The non-Kerr black hole with acceleration

The no-hair theorem can be tested in the strong gravity regime by using the top-bottom approach and the bottom-top approach. The non-Kerr spacetime of the later approach is an ideal framework to do the tests in the region very close to the black holes. In this work, we propose a non-Kerr black hole metric (and its charged extension) that is accelerating as well. These new objects are studied for their basic properties and thermodynamics.


Introduction
The astrophysical black holes are described uniquely by mass, charge and spin and are represented by the Kerr-Newman metric or the Kerr metric in the absence of charge. This is the no-hair theorem. Therefore, all the astrophysical black holes are supposed to be described by the Kerr metric. One can test this black hole hypothesis in strong gravitational field by employing the top-bottom approach and the bottom-top approach [1]. In the first case, there are theoretical models in which the Kerr metric does not describe the spacetime of an astrophysical black hole [2,3]. The second approach involves phenomenological modification of the metric. The metric in this approach can be some black hole solution in some theory of gravitation. There is no particular theory involved. The fundamental theory is unknown here, although it is assumed that particles will follow geodesics in this theory. The idea is to develop the understanding into the theory through observations. Many such spacetimes have been developed [4][5][6][7][8] from which probable observational signatures of the Kerr-like spacetimes can be studied. The degree of deviation from the Kerr varies from small to large in such Kerr-like spacetimes. The free parameters are employed to study the observable. Such modified spacetimes reduce to the Kerr metric when the new parameters are set to zero.
The Newman-Janis algorithm is an important solution generating technique [9]. Starting from a static spactime, this algorithm induces rotation in it. The Kerr and Kerr-Newman metrics have been derived by using this algorithm. The seeds metrics for the derivation of Kerr and Kerr-Newman black holes were taken as Schwarzschild and Reissner-Nordström metrics, respectively [9,10].
The non-Kerr metric proposed by Johannsen and Psaltis [4] is an important and extensively studied example of the bottom-top approach. It contains a set of free parameters besides mass and spin. These parameters measure deviation from the Kerr metric. This metric is asymptotically flat, axisymmetric and vacuum solution of some unknown field equations which are different from the Einstein field equations for the non-zero deviation parameters. It has been shown that for the case of a single deformation parameter, this non-Kerr spacetime is regular and does not contain unphysical properties outside the event horizon and can represent a black hole upto the maximum range of the spin parameter. It is also an ideal framework for the tests of the no-hair theorem in the region very close to the black hole and which do not have explicit dependence on the field equations. The metric has been extended to include the electric charge and two different deviation functions [11,12].
The charged non-Kerr (deformed) black hole spacetime is given by [11] Here is the deformation parameter, a, M , q denote the spin, mass and charge of the black hole, respectively. It reduces to the non-Kerr spacetime in the case, q = 0, and to the Kerr-Newman spacetime when the deformation is made to vanish. It is also valid in the super spinning cases in strong gravitational field. This metric was developed with a desire to create an analogue of the Kerr-Newman metric of GR in the alternate theories of gravity. Karl Shawarzchild and Johannes Droste gave a non-trivial solution of Einstein's field equations in 1916 for first time in history but this solution was misunderstood due to the presence of singularity. This confusion was cleared after the discovery of David Finklestein, who provided a comprehensive description of the singularity as a surface which can only be traversed in one direction. These solutions were mathematically good but were not good enough physically. Levi-Civita extended this work and improved the solutions [13]. After him, Ehlers and Kundt [14,16] classified the research work of Levi-Civita in A, B and C classes of metrics. The name of C-metric still exists in the literature and it was interpreted as the solution of accelerated black holes in 1970 by Kinnersely and Walker [17]. Later on, Plebański and Demiański discovered a new spacetime in 1976 which covers a large family of electro-vacuum type-D metrics in GR. Starting from a general form of the metric, Einstein-Maxwell field equations are employed to generate the solutions [18]. The accelerating and rotating black hole solution is an important member of the Plebański and Demiański family of spacetimes. Here, acceleration of the black hole is measured by the parameter α. The metric represents the gravitational field of a pair of uniformly accelerating Kerr-type black holes.
If the cosmological constant is taken to be zero, the charged accelerating black hole solution is [19][20][21][22][23] where Ω = 1 − αr cos θ, (1.4) Here α denotes the acceleration parameter. For the case where the cosmological constant Λ ≥ 0, with α = 0, the spacetime gives a single black hole, but for α = 0 it gives a pair of causally separated black holes that are accelerating away from each other in opposite directions [20,21,24]. For Λ < 0, the spacetime is a single black hole if α is small and a pair of black holes if it has a large value [25][26][27]. Putting α = 0 gives the Kerr-Newman metric. Setting α = 0 = q yields the Kerr metric.
In this article, we develop non-Kerr black holes (charged and neutral), that are accelerating, in some alternate theory of gravity. Here, the main concern lies with the form of the metric and its subsequent properties. When the deformation vanishes, these spacetimes reduce to the accelerating black holes.
The black hole evaporation and thermal radiation theory [28,29] has a significant role in the development of theoretical physics. It has provided a connection among the fields of GR, quantum physics and thermodynamics. The laws of black hole thermodynamics are similar to the laws of ordinary thermodynamics i.e. these massive objects obey the standard laws of thermodynamics. The field of black hole thermodynamics is an extensively studied field and can be considered as a gateway to understanding the quantum nature of black holes. This article introduces two new metrics and we have discussed their thermodynamic properties.
The article is arranged as follows: In Section 2 non-Kerr metrics with acceleration are given. The basic structure of these spacetimes is also studied in this section. In Section 3 thermodynamic analysis of the new metrics with acceleration is done. In section 4 the thermodynamic properties of the non-Kerr black hole have been studied. The work is concluded in Section 5 . Throughout this work, we take G = c = 1.

Non-Kerr black hole spacetime with acceleration
In this section we propose two non-Kerr accelerating black hole metrics and their certain basic characteristics. First is the charged non-Kerr accelerating black hole metric and the second one is the special case of the new metric in the case of zero electric charge.

The charged case
The charged non-Kerr accelerating spacetime is proposed as where Ω, ρ, P , Q and h are given in Eqs. So, we make the assumption that the above spacetime might be an electro vacuum solution to some unknown field equations which are different from the Einstein-Maxwell equations for non vanishing h(r, θ). To confirm the validity this black hole metric, we study its basic structure, that is, event horizon, ergosphere and the redshift. The equation g rr = 0, where g rr is the inverse of the g rr component of the metric, gives the event horizon, which will be denoted by r + . For metric (2.1), the event horizon equation is For h = 0, the above equation reduces to the horizon equation for charged accelerating solution. In this case, Eq. (2.2) gives the event and Cauchy horizons (which are same as in the Kerr-Newman case) and also acceleration horizons at r A = 1/α and r α = 1/α cos θ [30]. For h = 0, the horizon equation is which shows that the acceleration horizon r A = 1/α does not exist. The presence of h introduces the θ-dependence onto the event horizon. Such dependence has been observed for other spacetimes as well [4,11,31,32]. Due to the complexity of the equation, the event horizon equation is solved numerically. The numerical plots are shown in Fig. 1. From Fig. 1 it is clearly seen that the event horizon exhibits two kinds of behaviour. In the right panel, the horizon shows spherical topology while on the left panel, two disjoined horizons are obtained. Such disjoined event horizon has also been observed for some other modified Kerr black holes as well [31,33]. The metric has the determinant The Kretschmann scalar for metric (2.1) is a long and complex expression which shows divergent behavior at 1 + h = 0. Also, from Eq. (2.4), it is evident that the determinant vanishes at 1 + h = 0. Therefore, the metric has an intrinsic singularity at the surface given by 1 + h = 0. This is in contrast to the accelerating black hole metric which has a Kerr-like singularity at r = 0, θ = π/2. The infinite redshift surface is obtained by solving the equation which for the black hole metric of Eq. (2.1) is 6) It is clear from this equation that there are two candidates for the infinite redshift surface, 1 + h = 0 being one of them. But 1 + h = 0 is the singularity, so it is concluded that infinite redshift surface exists at the solution of Q−a 2 P sin 2 θ = 0. Thus the infinite redshift surface is same as in the charged accelerating black hole metric. The region between the outer infinite redshift surface and the event horizon is known as the ergosphere and it is plotted in Fig. 2.  The graphical representation of the event horizon, the infinite red shift surface and 1 + h = 0 for varying numerical values of parameters are plotted in Fig. 3.

The uncharged case
If we put q = 0 in spacetime (2.1), we get the non-Kerr accelerating black hole spacetime. It is given by We have taken the acceleration parameter as 0.009, θ = π/2 and q = 0.00001. Mass has been set to 1 in these plots.
where Ω and ρ 2 are same as in Section 1 and P and Q have the expressions given as P = 1 − 2αM cos θ + α 2 a 2 cos 2 θ, (2.8) As with the charged case, it is assumed here, that the above metric is a vacuum solution to the unknown field equations which are different from Einstein's field equations for non-zero h(r, θ). The numerical plots of the event horizon for this black hole are shown in the Fig.  4. As in the charged case, we again observe two kinds of behavior for the event horizon.

Thermodynamics of the non-Kerr accelerating spacetime
This section comprises the thermodynamic analysis of the two black hole metrics presented in the previous section.

The charged case
The study of thermodynamic features is an important property of a black hole. To satisfy the first law of thermodynamics for black holes, we need to find the Hawking temperature that becomes proportional to surface gravity near the horizon and an entropy which is proportional to the area of horizon. The surface gravity κ is given as where h ab denotes the inverse of the metric deduced from the t − r sector of the spacetime and h = det h ab . Since the metric is stationary, so h 00 = 0. Putting the values of h 11 and √ −h from Eq. (2.1), the surface gravity takes the form × (a 2 +q 2 −2M r+r 2 )(1−α 2 r 2 )−a 2 (1−2αM cos θ+α 2 a 2 +q 2 cos 2 θ sin 2 θ 1/2 .

(3.3)
At the event horizon r + , the surface gravity κ + takes the form The Hawking temperature, T = κ/2π, at the event horizon can be obtained by substituting the value of κ from the above equation as .  The Hawking temperature at the equatorial plane has been plotted in Fig. 7. The plot shows that temperature exhibits negative behavior for various mass values. Such negative temperature has also been observed in the case of accelerating black holes [25].
To discuss entropy of the black holes, the first thing to be determined is the angular velocity ω, which leads to the horizon area [34]. We have Substituting the required metric components, the angular velocity at the event horizon is given as The horizon area of a rotating black hole is defined as Now we use Eq. (3.7) to obtain The entropy of the black hole, S = A/4, by substituting from the above equation, takes the form . (3.10) The above entropy expression has dependence on the r + and θ. For the graphical purpose, the entropy has been plotted at the equitorial plane (θ = π/2) in Fig. 8. The figure clearly shows that S increases with M in accordance with the second law of black hole thermodynamics.  Figure 8. Plot of the entropy against mass for the charged non-Kerr accelerating black hole for a = 0.5, = 0.5, θ = π/2, α = 0.01 and q = 0.5. Now, the first law of thermodynamics can be written in the form of the law of conservation of mass by the following equation [35,36] where ϕ + denotes the electrostatic potential of the black hole Substituting the area expression in the above equation, we get . (3.13) By substituting the values of κ + , ω + and ϕ + from Eqs. (3.4), (3.7) and (3.13) in (3.11), the first law takes the form . (3.14)

The uncharged case
Now a study of thermodynamic properties of black hole in the case of q = 0 is made to compare with the results obtained for the charged case. For the spacetime (2.7), the surface gravity and the Hawking temperature at the r + are (3.16) The Hawking temperature at the equatorial plane is shown in Fig. 9. Again we get , (3.19) where the value of P is given in Eq. (2.8). Fig. 10 shows that entropy is an increasing function of M . The entropy behavior in Figs. 8 and 10 can be further elaborated from  Figure 10. Plot of the entropy against mass for the uncharged non-Kerr accelerating black hole for a = 0.5, = 0.5, θ = π/2, α = 0.01. In the first law of thermodynamics given in Eq. (3.11), the last term vanishes as our spacetime is uncharged. By substituting the values of κ + and ω + from Eqs. (3.15) and (3.17) in (3.11), the first law takes the form (3.20) The first law given in the Eq. (3.14) reduces to the Eq. (3.20) in case of vanishing charge.

Thermodynamics of the non-Kerr black hole
For completeness and comparison, in this section, we work out the thermodynamic quantities for the non-Kerr spacetime (1.1) with q = 0. The surface gravity and the Hawking temperature at r + are The Hawking temperature has been plotted in Fig. 11. We see that when the acceleration parameter is absent, the negative values of temperature do not appear. The expressions  Figure 11. Plot of the Hawking Temperature against mass for the non-Kerr black hole.
(4.4) S = π r 2 + + a 2 2 − a 2 sin 2 θ(a 2 − 2M r + + r 2 . (4.5) The entropy has been plotted in the Fig. 12 which shows increasing behaviour with varying mass. The first law of thermodynamics in this case is dA + a (r 2 + + a 2 ) − (a 2 − 2M r + + r 2 + ) (1 + h) r 2 + + a 2 2 − a 2 sin 2 θ(a 2 − 2M r + + r 2 + )(1 + h) dJ. (4.7) . We conclude this section on Table. 2 which highlights the plots of the Hawking temperature of this article in an effective way. It is observed that the presence of α not only decreases the temperature but is also responsible for the negative values of the temperature when compared with the case α = 0 = q.

Conculsion
In this work, we have formulated two deformed Kerr black hole spacetimes with acceleration and studied the effects of the deviation parameter and the acceleration parameter α on their structure and thermodynamics. The well-known accelerating black hole solution is the limiting case (when the deformation vanishes) of these new metrics. First, we have discussed their event horizon structure. We observe that the presence of the function h(r, θ) introduces special features into the event horizon which for accelerating black holes are represented by the same equation as in the Kerr-Newman and the Kerr solution for the charged and uncharged cases, respectively. The event horizons in our metrics showed spherical topology as well as the disjointed structure. The ergosphere for both the black holes is also discussed.
In the analysis on thermodynamics, we observe that the presence of the acceleration parameter α makes the Hawking temperature negative for various mass values. The negative value of the Hawking temperature has also been observed in literature. We have seen that the factor (1 − α 2 r 2 + ) makes the temperature negative for α > 1/r + , restricting us to take only small values of acceleration [25][26][27]. The negative values of temperature thus obtained are extremely small. Therefore, to avoid the violation of the third law of black hole thermodynamics, it is required to limit α in the range α < 1/r + . For small acceleration parameter, the temperature of our metrics showed positive behavior. The entropy profiles showed increasing values for both the spacetimes and thus satisfied the condition of the second law of thermodynamics.
The present work has also given an approach to study other kinds of deformations that are possible in the case of accelerating black holes. One may include other parameters like NUT charge and cosmological constant besides the acceleration parameter α to see the effect of deformations similar to the one presented here.