The geodesic principle is one of the central principles of General Relativity (GR). It states that free massive test point particles traverse timelike geodesics. In his book Physical Relativity, Harvey Brown argues that the geodesic principle has a special status in GR that is not shared by some other principles of the theory, such as those those connecting spacetime structure to length contraction or time dilation. This special status arises, he says, because the geodesic principle is a theorem, rather than a postulate, of the theory, and thus, “GR is the first in the long line of dynamical theories… that explains inertial motion”.
My goal will not be to engage with the details of Brown’s views. But the quoted remark suggests an interesting question regarding the precise status of the geodesic principle, and thus inertial motion, in Newtonian physics. In Newton’s own formulation of his theory, inertial motion certainly does appear to have the status of a postulate, as Brown suggests. It enters the theory as Newton’s first law, which states that in the absence of external forces, a body will travel at constant velocity along a straight line. But there is another formulation of Newtonian physics, originally due to Elie Cartan, in which the classical theory is expressed in a “generally covariant,” or coordinate-independent, way. On this version of the theory, gravity becomes geometrized as in GR, in the senses that (a) the geometrical structure of spacetime depends on the distribution of mass within spacetime, and conversely (b) gravitational effects are seen to be manifestations of the resulting geometry.
I will present a recent result to the effect that in geometrized Newton-Cartan theory, the geodesic principle can again be expressed as a theorem, rather than a postulate. I will also discuss the senses in which, given this theorem, Newtonian physics can be said to explain inertial motion, keeping in mind the relation between my theorem and its equivalent in GR. I believe the theorem I discuss is of independent interest; however, its philosophical payoff will be that the status of the geodesic principle in Newtonian physics is, mutatis mutandis, strikingly similar to the relativistic case in a way that can be made perfectly precise.
University of California, Irvine