Are Dynamic Shifts Dynamical Symmetries?

Abstract

Shifts are a well-known feature of the literature on spacetime symmetries. Recently, discussions have focused on so-called dynamic shifts, which by analogy with static and kinematic shifts enact arbitrary linear accelerations of all matter (as well as a change in the gravitational potential). But in mathematical formulations of these shifts, the analogy breaks down: while static and kinematic shift act on the matter field, the dynamic shift acts on spacetime structure instead. I formulate a different, ‘active’ version of the dynamic shift which does act on matter, and analyse the consequences of this reformulation for Newton-Cartan Theory and Maxwell Gravitation.

Are Maxwell Gravitation and Newton-Cartan Theory Theoretically Equivalent?

“…This would avoid the worry that Maxwell gravitation and Newton-Cartan theory are inequivalent because Newton-Cartan theory draws its distinctions finer than Maxwell gravitation. However, Jacobs (2023) has recently presented another argument that Newton-Cartan theory and Maxwell gravitation are inequivalent, on the basis that the two theories have different spacetime and dynamical symmetry groups. Jacobs begins his analysis by defining an "active" version of the dynamic shiftanalogous to the standard kinematic and static shifts -which produces a linear time-dependent acceleration of the matter content of the original solution.…”

“…I discuss the fact that the models of these two theories are not in one-to-one correspondence, and clarify how this relates to the issue of test particles and counterfactuals about unactualised matter. Section 4 aims to diffuse Jacobs's (2023) recent argument that Maxwell gravitation and Newton-Cartan theory have different spacetime and dynamical symmetry groups. Finally, in sections 5 and 6, I use the resources of category theory to discuss how this relates to the question of theoretical equivalent.…”

“…2. A recent selection of competing views on the subject: Saunders (2013), Knox (2014), Weatherall (2016), Dewar (2018), Wallace (2020), Jacobs (2023).…”