A Derivation of Einstein Gravity without the Axiom of Choice: Topology Hidden in GR

TL;DR

This paper derives Einstein's equations without using the Axiom of Choice by explicitly constructing a topological choice function, revealing a hidden topological structure in general relativity that enhances its background independence.

Contribution

It introduces a novel derivation of Einstein gravity that avoids the Axiom of Choice, emphasizing the topological aspects underlying the theory.

Findings

Q is given by 2T^3+3S^1xS^2, embedded in four dimensions

Solutions relate curvature effects to topology of Q and source terms

Links to holography are suggested

Abstract

A derivation of the equations of motion of general relativity is presented that does not invoke the Axiom of Choice, but requires the explicit construction of a choice function q for continuous three-space regions. The motivation for this (seemingly academic) endeavour is to take the background independence intrinsic to Einstein gravity one step further, and to assure that both the equations of motion and the way in which those equations of motion are derived are as self-consistent as possible. That is, solutions to the equations of motion of general relativity endow a three-space region with a physical and distinguishing geometry in four-dimensional space-time. However, in order to derive these equations of motion one should first be able to choose a three-space region without having any prior knowledge of its physically appropriate geometry. The expression of this choice process requires a three-dimensional topological manifold Q, to which all considered three-space regions belong, and that generates an equation of motion whose solutions are q. These solutions relate the effects of curvature to the source term through the topology of Q and constitute Einstein gravity. Q is given by 2T^3+3S^1xS^2, and is embedded in four dimensions. This points toward a hidden topological content for general relativity, best phrased as: Q and q provide a structure for how to choose a three-space region irrespective of what geometric properties it has, while at the same time Q and q determine that only GR can endow a three-space with those geometric properties. In this sense, avoiding the Axiom of Choice allows one to gain physical insight into GR. Possible links with holography are pointed out.