Would Two Dimensions be World Enough for Spacetime?

TL;DR

This paper explores the peculiar features and limitations of general relativity and relativistic field theory in two-dimensional spacetime, highlighting fundamental differences from higher dimensions.

Contribution

It provides a comprehensive analysis of the unique properties and challenges of formulating physics in two-dimensional spacetime, including issues with Einstein tensor and initial-value problems.

Findings

Einstein tensor vanishes in 2D

Initial-value formulation fails in vacuum spacetimes

No Newtonian limit in 2D gravity

Abstract

We consider various curious features of general relativity, and relativistic field theory, in two spacetime dimensions. In particular, we discuss: the vanishing of the Einstein tensor; the failure of an initial-value formulation for vacuum spacetimes; the status of singularity theorems; the non-existence of a Newtonian limit; the status of the cosmological constant; and the character of matter fields, including perfect fluids and electromagnetic fields. We conclude with a discussion of what constrains our understanding of physics in different dimensions.