# Black Holes, Cosmological Solutions, Future Singularities, and Their Thermodynamical Properties in Modified Gravity Theories

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## Abstract

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## 1. Introduction

## 2. $F\left(R\right)$ Theories of Gravity

## 3. Static and Spherically Symmetric Black Holes in $f\left(R\right)$ Gravities

#### 3.1. Spherically Symmetric and Static Constant Curvature Solutions: Generalities

#### 3.2. Spherically Symmetric and Static Constant Curvature Solutions

- Firstly, by comparison with Equation (19), one can see that the term with the power ${r}^{2-D}$ is absent. This fact will be studied in Section 3.3.
- Secondly, this solution has no constant curvature in the general case since, as we found above, the constant curvature requirement demands ${C}_{1}=1$. This issue just requires a constant fixing (or equivalently a time reparametrization) and does not affect the solution.

#### 3.3. $f\left(R\right)$ Solutions Combined with Electromagnetism

#### 3.4. Perturbations Around Schwarzschild–(anti)-de Sitter Solutions

## 4. Kerr–Newman Black Holes in $f\left(R\right)$ Theories

#### 4.1. Event Horizons

- ${r}_{-}$ is always a negative solution with no physical meaning,
- ${r}_{int}$ and ${r}_{ext}$ are the interior and exterior horizon respectively, and
- ${r}_{cosm}$ represents—provided that it arises—the cosmological event horizon for observers between ${r}_{ext}$ and ${r}_{cosm}$.

**Figure 1.**Graphics showing horizons positions as solutions of the equation ${\Delta}_{r}=0$ taken from [120,121]. On the left panel (${R}_{0}<0$) the presented cases are: $h>0$ (

**I**), BH with well-defined horizons, dashed with dots), $h=0$ (

**II**), extremal BH, continuous line) and $h<0$ (

**III**), naked singularity, dashed). On the right panel (${R}_{0}>0$) the represented cases are: $h<0$ (

**I**, BH with well-defined horizons, dashed with dots), $h=0$ (

**II**, $extremal$ BH and

**III**, $extremal$ $marginal$ BH, continuous line), and $h>0$ (

**IV**, naked singularity and

**V**, naked marginal singularity, dashed).

- Upper spin bound, $a={a}_{max}$ for which the BH turns $extremal$—the interior and exterior horizons have merged into a single horizon with a null surface gravity. This is the usual configuration for the BH to become extremal.
- Lower spin bound, $a={a}_{min}$, below which the BH turns into a marginal extremal BH. This value can be understood as the cosmological limit for which a BH preserves its exterior horizon without being “torn apart" due to the relative recession speed between two radially separated points induced by the cosmic expansion [120,121].

**Figure 2.**The shaded regions, delimited by the upper ${a}_{max}$ and lower ${a}_{min}$ curves, represent the values of $a/M$ for which the existence of BH is possible once ${R}_{0}\phantom{\rule{0.166667em}{0ex}}{M}^{2}$ value is fixed. Panels show for $\overline{Q}\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}M=0$ (left) and $\overline{Q}\phantom{\rule{0.166667em}{0ex}}/\phantom{\rule{0.166667em}{0ex}}M=0.75$ (right) on the left and right panels respectively. Note that ${R}_{0}$ has dimensions of [length]${}^{-2}$ when normalizing. Original plots at [120,121].

## 5. Black Holes Thermodynamics in $f\left(R\right)$ Theories

#### 5.1. BH Thermodynamics for AdS Configuration

#### 5.2. BH Thermodynamics for KN Configuration

- fast BH, without phase transitions and always positive heat capacity $C>0$.
- slow BH, presents two phase transitions for two determined values of ${r}_{ext}$.

**Figure 3.**For ${R}_{0}=-0.2$, we graphically display temperature (left) and heat capacity (right) of a BH as functions of the mass parameter M for the cases: (

**I**) $a=0.5$ y $Q=0$ : “slow” BH that shows a local maximum temperature ${T}_{max}$ and a local minimum temperature ${T}_{min}$ at the points where the heat capacity diverges, taking the latter negative values between ${T}_{max}$ y ${T}_{min}$; (

**II**) $a\approx 0.965$ y $Q=0$ : limit case where ${T}_{max}$ and ${T}_{min}$ merge, hence resulting in an inflection point in the temperature and an always positive heat capacity; (

**III**) $a=1.5$ y $Q=0$ : “fast” BH with both temperature and heat capacity monotonously growing (always positive too).

- Fast BH, with bigger values of the spin and the electric charge than the slow ones, shows a heat capacity always positive and a positive free energy up to a $M={M}^{\phantom{\rule{0.166667em}{0ex}}\mathit{\text{limit}}}$ value, and negative onwards. Thus, this BH is unstable against tunneling decay into radiation for mass parameter values of $M<{M}^{\phantom{\rule{0.166667em}{0ex}}\mathit{\text{limit}}}$. For $M>{M}^{\phantom{\rule{0.166667em}{0ex}}\mathit{\text{limit}}}$, free energy becomes negative, therefore smaller than that of pure radiation, that will tend to collapse to the BH configuration in equilibrium with thermal radiation.
- Slow BH shows a more complex thermodynamics, being necessary to distinguish between four regions delimited by the mass parameter values: ${M}^{min}<{M}^{\mathrm{I}}<{M}^{\mathrm{II}}<{M}^{\phantom{\rule{0.166667em}{0ex}}\mathit{\text{limit}}}$, as follows:
- −
- For ${M}^{min}<M<{M}^{\mathrm{I}}$ and for ${M}^{\mathrm{II}}<M<{M}^{\phantom{\rule{0.166667em}{0ex}}\mathit{\text{limit}}}$, both the heat capacity and the free energy are positive, which means that the BH is unstable and decays into radiation by tunneling.
- −
- If ${M}^{\mathrm{I}}<M<{M}^{\mathrm{II}}$, the heat capacity becomes negative but free energy remains positive, being therefore unstable and decays into pure thermal radiation or to larger values of mass.
- −
- Finally, for $M>{M}^{\phantom{\rule{0.166667em}{0ex}}\mathit{\text{limit}}}$ the heat capacity is positive whereas the free energy is now negative, thus tending pure radiation to tunnel to the BH configuration in equilibrium with thermal radiation.

## 6. Thermodynamics in AdS and Kerr–Newman: Particular Examples

#### 6.1. Thermodynamics in AdS

#### 6.1.1. Model I: $f\left(R\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\alpha {(-R)}^{\beta}$

**Figure 4.**Thermodynamical regions in the $\left(\right|\alpha |,|\beta \left|\right)$ plane for Model I in $D=4$ (left) and $D=5$ (right).

**Figure 5.**Thermodynamical regions in the $\left(\right|\alpha |,|\beta \left|\right)$ plane for Model I in $D=10$.

#### 6.1.2. Model II: $f\left(R\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}-\alpha \phantom{\rule{0.166667em}{0ex}}\frac{{\displaystyle \kappa {\left(\frac{R}{\alpha}\right)}^{n}}}{{\displaystyle 1+\beta {\left(\frac{R}{\alpha}\right)}^{n}}}$

#### 6.2. KN–AdS Thermodynamics

#### 6.2.1. Model I: $f\left(R\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}\alpha {(-R)}^{\beta}$

- Region 3 $\left\{\alpha <0,\phantom{\rule{0.166667em}{0ex}}\beta >2\right\}$. For this region, the value of ${a}_{max}$ decreases suddenly from its normal value to 0 near the frontier of the region, i.e., $\alpha \approx 0$ and $\beta \approx 2$; other values of α and β display a relatively low curvature and the existence of BH is assured.
- Region 4 $\left\{\alpha >0,\phantom{\rule{0.166667em}{0ex}}\beta <1\right\}$. This region only shows problems when $0<\beta <1$, where ${a}_{max}\to 0$, but, as β becomes more negative, the surface of ${a}_{max}$ slowly acquires higher values, recovering its usual value for $\beta =-2$.

**Figure 6.**

**Model I**: Region 3: $\left\{\alpha <0,\phantom{\rule{0.166667em}{0ex}}\beta >2\right\}$, and Region 4: $\left\{\alpha >0,\phantom{\rule{0.166667em}{0ex}}\beta <1\right\}$. BH with a well defined horizon structure will only exist if they have a spin parameter below the upper surface ${a}_{max}$. For the presented regions 3 and 4, the surface ${a}_{min}$ does not exist.

**Figure 7.**Thermodynamical regions for Model I. We distinguish between three different regions: (

**i**) $C<0$ and $F>0$, in black; (

**ii**) $C>0$ and $F>0$, in gray; (

**iii**) $C>0$ and $F<0$, in white.

#### 6.2.2. Model II: $f\left(R\right)\phantom{\rule{0.166667em}{0ex}}=\phantom{\rule{0.166667em}{0ex}}-\alpha \phantom{\rule{0.166667em}{0ex}}\frac{{\displaystyle \kappa {\left(\frac{R}{\alpha}\right)}^{n}}}{{\displaystyle 1+\beta {\left(\frac{R}{\alpha}\right)}^{n}}}$

**Figure 8.**

**Model II**. Region 1: $\left\{\kappa >1,\phantom{\rule{0.166667em}{0ex}}\gamma >0\right\}$, and Region 2: $\left\{\kappa >1,\phantom{\rule{0.166667em}{0ex}}\gamma <0\right\}$. BH with a well-defined horizon structure will only exist if they have a spin parameter below the upper surface ${a}_{max}$, and above a second surface ${a}_{min}$ (in case it exists, only in region 1 for this model) for certain values of κ and γ.

**Figure 9.**Thermodynamical regions with negative scalar curvature ${R}_{0}<0$ of model I. We distinguish between three different regions: (

**i**) $C<0$ and $F>0$, in black; (

**ii**) $C>0$ and $F>0$, in gray; (

**iii**) $C>0$ and $F<0$, in white.

## 7. Cosmological Solutions in Modified Gravity

## 8. First Law of Thermodynamics and FLRW Equations

## 9. Generalization of Cardy–Verlinde Formula

#### 9.1. Multicomponent Universe

#### 9.2. Inhomogeneous EoS Fluid and Modified Gravity

## 10. On the Cosmological Bounds Near Future Singularities

- Type I (“Big Rip”): For $t\to {t}_{s}$, $a\to \infty $ and $\rho \to \infty $, $\left|p\right|\to \infty $.
- Type II (“Sudden”): For $t\to {t}_{s}$, $a\to {a}_{s}$ and $\rho \to {\rho}_{s}$, $\left|p\right|\to \infty $.
- Type III: For $t\to {t}_{s}$, $a\to {a}_{s}$ and $\rho \to \infty $, $\left|p\right|\to \infty $.
- Type IV: For $t\to {t}_{s}$, $a\to {a}_{s}$ and $\rho \to {\rho}_{s}$, $p\to {p}_{s}$ but higher derivatives of Hubble parameter diverge (see [164]).

#### 10.1. Big Rip Singularity

#### 10.2. Sudden Singularity

#### 10.3. Type III Singularity

#### 10.4. Type IV Singularity

#### 10.5. Big Bang Singularity

## 11. Conclusions

## Acknowledgments

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