# Kerr Black Holes within a Modified Theory of Gravity

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## Abstract

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## 1. Introduction

## 2. The Modified Theory: Pseudo-Complex General Relativity

## 3. Event Horizons and Light-Rings: Phase Transitions

#### 3.1. Circular Orbital Motion

#### 3.2. Event Horizons

#### 3.3. Light-Rings

## 4. Conclusions

## Author Contributions

## Funding

## Conflicts of Interest

## References

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**Figure 1.**The dependence of the orbital frequency of a particle in a circular orbit on the radial distance. The upper curve shows the result for GR and the lower two curves for pc-GR. The one with its maximum further to the left corresponds to $n=3$ and the other one to $n=4$. Equation (21) was used with $b=\frac{64}{27}$ for $n=3$ and $b=\frac{81}{8}$ for $n=4$.

**Figure 2.**The position of the Innermost Stable Circular Orbit (ISCO) for $n=3$ (

**left panel**) and $n=4$ (

**right panel**) is plotted versus the rotational parameter a. The upper curve in each figure corresponds to GR and the lower curve to pc-GR. In the region to the left of the pc-GR curve, the orbits are unstable in pc-GR; above and to the right, the orbits are stable. For small values of a, the ISCO in pc-GR follows more or less the one of GR, but at smaller values of r. From a certain value of a on, stable orbits are allowed until the surface of the star. For $n=3$, this limit is approximately $a=0.4\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$ and, for $n=4$, it is above $a=0.5\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$. For larger values of a, all orbits are stable in pc-GR. For the construction of the curve for $n=3$, Equation (42) of [12] was used. This equation has to be modified for $n=4$, which is direct. It can be retrieved from [45], where all the routines used here are openly accessible.

**Figure 3.**Infinite, counter clockwise rotating geometrically thin accretion disc around a rotating compact object, viewed from an inclination of 80${}^{\circ}$. The disc model by [49] was used. The right panel is a simulation within pc-GR for $a=0$ and the right panel for $a=0.9\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$. Both figures are for $n=3$. (Figures taken from [14,16,17].) The figures were obtained using the open accessible 2014 version of the GYOTO routines [48].

**Figure 4.**The explanation is the same as in Figure 3. The left panel shows the simulation for pc-GR and $n=3$ (the same as the right panel in Figure 3), while the right panel is a simulation for $n=4$. As noted, the position of the dark and bright rings are shifted slightly to larger radial distances. The figures were obtained using the open accessible 2014 version of the GYOTO routines [48]. For $n=4$, the modified C++ rountines can be retrieved in [45].

**Figure 5.**Infinite, counterclockwise rotating geometrically thin accretion disc around static rotating compact objects viewed from an inclination of 70${}^{\circ}$. The upper row and the left panel show the result for a resolution of 0.5 $\mu as$ while the right panel corresponds to a resolution of 20$\mu as$. In the upper row, $a=0.6\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$. The lower figure is taken from the EHT results, rotated by 90${}^{\circ}$. The figures were obtained using the open accessible 2014 version of the GYOTO routines [48]. For $n=4$, the modified C++ routines can be retrieved in [45].

**Figure 6.**Simulations of accretion discs for an inclination of 60${}^{\circ}$ and $a=0.6$ m. A resolution of $20\phantom{\rule{3.33333pt}{0ex}}\mu as$ was assumed [50]. The left panel shows the result for GR and the right one for pc-GR. As seen, the GR and pc-GR cannot be distinguished clearly. The dark center in GR is slightly larger than in pc-GR, which is also noted in the calculation of fluxes (see main text). The figures were obtained using the open accessible 2014 version of the GYOTO routines [48]. For $n=4$, the modified C++ rountines can be retrieved in [45].

**Figure 7.**The intensity distribution for pc-GR (upper) curve and GR (lower curve). The rotational parameter a is changed from 0.6 ${m}_{0}$ to 0.9 ${m}_{0}$ ($m={m}_{0}$) starting on the left in the upper row and ending to the right in the lower row, in steps of $0.1\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$. The intensity in pc-GR is always larger. The peak shifts to the left (lower distances) as a increases. For $a=0.9\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$, the peak is around $r=0.4\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$, while the peak of the pc-GR curve is always at lower r. The intensities are obtained using (23). The relation between ${L}_{z}$ and E can be retrieved from [14].

**Figure 8.**Shown is the surface of allowed horizons for $n=3$ (

**left panel**) and $n=4$ (

**right panel**); also shown is the projected curve of the separatrix (blue curve). The vertical axis denotes r in units of ${m}_{0}=m$, the x-axis the a in units of ${m}_{0}$ and the y-axis the ${b}_{n}=b$ parameter. The figures are obtained using (25). The corresponding MATHEMATICA [44] tool can be retrieved from [45].

**Figure 9.**Shown are two sets for $n=4$ ($a=0$ and $a=0.5\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$, respectively), of the position of the extrema as a function of ${b}_{n}$. For each a, the upper curve corresponds to a maximum and the lower one to the position of the minimum. The two curves meet at a certain ${b}_{n}=b$ value, which increases with decreasing a. The dashed line corresponds to $a=0.5\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$, while the continuous line is for $a=0$. Equation (25) was used and the corresponding MATHEMATICA routine can be retrieved from [45].

**Figure 10.**Shown are four cuts of the potential at $a=0.5\phantom{\rule{3.33333pt}{0ex}}{m}_{0}$ and, respectively from upper left to lower right, for the values of ($n=4$) ${b}_{n}$ = 2, 3.3, 4 and 7.5. For ${b}_{n}=2$, the deformed minimum is at negative values; for ${b}_{n}=3.3$, it is at the same height as for U at $y=0$, for b = 4, the deformed minimum is at positive values and, at 7.5, it is disappearing, being close to the point where the maximum and minimum join, as shown in Figure 9. For the potential, (27) was used. The corresponding MATHEMATICA routine can be retrieved from [45].

**Figure 11.**The critical surfaces for the horizon (inner surface) and the light-ring (outer surface). The critical surface of the light-ring follows the one of the horizon, only further out. Note the region where the light-ring does not exist. The b value is equal to ${b}_{n}$. For the construction of the light ring surface, (38) was used. The corresponding MATHEMATICA routine can can be retrieved from [13].

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**MDPI and ACS Style**

Hess, P.O.; López-Moreno, E.
Kerr Black Holes within a Modified Theory of Gravity. *Universe* **2019**, *5*, 191.
https://doi.org/10.3390/universe5090191

**AMA Style**

Hess PO, López-Moreno E.
Kerr Black Holes within a Modified Theory of Gravity. *Universe*. 2019; 5(9):191.
https://doi.org/10.3390/universe5090191

**Chicago/Turabian Style**

Hess, Peter O., and Enrique López-Moreno.
2019. "Kerr Black Holes within a Modified Theory of Gravity" *Universe* 5, no. 9: 191.
https://doi.org/10.3390/universe5090191