravity, respectively. The nonrelativistic Chern-Simons theories admit various extensions according to the symmetry algebras [17,18,19]. As we shall show later, these doubly extended nonrelativistic algebras can be obtained from the relativistic ones with two uð1Þ generators after suitable contractions The uð1Þ generators become central elements, and the resulting algebra cannot be arranged as a direct sum anymore This is the key mechanism for a nondegenerate bilinear form—and for the action principle.
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