THE GEOMETRIZATION OF PHYSICS

Studies in History and Philosophy of Modern Physics, 2011

A theorem due to Bob Geroch and Pong Soo Jang ["Motion of a Body in General Relativity." Journal of Mathematical Physics 16(1), (1975)] provides a sense in which the geodesic principle has the status of a theorem in General Relativity (GR). I have recently shown that a similar theorem holds in the context of geometrized Newtonian gravitation (Newton-Cartan theory) [Weatherall, J. O. "The Motion of a Body in Newtonian Theories." Journal of Mathematical Physics 52(3), (2011)]. Here I compare the interpretations of these two theorems. I argue that despite some apparent differences between the theorems, the status of the geodesic principle in geometrized Newtonian gravitation is, mutatis mutandis, strikingly similar to the relativistic case.

Arxiv preprint arXiv:0704.3003, 2007

The space-time geometry is considered to be a physical geometry, i.e. a geometry described completely by the world function. All geometrical concepts and geometric objects are taken from the proper Euclidean geometry. They are expressed via the Euclidean world function σ E and declared to be concepts and objects of any physical geometry, provided the Euclidean world function σ E is replaced by the world function σ of the physical geometry in question. The set of physical geometries is more powerful, than the set of Riemannian geometries, and one needs to choose a true space-time geometry. In general, the physical geometry is multivariant (there are many vectors Q 0 Q 1 , Q 0 Q ′ 1 ,... which are equivalent to vector P 0 P 1 , but are not equivalent between themselves). The multivariance admits one to describe quantum effects as geometric effects and to consider existence of elementary particles as a geometrical problem, when the possibility of the physical existence of an elementary geometric object in the form of a physical body is determined by the space-time geometry. Multivariance admits one to describe discrete and continuous geometries, using the same technique. A use of physical geometry admits one to realize the geometrical approach to the quantum theory and to the theory of elementary particles.

Assume that two particles on the sphere leave the equator moving due south and travel at a constant and equal speed along a geodesic colliding at the south pole. An observer who is unaware of the curvature of the space will conclude that there is an attractive force acting between the particles. On the other hand, if particles travel at the same speed (initially parallel) along geodesics in the hyperbolic plane, then the particle paths diverge. Imagine two particles in the hyperbolic plane that are bound together at a constant distance with their center of mass traveling along a geodesic path at a con- stant velocity, then the force due to the curvature of the space acts to break the bond and increases as the velocity increases. We will give the formula for the apparent force between the particles induced on dimensional space forms of non-zero curvature. AMS classic ation: 53A; 70E; 85.

A classic problem in general relativity, long studied by both physicists and philosophers of physics, concerns whether the geodesic principle may be derived from other principles of the theory, or must be posited independently. In a recent paper [Geroch & Weatherall, " The Motion of Small Bodies in Space-Time " , Comm. Math. Phys. (forthcoming)], Bob Geroch and I have introduced a new approach to this problem, based on a notion we call " tracking ". In the present paper, I situate the main results of that paper with respect to two other, related approaches, and then make some preliminary remarks on the interpreta-tional significance of the new approach. My main suggestion is that " tracking " provides the resources for eliminating " point particles " —a problematic notion in general relativity—from the geodesic principle altogether.

International Journal of Geometric Methods in Modern Physics 16 , 1950015, 2019

Riemann's principle ``force equals geometry" provided the basis for Einstein's General Relativity - the geometric theory of gravitation. In this paper, we follow this principle to derive the dynamics for any static, conservative force. The geometry of spacetime of a moving object is described by a metric obtained from the potential of the force field acting on it. We introduce a generalization of Newton's First Law - the Generalized Principle of Inertia stating that: An inanimate object moves inertially, that is, with constant velocity, in its own spacetime whose geometry is determined by the forces affecting it}. Classical Newtonian dynamics is treated within this framework, using a properly defined Newtonian metric with respect to an inertial lab frame. We reveal a physical deficiency of this metric (responsible for the inability of Newtonian dynamics to account for relativistic behavior), and remove it. The dynamics defined by the corrected Newtonian metric leads to a new \emph{Relativistic Newtonian Dynamics} for both massive objects and massless particles moving in any static, conservative force field, not necessarily gravitational. This dynamics reduces in the weak field, low velocity limit to classical Newtonian dynamics and also exactly reproduces the classical tests of General Relativity, as well as the post-Keplerian precession of binaries.

Foundations of Physics, 1995

We give a precise and modern mathematical characterization of the Newtonian spacetOne structure (N). Our formulation clarifies the concepts of absohtte space, Newton's relative spaces, and absolute tone. The concept of reference frames (which are "timelike" vector fieMs on N) plays a fundamental role in our approach, and the classification of all possible reference frames on ~ is O,vestigated in detail. We succeed #i identifying a Lorentzian structure on ~1 and we study the classical electrodynamics of Maxwell and Lorentz relative to this structure, obtahffng the #nportant result that there exists only one #1trinsic generalization of the Lorentz force law which is compatible with Maxwell equations. This is at variance with other proposed #1trinsic generalizations of the Loreutz force law appearing #z the literature. We present also a formulation of Newtonian gravitational theoo, as a curve spacetime theory and discuss its meaning. 2.1. Geometrical Structure 2.2. Newtonian Dynamics 3. Reference and Moving Frames 3.1. Reference Frames 3.2. Moving Frames 3.3. Newtonian Space and Newtonian Time. Relative Rest Spaces 3.4. Galileo's Principle of Relativity

Erkenntnis, 1995

Arxiv preprint arXiv:1001.5362, 2010

It is shown, that a free motion of microparticles (elementary particles) in the gravitational field is multivariant (stochastic). This multivariance is conditioned by multivariant physical space-time geometry. The physical geometry is described completely by a world function. The Riemannian geometries form a small part of possible physical geometries. The contemporary theory of gravitation ignores existence of physical geometries. It supposes, that any space-time geometry is a Riemannian geometry. It is a mistake. As a result the contemporary theory of gravitation needs a revision. Besides, the Riemannian geometry is inconsistent, and conclusions of the gravitational theory, based on inconsistent geometry may be invalid. Free motion of macroparticles (planets), consisting of many connected microparticles, is deterministic, because connection of microparticles inside the macroparticle averages stochastic motion of single microparticles.