Abstract
Comparisons of gravitational theories and the structures they posit have a long and fruitful history in the philosophy of physics literature. Studying the relation between General Relativity (GR) and Newton-Cartan theory (NCT), for example, has been a valuable means to deepen our understanding of the ontology and structures each theory posits. Similarly, investigation of the relation between GR and its modified gravity counterparts have been of recent interest (see, e.g., Knox (2011) for a comparison of GR and Teleparallel Gravity, a relativistic theory of gravity that allows for non-vanishing torsion, as well as Duerr (2021) for a comparison of GR and f(R) gravity, a theory that is arguable the most natural extension of GR). Here, I investigate 1) the relationship between NCT and a classical theory of gravity with possibly non-vanishing torsion and 2) the relation between such a classical theory of gravity and Teleparallel Gravity.
I first develop a theory of Newtonian Gravity with possibly non-vanishing torsion. This is done by following the procedure Trautman Recovery Theorem---a theorem that, in the torsion-free context, establishes the relation between NCT and Newtonian Gravity. Here, by relaxing the conditions on the possible derivative operators to allow those with torsion, I recover a theory of gravity with possibly non-vanishing torsion from NCT. For the second part of the project, I consider the classical limit of Teleparallel Gravity, i.e., as the speed of light becomes unbounded. Knowing that GR reduces to NCT as the light cones are "opened up," I consider what the result of a similar procedure is on Teleparallel Gravity. The spacetime I recover is a classical spacetime with possibly non-vanishing torsion.
Overall, this project is in the spirit of Read and Teh (2018). However, while they adopt the tetrad formulations of the theories and employ the teleparallel Bargmann–Eisenhart solution to show the reduction relation, I remain closer to typical formulations of GR and NCT. This methodology, I argue, allows for a more straightforward comparison of the theories. This project not only helps us better understand the concepts in gravitational theories, it also presses us to consider how theoretical terms operate within/amongst theories of gravity. In each theory, terms like "curvature'' and "force'' are mathematically and conceptually redefined. However, the limiting relations amongst the theories (established through, e.g., recovery or by taking the classical limit) trouble the idea that the terms operate only within the limited context of each theory.
