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Asymptotic evolution of bulk-viscous, spherically symmetric spacetimes

Balázs Endre Szigeti, Imre Ferenc Barna, Gergely Gábor Barnaföldi

TL;DR

This work systematically classifies kinematic self-similar bulk-viscous solutions in general spherically symmetric spacetimes within the Landau–Lifshitz–Eckart framework. By enforcing $oldsymbol{\xi}$-based scaling through the indices $\alpha$ and $\delta$, and constraining the bulk viscosity via $\zeta(\rho)$ or $\zeta(\theta)$, the authors derive reduced ODE systems and map out the existence of tilted, parallel, and orthogonal cases across the Second, First, Zeroth, and Infinite kinds. They find that viscosity largely vanishes in parallel/orthogonal configurations, while tilted self-similarity admits nontrivial bulk-viscous solutions that connect to Kantowski–Sachs and FRW-like geometries; many solutions reduce to known cosmologies in appropriate limits. The results highlight that self-similar dissipative dynamics often approach asymptotically perfect-fluid behavior, suggesting bulk viscosity leaves a subtle imprint on late-time evolution. The authors propose extending the analysis to causal, second-order theories (e.g., Müller–Israel–Stewart) and incorporating shear effects for a more complete description of relativistic dissipation in strongly gravitating systems.

Abstract

The scale-free nature of gravitational interaction in both Newtonian gravity and the general theory of relativity gives rise to the concept of self-similarity, where solutions are scale invariant. As a result of this property, the governing partial differential equations are greatly simplified and can be transformed into ordinary ones. These solutions function as attractors, characterizing the asymptotic dynamics of more general solutions. There exist situations in which self-similarity is only partially realized, giving rise to kinematic self-similar solutions. Our study provides a systematic classification of kinematic self-similar solutions corresponding to the most general spherically symmetric spacetime in the presence of bulk viscous flows.

Asymptotic evolution of bulk-viscous, spherically symmetric spacetimes

TL;DR

This work systematically classifies kinematic self-similar bulk-viscous solutions in general spherically symmetric spacetimes within the Landau–Lifshitz–Eckart framework. By enforcing -based scaling through the indices and , and constraining the bulk viscosity via or , the authors derive reduced ODE systems and map out the existence of tilted, parallel, and orthogonal cases across the Second, First, Zeroth, and Infinite kinds. They find that viscosity largely vanishes in parallel/orthogonal configurations, while tilted self-similarity admits nontrivial bulk-viscous solutions that connect to Kantowski–Sachs and FRW-like geometries; many solutions reduce to known cosmologies in appropriate limits. The results highlight that self-similar dissipative dynamics often approach asymptotically perfect-fluid behavior, suggesting bulk viscosity leaves a subtle imprint on late-time evolution. The authors propose extending the analysis to causal, second-order theories (e.g., Müller–Israel–Stewart) and incorporating shear effects for a more complete description of relativistic dissipation in strongly gravitating systems.

Abstract

The scale-free nature of gravitational interaction in both Newtonian gravity and the general theory of relativity gives rise to the concept of self-similarity, where solutions are scale invariant. As a result of this property, the governing partial differential equations are greatly simplified and can be transformed into ordinary ones. These solutions function as attractors, characterizing the asymptotic dynamics of more general solutions. There exist situations in which self-similarity is only partially realized, giving rise to kinematic self-similar solutions. Our study provides a systematic classification of kinematic self-similar solutions corresponding to the most general spherically symmetric spacetime in the presence of bulk viscous flows.

Paper Structure

This paper contains 26 sections, 136 equations, 2 figures, 1 table.

Table of Contents

  1. Introduction
  2. Kinematic self-similarity of general spherical symmetric space-time
  3. Kinematic self-similar solutions
  4. Viscous Model
  5. Results
  6. Tilted Case
  7. Self-similarity of Second Kind
  8. Self-similarity of First Kind (Homothetic solution)
  9. Self-similarity of Zeroth Kind
  10. Self-similarity of Infinite Kind
  11. Parallel Case
  12. Self-similarity of the First, Second, and Zeroth Kind
  13. Self-similarity of Infinite Kind
  14. Orthogonal Case
  15. Self-similarity of the First, Second, and Zeroth Kind
  16. ...and 11 more sections

Figures (2)

  • Figure 1: Schematic view of the kinematic self-similar vector fields, with local Killing vector fields, where $\xi_t$ and $\xi_r$ represent the local basis.
  • Figure 2: The figure shows the possible 'tilted' kinematic self-similar solutions for spherically symmetric bulk-viscous space-times, classified according to the known geometries. KS(A) and KS(B) indicate two types of Kantowski -- Sach solutions.