Abstract
Non-Riemannian gravitational theories suggest alternative avenues to understand properties of quantum gravity and provide a concrete setting to study condensed matter systems with non-relativistic symmetry. Derivation of an action principle for these theories generally proved challenging for various reasons. In this technical note, we employ the formulation of double field theory to construct actions for a variety of such theories. This formulation helps removing ambiguities in the corresponding equations of motion. In particular, we embed Torsional Newton-Cartan gravity, Carrollian gravity and String Newton-Cartan gravity in double field theory, derive their actions and compare with the previously obtained results in literature.
Highlights
Introduction and summaryNon-Riemannian gravitational theories have attracted renewed interest [1,2,3,4] mainly due to the role they might play as an alternative route to understanding properties of quantum gravity [5,6,7]
Importance of understanding dynamics of non-Riemannian gravity is underpinned by the far-reaching possibilities this entails: theories based on Galilean symmetry play a role on truncations of string theory [10,11,12], post-Newtonian physics [13,14,15,16], and provide a natural setting to study response in condensed matter systems with Galilean symmetry [17,18,19,20]
In particular we studied three spacetimes which are related to each other: Type I Torsional Newton Cartan, String Newton-Cartan and Carroll
Summary
Non-Riemannian gravitational theories have attracted renewed interest [1,2,3,4] mainly due to the role they might play as an alternative route to understanding properties of quantum gravity [5,6,7]. When T-duality is applied in the null direction of the parent relativistic theory a mapping between non-Riemannian geometries, in particular Torsional NewtonCartan (TNC) or Carroll type, can be established [10, 30, 32] We note that this can be done for algebras obtained via group contractions of the Poincaré group, while algebras obtained from large speed of light expansions, i.e. expansions in 1/c, will fall out of the scope of this construction [2, 6]. The generalized metric is not restricted to this form and in particular admits non-Riemannian parametrizations, where the TNC and Carroll limits appear as particular cases [32] This means that the gravitional dynamics of such non-Riemannian geometries can be obtained by considering the double field equations of motion.
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