The spacetime geodesy of perfect fluid spheres

TL;DR

This paper advocates for a 'spacetime geodesy' approach that emphasizes symmetries and geometry over dynamical equations, providing new insights into perfect fluid spheres and their properties in general relativity.

Contribution

It introduces the concept of spacetime geodesy as a tool to analyze perfect fluid spheres without relying on specific dynamical equations, highlighting the use of symmetries and geometric features.

Findings

Analyzes the implications of spatially isotropic Ricci tensors for perfect fluid conditions.

Examines the structure of the Weyl tensor in spherical symmetry.

Discusses potential generalizations to axisymmetric spacetimes.

Abstract

Herein we shall argue for the utility of "spacetime geodesy", a point of view where one delays as long as possible worrying about dynamical equations, in favour of the maximal utilization of both symmetries and geometrical features. This closely parallels Weinberg's distinction between "cosmography" and "cosmology", wherein maximal utilization of both the symmetries and geometrical features of Friedmann--Lemaitre--Robertson--Walker (FLRW) spacetimes is emphasized. This "spacetime geodesy" point of view is particularly useful in those situations where, for one reason or another, the dynamical equations of motion are either uncertain or completely unknown. Several specific examples are discussed -- we shall illustrate what can be done by considering the physics implications of demanding spatially isotropic Ricci tensors as a way of automatically implementing the (isotropic) perfect fluid condition, without committing to a specific equation of state. We also consider the structure of the Weyl tensor in spherical symmetry, with and without the (isotropic) perfect fluid condition, and relate this to the notion of "complexity". In closing, we indicate some ways in which these considerations might be further generalized to more physically complicated (and technically very much more complicated) situations such as axisymmetric spacetimes.