# Expansion-Free Dissipative Fluid Spheres: Analytical Solutions

^{1}

^{2}

^{3}

^{*}

^{†}

## Abstract

**:**

## 1. Introduction

## 2. Relevant Physical and Geometric Variables, Field Equations and Junction Conditions

#### 2.1. The Exterior Spacetime and Junction Conditions

#### 2.2. The Weyl Tensor and the Complexity Factor

## 3. The Transport Equation

## 4. The Homologous and Quasi-Homologous Conditions

## 5. Shearing Expansion-Free Motion

## 6. Solutions

#### 6.1. Non-Geodesic, ${Y}_{TF}=0$, Quasi-Homologous Evolution and $\mathsf{\Theta}=0$ Solutions

#### 6.2. Non-Geodesic, ${Y}_{TF}=0$, $\mathsf{\Theta}=0$, $A=\gamma B$ and $\gamma =Constant$ Solutions

#### 6.3. Non-Geodesic, ${Y}_{TF}=0$, $\mathsf{\Theta}=0$, $A=A\left(r\right)$ and $R={R}_{1}\left(t\right){R}_{2}\left(r\right)$ Solutions

#### 6.4. Geodesic Models

## 7. Discusion

- The analytical models presented here have the main advantage of simplicity, which allows one to use them as test models for describing the evolution of voids. However, they were obtained under specific restrictions, some of which are of a purely heuristic nature. In order to get closer to a physically meaningful scenario, one should use some observational data as input for solving the field equations. At this point, the best candidate for that purpose appears to be the luminosity profile produced by the dissipative processes within the fluid. Afterward, it seems unavoidable to resort to a numerical approach in order to solve the field equations.
- In the first two models, the vanishing complexity factor condition leads to two differential equations (Equations (61) and (77)) which have been solved analytically, resorting to the heuristic ansatz in Equations (62) and (78), respectively. Of course, a much more satisfactory procedure would be to solve those equations using numerical methods. However, this would be out of the scope of this work.

## Author Contributions

## Funding

## Data Availability Statement

## Conflicts of Interest

## Appendix A. Einstein Equations

## Appendix B. Dynamical Equations

## References

- Skripkin, V.A. Point explosion in an ideal incompressible fluid in the general theory of relativity. Sov.-Phys.-Dokl.
**1960**, 135, 1072. [Google Scholar] - Stephani, H.; Kramer, D.; MacCallum, M.; Honselaers, C.; Herlt, E. Exact Solutions to Einsteins Field Equations, 2nd ed.; Cambridge University Press: Cambridge, UK, 2003. [Google Scholar]
- Herrera, L.; Santos, N.O.; Wang, A. Shearing expansion-free spherical anisotropic fluid evolution. Phys. Rev. D
**2008**, 78, 084026-10. [Google Scholar] [CrossRef] [Green Version] - Sherif, A.; Goswami, R.; Maharaj, S. Nonexistence of expansion-free dynamical stars with rotation and spatial twist. Phys. Rev. D
**2020**, 101, 104015. [Google Scholar] [CrossRef] - Sherif, A.; Goswami, R.; Maharaj, S. Properties of expansion-free dynamical stars. Phys. Rev. D
**2019**, 100, 044039. [Google Scholar] [CrossRef] [Green Version] - Sharif, M.; Yousaf, Z. Dynamical instability of the charged expansion-free spherical collapse in f(R) gravity. Phys. Rev. D
**2013**, 88, 024020. [Google Scholar] [CrossRef] - Zubair, M.; Asmat, H.; Noureen, I. Anisotropic stellar filaments evolving under expansion-free condition in f(R,T) gravity. Int. J. Mod. Phys. D
**2018**, 27, 1850047. [Google Scholar] [CrossRef] - Sharif, M.; Yousaf, Z. Expansion-free cylindrically symmetric models. Can. J. Phys.
**2012**, 90, 865. [Google Scholar] [CrossRef] [Green Version] - Sharif, M.; Ul Haq Bhatti, M.Z. Stability of the expansion-free charged cylinder. JCAP
**2013**, 10, 056. [Google Scholar] [CrossRef] - Manzoor, R.; Mumtaz, S.; Intizar, D. Dynamics of evolving cavity in cluster of stars. Eur. Phys. J. C
**2022**, 82, 739. [Google Scholar] [CrossRef] - Manzoor, R.; Ramzan, K.; Farooq, M.A. Evolution of expansion-free massive stellar object in f(R,T) gravity. Eur. Phys. J. Plus
**2023**, 138, 134. [Google Scholar] [CrossRef] - Herrera, L.; Le Denmat, G.; Santos, N.O. Dynamical instability and the expansion-free condition. Gen. Relativ. Gravit.
**2012**, 44, 1143. [Google Scholar] [CrossRef] [Green Version] - Sharif, M.; Yousaf, Z. Stability analysis of cylindrically symmetric self-gravitating systems in R+ϵR
^{2}gravity. Mon. Not. R. Astron. Soc.**2014**, 440, 3479. [Google Scholar] [CrossRef] [Green Version] - Noureen, I.; Zubair, M. Dynamical instability and expansion-free condition in f(R,T) gravity. Eur. Phys. J. C
**2015**, 75, 62. [Google Scholar] [CrossRef] - Sharif, M.; Ul Haq Bhatti, M.Z. Role of adiabatic index on the evolution of spherical gravitational collapse in Palatini f(R) gravity. Astrophys. Space Sci.
**2015**, 355, 317. [Google Scholar] [CrossRef] - Sharif, M.; Yousaf, Z. Stability analysis of expansion-free charged planar geometry. Astrophys. Space Sci.
**2015**, 355, 389. [Google Scholar] [CrossRef] - Yousaf, Z.; Ul Haq Bhatti, M.Z. Cavity evolution and instability constraints of relativistic interiors. Eur. Phys. J. C
**2016**, 76, 267. [Google Scholar] [CrossRef] [Green Version] - Tahir, M.; Abbas, G. Instability of collapsing source under expansion-free condition in Einstein-Gauss-Bonnet gravity. Chin. J. Phys.
**2019**, 61, 8. [Google Scholar] [CrossRef] - Di Prisco, A.; Herrera, L.; Ospino, J.; Santos, N.O.; Viña–Cervantes, V.M. Expansion-free cavity evolution: Some exact analytical models. Int. J. Mod. Phys. D
**2011**, 20, 2351. [Google Scholar] [CrossRef] [Green Version] - Sharif, M.; Nasir, Z. Evolution of Dissipative Anisotropic Expansion-Free Axial Fluids. Commun. Theor. Phys.
**2015**, 64, 139. [Google Scholar] [CrossRef] - Yousaf, Z. Spherical relativistic vacuum core models in a Λ dominated era. Eur. Phys. J. Plus
**2017**, 132, 71. [Google Scholar] [CrossRef] - Yousaf, Z. Stellar filaments with Minkowskian core in the Einstein-Λ gravity. Eur. Phys. J. Plus
**2017**, 132, 276. [Google Scholar] [CrossRef] - Kumar, R.; Srivastava, S. Evolution of expansion-free spherically symmetric self-gravitating non-dissipative fluids and some analytical solutions. Int. J. Geom. Methods Mod. Phys.
**2018**, 15, 1850058. [Google Scholar] [CrossRef] - Kumar, R.; Srivastava, S. Expansion-free self-gravitating dust dissipative fluids. Gen. Relativ. Gravit.
**2018**, 50, 95. [Google Scholar] [CrossRef] - Kumar, R.; Srivastava, S. Dynamics of an Expansion-Free Spherically Symmetric Radiating Star. Gravit. Cosmol.
**2021**, 27, 163. [Google Scholar] [CrossRef] - Herrera, L.; Santos, N.O. Local anisotropy in self–gravitating systems. Phys. Rep.
**1997**, 286, 53–130. [Google Scholar] [CrossRef] - Herrera, L. Stability of the isotropic pressure condition. Phys. Rev. D
**2020**, 101, 104024. [Google Scholar] [CrossRef] - Misner, C.; Sharp, D. Relativistic Equations for Adiabatic, Spherically Symmetric Gravitational Collapse. Phys. Rev.
**1964**, 136, B571. [Google Scholar] [CrossRef] - Cahill, M.; McVittie, G. Spherical Symmetry and Mass-Energy in General Relativity. I. General Theory. J. Math. Phys.
**1970**, 11, 1382. [Google Scholar] [CrossRef] - Herrera, L. New definition of complexity for self–gravitating fluid distributions: The spherically symmetric static case. Phys. Rev. D
**2018**, 97, 044010. [Google Scholar] [CrossRef] [Green Version] - Herrera, L.; Di Prisco, A.; Ospino, J. Definition of complexity for dynamical spherically symmetric dissipative self–gravitating fluid distributions. Phys. Rev. D
**2018**, 98, 104059. [Google Scholar] [CrossRef] [Green Version] - Herrera, L.; Ospino, J.; Di Prisco, A.; Fuenmayor, E.; Troconis, O. Structure and evolution of self–gravitating objects and the orthogonal splitting of the Riemann tensor. Phys. Rev. D
**2009**, 79, 064025. [Google Scholar] [CrossRef] [Green Version] - Israel, W. Nonstationary irreversible thermodynamics: A causal relativistic theory. Ann. Phys.
**1976**, 100, 310–331. [Google Scholar] [CrossRef] - Israel, W.; Stewart, J.M. Thermodynamics of nonstationary and transient effects in a relativistic gas. Phys. Lett. A
**1976**, 58, 213–215. [Google Scholar] [CrossRef] - Israel, W.; Stewart, J.M. Transient relativistic thermodynamics and kinetic theory. Ann. Phys.
**1979**, 118, 341–372. [Google Scholar] [CrossRef] - Triginer, J.; Pavón, D. Heat transport in an inhomogeneous spherically symmetric universe. Class. Quantum Gravity
**1995**, 12, 689–698. [Google Scholar] [CrossRef] - Schwarzschild, M. Structure and Evolution of the Stars; Dover: New York, NY, USA, 1958. [Google Scholar]
- Hansen, C.; Kawaler, S. Stellar Interiors: Physical Principles, Structure and Evolution; Springer: Berlin/Heidelberg, Germany, 1994. [Google Scholar]
- Kippenhahn, R.; Weigert, A. Stellar Structure and Evolution; Springer: Berlin/Heidelberg, Germany, 1990. [Google Scholar]
- Herrera, L.; Di Prisco, A.; Ospino, J. Quasi-homologous evolution of self-gravitating systems with vanishing complexity factor. Eur. Phys. J. C
**2020**, 80, 631. [Google Scholar] [CrossRef] - Liddle, A.R.; Wands, D. Microwave background constraints on extended inflation voids. Mon. Not. R. Astron. Soc.
**1991**, 253, 637. [Google Scholar] [CrossRef] [Green Version] - Peebles, P.J.E. The Void Phenomenon. Astrophys. J.
**2001**, 557, 495. [Google Scholar] [CrossRef]

Disclaimer/Publisher’s Note: The statements, opinions and data contained in all publications are solely those of the individual author(s) and contributor(s) and not of MDPI and/or the editor(s). MDPI and/or the editor(s) disclaim responsibility for any injury to people or property resulting from any ideas, methods, instructions or products referred to in the content. |

© 2023 by the authors. Licensee MDPI, Basel, Switzerland. This article is an open access article distributed under the terms and conditions of the Creative Commons Attribution (CC BY) license (https://creativecommons.org/licenses/by/4.0/).

## Share and Cite

**MDPI and ACS Style**

Herrera, L.; Di Prisco, A.; Ospino, J.
Expansion-Free Dissipative Fluid Spheres: Analytical Solutions. *Symmetry* **2023**, *15*, 754.
https://doi.org/10.3390/sym15030754

**AMA Style**

Herrera L, Di Prisco A, Ospino J.
Expansion-Free Dissipative Fluid Spheres: Analytical Solutions. *Symmetry*. 2023; 15(3):754.
https://doi.org/10.3390/sym15030754

**Chicago/Turabian Style**

Herrera, Luis, Alicia Di Prisco, and Justo Ospino.
2023. "Expansion-Free Dissipative Fluid Spheres: Analytical Solutions" *Symmetry* 15, no. 3: 754.
https://doi.org/10.3390/sym15030754