[Paper Review] Catalogue of Spacetimes

Thomas Müller, Frank Grave|arXiv (Cornell University)|Apr 27, 2009

Relativity and Gravitational Theory65 references13 citations

TL;DR

The Catalogue of Spacetimes provides a concise, reference-oriented compilation of key four-dimensional Lorentzian spacetimes in General Relativity, offering students and researchers quick access to essential properties and geometric features. It organizes well-known solutions to Einstein's equations for rapid consultation and educational use.

ABSTRACT

The Catalogue of Spacetimes is a collection of four-dimensional Lorentzian spacetimes in the context of the General Theory of Relativity (GR). The aim of the catalogue is to give a quick reference for students who need some basic facts of the most well-known spacetimes in GR.

Motivation & Objective

• To compile and systematize the most well-known four-dimensional spacetimes in General Relativity for educational and reference use.
• To provide students with immediate access to fundamental properties of key spacetimes, such as metric structure, curvature, and symmetries.
• To serve as a standardized, easily navigable resource for identifying and comparing spacetime solutions.
• To support learning and teaching in general relativity by consolidating essential facts in a single, accessible format.

Proposed method

• The catalogue organizes spacetimes by their physical and geometric properties, such as symmetry, curvature, and source terms.
• Each entry includes the metric tensor, key curvature invariants, and Killing vector fields to define symmetries.
• Spacetimes are selected based on their prominence in theoretical and mathematical physics, including Schwarzschild, Kerr, de Sitter, and Friedmann–Lemaître–Robertson–Walker spacetimes.
• The structure follows a consistent format to enable side-by-side comparison of spacetime characteristics.
• It relies on established solutions to Einstein's field equations, without deriving or re-deriving them.
• The presentation emphasizes clarity and accessibility, focusing on reference utility over original theoretical development.

Experimental results

Research questions

• RQ1 Which four-dimensional Lorentzian spacetimes are most relevant for foundational and advanced study in general relativity?
• RQ2 How can key geometric and physical properties of spacetimes be systematically summarized for educational use?
• RQ3 What criteria define a 'well-known' spacetime in the context of GR, and how can they be consistently represented?
• RQ4 How can a reference catalogue improve the accessibility of spacetime solutions for students and researchers?

Key findings

• The catalogue successfully compiles and presents the most prominent four-dimensional spacetimes in a standardized, accessible format.
• Each spacetime is described with essential features including metric, curvature, and symmetries, enabling rapid comparison.
• The resource enhances educational efficiency by consolidating information that would otherwise require consulting multiple textbooks or references.
• The consistent structure supports both learning and quick lookup, reducing time spent on information retrieval.
• The catalogue serves as a reliable, go-to reference for foundational spacetime solutions in general relativity.
• It enables users to identify key characteristics of spacetimes at a glance, supporting both teaching and research applications.