Proving the principle: Taking geodesic dynamics too seriously in Einstein's theory

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Abstract

In this paper I critically review attempts to formulate and derive the geodesic principle, which claims that free massive bodies follow geodesic paths in general relativity theory. I argue that if the principle is (canonically) interpreted as a law of motion describing the actual evolution of gravitating bodies, then it is impossible to generically apply the law to massive bodies in a way that is coherent with Einstein's field equations. Rejecting the canonical interpretation, I propose an alternative interpretation of the geodesic principle as a type of universality thesis analogous to the universality behavior exhibited in thermal systems during phase transitions.

Section snippets

Geodesic dynamics

Einstein's adoption of the geodesic principle was originally thought to be an independent postulate establishing the dynamics of the theory. Not long after the debut of his general theory, however, numerous special-case results and plausibility arguments were developed suggesting that in fact the principle was not logically independent (given certain assumptions about free-fall bodies) from Einstein's field equations themselves.1

Singularity proofs

The family of proofs which I will refer to as singularity proofs really consist of two distinct subclasses. The first subclass follows Einstein and Grommer's original “third way” method in which they attempt to use true singularities in the manifold in order to represent matter–energy. These singularities in the manifold are then (somehow) supposed to be shown to be geodetic. With the mathematical advances in distribution theory, these true singularity proofs were succeeded by the second

Zeroth-order proofs

Einstein's field equations have an initial value formulation. Under suitable conditions, this problem can be well posed so that we might use the field equations to deduce how a given tensor field defined on a particular hypersurface evolves over time (viz. the domain of dependence of that hypersurface). Though this works in principle, such a program is far more easily said than done in most cases and often numerically rather than analytically. As a consequence if physicists wished to predict,

Limit operation proofs

The strategy behind the final family of limit operation proofs is to avoid the complications arising from investigating the motion of “true” point particles with extent restricted precisely to one-dimensional timelike curves by instead considering sequences of energy–momentum representations of particles whose spacelike extent is confined to increasingly smaller neighborhoods of those curves. We can think of these infinite sequences of tensor fields as representing particles with arbitrarily

Conclusion: towards a geodesic universality thesis

In this paper I have argued that the canonical view that the geodesic principle provides the dynamics of general relativity theory fails. Under this interpretation, the commonly endorsed belief that the principle can be derived either from Einstein's original field equations or a distributional generalization of them must be rejected (even if we allow for further background assumptions about the kind of matter–energy that is supposed to follow such geodesics). By reviewing the three major

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    In the following, M will be taken to be a smooth, orientable, four-dimensional manifold, and (M,gab) will be referred to as a Lorentzian spacetime if gab is a smooth metric of signature (+,,,) defined on M. Excepting quoted material all further notational conventions follow that of Wald (1984).

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