Spacetimes with Torsion

null, Helen Meskhidze

Abstract: General Relativity teaches us that gravitational influence should be understood as the manifestation of spacetime curvature. This lesson, however, is complicated by the existence of a gravitational theory that is empirically equivalent to General Relativity but represents gravitational influence through forces: Teleparallel Gravity (TPG). TPG raises both questions regarding underdetermination and more fundamental conceptual questions: Which theory, General Relativity or Teleparallel Gravity, describes our world? And how should we understand the torsional forces posited by Teleparallel Gravity?

To tackle these questions, I first consider a context in which gravitational force is better understood: classical spacetimes. I will show how one can incorporate torsion into the classical spacetime context so that it yields a classical theory of gravity with a closed temporal metric and spacetime torsion. This proves a result analogous to the Trautman degeometrization theorem, that every model of Newton-Cartan theory gives rise, non-uniquely, to a model of torsional, classical gravity. I will then turn to consider the non-relativistic limit of Teleparallel Gravity. I will propose a method of "opening up the lightcones" of TPG to allow the speed of light to become unbounded while employing the tetrad formalism. With this method, I prove that TPG reduces not to the previously outlined non-relativistic, torsional theory but, rather, to standard Newtonian Gravity. This result is somewhat analogous to the classical limit of General Relativity. In that context, it has been shown that taking the classical limit “squeezes out” the spatial curvature. Here, I show that taking the classical limit of TPG “squeezes out” the torsion.