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Complexity hierarchies in Euclidean stars

L. Herrera, A. Di Prisco, J. Ospino

TL;DR

This work defines Euclidean stars as relativistic fluids with the Euclidean condition $B=R'$ and equal areal/proper radii, and introduces a hierarchy of solutions by complexity using the complexity factor $Y_{TF}$ alongside evolution patterns $H$ and $QH$, covering both non-dissipative and dissipative regimes. The authors derive the governing equations, transport relations from causal thermodynamics, and exterior matching to a Vaidya spacetime, then classify solutions from the simplest FRW/LTB-like cases to more general anisotropic, dissipative families constrained by $Y_{TF}=0$ and either $QH$ or shear-free conditions. A central result is that the FRW spacetime emerges as the simplest non-dissipative Euclidean star under $H$ and $Y_{TF}=0$, while more complex configurations arise when relaxing these constraints, including explicit toy models and a conformally flat shear-free branch. The framework provides analytic methods to model evolving stellar configurations with controlled complexity, potentially linking arbitrary radial/time functions to observables such as surface redshift and emitted energy in realistic astrophysical scenarios.

Abstract

We establish a hierarchy of Euclidean stars according to their degree of complexity, as measured by the complexity factor and the complexity of the pattern of evolution. We consider both, nondissipative and dissipative systems. Solutions are ranged from the simplest one, in order of increasing complexity. Some specific models are found and analyzed in detail.

Complexity hierarchies in Euclidean stars

TL;DR

This work defines Euclidean stars as relativistic fluids with the Euclidean condition and equal areal/proper radii, and introduces a hierarchy of solutions by complexity using the complexity factor alongside evolution patterns and , covering both non-dissipative and dissipative regimes. The authors derive the governing equations, transport relations from causal thermodynamics, and exterior matching to a Vaidya spacetime, then classify solutions from the simplest FRW/LTB-like cases to more general anisotropic, dissipative families constrained by and either or shear-free conditions. A central result is that the FRW spacetime emerges as the simplest non-dissipative Euclidean star under and , while more complex configurations arise when relaxing these constraints, including explicit toy models and a conformally flat shear-free branch. The framework provides analytic methods to model evolving stellar configurations with controlled complexity, potentially linking arbitrary radial/time functions to observables such as surface redshift and emitted energy in realistic astrophysical scenarios.

Abstract

We establish a hierarchy of Euclidean stars according to their degree of complexity, as measured by the complexity factor and the complexity of the pattern of evolution. We consider both, nondissipative and dissipative systems. Solutions are ranged from the simplest one, in order of increasing complexity. Some specific models are found and analyzed in detail.

Paper Structure

This paper contains 19 sections, 107 equations, 3 figures.

Table of Contents

  1. Introduction
  2. The Euclidean condition and its consequences
  3. Basic equations and variables
  4. The Weyl tensor
  5. The transport equation
  6. The exterior spacetime and junction conditions
  7. The exterior spacetime and junction conditions
  8. The conditions for classifying Euclidean stars
  9. The complexity factor
  10. The homologous ($H$) and the quasi-homologous ($QH$) condition
  11. Euclidean stars and their properties
  12. A hierarchy of Euclidean stars according to their complexity
  13. Evolution with $q=0$
  14. Models satisfying $H$ condition and $Y_{TF}=0$.
  15. Evolution with $q\neq 0$
  16. ...and 4 more sections

Figures (3)

  • Figure 1: $m^\ast$ as function of $z$ in the interval $[0,1]$, and $t^\ast$ in the interval $[2,10]$
  • Figure 2: $72\pi \mu r^2_{\Sigma^{(e)}}$, $72\pi P_r r^2_{\Sigma^{(e)}}$ and $72\pi P_\bot r^2_{\Sigma^{(e)}}$ as functions of $z$ in the interval $[0,1]$, and $t^\ast$ in the interval $[2,10]$.
  • Figure 3: $8\pi \mu/\omega^2$, $8\pi P_r /3\omega^2$ and $2\pi q/3\omega^2$ as functions of $z$ in the interval $[0,1]$, and $t^\ast$ in the interval $[0,30]$.