Nordström’s scalar theory of gravity and the equivalence principle

Abstract

General Relativity obeys the three equivalence principles, the “weak” one (all test bodies fall the same way in a given gravitational field), the “Einstein” one (gravity is locally effaced in a freely falling reference frame) and the “strong” one (the gravitational mass of a system equals its inertial mass to which all forms of energy, including gravitational energy, contribute). The first principle holds because matter is minimally coupled to the metric of a curved spacetime so that test bodies follow geodesics. The second holds because Minkowskian coordinates can be used in the vicinity of any event. The fact that the latter, strong, principle holds is ultimately due to the existence of superpotentials which allow to define the inertial mass of a gravitating system by means of its asymptotic gravitational field, that is, in terms of its gravitational mass. Nordström’s theory of gravity, which describes gravity by a scalar field in flat spacetime, is observationally ruled out. It is however the only theory of gravity with General Relativity to obey the strong equivalence principle. I show in this paper that this remarkable property is true beyond post-newtonian level and can be related to the existence of a “Nordström-Katz” superpotential.