Black Holes, Cosmological Solutions, Future Singularities, and Their Thermodynamical Properties in Modified Gravity Theories
Abstract
:1. Introduction
2. Theories of Gravity
3. Static and Spherically Symmetric Black Holes in 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 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 . This issue just requires a constant fixing (or equivalently a time reparametrization) and does not affect the solution.
3.3. Solutions Combined with Electromagnetism
3.4. Perturbations Around Schwarzschild–(anti)-de Sitter Solutions
4. Kerr–Newman Black Holes in Theories
4.1. Event Horizons
- is always a negative solution with no physical meaning,
- and are the interior and exterior horizon respectively, and
- represents—provided that it arises—the cosmological event horizon for observers between and .
- Upper spin bound, for which the BH turns —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, , 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].
5. Black Holes Thermodynamics in 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 .
- slow BH, presents two phase transitions for two determined values of .
- 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 value, and negative onwards. Thus, this BH is unstable against tunneling decay into radiation for mass parameter values of . For , 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: , as follows:
- −
- For and for , both the heat capacity and the free energy are positive, which means that the BH is unstable and decays into radiation by tunneling.
- −
- If , 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 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:
6.1.2. Model II:
6.2. KN–AdS Thermodynamics
6.2.1. Model I:
- Region 3 . For this region, the value of decreases suddenly from its normal value to 0 near the frontier of the region, i.e., and ; other values of α and β display a relatively low curvature and the existence of BH is assured.
- Region 4 . This region only shows problems when , where , but, as β becomes more negative, the surface of slowly acquires higher values, recovering its usual value for .
6.2.2. Model II:
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 , and , .
- Type II (“Sudden”): For , and , .
- Type III: For , and , .
- Type IV: For , and , 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
References
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