# Expansion-Free Dissipative Fluid Spheres: Analytical Solutions

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## 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

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**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