# Evolving Black Hole Horizons in General Relativity and Alternative Gravity

## Abstract

**:**

## 1. Introduction

**Figure 1.**The conformal diagram of a hypothetical cosmological black hole. The bottom horizontal (dashed) line represents a Big Bang singularity. The top horizontal line (dashed) is a spacelike black hole singularity. An apparent horizon (marked AH) can change from timelike, to null, to spacelike, and it can be located inside or outside the event horizon (the forty-five-degree line marked EH), according to whether the energy conditions are satisfied or not. If ${R}_{ab}{l}^{a}{l}^{b}\ge 0$ for all null vectors, ${l}^{a}$, then the apparent horizon lies inside the event horizon ([3], p. 311).

## 2. Various Notions of Horizon

#### 2.1. Null Geodesic Congruences and Trapped Surfaces

- A normal surface corresponds to ${\theta}_{l}>0$ and ${\theta}_{n}<0$ (for example, a two-sphere in Minkowski space satisfies this property).
- A trapped surface [83] corresponds to ${\theta}_{l}<0$ and ${\theta}_{n}<0$. The outgoing, in addition to the ingoing, future-directed null rays converge here instead of diverging, and outward-propagating light is dragged back by strong gravity.
- A marginally outer trapped (or marginal) surface (MOTS) corresponds to ${\theta}_{l}=0$ (where ${l}^{a}$ is the outgoing null normal to the surface) and ${\theta}_{n}<0$.
- An untrapped surface is one with ${\theta}_{l}{\theta}_{n}<0$.
- An anti-trapped surface corresponds to ${\theta}_{l}>0$ and ${\theta}_{n}>0$ (both outgoing and ingoing future-directed null rays are diverging).
- A marginally outer trapped tube (MOTT) is a three-dimensional surface, which can be foliated entirely by marginally outer trapped (two-dimensional) surfaces.

#### 2.2. Event Horizons

#### 2.3. Killing Horizons

#### 2.4. Apparent Horizons

#### 2.5. Trapping Horizons

#### 2.6. Isolated, Dynamical and Slowly Evolving Horizons

#### 2.7. Kodama Vector

#### 2.8. Surface Gravities

#### 2.9. Spherical Symmetry

## 3. Evolving Horizons, Cosmological Black Holes and Naked Singularities in GR

#### 3.1. The Schwarzschild-de Sitter-Kottler Spacetime

#### 3.2. The McVittie Solution

- For $t<{t}_{*}$, it is ${m}_{0}>\frac{1}{3\sqrt{3}\phantom{\rule{0.166667em}{0ex}}H\left(t\right)}$, and both ${R}_{1}\left(t\right)$ and ${R}_{2}\left(t\right)$ are complex. There are no apparent horizons.
- The critical time $t={t}_{*}$ corresponds to ${m}_{0}=\frac{1}{3\sqrt{3}\phantom{\rule{0.166667em}{0ex}}H\left(t\right)}$. ${R}_{1}\left(t\right)$ and ${R}_{2}\left(t\right)$ coincide at a real value, and there is a single apparent horizon at ${R}_{*}=\frac{1}{\sqrt{3}\phantom{\rule{0.166667em}{0ex}}H\left({t}_{*}\right)}$.
- For $t>{t}_{*}$, it is ${m}_{0}<\frac{1}{3\sqrt{3}\phantom{\rule{0.166667em}{0ex}}H\left(t\right)}$, and there are two apparent horizons of real positive radii, ${R}_{1}\left(t\right)$ and ${R}_{2}\left(t\right)$.

**Figure 2.**The McVittie cosmological (dashed) and black hole (solid) apparent horizons in a dust-dominated background universe. Time t (on the horizontal axis) and radius R (on the vertical axis) are in units of ${m}_{0}$, and we arbitrarily fix ${m}_{0}=1$.

#### 3.2.1. A Phantom Background

**Figure 3.**The radii of the McVittie apparent horizons (vertical axis) versus time (horizontal axis) in a phantom-dominated universe (here, $w=-1.5$ and ${t}_{\text{rip}}=0$).

#### 3.3. Area Quantization and McVittie Solutions as Toy Models

#### 3.4. Generalized McVittie Solutions

#### 3.4.1. Single Perfect Fluid

#### 3.4.2. Imperfect Fluid and No Radial Mass Flow

#### 3.4.3. Imperfect Fluid and Radial Mass Flow

#### 3.4.4. The “Comoving Mass” Solution

#### 3.4.5. The General Class of Solutions

#### 3.4.6. Attractor Behaviour of the “Comoving Mass” Solution

#### 3.5. The Sultana-Dyer Solution

#### 3.6. The Husain-Martinez-Nuñez Solution

**Figure 5.**The radii of the apparent horizons of the Husain-Martinez-Nuñez spacetime (vertical axis) versus comoving time (horizontal axis) for $\alpha =\sqrt{3}/2$ [t and R are measured in arbitrary units of length, and the parameter values are chosen so that ${\left(C{a}_{0}\right)}^{3/2}={10}^{3}$ in Equation (175)].

**Figure 6.**The radius of the Husain-Martinez-Nuñez apparent horizon (vertical axis) versus comoving time (horizontal axis) for $\alpha =-\sqrt{3}/2$. There is always only one expanding cosmological apparent horizon, and there is a naked singularity at $R=0$.

#### 3.7. The Fonarev and Generalized Fonarev Solutions

#### 3.7.1. A Generalized Fonarev Solution

#### 3.8. The Swiss-Cheese Model

#### 3.9. Other GR Solutions

## 4. Some Cosmological Black Holes and Naked Singularities in Alternative Gravity

#### 4.1. The Conformal Cousin of the Husain-Martinez-Nuñez Solution

#### 4.2. The Brans-Dicke Solutions of Clifton, Mota and Barrow

#### 4.3. Clifton’s Solution of $f\left({{R}^{c}}_{c}\right)$ Gravity

#### 4.4. Other Solutions

## 5. Conclusions

- Based on the type of matter filling the background FLRW universe (e.g., dust, general perfect fluid, imperfect fluid or scalar field);
- Based on the phenomenology of the apparent horizons.

## Acknowledgments

## Conflicts of Interest

## References

- Poisson, E. A Relativist’s Toolkit: The Mathematics of Black-Hole Mechanics; Cambridge University Press: Cambridge, UK, 2004. [Google Scholar]
- Frolov, V.P.; Novikov, I.D. Black Hole Physics, Basic Concepts and New Developments; Kluwer Academic Publishing: Dordrecht, The Netherlands, 1998. [Google Scholar]
- Wald, R.M. General Relativity; Chicago University Press: Chicago, IL, USA, 1984. [Google Scholar]
- Rindler, W. Visual horizons in world-models. Mon. Not. R. Astron. Soc.
**1956**, 116, 662–677, reprinted in Gen. Relativ. Gravit.**2002**, 34, 133–153. [Google Scholar] [CrossRef] - Booth, I. Black hole boundaries. Can. J. Phys.
**2005**, 83, 1073–1099. [Google Scholar] [CrossRef] - Nielsen, A.B. Black holes and black hole thermodynamics without event horizons. Gen. Relativ. Gravit.
**2009**, 41, 1539–1584. [Google Scholar] [CrossRef] - Ashtekar, A.; Krishnan, B. Isolated and dynamical horizons and their applications. Living Rev. Relativ.
**2004**, 7, 10. [Google Scholar] [CrossRef] - Gourghoulhon, E.; Jaramillo, J.L. New theoretical approaches to black holes. New Astron. Rev.
**2008**, 51, 791–798. [Google Scholar] [CrossRef] - Ben-Dov, I. Penrose inequality and apparent horizons. Phys. Rev. D
**2005**, 70, 124031. [Google Scholar] [CrossRef] - Ashtekar, A.; Galloway, G.J. Some uniqueness results for dynamical horizons. Adv. Theor. Math. Phys.
**2005**, 9, 1–30. [Google Scholar] [CrossRef] - Thornburg, J. Event and apparent horizon finders for 3 + 1 numerical relativity. Living Rev. Relativ.
**2007**, 10, 3. [Google Scholar] [CrossRef] - Baumgarte, T.W.; Shapiro, S.L. Numerical relativity and compact binaries. Phys. Rep.
**2003**, 376, 41–131. [Google Scholar] [CrossRef] - Chu, T.; Pfeiffer, H.P.; Cohen, M.I. Horizon dynamics of distorted rotating black holes. Phys. Rev. D
**2011**, 83, 104018. [Google Scholar] [CrossRef] - Booth, I. Two physical characteristics of numerical apparent horizons. Can. J. Phys.
**2008**, 86, 669–673. [Google Scholar] [CrossRef] - Kolb, E.W.; Turner, M.S. The Early Universe; Addison-Wesley: Reading, MA, USA, 1990. [Google Scholar]
- Liddle, A.R.; Lyth, D.H. Cosmological Inflation and Large Scale Structure; Cambridge University Press: Cambridge, UK, 2000. [Google Scholar]
- Mukhanov, V. Physical Foundations of Cosmology; Cambridge University Press: Cambridge, UK, 2005. [Google Scholar]
- Hawking, S.W. Black hole explosions. Nature
**1970**, 248, 30–31. [Google Scholar] [CrossRef] - Hawking, S.W. Particle creation by black holes. Commun. Math. Phys.
**1975**, 43, 199–220, erratum in**1976**, 46, 206. [Google Scholar] [CrossRef] - Gibbons, W.; Hawking, S.W. Cosmological event horizon, thermodynamics, and particle creation. Phys. Rev. D
**1977**, 15, 2738–2751. [Google Scholar] [CrossRef] - Collins, W. Mechanics of apparent horizons. Phys. Rev. D
**1992**, 45, 495–498. [Google Scholar] [CrossRef] - Hayward, S.A. General laws of black hole dynamics. Phys. Rev. D
**1994**, 49, 6467–6474. [Google Scholar] [CrossRef] - Faraoni, V. Black hole entropy in scalar-tensor and f(R) gravity: An overview. Entropy
**2010**, 12, 1246–1263. [Google Scholar] [CrossRef] - Afshordi, N. Where will Einstein fail? Lessons for gravity and cosmology. ArXiv E-Prints
**2012**. arXiv:1203.3827. [Google Scholar] - Sotiriou, T.P.; Faraoni, V. f(R) theories of gravity. Rev. Mod. Phys.
**2010**, 82, 451–497. [Google Scholar] [CrossRef] - De Felice, A.; Tsujikawa, S. f(R) theories. Living Rev. Relativ.
**2010**, 13, 3. [Google Scholar] [CrossRef] - Capozziello, S.; Faraoni, V. Beyond Einstein Gravity; Springer: New York, NY, USA, 2010. [Google Scholar]
- Amendola, L.; Tsujikawa, S. Dark Energy, Theory and Observations; Cambridge University Press: Cambridge, UK, 2010. [Google Scholar]
- Wall, A.C. Testing the generalized second law in 1 + 1 dimensional conformal vacua: An argument for the causal horizon. Phys. Rev. D
**2012**, 85, 024015. [Google Scholar] [CrossRef] - Brans, C.; Dicke, R.H. Mach’s principle and a relativistic theory of gravitation. Phys. Rev.
**1961**, 124, 925–935. [Google Scholar] [CrossRef] - Bergmann, P.G. Comments on the scalar tensor theory. Int. J. Theor. Phys.
**1968**, 1, 25–36. [Google Scholar] [CrossRef] - Wagoner, R.V. Scalar-tensor theory and gravitational waves. Phys. Rev. D
**1970**, 1, 3209–3216. [Google Scholar] [CrossRef] - Nordtvedt, K. Post-Newtonian metric for a general class of scalar tensor gravitational theories and observational consequences. Astrophys. J.
**1970**, 161, 1059–1067. [Google Scholar] [CrossRef] - McVittie, G.C. The mass-particle in an expanding universe. Mon. Not. R. Astron. Soc.
**1933**, 93, 325–339. [Google Scholar] [CrossRef] - Carrera, M.; Giulini, D. Influence of global cosmological expansion on local dynamics and kinematics. Rev. Mod. Phys.
**2010**, 82, 169–208. [Google Scholar] [CrossRef] - Kaloper, N.; Kleban, M.; Martin, D. McVitties legacy: Black holes in an expanding universe. Phys. Rev. D
**2010**, 81, 104044. [Google Scholar] [CrossRef] - Lake, K.; Abdelqader, M. More on McVittie’s legacy: A Schwarzschild-de Sitter black and white hole embedded in an asymptotically ΛCDM cosmology. Phys. Rev. D
**2011**, 84, 044045. [Google Scholar] [CrossRef] - Anderson, M. Horizons, singularities and causal structure of the generalized McVittie space-times. J. Phys. Conf. Ser.
**2011**, 283, 012001. [Google Scholar] [CrossRef] - Nandra, R.; Lasenby, A.N.; Hobson, M.P. The effect of a massive object on an expanding universe. Mon. Not. R. Astron. Soc.
**2012**, 422, 2931–2944. [Google Scholar] [CrossRef] - Nandra, R.; Lasenby, A.N.; Hobson, M.P. The effect of an expanding universe on massive objects. Mon. Not. R. Astron. Soc.
**2012**, 422, 2945–2959. [Google Scholar] [CrossRef] - Faraoni, V.; Zambrano Moreno, A.F.; Nandra, R. Making sense of the bizarre behavior of horizons in the McVittie spacetime. Phys. Rev. D
**2012**, 85, 083526. [Google Scholar] [CrossRef] - Da Silva, A.; Fontanini, M.; Guariento, D.C. How the expansion of the universe determines the causal structure of McVittie spacetimes. Phys. Rev. D
**2013**, 87, 064030. [Google Scholar] [CrossRef] - Krasiński, A. Inhomogeneous Cosmological Models; Cambridge University Press: Cambridge, UK, 1997. [Google Scholar]
- Buchert, T. On average properties of inhomogeneous fluids in general relativity. 1. Dust cosmologies. Gen. Relativ. Gravit.
**2000**, 32, 105–125. [Google Scholar] [CrossRef] - Buchert, T.; Carfora, M. Regional averaging and scaling in relativistic cosmology. Class. Quantum Gravity
**2002**, 19, 6109–6105. [Google Scholar] [CrossRef] - Kolb, E.W.; Matarrese, S.; Riotto, A. On cosmic acceleration without dark energy. New J. Phys.
**2006**, 8, 322–353. [Google Scholar] [CrossRef] - Larena, J.; Buchert, T.; Alimi, J.-M. Correspondence between kinematical backreaction and scalar field cosmologies: The “Morphon field”. Class. Quantum Gravity
**2006**, 23, 6379–6408. [Google Scholar] - Paranjape, A.; Singh, T.P. The spatial averaging limit of covariant macroscopic gravity: Scalar corrections to the cosmological equations. Phys. Rev. D
**2007**, 76, 044006. [Google Scholar] [CrossRef] - Li, N.; Schwarz, D.J. Onset of cosmological backreaction. Phys. Rev. D
**2007**, 76, 083011. [Google Scholar] [CrossRef] - Wiltshire, D.L. Cosmic clocks, cosmic variance and cosmic averages. New J. Phys.
**2007**, 9, 377–449. [Google Scholar] [CrossRef] - Wiltshire, D.L. Exact solution to the averaging problem in cosmology. Phys. Rev. Lett.
**2007**, 99, 251101. [Google Scholar] [CrossRef] [PubMed] - Buchert, T. Dark Energy from structure: A status report. Gen. Relativ. Gravit.
**2008**, 40, 467–527. [Google Scholar] [CrossRef] - Li, N.; Schwartz, D.J. Scale dependence of cosmological backreaction. Phys. Rev. D
**2008**, 78, 083531. [Google Scholar] [CrossRef] - Larena, J.; Alimi, J.-M.; Buchert, T.; Kunz, M.; Corasaniti, P. Testing backreaction effects with observations. Phys. Rev. D
**2009**, 79, 083011. [Google Scholar] [CrossRef] - Tsagas, C.G.; Challinor, A.; Maartens, R. Relativistic cosmology and large-scale structure. Phys. Rep.
**2008**, 465, 61–147. [Google Scholar] [CrossRef][Green Version] - Vitagliano, V.; Liberati, S.; Faraoni, V. Averaging inhomogeneities in scalar-tensor cosmology. Class. Quantum Grav.
**2009**, 26, 215005. [Google Scholar] [CrossRef] - Green, S.R.; Wald, R.M. A new framework for analyzing the effects of small scale inhomogeneities in cosmology. Phys. Rev. D
**2011**, 83, 084020. [Google Scholar] [CrossRef] - Boleiko, K.; Célérier, M.-N.; Krasiński, A. Inhomogeneous cosmological models: Exact solutions and their applications. Class. Quantum Gravity
**2011**, 28, 164002. [Google Scholar] [CrossRef] - Babichev, E.; Dokuchaev, V.; Eroshenko, Yu. Black hole mass decreasing due to phantom energy accretion. Phys. Rev. Lett.
**2004**, 93, 021102. [Google Scholar] [CrossRef] [PubMed] - Chen, S.; Jing, J. Quasinormal modes of a black hole surrounded by quintessence. Class. Quantum Gravity
**2005**, 22, 4651–4657. [Google Scholar] [CrossRef] - Izquierdo, G.; Pavon, D. The generalized second law in phantom dominated universes in the presence of black holes. Phys. Lett. B
**2006**, 639, 1–4. [Google Scholar] [CrossRef] - De Freitas Pacheco, J.A.; Horvath, J.E. Generalized second law and phantom cosmology: Accreting black holes. Class. Quantum Gravity
**2007**, 24, 5427–5434. [Google Scholar] [CrossRef] - Maeda, H.; Harada, T.; Carr, B.J. Self-similar cosmological solutions with dark energy. II. Black holes, naked singularities and wormholes. Phys. Rev. D
**2008**, 77, 024023. [Google Scholar] [CrossRef] - Gao, G.; Chen, X.; Faraoni, V.; Shen, Y.-G. Does the mass of a black hole decrease due to accretion of phantom energy? Phys. Rev. D
**2008**, 78, 024008. [Google Scholar] [CrossRef] - Guariento, D.C.; Horvath, J.E.; Custodio, P.S.; de Freitas Pacheco, J.A. Evolution of primordial black holes in a radiation and phantom energy environment. Gen. Rel. Grav.
**2008**, 40, 1593–1602. [Google Scholar] [CrossRef] - Lima, J.A.S.; Pereira, S.H.; Horvath, J.E.; Guariento, D.C. Phantom accretion by black holes and the generalized second law of thermodynamics. Astropart. Phys.
**2010**, 33, 292–295. [Google Scholar] [CrossRef] - De Lima, J.A.S.; Guariento, D.C.; Horvath, J.E. ; Analytical solutions of accreting black holes immersed in a Lambda-CDM model. Phys. Lett. B
**2010**, 693, 218–220. [Google Scholar] [CrossRef] - Guariento, D.C.; Horvath, J.E. Consistency of the mass variation formula for black holes accreting cosmological fluids. Gen. Relativ. Gravit.
**2012**, 44, 985–992. [Google Scholar] [CrossRef] - Guariento, D.C.; Fontanini, M.; da Silva, A.M.; Abdalla, E. Realistic fluids as source for dynamically accreting black holes in a cosmological background. Phys. Rev. D
**2012**, 86, 124020. [Google Scholar] [CrossRef] - Le Delliou, M.; Mimoso, J.P.; Mena, F.C.; Fontanini, M.; Guariento, D.C.; Abdalla, E. Separating expansion and collapse in general fluid models with heat flux. Phys. Rev. D
**2013**, 88, 027301. [Google Scholar] [CrossRef] - Sun, C.-Y. Phantom energy accretion onto black holes in a cyclic universe. Phys. Rev. D
**2008**, 78, 064060. [Google Scholar] [CrossRef] - Sun, C.-Y. Dark Energy accretion onto a black hole in an expanding universe. Commun. Theor. Phys.
**2009**, 52, 441–444. [Google Scholar] - Gonzalez, J.A.; Guzman, F.S. Accretion of phantom scalar field into a black hole. Phys. Rev. D
**2009**, 79, 121501. [Google Scholar] [CrossRef] - He, X.; Wang, B.; Wu, S.-F.; Lin, C.Y. Quasinormal modes of black holes absorbing dark energy. Phys. Lett. B
**2009**, 673, 156–160. [Google Scholar] [CrossRef] - Babichev, E.O.; Dokuchaev, V.I.; Eroshenko, Yu.N. Perfect fluid and scalar field in the Reissner-Nordstrom metric. J. Exp. Theor. Phys.
**2011**, 112, 784–793. [Google Scholar] [CrossRef] - Nouicer, K. Hawking radiation and thermodynamics of dynamical black holes in phantom dominated universe. Class. Quantum Gravity
**2011**, 28, 015005. [Google Scholar] [CrossRef] - Chadburn, S.; Gregory, R. Time dependent black holes and scalar hair. ArXiv E-Prints
**2013**. arXiv:1304.6287. [Google Scholar] [CrossRef] - Hawking, S.W.; Ellis, G.F.R. The Large Scale Structure of Space-Time; Cambridge University Press: Cambridge, UK, 1973. [Google Scholar]
- Nielsen, A.B.; Yoon, J.H. Dynamical surface gravity. Class. Quantum Gravity
**2008**, 25, 085010. [Google Scholar] [CrossRef] - Pielahn, M.; Kunstatter, G.; Nielsen, A.B. Dynamical surface gravity in spherically symmetric black hole formation. Phys. Rev. D
**2011**, 84, 104008. [Google Scholar] [CrossRef] - Booth, I.; Brits, L.; Gonzalez, J.A.; van den Broeck, V. Marginally trapped tubes and dynamical horizons. Class. Quantum Gravity
**2006**, 23, 413. [Google Scholar] [CrossRef] - Ben-Dov, I. Outer trapped surfaces in Vaidya spacetimes. Phys. Rev. D
**2007**, 75, 064007. [Google Scholar] [CrossRef] - Penrose, R. Gravitational collapse and spacetime singularities. Phys. Rev. Lett.
**1965**, 14, 57–59. [Google Scholar] [CrossRef] - Eardley, D. Black hole boundary conditions and coordinate conditions. Phys. Rev. D
**1998**, 57, 2299–2304. [Google Scholar] [CrossRef] - Andersson, L.; Mars, M.; Simon, W. Local existence of dynamical and trapping horizons. Phys. Rev. Lett.
**2005**, 95, 11102. [Google Scholar] [CrossRef] - Hawking, S.W. Black holes in general relativity. Commun. Math. Phys.
**1972**, 25, 152–166. [Google Scholar] [CrossRef] - Bengtsson, I.; Senovilla, J.M.M. Region with trapped surfaces in spherical symmetry, its core, and their boundaries. Phys. Rev. D
**2011**, 83, 044012. [Google Scholar] [CrossRef] - Bengtsson, I. Some examples of trapped surfaces. ArXiv E-Prints
**2011**. arXiv:1112.5318. [Google Scholar] - Chriusciel, P.T. Uniqueness of stationary, electro-vacuum black holes revisited. Helv. Phys. Acta
**1996**, 69, 529–552. [Google Scholar] - Wald, R.M. The thermodynamics of black holes. Living Rev. Relativ.
**2001**, 4, 6. [Google Scholar] [CrossRef] - Dyer, C.C.; Honig, E. Conformal Killing horizons. J. Math. Phys.
**1979**, 20, 409–412. [Google Scholar] [CrossRef] - Sultana, J.; Dyer, C.C. Conformal Killing horizons. J. Math. Phys.
**2004**, 45, 4764–4776. [Google Scholar] [CrossRef] - Sultana, J.; Dyer, C.C. Cosmological black holes: A black hole in the Einstein-de Sitter universe. Gen. Relativ. Gravit.
**2005**, 37, 1349–1370. [Google Scholar] [CrossRef] - McClure, M.L.; Dyer, C.C. Asymptotically Einstein-de Sitter cosmological black holes and the problem of energy conditions. Class. Quantum Gravity
**2006**, 23, 1971–1987. [Google Scholar] [CrossRef] - McClure, M.L.; Anderson, K.; Bardahl, K. Cosmological versions of Vaidya’s radiating stellar exterior, an accelerating reference frame, and Kinnersley’s photon rocket. ArXiv E-Prints
**2008**. arXiv:0709.3288. [Google Scholar] - McClure, M.L.; Anderson, K.; Bardahl, K. Nonisolated dynamical black holes and white holes. Phys. Rev. D
**2008**, 77, 104008. [Google Scholar] [CrossRef] - McClure, M.L. Cosmological Black Holes as Models of Cosmological Inhomogeneities. Ph.D. Thesis, University of Toronto, Toronto, Canada, 2005. [Google Scholar]
- Wald, R.M.; Iyer, V. Trapped surfaces in the Schwarzschild geometry and cosmic censorship. Phys. Rev. D
**1991**, 44, R3719–R3722. [Google Scholar] [CrossRef] - Schnetter, E.; Krishnan, B. Non-symmetric trapped surfaces in the Schwarzschild and Vaidya spacetimes. Phys. Rev. D
**2006**, 73, 021502. [Google Scholar] [CrossRef] - Figueras, P.; Hubeny, V.E.; Rangamani, M.; Ross, S.F. Dynamical black holes and expanding plasmas. J. High Energy Phys.
**2009**, 2009, 137. [Google Scholar] [CrossRef] - Kavanagh, W.; Booth, I. Spacetimes containing slowly evolving horizons. Phys. Rev. D
**2006**, 74, 044027. [Google Scholar] [CrossRef] - Visser, M. Gravitational vacuum polarization. I. Energy conditions in the Hartle-Hawking vacuum. Phys. Rev. D
**1996**, 54, 5103–5115. [Google Scholar] [CrossRef] - Scheel, M.A.; Shapiro, S.L.; Teukolsky, S.A. Collapse to black holes in Brans-Dicke theory. 2. Comparison with general relativity. Phys. Rev. D
**1995**, 51, 4236–4249. [Google Scholar] [CrossRef] - Hawking, S.W. Black holes in the Brans-Dicke theory of gravitation. Commun. Math. Phys.
**1972**, 25, 167–171. [Google Scholar] [CrossRef] - Sotiriou, T.P.; Faraoni, V. Black holes in scalar-tensor gravity. Phys. Rev. Lett.
**2012**, 108, 081103. [Google Scholar] [CrossRef] [PubMed] - Nielsen, A.B.; Visser, M. Production and decay of evolving horizons. Class. Quantum Gravity
**2006**, 23, 4637. [Google Scholar] [CrossRef] - Haijcek, P. On the origin of Hawking radiation. Phys. Rev. D
**1987**, 36, 1065–1107. [Google Scholar] [CrossRef] - Hiscock, W.A. Gravitational entropy of nonstationary black holes and spherical shells. Phys. Rev. D
**1989**, 40, 1336–1339. [Google Scholar] [CrossRef] - Sorkin, R.D. How wrinkled is the surface of a black hole? ArXiv E-Prints
**1997**. arXiv:gr-qc/9701056. [Google Scholar] - Corichi, A.; Sudarsky, D. When is S = A/4? Mod. Phys. Lett. A
**2002**, 17, 1431–1444. [Google Scholar] [CrossRef] - Nielsen, A.B.; Firouzjaee, J.T. Conformally rescaled spacetimes and hawking radiation. ArXiv E-Prints
**2012**. arXiv:1207.0064. [Google Scholar] [CrossRef] - Parikh, M.K.; Wilczek, F. Hawking radiation as tunneling. Phys. Rev. Lett.
**2000**, 85, 5042. [Google Scholar] [CrossRef] [PubMed] - Visser, M. Essential and inessential features of Hawking radiation. Int. J. Mod. Phys. D
**2003**, 12, 649–661. [Google Scholar] [CrossRef] - Di Criscienzo, R.; Nadalini, M.; Vanzo, L.; Zerbini, S.; Zoccatelli, G. On the Hawking radiation as tunneling for a class of dynamical black holes. Phys. Lett. B
**2007**, 657, 107–111. [Google Scholar] [CrossRef] - Clifton, T. Properties of black hole radiation from tunnelling. Class. Quantum Gravity
**2008**, 25, 175022. [Google Scholar] [CrossRef] - Nielsen, A.B.; Yeom, D.-H. Spherically symmetric trapping horizons, the Misner-Sharp mass and black hole evaporation. Int. J. Mod. Phys. A
**2009**, 24, 5261–5285. [Google Scholar] [CrossRef] - Jang, K.-X.; Feng, T.; Peng, D.-T. Hawking radiation of apparent horizon in a FRW universe as tunneling beyond semiclassical approximation. Int. J. Theor. Phys.
**2009**, 48, 2112–2121. [Google Scholar] - Nielsen, A.B. Black holes without boundaries. Int. J. Mod. Phys. D
**2009**, 17, 2359–2366. [Google Scholar] [CrossRef] - Angheben, M.; Nadalini, M.; Vanzo, L.; Zerbini, S. Hawking radiation as tunneling for extremal and rotating black holes. J. High Energy Phys.
**2005**, 2005, 014. [Google Scholar] [CrossRef] - Hayward, S.A.; Di Criscienzo, R.; Nadalini, M.; Vanzo, L.; Zerbini, S. Local Hawking temperature for dynamical black holes. Class. Quantum Gravity
**2009**, 26, 062001. [Google Scholar] [CrossRef] - Barcelo, C.; Liberati, S.; Sonego, S.; Visser, M. Hawking-like radiation does not require a trapped region. Phys. Rev. Lett.
**2006**, 97, 171301. [Google Scholar] [CrossRef] [PubMed] - Roman, T.A.; Bergmann, P.G. Stellar collapse without singularities? Phys. Rev. D
**1983**, 28, 1265–1277. [Google Scholar] [CrossRef] - Hayward, S.A. Formation and evaporation of nonsingular black holes. Phys. Rev. Lett.
**2006**, 96, 031103. [Google Scholar] [CrossRef] [PubMed] - Nielsen, A.B. The spatial relation between the event horizon and trapping horizon. Class. Quantum Gravity
**2010**, 27, 245016. [Google Scholar] [CrossRef] - Ashtekar, A.; Beetle, C.; Fairhurst, S. Isolated horizons: A generalization of black hole mechanics. Class. Quantum Grav.
**1999**, 16, L1–L7. [Google Scholar] [CrossRef] - Ashtekar, A.; Beetle, C.; Fairhurst, S. Mechanics of isolated horizons. Class. Quantum Gravity
**2000**, 17, 253–298. [Google Scholar] [CrossRef] - Ashtekar, A.; Beetle, C.; Dreyer, O.; Fairhurst, S.; Krishnan, B.; Lewandowski, J.; Wiśnieski, J. Isolated horizons and their applications. Phys. Rev. Lett.
**2000**, 85, 3564–3567. [Google Scholar] [CrossRef] [PubMed] - Ashtekar, A.; Corichi, A.; Krasnov, K. Isolated horizons: The classical phase space. Adv. Theor. Math. Phys.
**2000**, 3, 419–478. [Google Scholar] - Ashtekar, A.; Corichi, A. Laws governing isolated horizons: Inclusion of dilaton couplings. Class. Quantum Gravity
**2000**, 17, 1317–1332. [Google Scholar] [CrossRef] - Fairhurst, S.; Krishnan, B. Distorted black holes with charge. Int. J. Mod. Phys. D
**2001**, 10, 691–710. [Google Scholar] [CrossRef] - Ashtekar, A.; Beetle, C.; Lewandowski, J. Geometry of generic isolated horizons. Class. Quantum Gravity
**2002**, 19, 1195–1225. [Google Scholar] [CrossRef] - Ashtekar, A.; Beetle, C.; Lewandowski, J. Mechanics of rotating isolated horizons. Phys. Rev. D
**2002**, 64, 044016. [Google Scholar] [CrossRef] - Booth, I.; Fairhurst, S. The first law for slowly evolving horizons. Phys. Rev. Lett.
**2004**, 92, 011102. [Google Scholar] [CrossRef] [PubMed] - Booth, I.; Fairhurst, S. Isolated, slowly evolving, and dynamical trapping horizons: Geometry and mechanics from surface deformations. Phys. Rev. D
**2007**, 75, 084019. [Google Scholar] [CrossRef] - Kodama, H. Conserved energy flux from the spherically symmetric system and the back reaction problem in the black hole evaporation. Prog. Theor. Phys.
**1980**, 63, 1217–1228. [Google Scholar] [CrossRef] - Tung, R.-S. Stationary untrapped boundary conditions in general relativity. Class. Quantum Gravity
**2008**, 25, 085005. [Google Scholar] [CrossRef] - Abreu, G.; Visser, M. Kodama time: Geometrically preferred foliations of spherically symmetric spacetimes. Phys. Rev. D
**2010**, 82, 044027. [Google Scholar] [CrossRef] - Rácz, I. On the use of the Kodama vector field in spherically symmetric dynamical problems. Class. Quantum Gravity
**2006**, 23, 115. [Google Scholar] [CrossRef] - Misner, C.W.; Sharp, D.H. Relativistic equations for adiabatic, spherically symmetric gravitational collapse. Phys. Rev.
**1964**, 136, B571–B576. [Google Scholar] [CrossRef] - Hernandez, W.C.; Misner, C.W. Observer time as a coordinate in relativistic spherical hydrodynamics. Astrophys. J.
**1966**, 143, 452–464. [Google Scholar] [CrossRef] - Hayward, S.A. Gravitational energy in spherical symmetry. Phys. Rev. D
**1996**, 53, 1938–1949. [Google Scholar] [CrossRef] - Szabados, L. Quasi-local energy-momentum and angular momentum in GR: A review article. Living Rev. Relativ.
**2004**, 7, 4. [Google Scholar] [CrossRef] - Hayward, S.A. Unified first law of black-hole dynamics and relativistic thermodynamics. Class. Quantum Gravity
**1998**, 15, 3147–3162. [Google Scholar] [CrossRef] - Di Criscienzo, R.; Hayward, S.A.; Nadalini, M.; Vanzo, L.; Zerbini, S. Hamilton-Jacobi method for dynamical horizons in different coordinate gauges. Class. Quantum Gravity
**2010**, 27, 015006. [Google Scholar] [CrossRef] - Vanzo, L.; Acquaviva, G.; Di Criscienzo, R. Tunnelling methods and Hawking’s radiation: Achievements and prospects. Class. Quantum Gravity
**2011**, 28, 183001. [Google Scholar] [CrossRef] - Fodor, G.; Nakamura, K.; Oshiro, Y.; Tomimatsu, A. Surface gravity in dynamical spherically symmetric space-times. Phys. Rev. D
**1996**, 54, 3882–3891. [Google Scholar] [CrossRef] - Ashtekar, A.; Fairhurst, S.; Krishnan, B. Isolated horizons: Hamiltonian evolution and the first law. Phys. Rev. D
**2000**, 62, 104025. [Google Scholar] [CrossRef] - Mukohyama, S.; Hayward, S.A. Quasilocal first law of black hole dynamics. Class. Quantum Gravity
**2000**, 17, 2153–2157. [Google Scholar] [CrossRef] - Hawking, S.W. Gravitational radiation in an expanding universe. J. Math. Phys.
**1968**, 9, 598–604. [Google Scholar] [CrossRef] - Hayward, S.A. Quasilocal gravitational energy. Phys. Rev. D
**1994**, 49, 831–839. [Google Scholar] [CrossRef] - Kottler, F. Über die physikalischen ndlagen der Einsteinschen gravitationstheorie [in German]. Annalen der Physik
**1918**, 361, 401–462. [Google Scholar] [CrossRef] - Bousso, R. Adventures in de Sitter Space. ArXiv E-Prints
**2002**. arXiv:hep-th/0205177. [Google Scholar] - Griffiths, J.B.; Podolsky, J. Exact Space-Times in Einstein’s General Relativity; Cambridge University Press: Cambridge, UK, 2009. [Google Scholar]
- Hubeny, V. The fluid/gravity correspondence: A new perspective on the membrane paradigm. Class. Quantum Gravity
**2011**, 28, 114007. [Google Scholar] [CrossRef] - Sussman, R. Conformal structure of a Schwarzschild black hole immersed in a Friedman universe. Gen. Relativ. Gravit.
**1985**, 17, 251–291. [Google Scholar] [CrossRef] - Nolan, B.C. A point mass in an isotropic universe: Existence, uniqueness and basic properties. Phys. Rev. D
**1998**, 58, 064006. [Google Scholar] [CrossRef][Green Version] - Nolan, B.C. A point mass in an isotropic universe. 2. Global properties. Class. Quantum Gravity
**1999**, 16, 1227–1254. [Google Scholar] [CrossRef] - Nolan, B.C. A point mass in an isotropic universe. 3. The region R ≤ 2m. Class. Quantum Gravity
**1999**, 16, 3183–3191. [Google Scholar] [CrossRef] - Landry, P.; Abdelqader, M.; Lake, K. McVittie solution with a negative cosmological constant. Phys. Rev. D
**2012**, 86, 084002. [Google Scholar] [CrossRef] - Gao, C.J.; Zhang, S.N. Reissner-Nordström metric in the Friedman-Robertson-Walker universe. Phys. Lett. B
**2004**, 595, 28–35. [Google Scholar] [CrossRef] - Einstein, A.; Straus, E.G. The influence of the expansion of space on the gravitation fields surrounding the individual stars. Rev. Mod. Phys.
**1945**, 17, 120–124. [Google Scholar] [CrossRef] - Einstein, A.; Straus, E.G. Corrections and additional remarks to our paper: The influence of the expansion of space on the gravitation fields surrounding the individual stars. Rev. Mod. Phys.
**1946**, 18, 148–149. [Google Scholar] [CrossRef] - Ferraris, M.; Francaviglia, M.; Spallicci, A. Physical limitations of the McVittie metric. Nuovo Cimento
**1996**, 111B, 1031–1037. [Google Scholar] [CrossRef] - McClure, M.L.; Dyer, C.C. Matching radiation-dominated and matter-dominated Einstein-de Sitter universes and an application for primordial black holes in evolving cosmological backgrounds. Gen. Relativ. Gravit.
**2006**, 38, 1347–1354. [Google Scholar] [CrossRef] - Faraoni, V.; Jacques, A. Cosmological expansion and local physics. Phys. Rev. D
**2007**, 76, 063510. [Google Scholar] [CrossRef] - Li, Z.-H.; Wang, A. Existence of black holes in Friedmann-Robertson-Walker universe dominated by dark energy. Mod. Phys. Lett. A
**2007**, 22, 1663–1676. [Google Scholar] [CrossRef] - Barrow, J.D. Unusual features of varying speed of light cosmologies. Phys. Lett. B
**2003**, 564, 1–7. [Google Scholar] [CrossRef] - Caldwell, R.R.; Kamionkowski, M.; Weinberg, N.N. Phantom energy and cosmic doomsday. Phys. Rev. Lett.
**2003**, 91, 071301. [Google Scholar] [CrossRef] [PubMed] - Nolan, B.C. Sources for McVitties mass particle in an expanding universe. J. Math. Phys.
**1993**, 34, 178–185. [Google Scholar] [CrossRef] - Larsen, F. String model of black hole microstates. Phys. Rev. D
**1997**, 56, 1005–1008. [Google Scholar] [CrossRef] - Cvetic, M.; Larsen, F. General rotating black holes in string theory: Grey body factors and event horizons. Phys. Rev. D
**1997**, 56, 4994–5007. [Google Scholar] [CrossRef] - Ansorg, M.; Hennig, J. The Inner Cauchy horizon of axisymmetric and stationary black holes with surrounding matter. Class. Quantum Gravity
**2008**, 25, 222001. [Google Scholar] [CrossRef] - Ansorg, M.; Hennig, J. The Inner Cauchy horizon of axisymmetric and stationary black holes with surrounding matter in Einstein-Maxwell theory. Phys. Rev. Lett.
**2009**, 102, 221102. [Google Scholar] [CrossRef] [PubMed] - Castro, A.; Rodriguez, M.J. Universal properties and the first law of black hole inner mechanics. Phys. Rev. D
**2012**, 86, 024008. [Google Scholar] [CrossRef] - Castro, A.; Dehmami, N.; Giribet, G.; Kastor, D. On the universality of inner black hole mechanics and higher curvature gravity. ArXiv E-Prints
**2013**. arXiv:1304.1696. [Google Scholar] [CrossRef] - Visser, M. Quantization of area for event and Cauchy horizons of the Kerr-Newman black hole. J. High Energy Phys.
**2012**, 1206, 023. [Google Scholar] [CrossRef] - Visser, M. Area products for black hole horizons. Phys. Rev. D
**2013**, 88, 044014. [Google Scholar] [CrossRef] - Faraoni, V.; Zambrano Moreno, A.F. Are quantization rules for horizon areas universal? Phys. Rev. D
**2013**, 88, 044011. [Google Scholar] [CrossRef] - Carrera, M.; Giulini, D. Generalization of McVitties model for an inhomogeneity in a cosmological spacetime. Phys. Rev. D
**2010**, 81, 043521. [Google Scholar] [CrossRef] - Faraoni, V.; Gao, C.; Chen, X.; Shen, Y.-G. What is the fate of a black hole embedded in an expanding universe? Phys. Lett. B
**2009**, 671, 7–9. [Google Scholar] [CrossRef] - Saida, H.; Harada, T.; Maeda, H. Black hole evaporation in an expanding universe. Class. Quantum Gravity
**2007**, 24, 4711. [Google Scholar] [CrossRef] - Faraoni, V. Hawking temperature of expanding cosmological black holes. Phys. Rev. D
**2007**, 76, 104042. [Google Scholar] [CrossRef] - Husain, V.; Martinez, E.A.; Nuñez, D. Exact solution for scalar field collapse. Phys. Rev. D
**1994**, 50, 3783–3786. [Google Scholar] [CrossRef] - Fisher, I.Z. Scalar mesostatic field with regard for gravitational effects. Zh. Eksp. Teor. Fiz.
**1948**, 18, 636–640, translated in arXiv:gr-qc/9911008. [Google Scholar] - Bergman, O.; Leipnik, R. Space-time structure of a static spherically symmetric scalar field. Phys. Rev.
**1957**, 107, 1157–1161. [Google Scholar] [CrossRef] - Janis, A.I.; Newman, E.T.; Winicour, J. Reality of the Schwarzschild singularity. Phys. Rev. Lett.
**1968**, 20, 878–880. [Google Scholar] [CrossRef] - Buchdahl, H.A. Static solutions of the Brans-Dicke equations. Int. J. Theor. Phys.
**1972**, 6, 407–412. [Google Scholar] [CrossRef] - Wyman, M. Static spherically symmetric scalar fields in general relativity. Phys. Rev. D
**1981**, 24, 839–841. [Google Scholar] [CrossRef] - Agnese, A.G.; La Camera, M. Gravitation without black holes. Phys. Rev. D
**1985**, 31, 1280–1286. [Google Scholar] [CrossRef] - Virbhadra, K.S. Janis-Newman-Winicour and Wyman solutions are the same. Int. J. Mod.Phys. A
**1997**, 12, 4831–4835. [Google Scholar] [CrossRef] - Roberts, M.D. Massless scalar static spheres. Astrophys. Space Sci.
**1993**, 200, 331–335. [Google Scholar] [CrossRef] - Abe, S. Stability of a collapsed scalar field and cosmic censorship. Phys. Rev. D
**1988**, 38, 1053–1055. [Google Scholar] [CrossRef] - Fonarev, O.A. Exact Einstein scalar field solutions for formation of black holes in a cosmological setting. Class. Quantum Gravity
**1995**, 12, 1739–1752. [Google Scholar] [CrossRef] - Maeda, H. Global structure and physical interpretation of the Fonarev solution for a scalar field with exponential potential. ArXiv E-Prints
**2007**. arXiv:0704.2731. [Google Scholar] - Mars, M.; Mena, F.C.; Vera, R. Review on exact and perturbative deformations of the Einstein-Straus model: Uniqueness and rigidity results. ArXiv E-Prints
**2013**. arXiv:1307.4371. [Google Scholar] [CrossRef][Green Version] - Eisenstaedt, J. Density constraint on local inhomogeneities of a Robertson-Walker cosmological universe. Phys. Rev. D
**1977**, 16, 927–928. [Google Scholar] [CrossRef] - Bonnor, W.B. Size of a hydrogen atom in the expanding universe. Class. Quantum Gravity
**1999**, 16, 1313–1321. [Google Scholar] [CrossRef] - Senovilla, J.M.M.; Vera, R. Impossibility of the cylindrically symmetric Einstein-Straus model. Phys. Rev. Lett.
**1997**, 78, 2284–2287. [Google Scholar] [CrossRef] - Mars, M. Axially symmetric Einstein-Straus models. Phys. Rev. D
**1998**, 57, 3389–3400. [Google Scholar] [CrossRef] - Mena, F.C.; Tavakol, R.; Vera, R. Generalization of the Einstein-Straus model to anisotropic settings. Phys. Rev. D
**2002**, 66, 044004. [Google Scholar] [CrossRef] - Balbinot, R.; Bergamini, R.; Comastri, A. Solution of the Einstein-Strauss problem with a Λ term. Phys. Rev. D
**1988**, 38, 2415–2418. [Google Scholar] [CrossRef] - Bona, C.; Stela, J. “Swiss cheese” models with pressure. Phys. Rev. D
**1987**, 36, 2915–2918. [Google Scholar] [CrossRef] - Bonnor, W.B. A generalization of the Einstein-Straus vacuole. Class. Quantum Gravity
**2000**, 17, 2739–2748. [Google Scholar] [CrossRef] - Saida, H. Hawking radiation in the Swiss-cheese universe. Class. Quantum Gravity
**2002**, 19, 3179–3205. [Google Scholar] [CrossRef] - Goncalves, S.M.C.V. Shell crossing in generalized Tolman-Bondi spacetimes. Phys. Rev. D
**2001**, 63, 124017. [Google Scholar] [CrossRef] - Gao, C.; Chen, X.; Shen, Y.-G.; Faraoni, V. Black holes in the universe: Generalized Lemaitre-Tolman-Bondi solutions. Phys. Rev. D
**2011**, 84, 104047. [Google Scholar] [CrossRef] - Firouzjaee, J.T.; Mansouri, R. Asymptotically FRW black holes. Gen. Relativ. Gravit.
**2010**, 42, 2431–2452. [Google Scholar] [CrossRef] - Firouzjaee, J.T.; Parsi Mood, N. Do we know the mass of a black hole? Mass of some cosmological black hole models. Gen. Relativ. Gravit.
**2012**, 44, 639–656. [Google Scholar] [CrossRef] - Firouzjaee, J.T. The spherical symmetry black hole collapse in expanding universe. Int. J. Mod. Phys. D
**2012**, 21, 1250039. [Google Scholar] [CrossRef] - Firouzjaee, J.T.; Mansouri, M. Radiation from the LTB black hole. Europhys. Lett.
**2012**, 97, 29002. [Google Scholar] [CrossRef] - Moradi, R.; Firouzjaee, J.T.; Mansouri, R. The spherical perfect fluid collapse in the cosmological background. ArXiv E-Prints
**2013**. arXiv:1301.1480. [Google Scholar] - Krasiński, A.; Hellaby, C. Formation of a galaxy with a central black hole in the Lemaître-Tolman model. Phys. Rev. D
**2004**, 69, 043502. [Google Scholar] [CrossRef] - Oppenheimer, J.R.; Snyder, H. On continued gravitational contraction. Phys. Rev.
**1939**, 56, 455–459. [Google Scholar] [CrossRef] - Barnes, A. On shear free normal flows of a perfect fluid. Gen. Relativ. Gravit.
**1973**, 2, 105–129. [Google Scholar] [CrossRef] - Roberts, M.D. Scalar field counterexamples to the Cosmic Censorship hypothesis. Gen. Relativ. Gravit.
**1989**, 21, 907–939. [Google Scholar] [CrossRef] - Burko, L.M. Comment on the Roberts solution for the spherically-symmetric Einstein-scalar field equations. Gen. Relativ. Grav.
**1997**, 29, 259–262. [Google Scholar] [CrossRef] - Patel, L.K.; Trivedi, H.B. Kerr-Newman metric in cosmological background. J. Astrophys. Astron.
**1982**, 3, 63–67. [Google Scholar] [CrossRef] - Vaidya, P.C. The Kerr metric in cosmological background. Pramana
**1977**, 8, 512–517. [Google Scholar] [CrossRef] - Balbinot, R. Hawking radiation and the back reaction—A first approach. Class. Quantum Gravity
**1984**, 1, 573–577. [Google Scholar] [CrossRef] - Nayak, K.R.; MacCallum, M.A.H.; Vishveshwara, C.V. Black holes in nonflat backgrounds: The Schwarzschild black hole in the Einstein universe. Phys. Rev. D
**2000**, 63, 024020. [Google Scholar] [CrossRef] - Cox, D.P.G. Vaidya’s “Kerr-Einstein” metric cannot be matched to the Kerr metric. Phys. Rev. D
**2003**, 68, 124008. [Google Scholar] [CrossRef] - Lindesay, J. Coordinates with non-singular curvature for a time dependent black hole horizon. Found. Phys.
**2007**, 37, 1181–1196. [Google Scholar] [CrossRef] - Brown, B.A.; Lindesay, J. Construction of a Penrose diagram for a spatially coherent evaporating black hole. Class. Quantum Gravity
**2008**, 25, 105026. [Google Scholar] [CrossRef] - Brown, B.A.; Lindesay, J. Radial photon trajectories near an evaporating black hole. ArXiv E-Prints
**2008**. arXiv:0802.1660. [Google Scholar] - Lindesay, J. Quantum behaviors on an excreting black hole. Class. Quantum Gravity
**2009**, 26, 125014. [Google Scholar] [CrossRef] - Brown, B.A.; Lindesay, J. Construction of a Penrose diagram for an accreting black hole. Class. Quantum Gravity
**2009**, 26, 045010. [Google Scholar] [CrossRef] - Lindesay, J.; Sheldon, P. Penrose diagram for a transient black hole. Class. Quantum Gravity
**2010**, 27, 215015. [Google Scholar] [CrossRef] - Lindesay, J.; Finch, T. Global geometry of a transient black hole in a dynamic de Sitter cosmology. In Classical and Quantum Gravity: Theory and Applications; Frignanni, V.R., Ed.; Nova Science: New York, NY, USA, 2012; Chapter 16. [Google Scholar]
- Lindesay, J.; Finch, T. Global causal structure of a transient black object. ArXiv E-Prints
**2011**. arXiv:1110.6928. [Google Scholar] - Adler, R.J.; Bjorken, J.D.; Chen, P.; Liu, J.S. Simple analytical models of gravitational collapse. Am. J. Phys.
**2005**, 73, 1148–1159. [Google Scholar] [CrossRef] - Zilhão, M.; Cardoso, V.; Gualtieri, L.; Herdeiro, C.; Sperhake, U.; Witek, H. Dynamics of black holes in de Sitter spacetimes. Phys. Rev. D
**2012**, 85, 104039. [Google Scholar] [CrossRef] - Kastor, D.; Traschen, J. Cosmological multi-black-hole solutions. Phys. Rev. D
**1993**, 47, 5370–5375. [Google Scholar] [CrossRef] - Brill, D.R.; Horowitz, G.T.; Kastor, D.; Traschen, J. Testing cosmic censorship with black hole collisions. Phys. Rev. D
**1994**, 49, 840–852. [Google Scholar] [CrossRef] - Koberlein, B.D.; Mallett, R.L. Charged, radiating black holes, inflation, and cosmic censorship. Phys. Rev. D
**1994**, 49, 5111–5114. [Google Scholar] [CrossRef] - Husain, V. Exact solutions for null fluid collapse. Phys. Rev. D
**1996**, 53, R1759–R1762. [Google Scholar] [CrossRef] - Dawood, A.K.; Ghosh, S.G. Generating dynamical black hole solutions. Phys. Rev. D
**2004**, 70, 104010. [Google Scholar] [CrossRef] - Conboy, S.; Lake, K. Smooth transitions from the Schwarzschild vacuum to de Sitter space. Phys. Rev. D
**2005**, 71, 124017. [Google Scholar] [CrossRef] - Kyo, M.; Harada, T.; Maeda, H. Asymptotically Friedmann self-similar scalar field solutions with potential. Phys. Rev. D
**2008**, 77, 124036. [Google Scholar] [CrossRef] - Meissner, K.A. Horizons and the cosmological constant. ArXiv E-Prints
**2009**. arXiv:0901.0640. [Google Scholar] - Gibbons, G.W.; Maeda, K. Black holes in an expanding universe. Phys. Rev. Lett.
**2010**, 104, 131101. [Google Scholar] [CrossRef] [PubMed] - Maeda, H. Exact dynamical AdS black holes and wormholes with a Klein-Gordon field. Phys. Rev. D
**2012**, 86, 044016. [Google Scholar] [CrossRef] - Culetu, H. Time-dependent embedding of a spherically symmetric Rindler-like spacetime. Class. Quantum Gravity
**2012**, 29, 235021. [Google Scholar] [CrossRef] - Clifton, T.; Mota, D.F.; Barrow, J.D. Inhomogeneous gravity. Mon. Not. R. Astron. Soc.
**2005**, 358, 601–613. [Google Scholar] [CrossRef] - Faraoni, V.; Zambrano Moreno, A.F. Interpreting the conformal cousin of the Husain-Martinez-Nuñez solution. Phys. Rev. D
**2012**, 86, 084044. [Google Scholar] [CrossRef] - Faraoni, V.; Vitagliano, V.; Sotiriou, T.P.; Liberati, S. Dynamical apparent horizons in inhomogeneous Brans-Dicke universes. Phys. Rev. D
**2012**, 86, 064040. [Google Scholar] [CrossRef] - Campanelli, M.; Lousto, C. Are black holes in Brans-Dicke theory precisely the same as in general relativity? Int. J. Mod. Phys. D
**1993**, 2, 451–462. [Google Scholar] [CrossRef] - Lousto, C.; Campanelli, M. On Brans-Dicke Black Holes. In Proceedings of the Origin of Structure in the Universe, Pont d’Oye, Belgium, 27 April–2 May 1992; Gunzig, E., Nardone, P., Eds.; Kluwer Academic: Dordrecht, The Netherlands, 1993; pp. 123–130. [Google Scholar]
- Vanzo, L.; Zerbini, S.; Faraoni, V. Campanelli-Lousto and veiled spacetimes. Phys. Rev. D
**2012**, 86, 084031. [Google Scholar] [CrossRef] - Clifton, T. Spherically symmetric solutions to fourth-order theories of gravity. Class. Quantum Gravity
**2006**, 23, 7445–7453. [Google Scholar] [CrossRef] - Faraoni, V. Clifton’s spherical solution in f(R) vacuum harbours a naked singularity. Class. Quantum Gravity
**2009**, 26, 195013. [Google Scholar] [CrossRef] - Faraoni, V. Jebsen-Birkhoff theorem in alternative gravity. Phys. Rev. D
**2010**, 81, 044002. [Google Scholar] [CrossRef] - Clifton, T.; Barrow, J.D. The power of general relativity. Phys. Rev. D
**2005**, 72, 103005. [Google Scholar] [CrossRef] - Clifton, T.; Barrow, J.D. Further exact cosmological solutions to higher-order gravity theories. Class. Quantum Grav.
**2006**, 23, 2951–2962. [Google Scholar] [CrossRef] - Dolgov, A.D.; Kawasaki, M. Can modified gravity explain accelerated cosmic expansion? Phys. Lett. B
**2003**, 573, 1–4. [Google Scholar] [CrossRef] - Faraoni, V. Matter instability in modified gravity. Phys. Rev. D
**2006**, 74, 104017. [Google Scholar] [CrossRef] - Faraoni, V. De Sitter space and the equivalence between f(R) and scalar-tensor gravity. Phys. Rev. D
**2007**, 75, 067302. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S.D. Modified gravity with negative and positive powers of curvature: Unification of inflation and cosmic acceleration. Phys. Rev. D
**2003**, 68, 123512. [Google Scholar] [CrossRef] - Sakai, N.; Barrow, J.D. Cosmological evolution of black holes in Brans-Dicke gravity. Class. Quantum Gravity
**2001**, 18, 4717–4724. [Google Scholar] [CrossRef] - Sakai, N.; Barrow, J.D. Evolution of black holes in Brans-Dicke cosmology. ArXiv E-Prints
**2000**. arXiv:gr-qc/0012067. [Google Scholar] - Nozawa, M.; Maeda, H. Dynamical black holes with symmetry in Einstein-Gauss-Bonnet gravity. Class. Quantum Gravity
**2008**, 25, 055009. [Google Scholar] [CrossRef] - Charmousis, C. Higher order gravity theories and their black hole solutions. ArXiv E-Prints
**2008**. arXiv:0805.0568. [Google Scholar] - Maeda, H.; Willison, S.; Ray, S. Lovelock black holes with maximally symmetric horizons. Class. Quantum Gravity
**2011**, 28, 165005. [Google Scholar] [CrossRef] - Horowitz, G.T. Black Holes in Higher Dimensions; Cambridge University Press: Cambridge, UK, 2012. [Google Scholar]
- Maeda, H.; Nozawa, M. Generalized Misner-Sharp quasilocal mass in Einstein-Gauss-Bonnet gravity. Phys. Rev. D
**2008**, 77, 064031. [Google Scholar] [CrossRef] - Cai, R.-G.; Cao, L.-M.; Hu, Y.-P.; Ohta, N. Generalized Misner-Sharp energy in f(R) gravity. Phys. Rev. D
**2009**, 80, 104016. [Google Scholar] [CrossRef] - Christodolou, D. Examples of naked singularity formation in the gravitational collapse of a scalar field. Ann. Math.
**1994**, 140, 607–653. [Google Scholar] [CrossRef] - Christodolou, D. The instability of naked singularities in the gravitational collapse of a scalar field. Ann. Math.
**1999**, 149, 183–217. [Google Scholar] [CrossRef]

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Faraoni, V. Evolving Black Hole Horizons in General Relativity and Alternative Gravity. *Galaxies* **2013**, *1*, 114-179.
https://doi.org/10.3390/galaxies1030114

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Faraoni V. Evolving Black Hole Horizons in General Relativity and Alternative Gravity. *Galaxies*. 2013; 1(3):114-179.
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Faraoni, Valerio. 2013. "Evolving Black Hole Horizons in General Relativity and Alternative Gravity" *Galaxies* 1, no. 3: 114-179.
https://doi.org/10.3390/galaxies1030114