# Weak cosmic censorship conjecture in Kerr black holes of modified gravity

###### Abstract

By neglecting the effects of self-force and radiation, we investigate the possibility of destroying the Kerr-MOG black hole through the point particle absorption process. Using the instability of the event horizon and the condition that Bekenstein-Hawking entropy never decreases, we get the upper and lower energy bounds allowed for a matter particle to produce the naked singularity. We find that the upper and lower energy bounds always coincide with each other for the extremal black hole case, which means that the extremal Kerr-MOG black hole cannot be overspun. While it is indeed possible for the near-extremal black hole to be destroyed, which leads to the violation of the weak cosmic censorship conjecture. Furthermore, when considering the effect of the adiabatic process, the weak cosmic censorship conjecture will be restored.

###### pacs:

04.20.Dw, 04.50.Kd, 04.70.Dy## I Introduction

It is widely acknowledged that spacetime singularity will be formed at the end of the gravitational collapse of regular matter Hawking (). And all physics we have known will be invalid at the singularity. But spacetime singularity cannot be seen by distant observers, which means that it is always hidden inside the event horizon of a black hole Penrose (). This situation is guaranteed by the weak cosmic censorship conjecture (WCCC). It is the foundation of black hole physics, especially the theorems of black hole thermodynamics. Although WCCC is believed to be correct at some level, but the test of its validity is still a huge challenge in astrophysics. For example, it was shown in Ref. Joshi () that naked singularity can be produced during the collapse of a massive matter cloud with regular initial data.

The first work of testing the validity of WCCC can be traced back to the gedanken experiment proposed by Wald Wald (). In the case, he envisaged that a matter particle with various quantities such as the mass, electric charge, and angular momentum is dropped into an extremal Kerr-Newman black hole. Result indicates that such particle which can destroy the horizon of the black hole either miss, or is repelled by the black hole. Later, the test of WCCC was revisited by many authors. Hubeny showed that it is indeed possible to overcharge a near-extremal Reissner-Nordstrom black hole by tossing a charged test particle, which obviously leads to the violation Hubeny (). Also, Jacobson and Sotiriou found that a near-extremal Kerr black hole can be overspun by a test particle with tailored angular momentum Jacobson (). But once considering the effects of radiation and self-force, it is inspiring that Kerr black holes are prohibited to be overspun so that WCCC can retain. Radiative effects occur for some special orbits but self-force always makes effects during the process that particles fall into the black holes Barausse (); Barausse2 (); Colleoni (); Colleoni2 (); Hod (); Zimmerman (). There are also many other works discussing about the validity of WCCC in various black hole backgrounds Siahaan (); Gwak (); Gao (); Gwak2 (); Rocha (); Crisford (); Gwak3 (); Husain (); Revelar (); An (); Ge (); Fairoos (); Hod2 (); Gao2 (); Crisford2 (); Natario (); Duztas (); Gwak4 (); Sorce (); Yu (). Interestingly, an extremal Reissner-Nordstrom black hole may be turned into a Kerr-Newman naked singularity by capturing a neutral spinning body Saa (). However, it is worthwhile to note that due to the narrow allowed range of particle’s energy to destroy the black holes, taking the radiative and self-force effects into account may be a cure for the problem that thought experiments break down WCCC. In other situations, many works have studied the possibility of destroying the horizon of a black hole through fields rather than the matter particles Duztas2 (); Duztas3 (); Duztas4 (); Duztas5 (); Casals (); Matsas (). Although most fields can encounter the phenomenon of superradiance which might prevent the violation of WCCC, there are still some fields like massless fermions that do not exhibit superradiance Unruh (). Thus, event horizon can be destructed if the fields are absorbed by the black hole Saa2 ().

On the other hand, among various modified gravitational (MOG) theories, the scalar-tensor-vector gravity Moffat () has gained an increasing attention in recent years. It can provide better interpretation for the astronomical observations, such as the solar system observations Rahvar (); Toth (), the dynamics of galactic clusters Brownstein (); Moffat2 (), as well as the detection of gravitational waves MoffatMoffat (); Moffat5 (). Recently, the black hole solution was obtained by solving the field equations in this gravity and some novel results were revealed Moffat3 (). The shape of the shadow for the black hole was investigated in Ref. Moffat4 (). Thermodynamical properties for the non-rotating and rotating black hole solutions were studied in Mureika (); Pradhan (). The property of the accretion for the black hole was examined in Refs. John (); Armengol (). And in Refs. Lee (); Sharif (); Zakria (), the geodesics were explored.

In this paper, by neglecting the effects of self-force and radiation, we use the thoughts of Wald’s gedanken experiment to investigate the possibility of destroying the Kerr-MOG black holes. We find that the test particle within narrow range of energy can produce the naked singularity from the near-extremal black hole but not the extremal one. Thus, it seems that the validity of WCCC can be broken down. Next we introduce an adiabatic process that indicates the absorption of test particles happens during a period rather than at a moment. It is very inspiring that the violation can be reduced and the near-extremal black holes remain stable during the particle absorption process. So it can be concluded that WCCC is always valid for Kerr-MOG black holes. Our method can be generalized to other cases.

The paper is organized as follows. In Sec. II, we briefly review the Kerr-MOG black hole. In Sec. III, we study the instability of the black hole event horizon and the constraint that the Bekenstein-Hawking entropy never decreases. Then we get the upper and lower bounds of energy needed by test particles to destroy the Kerr-MOG black hole. In Sec. IV, we introduce the adiabatic process and restore WCCC. Finally, the discussions and conclusions are presented in Sec. V.

## Ii Kerr-MOG black hole

The field equation of matter-free MOG is

(1) |

where the parameter is related to the Newton’s gravitational constant with the relation . Here is a deformed parameter measuring the deviation from general relativity (GR). For a vector field , the energy-momentum tensor reads

(2) | |||||

(3) |

The positive gravitational charge of the vector field is , which indicates that should be positive and this MOG can reduce to GR when vanishes. Moreover, the field strength satisfies

(4) | |||

(5) |

By solving these fields equations, one can obtain black hole solutions. In Boyer-Lindquist coordinates, the metric of a Kerr-MOG black hole is given as Moffat3 ()

(6) |

where the metric functions are given by

(7) | |||||

(8) |

In the following, we will adopt for simplicity. According to Ref. Sheoran (), the ADM mass is and the angular momentum is . Thus, we get

(9) |

By solving , we can obtain the radius of the black hole horizon

(10) |

It is clear that such black hole can possess two horizons for , one degenerate horizon for , and no horizon related to naked singularity for .

In order to describe the process that a test particle is absorbed by the Kerr-MOG black hole, we necessarily show the geodesics of the test particle with unit mass just in terms of the radial and -directional momentum Moffat3 ()

(11) | |||||

(12) |

where

(13) | |||||

(14) |

The sign functions and are independent from each other, is the carter constant, and =1 and 0 for massive particle and photon, respectively. Then, by removing the separate constant , one gains the energy formula describing a future-forwarding particle when it passes through the event horizon of the black hole along the equator plane with :

(15) |

Due to the conserved quantities such as energy and angular momentum of the test particle, the Kerr-MOG black hole will undergo infinitesimal changes constrained by this relation.

## Iii Kerr-MOG black hole and Naked Singularity

Next, by neglecting the effects of radiation and self-force, we will investigate whether it is possible to destroy the horizon of the Kerr-MOG black hole. First, we introduce a small parameter

(16) |

which satisfies . In order to treat the matter particle as a test body, we assume that the energy and angular momentum of the particle satisfy , . When the test particle reaches the event horizon, the final change of the black hole energy should be constrained as

(17) |

It is natural that the violation of WCCC should be related to the instability of the event horizon. Thus, we investigate whether the minimum of the function can be positive when the test particle is absorbed by the black hole. First, we determine the location of the minimum under the initial condition

(18) | |||

(19) |

Particle absorption infinitesimally changes the charge of the black hole and the function , thus, the minimum point moves to a new vicinity , which should satisfy

(21) | |||||

We can easily get and :

(22) | |||||

(23) |

In order to produce the naked singularity which corresponds to no horizon solution, the minimum of the function at the new point should be positive

(24) |

Plunging Eqs. (22) and (23) into the above expression (24), we can obtain the upper energy bound

(25) |

On the other hand, the surface of the horizon should always cover the inside of the black hole when the test particle is captured. It means that the surface area of the black hole which is proportional to the Bekenstein-Hawking entropy never decreases. The change of the black hole entropy is described as

(26) |

Then, we need to determine the exact formula of . Similar to the situation of the minimum of the function given above, we have

(27) |

Solving , and substituting it into (26), we can get the lower bound

(28) |

Now let us study the extremal Kerr-MOG black hole case of . According to Eqs. (25) and (28), we gain the lower and upper energy bounds

(29) | |||||

(30) |

Obviously there is no energy solution of the test particle allowed to destroy the extremal Kerr-MOG black hole. The conclusion is very similar to the case of the extremal Kerr black hole. Actually, we can set or to restore the result of the extremal Kerr black hole.

Next we examine the near-extremal Kerr-MOG black hole case. Taking =1 and =0.99 for the Kerr-MOG black hole, we plot the lower and upper energy bounds as a function of for the test particle with angular momentum =0.01 and 0.001 in Fig. 1. Both the bounds decrease with . For =0, it reduces to an extremal black hole case. For the near-extremal black hole , we find that there always exists a region (denoted with dark green color) which satisfies . Thus, WCCC can be violated in the region. With the increase of , the energy gap first increases, and then decreases. Moreover, for smaller angular momentum , the energy region allowed to violate WCCC shrinks and the energy gap also decreases.

## Iv Restoration of Weak Cosmic Censorship Conjecture

As suggested in Refs. Gwak (); Chirco (), we would like to consider a continuous path called the adiabatic process during the particle absorption. The violation of WCCC is mostly due to the fact that the absorption of the test particle is assumed to happen at a moment, while it should take time to spread across the horizon of the black hole. Instead of jumping from the initial condition to the final state at a once, we split the process into steps.

When the black hole parameters change from and to and , the parameter will change to

(31) |

For simplicity, we define . Then substituting Eqs. (10), (16), and (17) into the above equation (31), we can clearly obtain

(32) | |||||

Here, is the radial momentum of the test particle, which is chosen as a constant. Then the continuous path is given in terms of steps

(33) | |||||

The final state of the black hole is still and , but the resulting parameter can always be positive if the step number is taken to be large enough. Thus, the event horizon will not be destroyed, which leads to the restoration of WCCC. In Table.1, we list the changes of the parameter and the horizon radius . The numerical values which we choose are , , and . In addition, the initial value and the step number is and . According to the relation (17), we set and .

Instead of destroying the event horizon of the near-extremal Kerr-MOG black hole at once, the black hole gets further away from the extremal condition with the step . Moreover, we find that the horizon radius becomes larger within this adiabatic process, which may mean that the black hole gets more stable during the absorption of the test particle rather than produces the naked singularity. Thus, we restore the validity of WCCC when considering effect of the adiabatic process.

0 | 1 | 2 | 3 | 4 | 5 | 6 | 7 | 8 | 9 | 10 | |
---|---|---|---|---|---|---|---|---|---|---|---|

1.00 | 2.64 | 4.18 | 5.65 | 7.05 | 8.42 | 9.74 | 11.03 | 12.28 | 13.51 | 14.71 | |

1.0099 | 1.0166 | 1.0213 | 1.0250 | 1.0283 | 1.0312 | 1.0339 | 1.0364 | 1.0387 | 1.0409 | 1.0430 |

## V Discussions and Conclusions

In this paper, we investigated whether the validity of WCCC still remains in Kerr-MOG black hole background. By using the equation of motion of a test particle with energy and angular momentum, we got the infinitesimal change of the corresponding charge of the black hole during the absorption process of the test particle.

Next, we studied the instability of the event horizon and the constraint condition that the Bekenstein-Hawking entropy never decreases during the absorption process. Thus, we gained the upper and lower energy bounds needed by a matter particle to destroy the Kerr-MOG black hole. It is interesting that the upper and lower bounds coincide with each other for the extremal black hole case, which indicates that the extremal black hole cannot be overspun. While for the near-extremal black hole case, there always exists a narrow energy range allowed to destroy the black hole.

Further, we also considered the effect of the adiabatic process. Instead of assuming the absorption process happens at a moment, the whole process is split into many steps during a period of time. In this case, the near-extremal black hole becomes larger and more stable rather than a naked singularity. Thus, WCCC is restored. In conclusion, the validity of WCCC is guaranteed in the Kerr-MOG black hole background.

## Acknowledgements

This work was supported by the National Natural Science Foundation of China (Grants No. 11675064, No. 11522541, No. 11375075, and No 11205074), and the Fundamental Research Funds for the Central Universities (Grants Nos. lzujbky-2016-115).

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