Eur. Phys. J. C (2018) 78:513 https://doi.org/10.1140/epjc/s10052-018-5991-x

Regular Article - Theoretical Physics

Kerr–Newman-AdS black hole surrounded by perfect fluid matter in Rastall gravity

Zhaoyi Xu1,2,3,4,a,XianHou1,3,4,b, Xiaobo Gong1,2,3,4,c, Jiancheng Wang1,2,3,4,d 1 Yunnan Observatories, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming 650216, People’s Republic of China 2 University of Chinese Academy of Sciences, Beijing 100049, People’s Republic of China 3 Key Laboratory for the Structure and Evolution of Celestial Objects, Chinese Academy of Sciences, 396 Yangfangwang, Guandu District, Kunming 650216, People’s Republic of China 4 Center for Astronomical Mega-Science, Chinese Academy of Sciences, 20A Datun Road, Chaoyang District, Beijing 100012, People’s Republic of China

Received: 1 May 2018 / Accepted: 13 June 2018 / Published online: 20 June 2018 © The Author(s) 2018

Abstract The Rastall gravity is the modified Einstein gen- 1 Introduction eral relativity, in which the energy-momentum conservation μν ,ν law is generalized to T ;μ = λR . In this work, we derive The dark matter and dark energy are two unsolved problems the Kerr–Newman-AdS (KN-AdS) black hole solutions sur- in cosmology and particle physics. Today many observa- rounded by the perfect fluid matter in the Rastall gravity tions, including TypeIa supernovae, cosmic microwave back- using the Newman–Janis method and Mathematica pack- ground (CMB), baryon acoustic oscillations (BAO), weak age. We then discuss the black hole properties surrounded lensing, rotation curve and larger scale structure, etc., have by two kinds of specific perfect fluid matter, the dark energy revealed that the dark energy accounts for 73%, the dark (ω =−2/3) and the perfect fluid dark matter (ω =−1/3). matter for 23% and ordinary baryonic matter only for 4% of Firstly, the Rastall parameter κλ could be constrained by the total mass-energy of the universe (e.g., [1,2]). Following the weak energy condition and strong energy condition. Sec- these observations, many theoretical models have been pro- ondly, by analyzing the number of roots in the horizon equa- posed (e.g., [3–6]). For example, the phenomenological dark tion, we get the range of the perfect fluid matter intensity energy models include quintessence, phantom, quintom and α, which depends on the black hole mass M and the Rastall other models. The dark matter includes Cold Dark Matter parameter κλ. Thirdly, we study the influence of the perfect (CDM) (e.g., [7–9]), Warm Dark Matter (WDM) and Scalar fluid dark matter and dark energy on the ergosphere. We find Field Dark Matter (SFDM) (e.g., [10–20]). The perfect fluid that the perfect fluid dark matter has significant effects on the dark matter (PFDM) model has also been proposed recently ergosphere size, while the dark energy has smaller effects. (e.g., [21–25]). For spiral galaxies, under the assumption of Finally, we find that the perfect fluid matter does not change the spherical symmetric mass distribution and perfect fluid the singularity of the black hole. Furthermore, we investigate matter, one can obtain the solution of the Einstein equation for the rotation velocity in the equatorial plane for the KN-AdS spherical symmetric gravitational field. When the equation black hole with dark energy and perfect fluid dark matter. of state for spiral galaxies is around − 1/3[21], the PFDM We propose that the rotation curve diversity in Low Surface model can explain the asymptotically flat rotation curves. Brightness galaxies could be explained in the framework of The covariant conservation of the energy-momentum ten- the Rastall gravity when both the perfect fluid dark matter sor plays an important role in Einstein’s general relativity halo and the baryon disk are taken into account. (GR). Taking this conservation as starting point, one can derive the conservation of some globally defined physical quantities through Noether symmetry theorem. These con- served quantities take the form of integrals of the components of the energy-momentum tensor over appropriate space-like a e-mail: [email protected] surfaces, which allow at least one of the Killing vectors of b e-mail: [email protected] the background space-time as their normal. Thus, the total c e-mail: [email protected] rest energy and rest mass of a physical system are conserved d e-mail: [email protected] in the context of GR [34]. On the other hand, some mod- 123 513 Page 2 of 12 Eur. Phys. J. C (2018) 78 :513

μν ,ν ified Einstein GR have relaxed conservation condition of T;μ = λR , (1) the energy-momentum tensor. One of the modified grav- where Tμν is the energy-momentum tensor, λ is the Rastall ity theory was introduced by Rastall [27,28], in which the parameter which represents the level of the energy-momentum conservation of the energy-momentum tensor changes to be μν ,ν conservation law in gravity theory and whether the Mach T = λR . When the space-time is flat, the conserva- ;μ principle is satisfied. In this theory, the generalized Einstein tion law becomes the usual formalism. This theory could field equation is given by be understood as Mach principle which represents the mass distribution depending on the energy and mass in the space- Gμν + κλgμν R = κTμν, (2) time [29], and has many interesting applications in cosmol- κ = π ogy and strong gravity space-time (e.g., [30–36]). However, where 8 G N is the gravitational constant of the Rastall the physical relevance of the Rastall gravity is under dispute. gravity and G N is the gravitational constant of the Newton λ → Recently, Visser [38] pointed out that the Rastall gravity is gravity. When 0, the field equation reduces to the Ein- equivalent to the Einstein GR, while Darabi et al. [37] sug- stein field equation in GR. gested that the two gravitational theories are not equivalent, In Heydarzade and Darabi [34], the Reissner–Nordstrom although both work admit that the parameter κλ always rep- black hole solution in perfect fluid matter is given by − resents the rearrangement of the matter sector of the Einstein ds2 =−f (r)dt2 + g 1(r)dr2 + r 2(dθ 2 + sin2θdφ2), (3) GR. In this context, it is still worth studying the Rastall grav- ity in both observational and theoretical aspects. where The physical effects of the dark matter and dark energy on 2M Q2 f (r) = g(r) = 1 − + the black hole properties have been investigated recently and r r 2 attracted increasing attention. The cosmological constant in 1 + 3ω − 6κλ(1 + ω) − Schwarzschild black hole space-time has been discussed in − κλ( + ω) − αr 1 3 1 . (4) Kottler [48], and later generalized to the rotational black hole by Carter [47]. The dynamics of the quintessence dark energy α is the perfect fluid matter intensity around the black hole, M in Schwarzschild space-time have been obtained by Kiselev is the mass of the black hole, Q is the charge of the black hole [49], in which the equation of state ω for perfect fluid matter and ω describes the equation of state defined by ω = p/ρ is constant. That work has been generalized to the Kerr black with p and ρ the pressure and density of the perfect fluid hole and Kerr–Newman-AdS (KN-AdS) black hole [25,50]. matter, respectively. In the case of spherical symmetry, the Reissner–Nordstrom The electromagnetic field is encoded by the gauge poten- black hole space-time surrounded by perfect fluid matter in tial A which is the Rastall gravity has been obtained by Heydarzade and Q Darabi [34]. In this work, we further generalize the Reissner– A = dt, (5) r Nordstrom black hole space-time to the rotational black hole Although the real dark matter or dark energy could not sat- (KN-AdS black hole) space-time in the Rastall gravity and isfy the perfect fluid condition, the perfect fluid is a good investigate accordingly the KN-AdS black hole properties. approximation to describe the behaviour of the dark matter The outline of the paper is as follows. In Sect. 2,weintro- and dark energy. When λ → 0, the black hole solution Eq. (3) duce the Reissner–Nordstrom black hole solution with per- reduces to the Kiselev formalism [49]. For −1 <ω<−1/3, fect fluid matter in the Rastall gravity. In Sect. 3, we derive the the solution represents the black hole surrounded by the dark KN-AdS black hole solution using Newman–Janis method energy [25,49,50]. For ω =−1/3, the solution represents and Mathematica package. In Sect. 4, we analyze the prop- the black hole surrounded by the perfect fluid dark matter erties of the KN-AdS black hole surrounded by perfect fluid [21–25]. matter in the Rastall gravity, including the energy condition, horizon structure, ergosphere and singularity. In Sect. 5,we calculate the rotation velocity in the equatorial plane and dis- 3 KN-AdS black hole in perfect fluid matter cuss its applications. The summary is given in Sect. 6. 3.1 Newman–Janis method and Kerr–Newman solution in 2 Reissner–Nordstrom black hole in perfect fluid matter perfect fluid matter

In this section, we introduce the Reissner–Nordstrom black Using the Newman–Janis method [51–54], we generalize the hole solution surrounded by perfect fluid matter in the Rastall Reissner–Nordstrom black hole to the Kerr–Newman black gravity. The Rastall gravity is based on the Rastall hypothesis hole in perfect fluid matter in the Rastall gravity. First, we which generalizes the energy-momentum conservation law perform transformation for the spherical symmetric black to the following formalism [27,28] hole space-time metric (Eq. 3) from Boyer–Lindquist (BL) 123 Eur. Phys. J. C (2018) 78 :513 Page 3 of 12 513

μ μ μ μ μ coordinates (t, r,θ,φ)to Eddington–Finkelstein (EF) coor- mμm = lμm = nμm = 0 and lμn =−mμm = 1. In dinates (u, r,θ,φ)through the plane of (u, r), the coordinate transformations are dr −→ − θ, du = dt − , u u iacos 1 + 3ω − 6κλ(1 + ω) − r −→ r − iacosθ. (12) 2M Q2 − κλ( + ω) 1 − + − αr 1 3 1 r r 2 We then perform the transformations of f (r) → F(r, a,θ), (6) g(r) → G(r, a,θ), and 2 = r 2 + a2cos2θ.Inthe(u, r) space, the null vectors become and the space-time metric (Eq. 3) becomes  ⎛ ⎞ 1 + 3ω − 6κλ(1 + ω) μ = δμ, μ = G δμ − 1 δμ, − l r n F r ⎜ 2M Q2 − κλ( + ω) ⎟ F 0 2 ds2 =−⎝1 − + − αr 1 3 1 ⎠ 2 μ μ i μ r r μ = √1 δ + θ(δ − δμ) + δ , m θ iasin r φ 2 0 sinθ × 2 − + 2 2. du 2dudr r d (7) 1 μ μ i μ mμ = √ δ − iasinθ(δ − δμ) − δ . θ 0 r θ φ (13) For the electromagnetic field part, the gauge potential A in 2 sin the (u, r) plane becomes From the definition of the null tetrad, the metric tensor gμν ⎛ ⎞ in the EF coordinates are given by ⎜ ⎟ ⎜ ⎟ 2 2θ 2 2θ Q ⎜ dr ⎟ uu a sin rr a sin A = ⎜du + ⎟ . g = , g = G + , r 1 + 3ω − 6κλ(1 + ω) 2 2 ⎝ − ⎠ 2M Q2 − κλ( + ω) 1 − + − αr 1 3 1 θθ 1 φφ 1 2 g = , g = , r r 2 2sin2θ (8)  G a2sin2θ gur = gru =− − , The non-zero components of the inverse space-time metric F 2 φ φ a φ φ a (Eq. 7)aregivenby gu = g u = , gr = g r =− . (14) 2 2 1 + 3ω − 6κλ(1 + ω) 2 − Through calculation, the covariant metric tensor in the EF rr = − 2M + Q − α 1 − 3κλ(1 + ω) , g 1 r coordinates are r r 2  θθ 1 g = , =− , = 2, = =− G , r 2 guu F gθθ gur gru  F φφ = 1 , ur = ru =− . g g g 1 (9) 2 2 2 F 2 r 2sin2θ gφφ = sin θ + a (2 − F)sin θ , G In the null (EF) frame, the metric matrix can be written as  μν μ ν ν μ μ ν ν μ F 2 g =−l n − l n + m m + m m , (10) guφ = gφu = a F − sin θ, G  where the corresponding components are 2 F grφ = gφr = asin θ . (15) lμ = δμ, G r ⎛ Using Eq. (12) we can obtain the electromagnetic field gauge 2 μ μ 1 ⎜ 2M Q n = δ − ⎝1 − + potential A in the EF coordinates as 0 2 r r 2 Qr ⎞ A = (du − asin2θdφ). (16) 1 + 3ω − 6κλ(1 + ω) 2 − ⎟ − αr 1 − 3κλ(1 + ω) ⎠ δμ, Finally we perform the coordinate transformation from the r EF to BL coordinates as = + λ( ) , φ = φ + ( ) , μ 1 μ i μ du dt r dr d d h r dr (17) m = √ δθ + √ δφ , 2r 2rsinθ where μ 1 μ i μ m = √ δ − √ δ . (11) r 2 + a2 a θ θ φ λ(r) =− , h(r) =− , 2r 2rsin r 2g(r) + a2 r 2g(r) + a2 For any point in the black hole space-time, the null vectors r 2g(r) + a2cos2θ μ = μ = F(r,θ) = G(r,θ)= . (18) of the null tetrad satisfy the relations of lμl nμn 2 123 513 Page 4 of 12 Eur. Phys. J. C (2018) 78 :513

 2 In the BL coordinates, the Kerr–Newman black hole solution θ sin2θ dt dφ + a − r 2 + a2 in perfect fluid matter is therefore given by 2   ⎛ ⎞ 1 − 3ω  φ 2 r dt 2 d ⎜ − κλ( + ω) ⎟ − − asin , (23) ⎜ 2Mr − Q2 + αr 1 3 1 ⎟ 2   ds2 =−⎜1 − ⎟ dt2 ⎝ 2 ⎠ where ⎛ ⎞ 1 − 3ω 1 − 3ω ⎜ − κλ( + ω) ⎟ 2asin2θ ⎝2Mr − Q2 + αr 1 3 1 ⎠ 2 2 2 − κλ( + ω) r = r − 2Mr + a + Q − αr 1 3 1  − dφdt 2 2 2 2 − r (r + a ), ⎛ 3   ⎜ 2 2 2 2 ⎜ θ = 1 + a cos θ,  = 1 + a . (24) + 2dθ 2 + dr2 + sin2θ ⎜r 2 + a2 + a2sin2 3 3 r ⎝ The solution should satisfy the Einstein–Maxwell field ⎞ 1 − 3ω equations which are − κλ( + ω) ⎟ 2Mr − Q2 + αr 1 3 1 ⎟ θ ⎟ dφ2, (19) 1 2 ⎠ Gμν = Rμν − Rgμν + κλgμν R + gμν = κTμν, 2 μν μν;α να;μ αμ;ν F;ν = 0, F + F + F = 0. (25) where 1 − 3ω Through calculations using the Mathematica package (inserting Eq. (23)intoGμν (Eq. 25)), we obtain the Ein-  = 2 − + 2 + 2 − α 1 − 3κλ(1 + ω) . r r 2Mr a Q r (20) stein tensors in the Rastall gravity as And the gauge potential A is accordingly  1 Q2 α(3κλ(1 + ω) − 3ω) G = + Qr tt 6 2 A = (dt − asin2θdφ). (21) 2r 2(1 − 3κλ(1 + ω)) 2 6κλ(1 + ω) − 3ω − 1   − κλ( + ω) From the Newman–Janis method, we know that this Kerr– r 1 3 1 r 4 − 2r 3 M + r 2 Q2 Newman black hole space-time metric in the Rastall grav- 6κλ(1 + ω) − 3ω + 3 ity (Eq. 19) satisfies the Einstein–Maxwell field equation. If − αr 1 − 3κλ(1 + ω) there is no perfect fluid matter around the black hole (α = 0),  ra2sin2θ + a2r 2 − a4sin2θcos2θ − this solution will reduce to the usual Kerr–Newman black 4  hole in the Rastall gravity. If κλ = 0, it will reduce to the 2 α[ κλ( + ω) − ω][ κλ( + ω) − ω − ] − Q + 3 1 3 6 1 3 1 Kerr–Newman black hole in perfect fluid matter in the Ein- r 3 2(1 − 3κλ(1 + ω))2 stein GR [25]. 9κλ(1 + ω) − 3ω − 2  r 1 − 3κλ(1 + ω) 3.2 KN-AdS solution in perfect fluid matter [ 4 − 3 + 2 2 − 4 2θ 2θ]  = 2 r 2r W a r a sin cos W 6 Now we extend the Kerr–Newman black hole to the KN-  2 2θ AdS black hole surrounded by the perfect fluid matter. Since − ra sin W , 4 the Newman–Janis method does not include the cosmologi-  1 − 3ω   1 3κλ(1 + ω) − 3ω cal constant , we employ other method to obtain the solu- G = −Q2 − α r 1 − 3κλ(1 + ω) rr 2 − κλ( + ω) tion with the presence of the cosmological constant, as pre- r 1 3 1 2  sented in Xu and Wang [25]. First, we rewrite the above =−2r W , 2 Kerr–Newman black hole metric (Eq. 19)as r 2asin2θ[(r 2 + a2)(a2cos2θ − r 2)] 2 2 G φ = sin θ t 6 ds2 = dr2 + 2dθ 2 + (adt − (r 2 + a2)dφ)2 ⎡ ⎤ 2 κλ( + ω) − ω − r 6 1 3 1 ⎢ Q2 α(3κλ(1 + ω) − 3ω) − κλ( + ω) ⎥ r ⎣ + r 1 3 1 ⎦ − (dt − asin2dφ)2. (22) 2r 2 2(1 − 3κλ(1 + ω)) 2  2 2θ( 2 + 2) 2 We then guess the KN-AdS black hole solution as − ra sin r a − Q 4 r 3 2 2 α[ κλ( + ω) − ω][ κλ( + ω) − ω − ] ds2 = dr2 + dθ 2 + 3 1 3 6 1 3 1 r θ 2(1 − 3κλ(1 + ω))2 123 Eur. Phys. J. C (2018) 78 :513 Page 5 of 12 513

κλ( + ω) − ω − 9 1 3 2  tensor. Analogically, in the Rastall gravity, the non-zero  1 − 3κλ(1 + ω) r should not appear in the explicit expression of the Einstein 2θ[( 2 + 2)( 2 2θ − 2)]  tensor (or energy-momentum tensor) components either, in = 2asin r a a cos r W 6 the case of the KN-AdS black hole space-time surrounded 2 2θ( 2 + 2)  by the perfect fluid matter. This is exactly what we obtained − ra sin r a W , 4  (Eq. 26). Therefore, our guessed solution (Eq. 23) satisfies 2a2cos2θ Q2 α(3κλ(1 + ω) − 3ω) the Einstein–Maxwell field equation (Eq. 25) with the per- Gθθ =− + 2 2r 2 2(1 − 3κλ(1 + ω)) fect fluid matter in the Rastall gravity. Similarly, the gauge 6κλ(1 + ω) − 3ω − 1   potential A in the KN-AdS black hole does not involve  and Q2 r 1 − 3κλ(1 + ω) −r − + r 3 has same expression as in the case of Kerr–Newman black α[3κλ(1 + ω) − 3ω][6κλ(1 + ω) − 3ω − 1] hole (Eq. 21), therefore it also satisfies the Einstein–Maxwell 2(1 − 3κλ(1 + ω))2 equation (Eq. 25). κλ( + ω) − ω − 9 1 3 2   In order to compute the energy-momentum tensor, we give 2a2cos2θW  r 1 − 3κλ(1 + ω) =− − rW , the following tetrad 2  2 2θ μ 1 a sin 2 2 2 2 2 =  ( 2 + 2, , , ), Gφφ =− (r + a )(a + (2r + a )cos2θ) et r a 0 0 a 6 2 2 ⎛ ⎞ √ r 3κλ(1 + ω) − 3ω  2 α μ r 3 2 ⎜ Q − κλ( + ω) ⎟ = √ ( , , , ), + 2r sin θ ⎝M − + r 1 3 1 )⎠ er 0 1 0 0 2r 2 2 √   2 α( κλ( + ω) − ω) μ θ Q + 3 1 3 eθ = √ (0, 0, 1, 0), 2r 2 2(1 − 3κλ(1 + ω)) 2 6κλ(1 + ω) − 3ω − 1  μ 1 2 2 rsin2θ(r 2 + a2)2 eφ =−√ (r + a , 0, 0, a). (27) r 1 − 3κλ(1 + ω) − 2 2 2θ 4 sin  2 α[ κλ( + ω) − ω][ κλ( + ω) − ω − ] The non-zero components of the energy-momentum tensor − Q + 3 1 3 6 1 3 1 r 3 2(1 − 3κλ(1 + ω))2 are given by 9κλ(1 + ω) − 3ω − 2  − κλ( + ω) 1 μ ν 1 tt 1 r 1 3 1 E =− e e Gμν =− g Rtt − Rgtt + κλgtt R + gtt , κ t t κ 2

2 2 2 2 2 2 2 3 2  a sin θ[(r + a )(a + (2r + a )cos2θ)+ 2r sin θW)]W 1 μ ν 1 rr 1 =− Pr = e e Gμν = g Rrr − Rgrr + κλgrr R + grr , 6 κ r r κ 2

2θ( 2 + 2)2  − rsin r a W , 1 μ ν 1 θθ 1 (26) Pθ = e eθ Gμν = g Rθθ − Rgθθ + κλgθθ R + gθθ , 4 κ θ κ 2 1 μ ν Pφ =− e e Gμν κ φ φ 3κλ(1 + ω) − 3ω 2 1 φφ 1 Q α − κλ( + ω) =− g Rφφ − Rgφφ + κλgφφ R + gφφ . (28) where W(r) = M − + r 1 3 1 . κ 2 2r 2 We then find that

6κλ(1 + ω) − 3ω − 1 r 2 α(3κλ(1 + ω) − 3ω)(1 − 4κλ) − κλ( + ω) ρ = E =−Pr = r 1 3 1 , κ(r 2 + a2cos2θ)2 (1 − 3κλ(1 + ω))2 6κλ(1 + ω) − 3ω − 1 α[3(κλ − ω) + 9ω2(1 − 5κλ) + 6ωκλ + 6κλ2(1 + ω)(6ω − 2 − 6ωκλ)] − κλ( + ω) Pθ = Pφ =−Pr − r 1 3 1 . 2κ(r 2 + a2cos2θ)(1 − 3κλ(1 + ω))2 (29)

It is clear that the distribution of the perfect fluid mat- Generically, the Kerr black hole in GR is a vacuum solu- ter around the black hole (Eq. 29) is determined by the tion and hence the energy-momentum tensor is zero. A non- parameters of a, , Q, α, ω and κλ. The space-time metric zero cosmological constant  in GR incorporates a static (Eq. 23) describes the perfect fluid matter around the KN- geometry which is known to only shift its vacuum energy AdS black hole in the Rastall gravity. The equation of state by a constant, while has no effect on the energy-momentum ω determines the material form. As described in Sect. 2,if 123 513 Page 6 of 12 Eur. Phys. J. C (2018) 78 :513

−1 <ω<−1/3, the perfect fluid matter represents the This equation describes the condition that α and κλ should dark energy. If ω =−1/3, the perfect fluid matter stands for satisfy when SEC is satisfied. From Eq. (31), we find that the perfect fluid dark matter. Using this space-time metric, ρ + Pr + Pθ + Pφ = 0 has one zero point, then Y = ρ + we can study the interaction between the perfect fluid dark Pr + Pθ + Pφ is in the range [r1, ∞) in the KN-AdS space- matter (or dark energy) and black hole. Such topic is very time with perfect fluid matter in the Rastall gravity (where r1 interesting in regard of the dark matter (dark energy)-black is the root of Eq. 31). At the same time, κλ must satisfy the hole systems. From Eq. (29), we can see that the Rastall condition (3κλ(1 + ω)− 3ω)(1 − 4κλ) ≥ 0 which is similar parameter κλ plays an important role in the distribution of to the WEC condition (Eq. 30). Therefore, the range of κλ perfect fluid matter around the black hole. Specifically, the under the SEC is consistent with the case of WEC. Rastall gravity makes the perfect fluid matter to be rearranged around the black hole. If the Rastall gravity is equivalent to 4.2 Horizon structure the Einstein GR, κλonly describes the re-arrangement of the perfect fluid matter. If the Rastall gravity is not equivalent Black hole properties are determined by the horizon structure to the Einstein GR, the meaning of κλ is the same, but the of the black hole. For stationary axisymmetric black hole, the physical origin of κλ is related to the Mach principle. horizon definition is 1 − 3ω  = 2 − + 2 + 2 − α 1 − 3κλ(1 + ω) 4 KN-AdS black hole properties in perfect fluid matter r r 2Mr a Q r  − r 2(r 2 + a2) = 0, (32) 4.1 Energy condition 3 where the property of horizon depends on α, a, Q,,κλ In this part, we study the energy condition of the KN-AdS and ω. Especially the parameters of κλ and ω change the black hole in perfect fluid matter in the Rastall gravity. First, horizon significantly. The number of horizon is determined we consider the diagonal energy-momentum tensor Tμν in the by ω. As examples, we discuss the cases of the dark energy standard locally non-rotating frame (LNRF, [55]). The weak (ω =−2/3) and dark matter (ω =−1/3). We can set the μ ν energy condition (WEC) can be written as Tμνu u ≥ 0, cosmological constant  = 0 given that it is small. where uν is the time-like vector. The physical meaning of Case I: for the dark energy, the horizon equation reduces the WEC is that the measured total energy density of all to matter fields for any observer traversing a time-like curve is 3 never negative. In the LNRF, the WEC corresponds to ρ ≥ 0 r 2 − 2Mr + a2 + Q2 − αr 1 − κλ = 0. (33) and ρ + Pi ≥ 0, where i = r,θ,φ.FromEq.(29) the WEC requires that Three horizons exist, including inner horizon (r−), event horizon (r+) and cosmological horizons (r ), where r is α(3κλ(1 + ω) − 3ω)(1 − 4κλ) ≥ 0, (30) q q determined by the dark energy. Then the above equation which is independent of the black hole spin a. Thus, the should have two extreme value points. Through calculation, WEC for the KN-AdS black hole in perfect fluid matter in the we find that α satisfies the condition of Rastall gravity is the same as that for the spherical symmetric −κλ black hole [34]. Since α is always positive, the condition (1 − κλ)2 0 <α≤ (2M)1 − κλ. (34) (3κλ(1 + ω) − 3ω)(1 − 4κλ) ≥ 0 will always be satisfied. 3(2 + κλ) If considering dark energy (ω =−2/3), then this condition When M = 1 and κλ = 0, the above condition reduces will result in −2 ≤ κλ ≤ 1/4. If considering dark matter to the Kiselev situation. In other cases, the maximum value (ω =−1/3), then −1/2 ≤ κλ ≤ 1/4. of α decreases with the increasing κλ when 0 <κλand The strong energy condition (SEC) requires that the Ray- κλ = 1, and increases with the increasing of | κλ | when 1 μ ν chaudhuri equation is (Tμν − Tgμν)u u ≥ 0. In the −2 <κλ<0. 2 LNRF, the Raychaudhuri equation becomes Y = ρ + Pr + Case II: for the perfect fluid dark matter, the horizon equa- Pθ + Pφ.FromEq.(29) the SEC implies that tion reduces to α 2 Y = ρ + Pr + Pθ + Pφ = κ 4(1 − 3κλ(1 + ω))2 r 2 − 2Mr + a2 + Q2 − αr 1 − 2κλ = 0. (35) [2r 2(3κλ(1 + ω) − 3ω)(1 − 4κλ) − 2(3(κλ − ω) Dark matter does not produce new horizon and only two + ω2( − κλ) + ωκλ 9 1 5 6 normal horizons exist, i.e., inner horizon r− and event hori- 2 + 6κλ (1 + ω)(6ω − 2 − 6ωκλ))]≥0. (31) zon r+. Then the horizon equation should have one extreme 123 Eur. Phys. J. C (2018) 78 :513 Page 7 of 12 513 value point. Through calculation, we find that α satisfies the (2) its size increases with the increasing κλ in the effective condition of range. These results imply that the Mach principle or the 4κλ re-arrangement of matter makes the rotational energy of the − black hole larger. 0 <α≤ (1 − κλ)(2M) 1 − 2κλ. (36)

When M = 1 and κλ = 0, the above condition reduces 4.4 Singularity to the Kiselev situation. Similarly, the maximum value of α decreases with the increasing κλ when 0 <κλ<1 and According to the Einstein GR, the stationary axisymmetric κλ = 1/2, and increases with the increasing | κλ | when black hole space-time has singularity. We consider whether κλ < 0. or not the singularity changes with the presence of the dark These results reveal that κλ would constrain α in the energy and/or the perfect fluid dark matter. From the defini- Rastall gravity. As mentioned in Sects. 1 and 3, the phys- tion of the singularity, the Kretsmann scalar for all ω is given ical origin could have two possibilities. If the Rastall gravity by is equivalent to the Einstein GR, κλ only describes the re- arrangement of the perfect fluid matter. If the Rastall gravity 2 μνρσ H(r,θ,a,ω,,Q ) is not equivalent to the Einstein GR, the meaning of κλis the R = R Rμνρσ = , (39) 12 same, but its physical origin is related to the Mach principle. Furthermore, the Mach principle would determine the value where H is a polynomial function. The singularity occurs at of α. r = 0 and θ = π/2 when 2 = r 2 + a2cos2θ = 0. In the Boyer–Lindquist coordinates, the singularity is a ring with a 4.3 Stationary limit surfaces radius of a in the equatorial plane. For different ω and α, there are different polynomial functions, but 2 is the same as in The stationary limit surface of the KN-AdS black hole is the usual Kerr black hole. So, the perfect fluid dark matter and important because the rotational energy of black hole relates dark energy do not change the singularity of the black hole. to the ergosphere size. From the black hole metric (Eq. 23), the ergosphere is defined by

1 2 2 g = (a sin θθ −  ) 5 Rotation velocities in the equatorial plane and tt 22 r ⎡ rotation curve diversity ⎢  = ⎣ r 2(r 2 + a2) − a2cos2θ + a4sin2θcos2θ − r 2 3 3 Basing on the KN-AdS solution in perfect fluid matter in the Rastall gravity (Sect. 3), we now discuss the rotation curves ⎤ π 1 − 3ω in the equatorial plane (θ = ). We first derive the rotation − κλ( + ω) ⎥ 1 2 + 2Mr − Q2 + αr 1 3 1 ⎦ = 0. velocity expression using ordinary method, then we calculate 22 the rotation velocities with the presence of the dark energy and perfect fluid dark matter. Especially for the case of per- (37) fect fluid dark matter, we can explain the diversity of rotation Through calculation, Eq. (37) becomes curve if considering both the perfect fluid dark matter and a ρ = [− / ]   baryon disk which is expressed as b 0exp r rd , 2 2 2 2 2 4 2 2 2 r (r + a ) − a cos θ + a sin θcos θ − r where 0 is the central surface density and rd is the scale 3 3 radius. 1 − 3ω In the four-dimensional space-time, the four-velocity sat- 2 1 − 3κλ(1 + ω) + 2Mr − Q + αr = 0. (38) isfies the normalized condition for zero angular momentum μ ν observer (ZAMO) as gμνu u =−1. In the axisymmetric This equation has two roots. There is an ergosphere space-time, the space-time symmetry leads to two conserved between the event horizon and the static limit surface. The quantities L and E. In our KN-AdS black hole space-time, parameters a,ω,Q,α,,θ and κλcould affect the shape of we write the normalized condition from the expressions of the ergosphere. In Figs. 1 and 2, we show the shapes of the uμ and uν as ergosphere for dark energy (ω =−2/3) and perfect fluid ω =− / dark matter ( 1 3), respectively. We find following properties of the ergosphere: (1) its size decreases with the dt 2 dt dφ dφ 2 dr 2 gtt + 2g φ + gφφ + grr =−1. increasing α, indicating that dark energy and perfect fluid dτ t dτ dτ dτ dτ dark matter reduce the rotational energy of the black hole; (40) 123 513 Page 8 of 12 Eur. Phys. J. C (2018) 78 :513

a=0.1,q=0.3,α=0.1,κλ =0.5 a=0.5,q=0.3,α=0.1,κλ =0.5 a=0.9,q=0.3,α=0.1,κλ =0.5 2 2 1.5

1.5 1.5 1

1 1

0.5 0.5 0.5

0 0 0 z/M z/M z/M -0.5 -0.5 -0.5

-1 -1

-1 -1.5 -1.5

-2 -2 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x/M x/M x/M

a=0.3,q=0.1,α=0.1,κλ =0.5 a=0.3,q=0.5,α=0.1,κλ =0.5 a=0.3,q=0.9,α=0.1,κλ =0.5 2 2 1.5

1.5 1.5 1

1 1

0.5 0.5 0.5

0 0 0 z/M z/M z/M -0.5 -0.5 -0.5

-1 -1

-1 -1.5 -1.5

-2 -2 -1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x/M x/M x/M

a=0.3,q=0.3,α=0.01,κλ =0.5 a=0.3,q=0.3,α=0.1,κλ =0.5 a=0.3,q=0.3,α=1,κλ =0.5 2 2 2.5

2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5

0 0 0 z/M z/M z/M -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2

-2 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x/M x/M x/M

a=0.3,q=0.3,α=0.1,κλ =-10 a=0.3,q=0.3,α=0.1,κλ =0.5 a=0.3,q=0.3,α=0.1,κλ =10 2 2 2

1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0 0 0 z/M z/M z/M -0.5 -0.5 -0.5

-1 -1 -1

-1.5 -1.5 -1.5

-2 -2 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x/M x/M x/M

Fig. 1 The ergosphere areas of the KN-AdS black hole surrounded by dark energy (ω =−2/3) in the Rastall gravity for different a, Q,α,κλ.The ergosphere region is between the event horizon (blue line) and the stationary limit surface (red line). The cosmological constant is set as  0

2 gφφ E2 + g φ EL + g L2 We then obtain dr =− 1 + 2 t  tt dτ grr 2 − gtφ gttgφφ grr dt dφ dr 2 − E + L + g =−1. (41) = 2 − 2. dτ dτ rr dτ E V (42)

Combining the equations of (40) and (41), we get 123 Eur. Phys. J. C (2018) 78 :513 Page 9 of 12 513

a=0.1,q=0.3,α=0.1,κλ =0.5 a=0.5,q=0.3,α=0.1,κλ =0.5 a=0.9,q=0.3,α=0.1,κλ =0.5 2 2 1.5

1.5 1.5 1

1 1

0.5 0.5 0.5

0 0 0 z/M z/M z/M

-0.5 -0.5 -0.5

-1 -1

-1 -1.5 -1.5

-2 -2 -1.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x/M x/M x/M

a=0.3,q=0.1,α=0.1,κλ =0.5 a=0.3,q=0.5,α=0.1,κλ =0.5 a=0.3,q=0.9,α=0.1,κλ =0.5 2 2 1.5

1.5 1.5 1

1 1

0.5 0.5 0.5

0 0 0 z/M z/M z/M

-0.5 -0.5 -0.5

-1 -1

-1 -1.5 -1.5

-2 -2 -1.5 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 x/M x/M x/M

a=0.3,q=0.3,α=0.01,κλ =0.5 a=0.3,q=0.3,α=0.1,κλ =0.5 a=0.3,q=0.3,α=1,κλ =0.5 2 2 2.5

2 1.5 1.5 1.5 1 1 1 0.5 0.5 0.5

0 0 0 z/M z/M z/M -0.5 -0.5 -0.5 -1 -1 -1 -1.5 -1.5 -1.5 -2

-2 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x/M x/M x/M

a=0.3,q=0.3,α=0.1,κλ =-10 a=0.3,q=0.3,α=0.1,κλ =0.5 a=0.3,q=0.3,α=0.1,κλ =10 2 2 2

1.5 1.5 1.5

1 1 1

0.5 0.5 0.5

0 0 0 z/M z/M z/M

-0.5 -0.5 -0.5

-1 -1 -1

-1.5 -1.5 -1.5

-2 -2 -2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 -2.5 -2 -1.5 -1 -0.5 0 0.5 1 1.5 2 2.5 x/M x/M x/M

Fig. 2 The ergosphere areas of the KN-AdS black hole surrounded by perfect fluid dark matter (ω =−1/3) in the Rastall gravity for different a, Q,α,κλ. The ergosphere region is between the event horizon (blue line) and the stationary limit surface (red line). The cosmological constant is set as  0

g + g φ φ In the black hole space-time, the general conditions for a E =± tt t , dr dV2 − − − 2 stable circular orbit are = 0 and = 0, we then get gtt 2gtφ φ gφφ φ dτ dr L E gtφ + gφφ φ the expressions of and as L =± , 2 −gtt − 2gtφ φ − gφφ φ

123 513 Page 10 of 12 Eur. Phys. J. C (2018) 78 :513

0.3 0.3 α=0.0003,Q=0.3,ω=-2/3,κλ =0.01 α=0.0003,Q=0.3,ω=-2/3, κλ =0.05

0.25 0.25

0.2 0.2

v 0.15 v 0.15

0.1 0.1

0.05 0.05

0 0 10 20 30 40 50 60 70 80 10 20 30 40 50 60 70 80 r/M r/M

0.3 0.3 α=5*10-6,Q=0.3,ω=-2/3,κλ =0.01 α=5*10-6,Q=0.3,ω=-2/3, κλ =0.05

0.25 0.25

0.2 0.2

v 0.15 v 0.15

0.1 0.1

0.05 0.05

0 0 0 100 200 300 400 500 600 700 800 900 1000 0 100 200 300 400 500 600 700 800 900 1000 r/M r/M

Fig. 3 Curves of the rotation velocity for the KN-AdS black hole surrounded by the dark energy in the Rastall gravity

 −g φ, + (g φ, )2 − g , gφφ, = t r t r tt r r . fluid matter around the black hole. From Sect. 3, for the per- φ (43) ω =− / gφφ,r fect fluid dark matter halo ( 1 3), we can calculate the energy density ρ through the Einstein equation. Because the From the definition of rotation velocity, we obtain motion velocity of the dark matter particle is much smaller

L 1 gtφ + gφφ φ than the speed of light, the energy density of the perfect fluid v = √ = √  . (44) dark matter then approximates to the mass density. On the gφφ gφφ − − − 2 gtt 2gtφ φ gφφ φ other hand, the rotation curve is usually observed at large distances from the black hole, so that the black hole spin Figures 3 and 4 show the curves of the rotation velocity a then can be approximated to zero. From Kamada et al. for the KN-AdS black hole surrounded by the dark energy [58], the baryon matter can be considered as an index disk, and perfect fluid dark matter, respectively, in the Rastall grav- i.e., ρ = exp[− r/r ]δ(z), in galaxies. Using the perfect ity. We find that: (1) in the case of dark energy, the rotation b 0 d fluid dark matter halo and the baryon disk, we can derive the velocity always decreases with the increasing distance from rotation curve expression. In the equatorial plane, the general the black hole and becomes flatter with the decreasing κλ; r r Mass function is M(r) = 4π r 2ρ dr + 2π rρ dr. (2) in the case of perfect fluid dark matter, the influence of 0 DM 0 b We then obtain the Mass function in our case as κλ on the rotation velocity is minor at small distances and significant at larger distances. Note that the effect at larger   α 1 − 4κλ 2κλ+1 −r distances is not shown in the figures, but can be inferred from M(r) = r 1−2κλ − 2π r exp (r + r). − κλ 0 d r d the explicit expression of the rotation velocity (Eq. 46). 2 1 2 d As an example of the application of the Rastall gravity, (45) we try to explain the rotation curve diversity of Low Sur- face Brightness (LSB) galaxies. In the Rastall gravity, the Thus the rotation velocity v in the equatorial plane is given parameter κλ represents the re-arrangement of the perfect by 123 Eur. Phys. J. C (2018) 78 :513 Page 11 of 12 513

0.3 0.3 0.3 α=5*10-6,Q=0.3,ω=-1/3,κλ =-20 α=5*10-6,Q=0.3,ω=-1/3, κλ =-12 α=5*10-6,Q=0.3,ω=-1/3, κλ =-4

0.25 0.25 0.25

0.2 0.2 0.2 v v 0.15 v 0.15 0.15

0.1 0.1 0.1

0.05 0.05 0.05

0 0 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 r/M r/M r/M

0.3 0.3 0.3 α=5*10-6,Q=0.3,ω=-1/3,κλ =4 α=5*10-6,Q=0.3,ω=-1/3,κλ =12 α=5*10-6,Q=0.3,ω=-1/3,κλ =20

0.25 0.25 0.25

0.2 0.2 0.2

v 0.15 v 0.15 v 0.15

0.1 0.1 0.1

0.05 0.05 0.05

0 0 0 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 0 200 400 600 800 1000 1200 1400 1600 1800 2000 r/M r/M r/M

Fig. 4 Curves of the rotation velocity for the KN-AdS black hole surrounded by the perfect fluid dark matter in the Rastall gravity

 ( ) v = GM r hole. By analyzing the horizon equation, we can constrain the α  r values of the parameter , which is determined by the equa- tion of state ω and the Rastall parameter κλ. In the case of Gα 1 − 4κλ 4κλ −r r − κλ d = r 1 2 − 2πG rd exp[ ]( + 1). the dark energy, the maximum value of α decreases with the 2 1 − 2κλ 0 r r d increasing κλ when 0 <κλand κλ = 1, and increases with (46) the increasing | κλ | when − 2 <κλ<0. In the case of the perfect fluid dark matter, the maximum value of α decreases α, , κλ The rotation velocity is determined by 0 rd and . with the increasing κλwhen 0 <κλ<1 and κλ = 1/2, and κλ v Varying would result in different expressions of , which increases with the increasing | κλ | when κλ < 0. The per- may explain the rotation velocity diversity in LSB galaxies. fect fluid dark matter and dark energy reduce the ergosphere We stress two points on the Rastall gravity. If the Rastall size, but they don’t change the singularity of the black hole. κλ gravity is equivalent to the Einstein GR, the parameter We also derive the rotation velocity for the KN-AdS black only describes the re-arrangement of the perfect fluid dark hole surrounded by the perfect fluid matter in the Rastall grav- matter. If the Rastall gravity is not equivalent to the Einstein ity. In the case of dark energy, the rotation velocity decreases κλ GR, the meaning of is the same, but the physical origin with the increasing distance from the black hole and becomes κλ of is related to the Mach principle. flatter with the decreasing κλ. In the case of perfect fluid dark matter, the effect at small distances is minor and become sig- nificant at galactic scale. Considering both the perfect fluid 6 Summary dark matter halo and the baryon disk, we have the possibility of explaining the rotation curve diversity in LSB galaxies. In this work, we obtain the KN-AdS black hole solutions sur- rounded by perfect fluid matter in the Rastall gravity using Acknowledgements We acknowledge the anonymous referee for a constructive report that has significantly improved this paper. We the Newman–Janis method and complex calculations. We acknowledge the financial support from the National Natural Sci- focus on the cases of the dark energy (ω =−2/3) and the ence Foundation of China through grants 11503078, 11573060 and perfect fluid dark matter (ω =−1/3). We discuss different 11661161010. black hole properties in the Rastall gravity with the pres- Open Access This article is distributed under the terms of the Creative ence of the perfect fluid matter. We find that if the condition Commons Attribution 4.0 International License (http://creativecomm (3κλ(1 + ω) − 3ω)(1 − 4κλ) ≥ 0 is satisfied, the WEC and ons.org/licenses/by/4.0/), which permits unrestricted use, distribution, SEC can always be hold everywhere in the KN-AdS black and reproduction in any medium, provided you give appropriate credit 123 513 Page 12 of 12 Eur. Phys. J. C (2018) 78 :513 to the original author(s) and the source, provide a link to the Creative 28. P. Rastall, Can. J. Phys. 54, 66 (1976) Commons license, and indicate if changes were made. 29. V. Majernik, L. Richterek, (2006). arXiv:gr-qc/0610070 Funded by SCOAP3. 30. C.E.M. Batista, M.H. Daouda, J.C. Fabris, O.F. Piattella, D.C. Rodrigues, Phys. Rev. D 85, 084008 (2012) 31. F. Darabi, K. Atazadeh, Y. Heydarzade, (2017). arXiv:1710.10429 32. J.C. Fabris, O.F. Piattella, D.C. Rodrigues, C.E.M. Batista, M.H. References Daouda, Int. J. Mod. Phys. Conf. Ser. 18, 67 (2012) 33. J.C. Fabris, O.F. Piattella, D.C. 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