MSc defense by Roberto Forbicia León

On non-Lorentzian geometry and the weak field limit of non-relativistic gravity

In recent years, the interest in non-relativistic physics has resurged following modern developments in non-Lorentzian geometry. This has led to a covariant formulation of non-relativistic gravity (NRG), obtained from an appropriate large speed of light expansion of GR. In this work, we study the weak field limit of NRG by exploring the two natural paths towards its description. The first one corresponds to a non-relativistic expansion of the well-known theory of linearised GR. We derive the resulting Lagrangians at LO and NLO, as well as the corresponding EOM. The second one amounts to a linearisation of the geometric fields of NRG around a flat Newton-Cartan background. 

We show explicitly that the two paths yield the same theory at LO, which suggests that our formulation renders the two approaches compatible. We argue that the weak field limit of NRG is already richer than Newtonian gravity in two senses: by allowing for small perturbations on the closedness of the clock-form and by allowing these to be time-dependent. Finally, building on the knowledge provided by the recently discovered covariant formulation of Carroll gravity as obtained from an ultra-local expansion of GR, we show that a truncated sector of the NLO theory in the non-relativistic expansion of GR can be seen as the non-relativistic magnetic limit of the latter.

Censor: Marta Orselli