The spacetime geodesy of perfect fluid spheres

TL;DR

The paper develops a spacetime geodesy framework that emphasizes geometry and symmetries over specific dynamics to study static, spherically symmetric perfect-fluid and related spacetimes. By using generalized Ricci eigenvalues and spatial Ricci isotropy, it provides a practical algorithm to construct exact PF-sphere geometries under one or two free functions across several coordinate choices, while delaying the equation of state. A key insight is that in spherical symmetry the Weyl tensor reduces to a single scalar $C_0(r,t)$ tied to the Misner–Sharp mass, linking it to Herrera's complexity factor via $Y_{TF} = 4\pi(p_r - p_t) + C_0(r)$ and enabling a geometrical distance measure to Schwarzschild's constant-density star through a quadratic complexity $Q$. The work also clarifies the status of PFDM as not a PF, discusses limitations for anisotropic fluids, and outlines future directions toward axisymmetric and rotating spacetimes, highlighting the broad utility and constraints of a geometry-first, cosmography-like approach to localized gravitating systems.

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.