# Universal Thermodynamics in the Context of Dynamical Black Hole

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## Abstract

**:**

## 1. Introduction

- outer if ${\mathcal{L}}_{-}{\theta}_{+}{|}_{\mathcal{H}}<0$, inner if ${\mathcal{L}}_{-}{\theta}_{+}{|}_{\mathcal{H}}>0$
- future if ${\theta}_{-}{|}_{\mathcal{H}}<0$, past if ${\theta}_{-}{|}_{\mathcal{H}}>0$.

- The ingoing light rays should be converging, i.e., ${\theta}_{-}{|}_{\mathcal{H}}<0$.
- The outgoing light rays should be instantaneously parallel on the horizon, i.e., ${\theta}_{-}{|}_{\mathcal{H}}=0$.
- They are diverging just outside the horizon and converging just inside, i.e., ${\mathcal{L}}_{-}{\theta}_{+}{|}_{\mathcal{H}}<0$.

## 2. Preliminaries on Dynamical Black Holes

## 3. Universal Thermodynamics in the FLRW Model

- trapped surface: ${\theta}_{l}{\theta}_{m}>0$,
- untrapped surface: ${\theta}_{l}{\theta}_{m}<0$,
- marginal surface: ${\theta}_{l}{\theta}_{m}=0$,

- future: ${\theta}_{m}<0$,
- past: ${\theta}_{m}>0$,
- bifurcating: ${\theta}_{m}=0$,
- outer: $\frac{\partial {\theta}_{l}}{\partial m}<0$,
- inner: $\frac{\partial {\theta}_{l}}{\partial m}>0$,
- degenerate: $\frac{\partial {\theta}_{l}}{\partial m}=0$.

- projecting along the vector $\left(\frac{2}{H{R}^{3}},0\right)$ and $\left(0,-\frac{2}{3{R}^{2}a}\right)$Projecting the UFL along the vector $\left(\frac{2}{H{R}^{3}},0\right)$, we obtain:$$\begin{array}{c}\hfill 2\dot{H}+3{H}^{2}+\frac{k}{{a}^{2}}=-8\pi p.\end{array}$$Projecting the UFL along the vector $\left(0,-\frac{2}{3{R}^{2}a}\right)$, we obtain:$$\begin{array}{c}\hfill {H}^{2}+\frac{k}{{a}^{2}}=\frac{8\pi G}{3}\rho .\end{array}$$
- Projecting along the vector ${U}^{\mu}$ and ${\overline{U}}^{\mu}$

## 4. Clausius Relation from the UFL

## 5. A Redefinition of Surface Gravity

## 6. Conclusions

## Author Contributions

## Acknowledgments

## Conflicts of Interest

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Bhattacharjee, S.; Chakraborty, S. Universal Thermodynamics in the Context of Dynamical Black Hole. *Universe* **2018**, *4*, 76.
https://doi.org/10.3390/universe4070076

**AMA Style**

Bhattacharjee S, Chakraborty S. Universal Thermodynamics in the Context of Dynamical Black Hole. *Universe*. 2018; 4(7):76.
https://doi.org/10.3390/universe4070076

**Chicago/Turabian Style**

Bhattacharjee, Sudipto, and Subenoy Chakraborty. 2018. "Universal Thermodynamics in the Context of Dynamical Black Hole" *Universe* 4, no. 7: 76.
https://doi.org/10.3390/universe4070076