# Yet another family of diagonal metrics for de Sitter and anti-de Sitter spacetimes

## TL;DR

This paper introduces a new class of diagonal, static, and axially symmetric coordinate metrics for de Sitter and anti-de Sitter spacetimes, based on a 2+2 foliation, expanding the understanding of their geometric and physical properties.

## Contribution

It presents a novel family of coordinate representations for (A)dS spacetimes using a 2+2 foliation, including explicit parametrizations, visualizations, and transformations to standard forms.

## Key findings

- New diagonal, static, axially symmetric metrics for (A)dS spacetimes.
- Explicit parametrizations and visualizations of the metrics.
- Physical interpretation of B-metrics as tachyon gravitational fields.

## Abstract

In this work we present and analyze a new class of coordinate representations of de Sitter and anti-de Sitter spacetimes for which the metrics are diagonal and (typically) static and axially symmetric. Contrary to the well-known forms of these fundamental geometries, that usually correspond to a 1+3 foliation with the 3-space of a constant spatial curvature, the new metrics are adapted to a 2+2 foliation, and are warped products of two 2-spaces of constant curvature. This new class of (anti-)de Sitter metrics depends on the value of cosmological constant Lambda and two discrete parameters +1,0,-1 related to the curvature of the 2-spaces. The class admits 3 distinct subcases for Lambda>0 and 8 subcases for Lambda<0. We systematically study all these possibilities. In particular, we explicitly present the corresponding parametrizations of the (anti-)de Sitter hyperboloid, visualize the coordinate lines and surfaces within the global conformal cylinder, investigate their mutual relations, present some closely related forms of the metrics, and give transformations to standard de Sitter and anti-de Sitter metrics. Using these results, we also provide a physical interpretation of B-metrics as exact gravitational fields of a tachyon.