# Black Hole Entropy for Two Higher Derivative Theories of Gravity

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^{2}

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## Abstract

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**PACS**04.70.-s, 04.70.Dy

## 1. Introduction

## 2. The Deser-Sarioglu-Tekin Solution

- $\sigma =0$ corresponds to Einstein-Hilbert action. In fact, $a\left(r\right)=1-\frac{\widehat{c}}{r}$ and $b\left(r\right)=\widehat{k}$ and for $\widehat{c},\widehat{k}$ positive constants, the Schwarzschild solution of general relativity is recovered;
- $\sigma =1$: only the trivial, physically unacceptable, solution $a\left(r\right)=0=b\left(r\right)$ exists;
- $\sigma =\frac{1}{4}$: then, for some positive constants $\tilde{k}$ and ${r}_{0}$:$$a\left(r\right)=ln\left(\frac{{r}_{0}}{r}\right)\phantom{\rule{2.em}{0ex}}\text{and}\phantom{\rule{2.em}{0ex}}b\left(r\right)=\frac{\tilde{k}}{r}\phantom{\rule{0.277778em}{0ex}};$$
- In all other cases, the general solution to (4) turns out to be$$a\left(r\right)=\frac{1-\sigma}{1-4\sigma}-c{r}^{-\frac{1-4\sigma}{1-\sigma}}\phantom{\rule{2.em}{0ex}}\text{and}\phantom{\rule{1.em}{0ex}}b\left(r\right)={\left(\frac{r}{k}\right)}^{\frac{3\sigma}{\sigma -1}}\phantom{\rule{0.277778em}{0ex}},$$

- for $p>1$ or $0<\sigma <\frac{1}{4}$, we have that ${g}_{00}\to 0$ and ${g}_{11}\to \frac{1}{p}$ as $r\to \infty $;
- for $0<p<1$ or $\sigma \in (-\infty ,0)\cup (1,+\infty )$, we have that ${g}_{00}\to \infty $ and ${g}_{11}\to \frac{1}{p}$ as $r\to \infty $.

- ${l}^{a}$ is s.t. on the horizon$${l}_{H}^{a}\equiv {\xi}^{a}\phantom{\rule{0.277778em}{0ex}};$$
- The normalization conditions hold$$l\xb7n=-1\phantom{\rule{4pt}{0ex}}\phantom{\rule{2.em}{0ex}}\&\phantom{\rule{2.em}{0ex}}m\xb7\overline{m}=1\phantom{\rule{0.277778em}{0ex}};$$
- All the other scalar products vanishes.

**Figure 1.**Wald’s entropy in units of ${\mathcal{A}}_{H}/4$ versus σ parameter for the Deser et al. black hole.

## 3. The Clifton-Barrow Solution

## 4. Conclusions

## Acknowledgements

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**MDPI and ACS Style**

Bellini, E.; Criscienzo, R.D.; Sebastiani, L.; Zerbini, S.
Black Hole Entropy for Two Higher Derivative Theories of Gravity. *Entropy* **2010**, *12*, 2186-2198.
https://doi.org/10.3390/e12102186

**AMA Style**

Bellini E, Criscienzo RD, Sebastiani L, Zerbini S.
Black Hole Entropy for Two Higher Derivative Theories of Gravity. *Entropy*. 2010; 12(10):2186-2198.
https://doi.org/10.3390/e12102186

**Chicago/Turabian Style**

Bellini, Emilio, Roberto Di Criscienzo, Lorenzo Sebastiani, and Sergio Zerbini.
2010. "Black Hole Entropy for Two Higher Derivative Theories of Gravity" *Entropy* 12, no. 10: 2186-2198.
https://doi.org/10.3390/e12102186