Overall signature of the metric and the cosmological constant

TL;DR

This paper explores how allowing the overall metric signature to change can lead to different signs of the cosmological constant, enabling novel classical solutions connecting de Sitter and Anti-de Sitter spaces.

Contribution

It demonstrates the classical consistency of signature change with varying cosmological constant signs across different formulations of general relativity.

Findings

Signature change can be consistent with sign changes in the cosmological constant.

Different Lorentzian signature sectors correspond to different signs of Λ.

Open de Sitter space can emerge from Anti-de Sitter space via classical signature change.

Abstract

We consider a little known aspect of signature change, where the overall sign of the metric is allowed to change, with physical implications. We show how, in different formulations of general relativity, this type of classical signature change across boundaries with a degenerate metric can be made consistent with a change in sign (and value) of the cosmological constant $\Lambda$. In particular, the separate "mostly plus" and "mostly minus" signature sectors of Lorentzian gravity are most naturally associated with different signs of $\Lambda$. We show how this general phenomenon allows for classical solutions where the open dS patch can arise from a portion of AdS space time. These can be interpreted as classical "imaginary space" extensions of the usual Lorentzian theory, with $a^2<0$.