# 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 $f\left(\right)open="("\; close=")">{{R}^{c}}_{c}$ 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

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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.
https://doi.org/10.3390/galaxies1030114

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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