Overspinning Kerr-MOG black holes by test fields and the third law of black hole dynamics

We evaluate the stability of the event horizons of Kerr-MOG black holes in the scattering process of scalar test fields. We show that both extremal and nearly extremal Kerr-MOG black holes can be overspun into naked singularities by scalar test fields with a frequency slightly above the superradiance limit. The overspinning becomes generic as the the modification parameter α increases, i.e. the black holes are rendered more and more vulnerable against perturbations as the space-time deviates from the Kerr solution. We also show that nearly extremal Kerr-MOG black holes can be continuously driven to extremality. The third law of black hole dynamics is violated for any non-zero value of the modification parameter α.


I. INTRODUCTION
The singularity theorems developed by Penrose and Hawking imply that the gravitational collapse of a body leads -inevitably-to the formation of singularities [1]. The presence of these singularities precludes the definition of a well-defined initial value problem and thereby ruins the smooth, deterministic structure of space-times in general relativity. The fact that the formation of singularities cannot be avoided led Penrose to propose the cosmic censorship conjecture, which -in its weak weak form (Wccc)-asserts that the gravitational collapse of a body always ends up in a black hole rather than a naked singularity [2]. The singularities should be hidden behind the event horizons of black holes which disable their causal contact with distant observers. This way, the observers at asymptotically flat spatial infinity do not encounter any effects propagating out of the singularity, and the smooth structure of space-times is preserved, at least locally.
As a concrete proof of the cosmic censorship conjecture has been elusive, it has become customary to attack the closely related -though not identical-problem of the stability of event horizons. In these problems one perturbs extremal or nearly extremal black holes with test particles and fields, and checks if the perturbations can lead to the destruction of event horizons which would imply that the singularities become naked. The first thought experiment in this vein was constructed by Wald [3]. There it was shown that test particles cannot overcharge or overspin an extremal Kerr-Newman black hole into a naked singularity. Following Wald many similar tests of Wccc were applied to black holes in vacuum and Einstein-Maxwell theory involving test particles [4][5][6][7][8][9][10][11][12][13], and fields [14][15][16][17][18][19][20][21][22][23][24]. The stability of event horizons in the asymptotically anti-de Sitter case was also evaluated by perturbing the black holes with test particles and fields [25][26][27][28][29][30].
The evolution of singularities indicate the failure of * koray.duztas@ozyegin.edu.tr general relativity at short length scales where quantum effects are expected to dominate. In addition, the fact that one needs to invoke the presence of dark components at large length scales motivated the quest for modified theories of gravity. One of the promising candidates to fill this gap is the Scalar-Tensor-Vector Gravity theory developed by Moffat [31]. This dark matter emulating theory of modified gravity has proved compatible with current observations regarding the rotation curves of galaxies and the dynamics of galactic clusters [32][33][34][35]. It also predicts the existence of gravitational waves which lends credence to its validity as an alternative theory of gravity [36,37]. The scalar-tensor-vector theory of modified gravity has a stationary and axi-symmetric black hole solution which is known as the Kerr-MOG black hole [38]. Kerr-MOG black holes are characterised by the mass parameter M , angular momentum J = M a and the dimensionless parameter α which determines the modification from the Kerr solution. The thermodynamics of Kerr-MOG black holes, their observable shadows, and the quasi-normal modes have been studied [39][40][41]. Recently, it was also shown that energy can be extracted from Kerr-MOG black holes by a Penrose process [42].
The validity of Wccc was tested for Kerr-MOG black holes in the process of the absorption of a point particle by Liang, Wei, and Liu [43]. It was found that -though the extremal black holes cannot-nearly-extremal black holes can be destroyed by point particles. However, the authors argued that the event horizon will be restored when one considers the effect of the adiabatic process. Another intriguing problem at this stage is to test the validity of Wccc in the case of test fields scattering off Kerr-MOG black holes. In this work we evaluate the stability of the event horizons of Kerr-MOG black holes as they are perturbed by test scalar fields. We consider the cases of both extremal and near-extremal black holes. Our analysis exploits the fact that superradiance occurs when scalar fields scatter off Kerr-MOG black holes, which was recently derived by Wondrak, Nicolini, and Moffat [44]. We also evaluate the validity of the third law of black hole dynamics which states that a nearly extremal black arXiv:1907.13435v1 [physics.gen-ph] 24 Jul 2019 hole cannot be driven to extremality by any continuous process.

II. KERR-MOG BLACK HOLES, SCALAR FIELDS, WCCC
In Boyer-Lindquist coordinates, the background geometry of the Kerr-MOG space-time is described by the metric where The MOG parameter α is a dimensionless measure of the difference between the Newtonian gravitational constant G N and the additional gravitational constant G The ADM mass and the angular momentum of the Kerr-MOG black hole are given by [45] The function ∆ can be re-written in terms of the ADM mass where we have set G N = 1 without loss of generality. The spatial locations of the horizons are the roots of ∆ Notice that the parameters of the Kerr-MOG space-time represent a black hole surrounded by an event horizon provided that where the equality corresponds to the case of an extremal black hole. In this work, we start with a Kerr-MOG black hole satisfying the main criterion (7), and perturb the space-time with a scalar field that is incident on the black hole from infinity. The interaction of the black hole with the test scalar field does not alter the structure of the background geometry, but leads to perturbations in the ADM mass and angular momentum parameters. At the end of the interaction the field decays away, leaving behind a space-time with perturbed parameters. If the final parameters of the space-time does not satisfy the inequality (7), one can conclude that the event horizon has been destroyed in the interaction of the scalar field with the black hole; i.e. Wccc is violated. The scattering of test scalar fields by Kerr-MOG black holes has recently been studied by Wondrak, Nicolini, and Moffat [44]. Analogous to the Kerr case, a neutral wave can be separated into variables in the form The contribution of the scattering wave to the mass and angular momentum parameters of the space-time are related by Superradiance occurs for scalar fields scattering off Kerr-MOG black holes as one would naively expect from Kerr analogy. If the frequency of the incoming wave is below the superradiance limit, the wave is reflected back with a larger amplitude, i.e. there is no net absorption of the wave by the black hole. The superradiance limit ω sl for Kerr-MOG black holes is also derived in [44] ω sl = mΩ = ma r 2 + + a 2 (10) where Ω is the angular velocity of the black hole and r + is the spatial location of the event horizon.

A. Overspinning extremal Kerr-MOG black holes
By definition, an extremal Kerr-MOG black hole satisfies where we have defined δ in . We perturb the extremal black hole with a scalar field to check if it is possible to overspin the black hole into a naked singularity. The contribution of the incoming wave to the energy and angular momentum parameters of the black hole are related by (9). A necessary condition for overspinning to occur is that one should be able to adjust the parameters of the incoming wave such that δ fin < 0 at the end of the interaction. To be more precise, we demand that By substituting δJ = (m/ω)δE, and using (11), the condition (12) can be simplified in the form We choose δE = M for the incoming field wit 1, so that the test field approximation is justified. With this choice we can derive the maximum frequency of an incoming wave, which can be used to overspin an extremal Kerr-MOG black hole If the frequency of a scalar field is below the maximum value determined in (14), the scalar field can overspin an extremal Kerr-MOG black hole into a naked singularity. However, this condition is not sufficient for overspinning to occur. For that purpose one should also demand that the incoming wave is absorbed by the black hole; i.e. the frequency of the wave is larger than the superradiance limit. These two conditions should be simultaneously satisfied for overspinning to occur. The superradiance limit for extremal black holes can be derived by substituting r + = M and and a = M/(1 + α) in (10) For overspinning to occur ω max should be larger than the superradiant limit ω sl , so that the frequencies in the range (ω sl , ω max ) can be used to overspin an extremal Kerr-MOG black hole. It is manifest in equations (14) and (15) that ω max will larger than ω sl , if α > . The extremal Kerr-MOG black holes can be overspun into naked singularities by scalar test fields provided that the deformation parameter α is larger than the small parameter ; i.e. as the spacetime considerably deviates from the Kerr solution.

B. Overspinning nearly-extremal Kerr-MOG black holes
In the last decade it was shown that though extremal Kerr black holes cannot be overspun, nearly extremal Kerr black holes can be overspun into naked singularities by a discrete jump by test particles [5] and fields [16]. Recently Sorce and Wald considered the second order variations which account for backreaction effects, and showed that overspinning is not possible in a complete second order analysis [22]. In this section we attempt to overspin nearly-extremal Kerr-MOG black holes by test scalar fields. We parametrise a nearly-extremal Kerr-MOG black hole in the form which implies that As in the case of extremal black holes, we send in a test field from infinity and demand that δ fin < 0 at the end of the interaction, so that the final parameters of the space-time represent a naked singularity.
where we have used that δJ = (m/ω)δE. Again we choose δE = M for the energy of the incident wave, and impose (17) to simplify (18). The condition that δ fin < 0 can be expressed as Using (19), one directly derives the maximum frequency ω max for a scalar field incident on a nearly extremal Kerr-MOG black hole parametrised as (17), that could overspin the black hole into a naked singularity As we mentioned in the case of extremal black holes, the condition (20) is not sufficient for overspinning to occur. We should also demand that the frequency of the incoming wave is larger than the limiting frequency for superradiance. If (ω sl < ω max ), there exists a range of frequencies (ω sl , ω max ) which can be chosen to overspin a nearly-extremal Kerr-MOG black hole. To compare ω sl and ω max , one has to express ω sl for a nearlyextremal black hole in terms of the small parameter .
Notice that for the nearly-extremal Kerr-MOG black hole parametrised as (16) and which leads to Though it is not quite manifest in equations (20) and (23), the maximum frequency for an incident wave to overspin a Kerr-MOG black hole is actually larger than the limiting frequency for superradiance. To clarify this we have set ω max = (m/2M)f (α) and ω sl = (m/2M)g(α) and plotted f (α) and g(α) for = 0.01, in the figure (1). The frequencies ω max and ω sl almost coincide for α = 0. However, as α increases the range of frequencies that can be used to overspin a Kerr-MOG black hole enlarges.
It would be appropriate to clarify the arguments above with a numerical example. Let us consider a nearly extremal Kerr-MOG black hole with α = 1, parametrised  FIG. 1. The graphs of f (α) and g(α) for = 0.01. ωmax is larger than ω sl for α > 0. ωmax and ω sl deviate from each other as α increases.
as (17), where = 0.01. Without loss of generality we take M = 1 for the initial mass of the black hole. The parametrization (17) implies that J in = 0.9999/ √ 2. We perturb this black hole with a test scalar field. We choose δE = M = 0.01 for the incoming field. The maximum value for the frequency of the incoming field to overspin the Kerr-MOG black hole, and the limiting value for superradiance can be calculated using (20) and (23).
Let us choose ω = m/(2M) for the frequency of the test scalar field. This field will be absorbed by the Kerr-MOG black hole since ω > ω sl . According to the analysis above this test field should overspin the Kerr-MOG black hole into a naked singularity. A straightforward calculation yields where δJ = δE(m/ω) = 0.02. The negative sign for δ fin indicates that the Kerr-MOG black hole is overspun into a naked singularity. The question at this stage is whether this overspinning can be fixed by the backreaction effects which were neglected in this analysis. The backreaction effects will bring second order corrections of magnitude ∼ M 2 2 to δ fin , which is not likely to compensate for the negative value in (25) of magnitude |δ fin | M 2 . The destruction of the horizon will become more generic as α increases. For example, if we were to perturb a Kerr-MOG black hole with α = 2 with the same scalar field (δE = 0.01, ω = m/2M), δ fin would read 1 where J in = 0.9999/ √ 3 according to the parametrization (17). The final parameters of the Kerr-MOG black hole represent a naked singularity. The destruction of the horizon cannot be fixed by any form of backreaction effects which will bring second order corrections of magnitude ∼ M 2 2 to δ fin , where δ fin ∼ −M 2 . Overspinning becomes more generic as α increases.

III. THE VALIDITY OF THE THIRD LAW FOR KERR-MOG BLACK HOLES
The laws of black hole dynamics which were proposed by Bardeen, Carter, and Hawking are based on a connection between thermodynamics and black hole dynamics [46]. In this manner the area of the event horizon and the surface gravity are analogous to the entropy and the temperature, respectively. The identification of the area of the event horizon with entropy entails that it should not be possible to decrease the area of the event horizon, which had been previously proved by Hawking assuming that no naked singularities exist in the outer region [47]. Accordingly, it should not be possible to drive a black hole to extremality which would be analogous to decreasing the temperature to absolute zero. After a decade Israel proved the third law of black hole dynamics which states a nearly extremal black hole cannot be driven to extremality in any continuous process [48]. An alternative approach by Dadhich and Karayan also justified the validity of the third law. They showed that the range of the allowed energy and angular momentum ratios to drive a Kerr black hole to extremality, pinches off as one gets arbitrarily close to extremality [49].
Currently, the validity of the third law is justified for Kerr, Kerr-Newman and Reissner-Nördstrom black holes. The derivations by Hubeny, Jacobson-Sotiriou, and Düztaş-Semiz that nearly-extremal black holes can be overcharged or overspun into naked singularities [4,5,16] should not be interpreted as counter-examples to the third law. These authors confirm that extremal black holes cannot be overcharged/overspun, which implies that nearly extremal black holes are driven beyond extremality by a discrete jump rather than a continuous process. As one gets arbitrarily close to extremality the allowed ranges of energy, angular momentum, and/or charge for the perturbation vanishes in accord with the derivations of Dadhich and Karayan. Moreover, Sorce and Wald recently showed that such violations of Wccc can be fixed by considering the backreaction effects in a full second order analysis [22].
The analysis for the nearly-extremal Kerr-MOG black holes in the previous section can be exploited to test the validity of the third law. Let us consider a Kerr-MOG black hole arbitrarily close to extremality, which corresponds to the case → 0. The maximum value for the frequency of an incoming scalar field to overspin this Kerr-MOG black hole is the → 0 limit of ω max derived in (20).
A Kerr-MOG black hole arbitrarily close to extremality would become extremal if it absorbed a test field with frequency ω = ω max given in (27), while it would be overspun if ω < ω max as discussed in the previous section.
The critical question at this stage is whether ω max is larger than the superradiant limit in the case → 0. The limiting frequency for superradiance in the case → 0 can be directly calculated using ω sl derived in (23). After some algebra one derives One can compare ω sl and ω max in the limiting case → 0, by (27) and (28) lim →0 ω max ω sl = 1 + α 2 (29) Contrary to the case of the Kerr family of solutions, the interval (ω sl , ω max ) does not pinch off as the black hole becomes arbitrarily close to extremality. Therefore Kerr-MOG black holes can be continuously driven to extremality by scalar test fields with frequency ω max , which is larger than the superradiance limit ω sl even in the → 0 limit. The third law of black hole dynamics is invalid for Kerr-MOG black holes which are characterised by a non-zero deformation parameter α.

IV. CONCLUSIONS
After the recent work of Sorce and Wald [22], the validity of Wccc in the interaction of Kerr and Kerr-Newman black holes with test particles and fields is established provided that the energy-momentum tensor of the perturbation satisfies the weak energy condition. In a very recent work we have shown that, if one removes the proviso that the perturbation obeys the weak energy condition as in the case of neutrino fields, Kerr and Kerr-Newman black holes can be generically overspun into naked singularities. (See [50] for details). However this is a different problem the solution of which would probably require a quantum treatment of the interaction beyond the semi-classical level.
In this work we applied a test of the weak cosmic censorship conjecture in the interaction of Kerr-MOG black holes with test fields. We restricted ourselves to the case of scalar fields the energy-momentum tensor of which obey the weak energy condition. Our analysis also exploits the fact that superradiance occurs for scalar fields scattering off Kerr-MOG black holes [44]. Superradiance is essential in a scattering process as it determines the lower limit for the frequency of a wave to ensure that it is absorbed by the black hole. In the absence of such a limit the modes carrying low energy and relatively high angular momentum can also be absorbed by the black hole which reinforces the overspinning of the black hole. We have shown that both extremal and nearly-extremal Kerr-MOG black holes can be overspun by test scalar fields with a frequency slightly above the superradiance limit. The range of the allowed frequencies for the incoming field is extended as the modification parameter α increases. Therefore the overspinning of Kerr-MOG black holes becomes more generic as the space-time deviates from the Kerr solution.
One would also expect the third law of black hole dynamics to hold for Kerr-MOG black holes analogous to the Kerr case. However our analysis for the nearlyextremal Kerr-MOG black holes imply that the allowed range of frequencies for overspinning to occur does not pinch off even in the → 0 limit. Thus, a nearly extremal Kerr-MOG black hole that is arbitrarily close to extremality can be continuously driven to extremality by absorbing a test field with frequency ω max , and beyond extremality if ω < ω max . It is manifest in (29) that the interval (ω sl , ω max ) will not vanish, unless α = 0. The introduction of the modification parameter α invalidates the third law of black hole dynamics for Kerr-MOG black holes interacting with test scalar fields.