Conservation, inertia, and spacetime geometry

Abstract

As Harvey Brown emphasizes in his book Physical Relativity, inertial motion in general relativity is best understood as a theorem, and not a postulate. Here I discuss the status of the "conservation condition", which states that the energy-momentum tensor associated with non-interacting matter is covariantly divergence-free, in connection with such theorems. I argue that the conservation condition is best understood as a consequence of the differential equations governing the evolution of matter in general relativity and many other theories. I conclude by discussing what it means to posit a certain spacetime geometry and the relationship between that geometry and the dynamical properties of matter.

General-Relativistic Covariance

“…• On a standard way of thinking about external symmetries-as automorphisms of bundles-the dynamical approach's central slogan (that dynamical symmetries 46 That said, there are ways in which one can see the role of spacetime structure in the field equations as being what gives rise to such equations of motion (Weatherall, 2017): one can use a variational analysis to ground a certain kind of conservation condition, and then employ that condition to prove an appropriate equation of motion. The best-known example of this kind of construction is the geodesic theorem in GR, but one can similarly prove a geodesic theorem in Newtonian theories (Weatherall, 2011), and the Lorentz force law in electromagnetism (Geroch and Weatherall, 2017).…”

Relativity without miracles

“…Here the problem is, as [40] note, that the 'dynamical behaviour of non-gravitational fields does not reflect the local (Poincaré) symmetries of the metric field -taken to represent spacetime." 49 From the perspective presented in this paper, this is a case where further constraints, beyond the one coming from the gravitational field equations, are imposed. This is not forbidden by the equivalence principle, as considered here, but it might be seen as unjustified.…”

The metaphysics of emergent spacetime theories

“…Perhaps emergent spacetime theories will tell us why geodesics are the way they are, and, in case of theories in which there is a link between matter and spacetime, it may be explained why matter follows these geodesics. It is often argued that inertial motion is already explained by the geodesic principle in General Relativity, which holds that inertial motion can be derived from Einstein's field equations (Brown, ; Ehlers & Geroch, ; Geroch & Jang, ; Geroch & Weatherall, manuscript; Weatherall, ). Whether the geodesic theorem indeed succeeds in removing the need of postulating inertial motion is controversial (Malament, ; Sus, manuscript; Tamir, ; Yang, ), but even if it does succeed, a deeper explanation may be gained from certain theories in which not only spacetime but also gravity itself is emergent.…”

“…Perhaps emergent spacetime theories will tell us why geodesics are the way they are, and, in case of theories in which there is a link between matter and spacetime, it may be explained why matter follows these geodesics. It is often argued that inertial motion is already explained by the geodesic principle in General Relativity, which holds that inertial motion can be derived from Einstein's field equations (Brown, 2005;Ehlers & Geroch, 2004;Geroch & Jang, 1975;Geroch & Weatherall, manuscript;Weatherall, 2017).…”