# Entropy Bounds and Field Equations

## Abstract

**:**

## 1. Motivation

## 2. Bound to the Entropy of an Element of Matter as Variation of the Clausius Entropy of the Hottest Horizon that is Going to Swallow It

**Figure 1.**(Piece of) the world line of an element of volume at $X=\frac{1}{\kappa}$, at rest in the inertial frame (see the text).

**Proposition**Given an element of matter of size l and energy-momentum tensor ${T}_{ab}$, its entropy $dS$ is bounded from above by the quantity $\frac{dE}{{T}_{H}}$, given by Equation (5), of a Rindler horizon at temperature ${T}_{H}=\frac{1}{\pi l}$ as perceived by an accelerating observer sitting instantaneously where matter is, with ${k}^{a}$ the tangent vector to the generators of the horizon, i.e., by the variation of Clausius entropy of a Rindler horizon at the perceived temperature ${T}_{H}=\frac{1}{\pi l}$, which engulfs the element of matter.

## 3. The Generalized Covariant Entropy Bound Extended to General Theories of Gravity

## 4. ${l}^{*}$ for Horizons

## 5. Concluding Remarks

- (i)
- $s>0$; implying $\mu <0$;
- (ii)
- $s=0$; implying $\mu n=0$.

## Acknowledgments

## Conflicts of Interest

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Pesci, A.
Entropy Bounds and Field Equations. *Entropy* **2015**, *17*, 5799-5810.
https://doi.org/10.3390/e17085799

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Pesci A.
Entropy Bounds and Field Equations. *Entropy*. 2015; 17(8):5799-5810.
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2015. "Entropy Bounds and Field Equations" *Entropy* 17, no. 8: 5799-5810.
https://doi.org/10.3390/e17085799