# Static Spherically Symmetric Black Holes in Weak f(T)-Gravity

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## Abstract

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## 1. Introduction

## 2. Black Holes in Weak $\mathbf{f}\left(\mathbb{T}\right)$ Gravity

#### 2.1. Covariant $f\left(\mathbb{T}\right)$ Gravity

#### 2.2. Static Spherically Symmetric Black Holes in Weak f($\mathbb{T}$) Gravity

#### 2.2.1. The General Static Spherically Symmetric Solution

#### 2.2.2. The Black Hole Solution

#### 2.3. The General Relativistic Perspective—Energy Conditions

## 3. Classical and Semi-Classical Properties

#### 3.1. Particle Propagation Effects: Photon Sphere, Perihelion Shift, Shapiro Delay and Light Deflection

**Photon sphere:**The photon sphere—a characterizing feature of black holes [64,65]— is derived from the geodesic Equation (31) by searching for orbits with $\dot{r}=0$ and $\sigma =0$. For this, we solve $V\left(r\right)=0$ and $V{\left(r\right)}^{\prime}=0$ to first order in $\beta $. In our case the photon sphere then lies at$$\begin{array}{c}\hfill {r}_{\mathrm{ph}}=\left(\frac{3}{2}+\beta \frac{(27ln\left(3\right)+32\sqrt{3}-80)}{18}\right){r}_{\mathrm{s}}\approx \left(1.5+\beta 0.282675\right){r}_{\mathrm{s}}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$This result is identical to the one found in [46].**Perihelion shift:**While an elliptic orbit in the Newtonian two-body problem would experience the perihelion always at the same angle, deviations from the two-body problem—either by adding more bodies or, as here, using different dynamics—forces this perihelion to move from orbit to orbit. For sufficiently small eccentricity, this is encoded in the quantity $\Delta \varphi $ of an orbit $r\left(\varphi \right)={r}_{\mathrm{c}}+{r}_{\varphi}\left(\varphi \right)$ which is a perturbation of an orbit with constant radius ${r}_{\mathrm{c}}$. It is derived from the effective potential as, see for example [46] for a derivation,$$\begin{array}{c}\hfill \Delta \varphi =2\pi \left(\frac{h}{{r}_{\mathrm{c}}^{2}\sqrt{{V}^{\prime \prime}\left({r}_{\mathrm{c}}\right)}}-1\right)\phantom{\rule{0.166667em}{0ex}}.\end{array}$$To first order in $\beta $, we find$$\begin{array}{c}\hfill \Delta \varphi =6\pi q+27\pi {q}^{2}+135\pi {q}^{3}+\frac{2835\pi}{4}{q}^{4}+\beta 32\pi {q}^{4}+\mathcal{O}\left({q}^{5}\right)\phantom{\rule{0.166667em}{0ex}},\end{array}$$This finding thus leads us to weaker constraints on the teleparallel coupling $\alpha =\beta {r}_{\mathrm{s}}^{2}$ from observations of the perihelion shift, for example, from the orbits of stars around the black hole in the center of our galaxy.**Shapiro time delay:**The Shapiro time delay is the time delay experienced by a radar signal between an emitter at ${r}_{\mathrm{e}}$ and a mirror at ${r}_{\mathrm{m}}$ due to the presence of a gravitational mass [66]. The time which passes until a light ray has travelled from an emitter $r={r}_{\mathrm{e}}$ to a point of closest encounter to the gravitational mass at ${r}_{0}$, respectively, from ${r}_{0}$ to a mirror at ${r}_{\mathrm{m}}$, is given by$$\begin{array}{cc}\hfill t({r}_{X},{r}_{0})\phantom{\rule{1.em}{0ex}}& ={\int}_{{r}_{0}}^{{r}_{X}}\frac{E}{\sqrt{-2V\left(r\right)}}d\phantom{\rule{-0.166667em}{0ex}}r,\phantom{\rule{2.em}{0ex}}X=\mathrm{e},\mathrm{m}\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =t{({r}_{X},{r}_{0})}_{\mathrm{GR}}\phantom{\rule{4pt}{0ex}}\left(1-\frac{{\zeta}_{1}}{2}\beta \right)+\mathcal{O}\left({\left(\frac{{r}_{\mathrm{s}}}{{r}_{X}}\right)}^{5}\right)\phantom{\rule{0.166667em}{0ex}}.\hfill \end{array}$$The last equality holds when the integrand is expanded in powers of the small parameter $\frac{{r}_{\mathrm{s}}}{r}$, i.e., assuming ${r}_{0}>{r}_{\mathrm{s}}$. The lowest orders can be integrated explicitly, but the higher orders do not allow for a (known) closed formula for the integral.We further assume that changes in the relative distances between emitter, mirror and mass can be neglected during this propagation. The total travel time for a return trip of the light signal is then $\Delta t=2(t({r}_{\mathrm{e}},{r}_{0})+t({r}_{\mathrm{m}},{r}_{0}))$. The Shapiro delay is given by$$\begin{array}{cc}\hfill \Delta {t}_{\mathrm{Shapiro}}\phantom{\rule{1.em}{0ex}}& =\Delta t-2\left(\sqrt{{r}_{\mathrm{e}}^{2}-{r}_{0}^{2}}+\sqrt{{r}_{\mathrm{m}}^{2}-{r}_{0}^{2}}\right))\left(1-\frac{{\zeta}_{1}}{2}\beta \right)\hfill \end{array}$$$$\begin{array}{cc}\hfill \phantom{\rule{1.em}{0ex}}& =\Delta {t}_{\mathrm{Shapiro},\mathrm{GR}}\phantom{\rule{4pt}{0ex}}\left(1-\frac{{\zeta}_{1}}{2}\beta \right)+\mathcal{O}\left({\left(\frac{{r}_{\mathrm{s}}}{{r}_{X}}\right)}^{5}\right)\phantom{\rule{0.166667em}{0ex}},\hfill \end{array}$$**Light deflection:**Another central quantity to investigate in this context is the deviation angle $\Delta \varphi $ between a null geodesic in the presence and the absence of a central gravitating mass. The point of closest encounter to the central object is again called ${r}_{0}$. Then the deflection angle is given by$$\begin{array}{c}\hfill \Delta \varphi =2{\int}_{{r}_{0}}^{\infty}\left(\frac{L}{{r}^{2}\sqrt{-2V\left(r\right)}}d\phantom{\rule{-0.166667em}{0ex}}r\right)-\pi ={\Delta}_{\varphi ,\mathrm{GR}}+\beta \frac{4}{15}\frac{{r}_{\mathrm{s}}^{5}}{{r}_{0}^{5}}\phantom{\rule{0.166667em}{0ex}}.\end{array}$$**Minimal photon impact parameter:**The impact parameter b is another closely related and only mildly different way to look at these matters of light deflection. It adds however, a simple way to characterize the photon capture cross-section of the black hole for an observer at infinity. Physically and more concretely, the minimal photon impact parameter ${b}_{\mathrm{min}}$ characterizes the closest encounter with the central mass a photon can have when it is scattered by this mass, and can still be received by a distant observer. It is defined as follows. Define $\ell =\frac{L}{E{r}_{\mathrm{s}}}$ and demand that the effective potential V as a function $V=V(r,E,L(E,\ell \left)\right)$ vanishes, which determines the ratio between E and L as a function of r, i.e., $\ell =\ell \left(r\right)$. This quantity possesses a minimum at $r={r}_{\mathrm{ph}}$ which identifies the minimal possible impact parameter for scattering as$$\begin{array}{c}\hfill {b}_{\mathrm{min}}=\sqrt{\ell \left({r}_{\mathrm{ph}}\right)}=\frac{3\sqrt{3}}{2}-\frac{1}{4}\beta \left(64+\frac{80}{\sqrt{3}}+3\sqrt{3}\frac{{C}_{2}}{\beta}\right)\phantom{\rule{0.166667em}{0ex}}.\end{array}$$Every geodesic with smaller impact parameter b will be gravitationally captured. We introduce ${b}_{\mathrm{min}}$ here, since it is needed for the discussion of the black hole sparsity in Section 3.4. Unlike in the previous cases, no additional small parameter was introduced.The minimal photon impact parameter ${b}_{\mathrm{min}}$ is the crucial quantity to describe the shadow of these black holes. The shadow corresponds exactly to the capture cross-section. Stricly speaking, the phrase “silhouette” would be a more apt description of this capture cross-section, but the phrase “shadow” is the established terminology, and we will abide by it.

#### 3.2. The Event Horizon

#### 3.3. Surface Gravity and Black Hole Temperature

#### 3.4. Sparsity

#### Comparing Weak $f\left(\mathbb{T}\right)$-Black Holes and a Schwarzschild Black Hole

## 4. Conclusions

## Author Contributions

## Funding

## Acknowledgments

## Conflicts of Interest

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**Figure 1.**The values of ${\mu}_{\mathrm{h}}$ (solid, blue curve) and of ${r}_{\mathrm{h}}/{r}_{\mathrm{s}}$ (dashed, orange curve) as functions of the perturbation parameter $\beta $. The horizon parameter is chosen to be ${\zeta}_{1}=0$. Both describe the location of the putative horizon of the teleparallel black hole, and are related to each other through ${r}_{\mathrm{h}}=\frac{{r}_{\mathrm{s}}}{1-{\mu}_{\mathrm{h}}^{2}}$. ${\mu}_{\mathrm{h}}$ has been determined numerically.

**Figure 2.**(

**a**): Surface gravity in units of the inverse Schwarzschild radius for different values of the perturbation parameter $\beta $ and ${\zeta}_{1}=0$. The position needed for the evaluation of the surface gravity has been determined numerically. (

**b**): the values of the zeroth $\left(\kappa \right(r\left)\right|\beta =0,r={r}_{h})$ and first $\left(\kappa \right({\partial}_{\beta k}\left(r\right)|\beta =0,r={r}_{h}))$ order of κ (46) plotted separately for a heuristic check of possible numerical issues.

**Figure 3.**(

**a**) The sparsity of a weak $f\left(\mathbb{T}\right)$ black hole compared to the sparsity of a Schwarzschild black hole as a function of $\beta $. (

**b**): A look at the terms of our perturbative approach to this ratio reveals a breakdown of this approach for sufficiently large $\beta $ somewhere below $\beta \approx 0.6$.

**Table 1.**Numerical values of the event horizon radius in units of the Schwarzschild radius for selected values of $\beta $.

$\mathit{\beta}$ | 0.001 | 0.005 | 0.01 | 0.02 | 0.05 | 0.1 | 0.5 | 1 |
---|---|---|---|---|---|---|---|---|

${\mu}_{\mathrm{h}}\left(\beta \right)$ | 0.0698 | 0.1301 | 0.1664 | 0.2091 | 0.2744 | 0.3285 | 0.4591 | 0.5134 |

$\frac{{r}_{\mathrm{h}}\left(\beta \right)}{{r}_{\mathrm{s}}}$ | 1.0049 | 1.0172 | 1.0285 | 1.0457 | 1.0814 | 1.1210 | 1.2670 | 1.3580 |

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Pfeifer, C.; Schuster, S.
Static Spherically Symmetric Black Holes in Weak *f*(*T*)-Gravity. *Universe* **2021**, *7*, 153.
https://doi.org/10.3390/universe7050153

**AMA Style**

Pfeifer C, Schuster S.
Static Spherically Symmetric Black Holes in Weak *f*(*T*)-Gravity. *Universe*. 2021; 7(5):153.
https://doi.org/10.3390/universe7050153

**Chicago/Turabian Style**

Pfeifer, Christian, and Sebastian Schuster.
2021. "Static Spherically Symmetric Black Holes in Weak *f*(*T*)-Gravity" *Universe* 7, no. 5: 153.
https://doi.org/10.3390/universe7050153