# An Alternative Study about the Geometry and the First Law of Thermodynamics for AdS Lovelock Gravity, Using the Definition of Conserved Charges

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## Abstract

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## 1. Introduction

#### 1.1. Change of the Cosmological Constant in Four Dimensions and Generalizations

#### 1.2. Thermodynamics

## 2. Phase Space and Charges

#### 2.1. Noether Charges

#### 2.2. The Presymplectic Form and Charges

#### 2.3. An Extended Covariant Phase Space Formalism

## 3. Lovelock Action Principle

#### 3.1. The Ground States and Regularization

#### 3.2. Problems and a Solution

## 4. Scale Transformations and an Improved Presymplectic Form

#### 4.1. Regularization in Even Dimensions

#### 4.2. Regularization in Odd Dimensions

## 5. Static Solution

#### 5.1. Even Dimensions

#### 5.2. Odd Dimensions

#### 5.3. Asymptotic Behavior

#### 5.4. Noether Charge in Even Dimensions

#### 5.5. Noether Charge in Odd Dimensions

#### 5.6. Variation along the Space of Solutions

#### 5.7. Hamiltonian Variation

#### 5.7.1. For $d=2n$

#### 5.7.2. For $d=2n+1$

#### 5.8. The Horizon

#### 5.9. Summary of First Law of Thermodynamics

## 6. Relevant Cases

#### 6.1. Einstein in d Dimensions

#### 6.2. Five Dimensional Einstein–Gauss–Bonnet Gravity

#### 6.3. Born-Infeld

#### 6.4. Pure Lovelock

#### 6.4.1. Even Dimensions with s Odd

#### 6.4.2. Odd Dimension with s Odd

#### 6.5. Chern–Simons Gravity

## 7. Conclusions and Prospects

## Author Contributions

## Funding

## Institutional Review Board Statement

## Informed Consent Statement

## Data Availability Statement

## Acknowledgments

## Conflicts of Interest

## Appendix A. Regulation

#### Appendix A.1. Even Dimensions

#### Appendix A.2. Odd Dimensions

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Aros, R.; Estrada, M.; Pereira, P.
An Alternative Study about the Geometry and the First Law of Thermodynamics for AdS Lovelock Gravity, Using the Definition of Conserved Charges. *Entropy* **2022**, *24*, 1197.
https://doi.org/10.3390/e24091197

**AMA Style**

Aros R, Estrada M, Pereira P.
An Alternative Study about the Geometry and the First Law of Thermodynamics for AdS Lovelock Gravity, Using the Definition of Conserved Charges. *Entropy*. 2022; 24(9):1197.
https://doi.org/10.3390/e24091197

**Chicago/Turabian Style**

Aros, Rodrigo, Milko Estrada, and Pablo Pereira.
2022. "An Alternative Study about the Geometry and the First Law of Thermodynamics for AdS Lovelock Gravity, Using the Definition of Conserved Charges" *Entropy* 24, no. 9: 1197.
https://doi.org/10.3390/e24091197