Canonical Relativity and the Dimensionality of the World

TL;DR

This paper explores the nature of spacetime's dimensionality using canonical formulations of relativity, highlighting how different frameworks influence our understanding of time and space, and emphasizing the need for a quantum gravity theory.

Contribution

It provides a canonical analysis of relativity frameworks to clarify the debate on whether spacetime is a mathematical space or a real four-dimensional world.

Findings

Canonical gravity offers a coordinate-independent way to analyze observable quantities.

Reintroducing Minkowski background clarifies the emergence of special relativity coordinates.

Current quantum gravity insights do not alter the role of time established by general relativity.

Abstract

Different aspects of relativity, mainly in a canonical formulation, relevant for the question "Is spacetime nothing more than a mathematical space (which describes the evolution in time of the ordinary three-dimensional world) or is it a mathematical model of a real four-dimensional world with time entirely given as the fourth dimension?" are presented. The availability as well as clarity of the arguments depend on which framework is being used, for which currently special relativity, general relativity and some schemes of quantum gravity are available. Canonical gravity provides means to analyze the field equations as well as observable quantities, the latter even in coordinate independent form. This allows a unique perspective on the question of dimensionality since the space-time manifold does not play a prominent role. After re-introducing a Minkowski background into the formalism, one can see how distinguished coordinates of special relativity arise, where also the nature of time is different from that in the general perspective. Just as it is of advantage to extend special to general relativity, general relativity itself has to be extended to some theory of quantum gravity. This suggests that a final answer has to await a thorough formulation and understanding of a fundamental theory of space-time. Nevertheless, we argue that current insights into quantum gravity do not change the picture of the role of time obtained from general relativity.