Gravitational Energy in Newtonian Gravity: A Response to Dewar and Weatherall

Abstract

The paper investigates the status of gravitational energy in Newtonian Gravity (NG), developing upon recent work by Dewar and Weatherall. The latter suggest that gravitational energy is a gauge quantity. This is potentially misleading: its gauge status crucially depends on the spacetime setting one adopts. In line with Møller-Nielsen's plea for a motivational approach to symmetries, we supplement Dewar and Weatherall's work by discussing gravitational energy-stress in Newtonian spacetime, Galilean spacetime, Maxwell-Huygens spacetime, and Newton-Cartan Theory (NCT). Although we ultimately concur with Dewar and Weatherall that the notion of gravitational energy is problematic in NCT, our analysis goes beyond their work. The absence of an explicit definition of gravitational energy-stress in NCT somewhat detracts from the force of Dewar and Weatherall's argument. We fill this gap by examining the supposed gauge status of prima facie plausible candidates-NCT analogues of gravitational energy-stress pseudotensors, the Komar mass, and the Bel-Robinson tensor. Our paper further strengthens Dewar and Weatherall's results. In addition, it sheds more light upon the subtle link between sufficiently rich inertial structure and the definability of gravitational energy in NG.

On the Status of Newtonian Gravitational Radiation

“…however, seek to break the symmetry in the Newtonian case with respect to gravitational radiation in that theory by considering the Newtonian limit of GR. To this, however, we would reply in the spirit of [11] as follows: the only way in which Dewar and Weatherall justify an interpretation of the Poisson equation as giving rise to gravitational waves is by reference to GR: but they thus do not establish that Newtonian gravity or NCT, considered on their own terms, in any clear and comprehensible manner features anything earning the name of a gravitational wave.…”

“…the functionalist take on gravitational stress-energy in GR presented in [27]. 11 The geometric Poisson equation (1) encodes information in 3-dimensional spacelike hypersurfaces; the question is to what extent such equations can (at least indirectly-see below) model evolution in the additional (temporal) direction.…”

“…theorems [35], and recovery theorems [31]; moreover, in both theories is it possible to define conformal structures [8], and in both theories is the notion of gravitational energy problematic [9,11]. With these parallels in mind, in this article we direct our attention towards the following question: to what extent do gravitational waves exist in NCT?…”

Energy in Newtonian Gravity

“…2. We add that other aspects of their paper have given rise to illuminating discussions [15]. In particular the Newton-Cartan formulation of the theory has been a subject for these discussions.…”