Abstract
We discuss non-relativistic limits of general relativity. In particular, we define a special fine-tuned non-relativistic limit, inspired by string theory, where the Einstein-Hilbert action has been supplemented by the kinetic term of a one-form gauge field. Taking the limit, a crucial cancellation takes place, in an expansion of the action in terms of powers of the velocity of light, between a leading divergence coming from the spin-connection squared term and another infinity that originates from the kinetic term of the one-form gauge field such that the finite invariant non-relativistic gravity action is given by the next subleading term. This non-relativistic action allows an underlying torsional Newton-Cartan geometry as opposed to the zero torsion Newton-Cartan geometry that follows from a more standard limit of General Relativity but it lacks the Poisson equation for the Newton potential. We will mention extensions of the model to include this Poisson equation.
Supported by the Dutch organization FOM/NWO.
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Notes
- 1.
String theory aims to unify gravity with quantum mechanics. At low energies, it gives rise to Einstein’s gravity coupled to certain matter fields as given in Eq. (3).
- 2.
A Wess-Zumino term is characterized by the fact that it is invariant under a gauge transformation only up to a total derivative.
- 3.
The letter J refers to the spatial rotation generators \(J_{A'B'}\) in the Bargmann algebra.
- 4.
For a related discussion where this was emphasized, see [6].
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Acknowledgments
The work of CŞ is part of the research programme of the Foundation for Fundamental Research on Matter (FOM), which is financially supported by the Netherlands Organisation for Science Research (NWO). The work of LR is supported by the FOM/NWO free program Scanning New Horizons.
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Bergshoeff, E., Lahnsteiner, J., Romano, L., Rosseel, J., Şimşek, C. (2021). Non-relativistic Limits of General Relativity. In: Nielsen, F., Barbaresco, F. (eds) Geometric Science of Information. GSI 2021. Lecture Notes in Computer Science(), vol 12829. Springer, Cham. https://doi.org/10.1007/978-3-030-80209-7_49
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