We delve into the rotating black hole solutions in de RhamGabadadzeTolley dRGT massive gravity. Employing an analytical approach we solve the field equations for scenarios both devoid of matter and in the presence of an electromagnetic field. Consequently we obtain black hole solutions that extend beyond the KerrNewman family. These solutions are characterized not only by mass M electric charge Q and angular momentum a but also by the graviton mass term which introduces the cosmological constant Lambda and the Stckelberg charge S into the black hole parameters. Instead of the Stckelberg field phia we utilize the matrix gamma2 to seek solutions. To derive the Stckelberg field phia from a given matrix gamma2 for axisymmetric metrics one must relax the constraint on the reference metric fab  etaab. These solutions potentially serve as candidates for astrophysical black holes.
['General Relativity and Quantum Cosmology']
Rotating black holes in de RhamGabadadzeTolley massive gravity
{'gravity': 'Ping Li 1, 2, ∗ and Jiang-he Yang 1, 3, † 1 College of Mathematics and Physics, Hunan University of Arts and Sciences, 3150 Dongting Dadao, Changde City, Hunan Province 415000, China 2 Hunan Province Key Laboratory Integration and Optical Manufacturing Technology, 3150 Dongting Dadao, Changde City, Hunan Province 415000, China 3  \nCenter for Astrophysics, Guangzhou University,  \n230 West Ring Road, Guangzhou, Guangdong Province 510006, China', 'Abstract': 'We delve into the rotating black hole solutions in de Rham-Gabadadze-Tolley (dRGT) massive gravity. Employing an analytical approach, we solve the field equations for scenarios both devoid of matter and in the presence of an electromagnetic field. Consequently, we obtain black hole solutions that extend beyond the Kerr-Newman family. These solutions are characterized not only by mass M , electric charge Q , and angular momentum a , but also by the graviton mass term, which introduces the cosmological constant Λ and the Stuckelberg charge S into the black hole parameters. Instead of the Stuckelberg field φ a , we utilize the matrix γ 2 to seek solutions. To derive the Stuckelberg field φ a from a given matrix γ 2 for axisymmetric metrics, one must relax the constraint on the reference metric f ab = η ab . These solutions potentially serve as candidates for astrophysical black holes.', 'I. INTRODUCTION': "The introduction of a mass term for gravity presents a formidable challenge. The linear mass term is plagued by the van Dam-Veltman-Zakharov (vDVZ) discontinuity [1]. On the other hand, the nonlinear mass term may propagate the sixth mode on a curved background, known as the Boulware-Deser (BD) ghost [2]. To circumvent these issues, Comelli and his cooperators try to introduce mass terms without Lorentz symmetry [3, 4]. We presented a detailed study of the spherically symmetric solutions in Lorentz breaking massive gravity [5]. Concurrently, de Rham, Gabadadze, and Tolley proposed a healthy Lorentz invariant massive gravity model [6] in 2011.  \nThe dRGT theory described a massive spin-2 particles that propagate 5 healthy degrees of freedom. A comprehensive review of massive gravity is available in [7]. If graviton possesses mass, the gravitational waves would travel at a speed different from light. The recent observation of GW150914 by LIGO [8] has placed an upper bound on the graviton mass m g ≤ 1 . 2 × 10 -22 eV . In terms of cosmology, the mass term of gravity might be responsible for the late-time cosmic acceleration, suggesting that the graviton mass should be comparable to the Hubble parameter m g ∼ H ∼ 10 -33 eV . Fortunately, the dRGT theory  \ncan produce a small graviton mass technically natural [10].  \nNumerous studies have explored black hole solutions within the dRGT theory. Nieuwenhuizen obtained the Schwarzschild-de Sitter and Reissner-Nordstrom-de Sitter black holes [11]. Cai et. al. also derived the static charged black hole solutions [12]. De Rham and his colleagues [13] highlighted the existence of a new basic invariant I ab = g αβ ∂ α φ a ∂ β φ b , whose singularities could pose a problem for fluctuations. In the unitary gauge, it is evident that I ab = g αβ δ a α δ b β would exhibit a singularity if g αβ contains an event horizon. Consequently, there are no self-consistent static spherically symmetric (SSS) solutions in the unitary gauge. In our previous work [14], we utilized the nonunitary gauge to search for SSS solutions and discovered seven solutions, including black hole with the Stuckelberg hair. We also derive the Vaidya solution in Ref. [15] and investigated the superradiant instabilities for charged furry black holes in dRGT theory [16]. Many papers have also studied the dRGT theory with degenerate reference metrics [17-22]. And Ref. [23] calculate the SSS solutions in arbitrary dimensions.  \nRecently, we extended Chandrasekhar's method of calculating rotating black holes into f ( R ) theory [24-26]. In this paper, we further extend this analytical method into the dRGT massive gravity. We solve the field equations for stationary axisymmetric metrics corresponding to both no matter and electromagnetic field cases. Similar to the SSS solutions [14], the rotating black holes also incorporate a cosmological constant term Λ and the Stuckelberg charge S , which can be interpreted as a hair of black hole.  \nThis paper is organized as follows: Sec. II review the field equation of dRGT theory [6]. In Sec. III, we give a general stationary axisymmetric metric and the ansatz for the matrix γ 2 . In Sec. IV, we derive the matrix square root γ by using the Cayley-Hamilton theorem and obtain T ( K ) α β . In Sec. V, we discussed the electromagnetic field equations in the stationary axisymmetric background and the energy-momentum of electromagnetic fields. In Sec. VI, we simplify the metric to a highly symmetric form by choosing the gauge freedom. In Sec. VII, we derive the Kerr-Ads metric and black hole solutions with the Stuckelberg charge S respectively. In Sec. VIII, we derive the Kerr-Newman-Ads metric and charged black holes with the Stuckelberg charge S respectively. The last section is a detailed conclusion and discussion.  \nIn this paper, the Greek letters α, β, σ, ... = 0 , 1 , 2 , 3 represent indices of the natural coordinate base, and the Latin letters a, b, c, d = 0 , 1 , 2 , 3 represent indices of moving frame.  \nThe Greek letters are lowered or upped by the metric g αβ , and the Latin letters are lowered or upped by the metric η ab = diag( -1 , 1 , 1 , 1).", 'II. THE MODIFIED EINSTEIN EQUATIONS IN DRGT THEORY': 'The action of the de Rham-Gabadadze-Tolley theory is given by  \nS = ∫ d 4 x √ -g ( R + m 2 U ( g, φ a )) + S m , (1)  \nwhere R is the Ricci scalar, and U is a potential if the graviton have mass m . S m is the action of matter and G = 1. The potential U in four-dimensional spacetime is composed of three parts,  \nU ( g, φ a ) = U 2 + α 3 U 3 + α 4 U 4 , (2)  \nwhere  \nU 2 = [ K ] 2 -[ K 2 ] , (3)  \nU 3 = [ K ] 3 -3[ K ][ K 2 ] + 2[ K 3 ] , (4)  \nU 4 = [ K ] 4 -6[ K ] 2 [ K 2 ] + 8[ K ][ K 3 ] + 3[ K 2 ] 2 -6[ K 4 ] , (5)  \nK α β = δ α β -γ α β , (6)  \nwith α 3 , α 4 being dimensionless parameters. Here the square brackets [ A ] denote the trace of matrix A αβ . The abbreviation of γ i stands for the contraction of i copies of γ ,  \nγ iα β = γ α α 1 γ α 1 α 2 ...γ α i -1 β . (7)  \nWe also set  \nγ 0 α β = δ α β . (8)  \nAnd the trace of γ iα β is defined as [ γ i ] = g αβ γ i αβ . In the original Ref. [6], the matrix γ 2 is defined by the expression  \nγ 2 α β = g ασ ∂ σ φ a ∂ β φ b f ab , (9)  \nwhere φ a are the Stuckelberg fields. γ α β is the matrix square root of γ 2 α β . The nondegenerate matrix f ab is named the reference (or fiducial) metric. Correspondingly, we also call g αβ the dynamic metric.  \nVariation of the action (1) with respect to metric g αβ leads to the modified Einstein equations  \nG αβ + m 2 T ( K ) αβ = 8 πT ( m ) αβ , (10)  \nwhere  \nand T ( m ) αβ represents the energy-momentum tensor of mater. Due to the Bianchi identity ∇ α G αβ = 0 and the conservation of energy-momentum tensor ∇ α T ( m ) αβ = 0, there is ∇ α T ( K ) αβ = 0. As is pointed in Ref.[20], the variation of [ γ i ] is given by the relationship  \nT ( K ) αβ = 1 √ -g δ ( √ -gU ) δg αβ , (11)  \nδ [ γ i ] = i 2 γ i αβ δg αβ . (12)  \nThen, one can easily derive  \nT ( K ) α β = ( -U 2 +3+6( α 3 + α 4 )) δ α β +([ γ ] -3) γ α β -γ 2 α β -3 α 3 2 ((6 -4[ γ ] + U 2 ) γ α β -2([ γ ] -2) γ 2 α β +2 γ 3 α β ) -2 α 4 ((6 -6[ γ ] + 3 U 2 -U 3 ) γ α β +3(2 -2[ γ ] + U 2 ) γ 2 α β +6(1 -[ γ ]) γ 3 α β +6 γ 4 α β ) , (13)  \nwhere  \nU 2 = [ γ ] 2 -[ γ 2 ] , (14)  \nU 3 = [ γ ] 3 -3[ γ ][ γ 2 ] + 2[ γ 3 ] , (15)  \nU 4 = [ γ ] 4 -6[ γ 2 ][ γ ] 2 +8[ γ ][ γ 3 ] + 3[ γ 2 ] 2 -6[ γ 4 ] . (16)  \nIn the actual calculation process, the following equations are also important  \nK nα β = n ∑ k =0 ( -1) k   n k   γ kα β , (17)  \nTherefore, the traces of K iα β are given by  \n[ K ] = 4 -[ γ ] , (18)  \n[ K 2 ] = 4 -2[ γ ] + [ γ 2 ] , (19)  \n[ K 3 ] = 4 -3[ γ ] + 3[ γ 2 ] -[ γ 3 ] , (20)  \n[ K 4 ] = 4 -4[ γ ] + 6[ γ 2 ] -4[ γ 3 ] + [ γ 4 ] . (21)  \nInsert these equations into Eqs. (2), we have  \nU 2 = 12 -6[ γ ] + U 2 , (22)  \nU 3 = 24 -18[ γ ] + 6 U 2 -U 3 , (23)  \nU 4 = 24 -24[ γ ] + 12 U 2 -4 U 3 + U 4 . (24)', 'III. THE AXISYMMETRIC ANS ATZ OF THE DYNAMIC METRIC AND THE MATRIX γ 2 α β': "To derive the rotating black hole solutions, we consider the dynamic metric g αβ to process a stationary axial symmetry. As demonstrated in Ref. [30], the most general form of stationary axisymmetric metric is  \nds 2 = -e 2 ν ( x 2 ,x 3 ) dt 2 + e 2 ψ ( x 2 ,x 3 ) ( dϕ -q ( x 2 , x 3 ) dt ) 2 + e 2 µ 2 ( x 2 ,x 3 ) ( dx 2 ) 2 + e 2 µ 3 ( x 2 ,x 3 ) ( dx 3 ) 2 , (25)  \nwhere ( t, ϕ, x 2 , x 3 ) are the coordinate axes. All unknown variables ν, ψ, µ 2 , µ 3 , q are the functions of ( x 2 , x 3 ). The frame 1-forms corresponding to metric (25) are ω a = e a α dx α , where  \ne a α =        e ν 0 0 0 -qe ψ e ψ 0 0 0 0 e µ 2 0 0 0 0 e µ 3        . (26)  \nGiven any tensor T α... β... , the tetrad components T a... b... can be obtained by the projection  \nT a... b... = T α... β... e α a · · · e β b · · · . (27)  \nAs is shown in Ref. [24-26], one can use the method of moving frame to derive the Einstein tensor G ab . Using the Cartan's equation of structure, one obtain nonzero components of G ab  \nas follows:  \nG 0 0 = e -2 µ 2 ( ψ , 2 , 2 + ψ 2 , 2 + ψ , 2 ( µ 3 -µ 2 ) , 2 ) + e -2 µ 3 ( ψ , 3 , 3 + ψ 2 , 3 + ψ , 3 ( µ 2 -µ 3 ) , 3 ) + e -µ 2 -µ 3 ( e µ 2 -µ 3 µ 2 , 3 ) , 3 +( e µ 3 -µ 2 µ 3 , 2 ) , 2 + 1 e 2( ψ -ν -µ 2 ) q 2 , 2 + 1 e 2( ψ -ν -µ 3 ) q 2 , 3 ,  \n( ) 4 4 (28) G 1 1 = e -2 µ 2 ( ν , 2 , 2 + ν 2 , 2 + ν , 2 ( µ 3 -µ 2 ) , 2 ) + e -2 µ 3 ( ν , 3 , 3 + ν 2 , 3 + ν , 3 ( µ 2 -µ 3 ) , 3 ) + e -µ 2 -µ 3 ( ( e µ 2 -µ 3 µ 2 , 3 ) , 3 +( e µ 3 -µ 2 µ 3 , 2 ) , 2 ) -3 4 e 2( ψ -ν -µ 2 ) q 2 , 2 -3 4 e 2( ψ -ν -µ 3 ) q 2 , 3 , (29) G 2 2 = e -2 µ 2 ( ψ , 2 ν , 2 +( ψ + ν ) , 2 µ 3 , 2 ) + e -2 µ 3 ( ( ψ + ν ) , 3 , 3 + ν 2 , 3 + ψ 2 , 3 + ψ , 3 ν , 3 -( ψ + ν ) , 3 µ 3 , 3 ) + 1 4 e 2( ψ -ν -µ 2 ) q 2 , 2 -1 4 e 2( ψ -ν -µ 3 ) q 2 , 3 , (30) G 3 3 = e -2 µ 3 ( ψ , 3 ν , 3 +( ψ + ν ) , 3 µ 2 , 3 ) + e -2 µ 2 ( ( ψ + ν ) , 2 , 2 + ν 2 , 2 + ψ 2 , 2 + ψ , 2 ν , 2 -( ψ + ν ) , 2 µ 2 , 2 ) -1 4 e 2( ψ -ν -µ 2 ) q 2 , 2 + 1 4 e 2( ψ -ν -µ 3 ) q 2 , 3 , (31) G 0 1 = -1 2 e -(2 ψ + µ 2 + µ 3 ) ( ( e 3 ψ -ν + µ 3 -µ 2 q , 2 ) , 2 +( e 3 ψ -ν + µ 2 -µ 3 q , 3 ) , 3 ) , (32) G 2 3 = -e -µ 2 -µ 3 (( ψ + ν ) , 2 , 3 + ν , 2 ν , 3 + ψ , 2 ψ , 3 -( ψ + ν ) , 2 µ 2 , 3 -( ψ + ν ) , 3 µ 3 , 2 ) + 1 2 e 2( ψ -ν ) -µ 2 -µ 3 q , 2 q , 3 . (33)  \nIn this paper, we directly use the matrix γ 2 instead of the Stuckelberg field φ a to search for solutions. In fact, if one uses the Stuckelberg field φ a , there exists a new basic invariant I ab = g αβ ∂ α φ a ∂ β φ b in massive gravity. A black hole with singularities in I ab would encounter a problem of fluctuations [13]. This bring more difficulty to searching for the black hole solution. In the case of static spherically symmetry, one need to release the unitary gauge φ a = x α δ a α to obtain a self-consistent solution [14]. We also attempted to use the nonunitary gauge to search for the stationary axisymmetric solutions but failed. Only choosing φ a does not guarantee that the degrees of freedom are in the correct component of γ . It is more likely that we need to release f ab = η ab at the same time. However, there are too many possibilities for the matrix γ 2 from the combination of the arbitrary reference metric f ab and the Stuckelberg field φ a (9). Therefore, we abandoned the relationship (9) and directly take the ansatz of matrix γ 2 as  \nγ 2 =   Ξ 1 0 0 Ξ 2   , (34)  \nwhere Ξ 1 and Ξ 2 are a general unknown 2 × 2 non-degenerate matrix and 0 is a 2 × 2 zero matrix.", 'IV. THE 2 × 2 MATRIX SQUARE ROOT AND T ( K ) αβ': "In fact, we require the matrix square root γ to derive T ( K ) αβ . Notice that the matrix square root γ can be expressed as  \nγ =   √ Ξ 1 0 0 √ Ξ 2   . (35)  \nThe square root of the 2 × 2 submatrices √ Ξ 1 and √ Ξ 2 can be derived using the CayleyHamilton theorem.  \nFor a general 2 × 2 matrix M , the Cayley-Hamilton theorem shows  \n[ M ] M = M 2 +(det M ) I 2 , (36)  \nwhere I 2 is 2 × 2 identity matrix. Taking the trace of the above equation, we have  \n[ M ] 2 = [ M 2 ] + 2det M. (37)  \nLet's say M 2 =   M 2 11 M 2 12 M 2 21 M 2 22   . Using the property (det M ) n = det M n , we obtain  \n[ M ] 2 = M 2 11 + M 2 22 +2det M, (38)  \n(det M ) 2 = M 2 11 M 2 22 -M 2 12 M 2 21 . (39)  \nThus, one can use the trace [ M ] and determinant det M to reexpress the diagonal component of M 2  \nM 2 11 = 1 2 ( [ M ] 2 -2det M -√ [ M ] 4 -4[ M ] 2 det M -4 M 2 12 M 2 21 ) , (40)  \nM 2 22 = 1 2 ( [ M ] 2 -2det M + √ [ M ] 4 -4[ M ] 2 det M -4 M 2 12 M 2 21 ) . (41)  \nGenerally speaking, a 2 × 2 matrix M 2 would have 4 components, each corresponding to a free function. We replace two diagonal components M 11 , M 22 with other unknowns [ M ] , det M . The matrix M 2 still introduces 4 free functions.  \nLet's define  \nT 1 = [ √ Ξ 1 ] , D 1 = det √ Ξ 1 , T 2 = [ √ Ξ 2 ] , D 2 = det √ Ξ 2 . (42)  \nAs is shown before, we can use T 1 , T 2 , D 1 , D 2 to re-express the submatrix Ξ 1 and Ξ 2 as  \nΞ 1 =   1 2 ( T 2 1 -2 D 1 -s 1 ) ε 01 ε 10 1 2 ( T 2 1 -2 D 1 + s 1 )   , (43)  \nΞ 2 =   1 2 ( T 2 2 -2 D 2 -s 2 ) ε 23 ε 32 1 2 ( T 2 2 -2 D 2 + s 2 )   , (44)  \ns 1 = √ T 4 1 -4 T 2 1 D 1 -4 ε 01 ε 10 , (45)  \ns 2 = √ T 4 2 -4 T 2 2 D 2 -4 ε 23 ε 32 . (46)  \nUsing the Cayley-Hamilton theorem (36), one can obtain the square root √ Ξ 1 and √ Ξ 2 as  \n√ Ξ 1 = 1 T 1   1 2 ( T 2 1 -s 1 ) ε 01 ε 10 1 2 ( T 2 1 + s 1 )   , (47)  \n√ Ξ 2 = 1 T 2   1 2 ( T 2 2 -s 2 ) ε 23 ε 32 1 2 ( T 2 2 + s 2 )   . (48)  \nEach matrix square root introduces 4 new functions. Thus, the matrix γ (35) can introduce up to 8 unknown functions.  \nWhin γ (35), the expression of T ( K ) α β is determined. Therefore, T ( K ) α β can introduce 8 degrees of freedom without considering (9). For brevity, we define new constant  \nc 3 = 3 α 3 +12 α 4 , (49)  \nc 4 = 1 + 6 α 3 +12 α 4 . (50)  \nwhere  \nInserting Eqs. (35) (47) and (48) into (13), one obtains  \nT ( K )0 0 = 1 2 T 1 ( T 1 ( -3 + 2 T 2 -2 c 3 ( T 2 -1) + c 4 ( -3 -2 D 2 +4 T 2 )) +( T 2 1 + s 1 )(1 + c 3 ( D 2 1) + c 4 (2 T 2 )) , (51)  \nT ( K )1 1 = 1 2 T 1 ( T 1 ( -3 + 2 T 2 -2 c 3 ( T 2 -1) + c 4 ( -3 -2 D 2 +4 T 2 )) +( T 2 1 s 1 )(1 + c 3 ( D 2 1) + c 4 (2 T 2 )) , (52)  \nT  \n(  \nK  \n)2  \n=  \n1  \n2  \nT  \n2  \n--)  \n---)  \nT  \n2  \n(  \n3 + 2  \nT  \n-  \n-  \n-  \n-  \n-  \n+( T 2 2 + s 2 )(1 + c 3 ( D 1 -1) + c 4 (2 -T 1 )) ) , (53)  \n(  \nT ( K )3 3 = 1 2 T 2 ( T 2 ( -3 + 2 T 1 -2 c 3 ( T 1 -1) + c 4 ( -3 -2 D 1 +4 T 1 ))  \nT ( K )0 1 = ε 01 T 1 (1 + c 3 ( D 2 -1) + c 4 (2 -T 2 )) , (55)  \n+( T 2 2 -s 2 )(1 + c 3 ( D 1 -1) + c 4 (2 -T 1 )) ) , (54)  \nT ( K )2 3 = ε 23 T 2 (1 + c 3 ( D 1 -1) + c 4 (2 -T 1 )) . (56)", 'V. THE ELECTROMAGNETIC FIELD IN STATIONARY AXISYMMETRIC SPACE-TIME': "In this section, we consider the electromagnetic field in stationary axisymmetric background. Given the action for electromagnetic fields S m = -1 16 π ∫ √ -gF αβ F αβ d 4 x , the energy-momentum is expressed as  \nT ( m ) αβ = 1 4 π ( F ασ F β σ -1 4 g αβ F σκ F σκ ) , (57)  \nwhere F αβ = ∂ α A β -∂ β A α , and A α is the vector potential of electromagnetic fields. In the stationary axisymmetric spacetime (25), we assume that the vector potentials A 2 , A 3 vanishes and A 0 , A 1 are functions of ( x 2 , x 3 ). Therefore, F 01 = F 23 = 0. The Maxwell's equations are  \n∇ λ F αλ = J α , (58)  \n∇ [ α F βλ ] = 0 , (59)  \n1  \n2  \nc  \n3  \n(  \nT  \n1  \n1) +  \nc  \n4  \n(  \n3  \n2  \nD  \n+4  \nT  \n))  \n2  \n1  \n1  \nwhere J α is the current vector. As the relationship F αβ = ∂ α A β -∂ β A α , the Eq. (59) is satisfied automatically.  \nLet's consider the Maxwell's equation (58) first. Since we are concerned with black hole solutions, we set the current vector to zero, J α = (0 , 0 , 0 , 0). For α = 0, we obtain  \n( e ψ -ν + µ 3 -µ 2 ( F 02 + qF 12 )) , 2 +( e ψ -ν + µ 2 -µ 3 ( F 03 + qF 13 )) , 3 = 0 . (60)  \nAs a simple solution, we define  \ne ψ -ν + µ 3 -µ 2 ( F 02 + qF 12 ) = -B , 3 , (61)  \ne ψ -ν + µ 2 -µ 3 ( F 03 + qF 13 ) = B , 2 , (62)  \nwhere B is a function of ( x 2 , x 3 ). In the stationary axisymmetric space-time, there are two new unknowns A 0 , A 1 for the electromagnetic field. We replace one of them with B , and the other one is still selected as A = A 1 . For α = 1, the Eq. (58) can be re-expressed as  \n( e ν -ψ + µ 3 -µ 2 A , 2 ) , 2 +( e ν -ψ + µ 2 -µ 3 A , 3 ) , 3 = q , 2 B , 3 -q , 3 B , 2 . (63)  \nFor Eq. (59), there exist extra equation for α = 0  \n( e ν -ψ + µ 3 -µ 2 B , 2 ) , 2 +( e ν -ψ + µ 2 -µ 3 B , 3 ) , 3 = q , 3 A , 2 -q , 2 A , 3 . (64)  \nWhen α = 1, Eq. (59) is automatically satisfied. Finally, we obtain the constraint equations (63) and (64) for A, B .  \nUsing A, B to express the energy-momentum of electromagnetic fields (57), we have  \nT ( m )0 0 = -T ( m )1 1 = -1 8 π ( e -2 ψ -2 µ 2 ( A 2 , 2 + B 2 , 2 ) + e -2 ψ -2 µ 3 ( A 2 , 3 + B 2 , 3 ) ) , (65)  \nT ( m )0 1 = 1 4 π e -(2 ψ + µ 2 + µ 3 ) ( B , 2 A , 3 -A , 2 B , 3 ) , (67)  \nT ( m )2 2 = -T ( m )3 3 = 1 8 π ( e -2 ψ -2 µ 2 ( A 2 , 2 + B 2 , 2 ) -e -2 ψ -2 µ 3 ( A 2 , 3 + B 2 , 3 ) ) , (66)  \nT ( m )2 3 = 1 4 π e -(2 ψ + µ 2 + µ 3 ) ( A , 2 A , 3 + B , 2 B , 3 ) , (68)  \nwhere the above expressions are the tetrad components T ( m ) a b = T ( m ) µ ν e a µ e ν b .", 'VI. GAUGE FREEDOM AND THE FIELD EQUATIONS': 'Analytical calculation of the field equations (10) is not an easy task. The original mathematical process is detailed in Refs. [27, 28]. A more streamlined calculation process is  \nprovided by Carter [29] or Chandrasekhar [30]. In this paper, we employ the methodology primarily adopted from Chandrasekhar. As a key strategy of Chandrasekhar, we require a highly symmetric metirc form [26]  \nds 2 = √ ∆ r ∆ µ ( -χdt 2 + 1 χ ( dϕ -qdt ) 2 ) + ρ 2 ( dr 2 ∆ r + dµ 2 ∆ µ ) , (69)  \nwhere χ = χ ( r, µ ) , q = q ( r, µ ). The function ∆ r depends only on r and ∆ µ depends only on µ . The new coordinates ( r, µ ) are related to ( x 2 , x 3 ) by the relationships  \ne 2 µ 2 ( x 2 ,x 3 ) = ρ 2 ( r, µ ) ∆ r , (70)  \ne 2 µ 3 ( x 2 ,x 3 ) = ρ 2 ( r, µ ) ∆ µ . (71)  \nIn the Boyer-Lindquist coordinate system, r is the radial coordinate and µ = cos θ . The metric form (69) can always be achieved by choosing the appropriate gauge freedom. Comparing (69) with (25), we have  \ne 2 ν = χ √ ∆ r ∆ µ , e 2 ψ = √ ∆ r ∆ µ χ , e 2 µ 2 = ρ 2 ∆ r , e 2 µ 3 = ρ 2 ∆ µ . (72)  \nIn this paper, we only concern the solution of (10) without matter or with electromagnetic fields. Since the Bianchi identity ∇ α G αβ = 0 is satisfied, we consider the following 5 equations,  \nG 2 2 + G 3 3 + m 2 ( T ( K )2 2 + T ( K )3 3 ) = 8 π ( T ( m )2 2 + T ( m )3 3 ) , (73)  \nG 2 2 -G 3 3 + m 2 ( T ( K )2 2 -T ( K )3 3 ) = 8 π ( T ( m )2 2 -T ( m )3 3 ) , (74)  \nG 1 1 + G 0 0 + m 2 ( T ( K )1 1 + T ( K )0 0 ) = 8 π ( T ( m )1 1 + T ( m )0 0 ) , (75)  \nG 1 1 -G 0 0 + m 2 ( T ( K )1 1 -T ( K )0 0 ) = 8 π ( T ( m )1 1 -T ( m )0 0 ) , (76)  \nG 0 1 + m 2 T ( K )0 1 = 8 πT ( m )0 1 . (77)  \nThe above equations uniquely decide the five unknowns χ, q, ρ 2 , ∆ r , ∆ µ . In this paper, the field equations (10) are solved in tetrad components or in Latin letters.', 'A. The Kerr-Ads solution': 'The Kerr-Ads black hole is a signature rotating solution in general relativity with a cosmological constant Λ. To derive the Kerr-Ads metric in dRGT theory, we define  \n1 + c 3 ( D 2 -1) + c 4 (2 -T 2 ) = 0 , (78)  \n1 + c 3 ( D 1 -1) + c 4 (2 -T 1 ) = 0 , (79)  \n-3 + 2 T 2 -2 c 3 ( T 2 -1) + c 4 ( -3 -2 D 2 +4 T 2 ) = 2Λ , (80)  \n-3 + 2 T 1 -2 c 3 ( T 1 -1) + c 4 ( -3 -2 D 1 +4 T 1 ) = 2Λ , (81)  \nwhere Λ is a constant. These four equations determine 4 free functions. Consequently, the Kerr-AdS metric requires 4 free functions introduced by the matrix γ to be determined, which are  \nD 1 = D 2 = -3 c 3 -2 c 2 3 -2 c 4 +5 c 3 c 4 -4 c 2 4 +2 c 3 Λ 2( -c 3 + c 2 3 -2 c 3 c 4 + c 2 4 ) , (82)  \nT 1 = T 2 = --2 + 4 c 3 -2 c 2 3 -5 c 4 +6 c 3 c 4 -5 c 2 4 +2 c 4 Λ 2( -c 3 + c 2 3 -2 c 3 c 4 + c 2 4 ) . (83)  \nThen, one can simplify  \nT ( K ) a b = T ( K ) µ ν e a µ e ν b = Λ δ a b . (84)  \nWithin Eqs. (69) and (84) , we can simplify Eqs. (73) - (77) to  \n∆ r,rr +∆ µ,µµ +4Λ m 2 ρ 2 = 0 , (85)  \n∆ r,rr -∆ µ,µµ -1 4 ∆ 2 r,r ∆ r + 1 4 ∆ 2 µ,µ ∆ µ -∆ r,r (ln ρ 2 ) ,r +∆ µ,µ (ln ρ 2 ) ,µ  \n+ ∆ r χ 2 ( χ 2 ,r -q 2 ,r ) -∆ µ χ 2 ( χ 2 ,µ -q 2 ,µ ) = 0 , (86)  \n∆ r,rr +∆ µ,µµ -1 4 ∆ 2 r,r ∆ r -1 4 ∆ 2 µ,µ ∆ µ +2∆ r (ln ρ 2 ) ,rr +2∆ µ (ln ρ 2 ) ,µµ  \n+∆ r,r (ln ρ 2 ) ,r +∆ µ,µ (ln ρ 2 ) ,µ + ∆ r χ 2 ( χ 2 ,r -q 2 ,r ) + ∆ µ χ 2 ( χ 2 ,µ -q 2 ,µ ) + 4Λ m 2 ρ 2 = 0 , (87)  \n1 χ 2 (∆ r q 2 ,r +∆ µ q 2 ,µ ) -( ∆ r χ χ ,r ) ,r -( ∆ µ χ χ ,µ ) ,µ = 0 , (88)  \n( ∆ r χ 2 q ,r ) ,r + ( ∆ µ χ 2 q ,µ ) ,µ = 0 . (89)  \nWe find that r and µ always appear symmetrically in these equations (85) - (89), indicating that the dependence of ∆ r on r is similar to that of ∆ µ on µ . Thus, one can try to solve three relatively simple Eqs. (85) (88) and (89) simultaneously. The detailed solution procedure can be found in the literature [26]. One need to first solve homogeneous Eq. (85), which corresponding to Λ = 0. Then, providing the solution for the non-homogeneous Eq. (85) and combining the solution of two parts, one can obtain the final solution. By repeating the solving process in Ref. [26], we can finally derive  \n∆ r = r 2 -2 Mr + a 2 -Λ m 2 3 r 2 ( r 2 + a 2 ) , (90)  \n∆ µ = 1 -µ 2 -Λ m 2 3 a 2 µ 2 ( µ 2 -1) , (91)  \nρ 2 = r 2 + a 2 µ 2 , (92)  \nχ = ( r 2 + a 2 µ 2 )∆ 1 2 r ∆ 1 2 µ ( r 2 + a 2 ) 2 ∆ µ -a 2 (1 -µ 2 ) 2 ∆ r , (93)  \nq = a ( r 2 + a 2 )∆ µ -(1 -µ 2 )∆ r ( r 2 + a 2 ) 2 ∆ µ -a 2 (1 -µ 2 ) 2 ∆ r , (94)  \nwhere a is a constant and can be interpreted as the angular momentum per unit mass.', 'B. The solution with the Stuckelberg charge S': "By carefully comparing T ( K ) α β (51) - (56) and T ( m ) α β (65) - (68), we find that it is possible to construct the Kerr-Newman type solutions without matter. Although we did not use the constraint Eq. (9), new charges can only be introduced by the Stuckelberg field. Thus, we call it the Stuckelberg charge S . Let's try to take T ( K ) α β as  \nT ( K )0 0 = -T ( K )1 1 = -S 2 ( -r 2 + a 2 ( -2 + µ 2 )) ( r 2 + a 2 µ 2 ) 3 +Λ , (95)  \nT ( K )2 2 = -T ( K )3 3 = S 2 ( r 2 + a 2 µ 2 ) 2 +Λ , (96)  \nT ( K )0 1 = 2 a ( a 2 + r 2 ) S 2 ( -1 + µ 2 ) ( r 2 + a 2 µ 2 ) 3 , (97)  \nT ( K )2 3 = 0 . (98)  \nWe found there exist solutions  \n∆ r = r 2 -2 Mr + a 2 + m 2 S 2 -Λ m 2 3 r 2 ( r 2 + a 2 ) , (99)  \nwhere functions ∆ µ , ρ 2 , χ, q are expressed as (91) - (94).  \nThis solution requires all 8 functions introduced from γ to be determined. Compared Eqs. (95) - (98) with (51) - (56), one can obtain all 8 unknowns T 1 , D 1 , T 2 , D 2 , ε 01 , ε 10 , ε 23 , ε 32 . These expressions are complex, we would not list them here.", 'A. the Kerr-Newman-Ads solution': "In this section, we seek solutions with electromagnetic fields. For simplicity, let's first consider the case where the tensor T ( K ) αβ can be reduced to a cosmological constant term. The matrix γ has the same form in the Kerr-Ads solution. Using the expression (84) for T ( K ) αβ , Eqs. (73) (76) and (77) can be rewritten as  \n2 2  \n∆ r,rr +∆ µ,µµ +4Λ m ρ = 0 , (100) 1 χ 2 (∆ r q 2 ,r +∆ µ q 2 ,µ ) -( ∆ r χ χ ,r ) ,r -( ∆ µ χ χ ,µ ) ,µ +2 ( χ √ ∆ µ ∆ r ( A 2 ,µ + B 2 ,µ ) + χ √ ∆ r ∆ µ ( A 2 ,r + B 2 ,r ) ) = 0 , (101) ∆ r χ 2 q ,r ) ,r + ( ∆ µ χ 2 q ,µ ) ,µ +4 ( B ,r A ,µ -B ,µ A ,r ) = 0 . (102)  \n(  \nThe constraint equations for A, B (64) and (63) are rewritten as  \n( χ √ ∆ r ∆ µ B ,r ) ,r + ( χ √ ∆ µ ∆ r B ,µ ) ,µ = q ,µ A ,r -q ,r A ,µ , (103) ( χ √ ∆ r ∆ µ A ,r ) ,r + ( χ √ ∆ µ ∆ r A ,µ ) ,µ = q ,r B ,µ -q ,µ B ,r . (104)  \n/negationslash  \nIn the case of Λ = 0, the reference [30] provides a general mathematical solving process. We can extend the solution to Λ = 0 by using a similar strategy in Ref. [26]. The other route was given by Ref. [29], where Carter use an additional physical condition to simplify the line element to the separable canonical metric form. After some calculations, we obtain  \nthe solution  \nA = -Qra (1 -µ 2 ) r 2 + a 2 µ 2 , (105)  \nB = -Qµ ( r 2 + a 2 ) r 2 + a 2 µ 2 , (106)  \n∆ r = r 2 -2 Mr + a 2 + Q 2 -Λ m 2 3 r 2 ( r 2 + a 2 ) , (107)  \nwhere ∆ µ and ρ 2 are also expressed as Eqs. (91) and (92).", 'B. The solution with charges S and Q': 'Similarly, if we choose T ( K ) αβ as the form (95) - (98), one can derive  \n∆ r = r 2 -2 Mr + a 2 + Q 2 + m 2 S 2 -Λ m 2 3 r 2 ( r 2 + a 2 ) , (108)  \nwhere the rest of the quantities have the same form as in the case of the Kerr-Newman-AdS solution.', 'IX. CONCLUSION AND DISCUSSION': 'In this paper, we have derived the rotating black hole in dRGT massive gravity. The graviton mass may introduces both cosmological constant term Λ and the Stuckelberg charge S into rotating solutions. All metrics can be expressed in a uniform form. By choosing µ = cos θ , we can transform the solution into the Boyer-Lindquist coordinate, which is given by  \nds 2 = -ρ 2 ∆ r ∆ θ Σ 2 dt 2 + Σ 2 ρ 2 ( dϕ -( r 2 + a 2 )∆ θ -sin 2 θ ∆ r Σ 2 adt ) 2 + ρ 2 ∆ r dr 2 + ρ 2 sin 2 θ ∆ θ dθ 2 , (109)  \nwhere  \nρ 2 = r 2 + a 2 cos 2 θ, (110)  \nΣ 2 = ( r 2 + a 2 ) 2 ∆ θ -a 2 sin 4 θ ∆ r , (111)  \n∆ r = -Λ m 2 3 ( r 4 + a 2 r 2 ) + r 2 -2 Mr + a 2 + Q 2 + m 2 S 2 , (112)  \n∆ θ = sin 2 θ ( 1 + Λ m 2 3 a 2 cos 2 θ ) . (113)  \nAnd the vector potentials of electromagnetic fields are  \nA 0 = Qr r 2 + a 2 cos 2 θ , (114)  \nA 1 = -Qra sin 2 θ r 2 + a 2 cos 2 θ . (115)  \nSince we used the matrix γ 2 instead of the Stuckelberg field φ a to search for solutions, one may wonder whether it is possible to derive the Stuckelberg field φ a through a given γ 2 . The analysis of the required degree of freedom suggests that the rotating black hole have to release the constraint on the reference metric f ab = η ab . As the degrees of freedom for f ab equal to those for γ 2 , we believe that there exists at least one possible combination of φ a and f ab that can derive a known matrix γ 2 . As an example, we derive the Stuckelberg field φ a and the reference metric f ab for the Kerr-Ads solution. We simply let the Stuckelberg field φ a = x α δ a α and  \nf ab =        -ξ Z 1 0 0 Z 1 ζ 0 0 0 0 ξ -Z 2 0 0 -Z 2 ζ        , (116)  \nwhere all four unknowns ξ, ζ, Z 1 , Z 2 are functions of ( x 2 , x 3 ). Four Eqs. (82) - (83) determined four unknowns ξ, ζ, Z 1 , Z 2 . By direct mathematical calculation, one can obtain φ a and f ab from a given matrix γ 2 which satisfied (82) - (83).  \nAcknowledgement This work is partially supported by the National Natural Science Foundation of China (NSFC U2031112) and the Scientific Research Foundation of Hunan University of Arts and Sciences (23BSQD237). We also acknowledge the science research grants from the China Manned Space Project with NO. CMS-CSST-2021-A06.  \n- [1] H. van Dam and M. J. G. Veltman, Massive and mass-less Yang-Mills and gravitational fields, Nucl. Phys. B 22 , 379 (1970); V. I. Zakharov, Linearized gravitation theory and the graviton mass, JETP Lett. 12 , 312 (1970).\n- [2] D.G. Boulware and S. Deser, Can Gravitation Have a Finite Range? Phys. Rev. D 6 , 3368 (1972).\n- [3] Comelli D, Crisostomi M, Nesti F and Pilo L, Degrees of Freedom in Massive Gravity, Phys. Rev. D 86 , 101502 (2012).  \n- [4] Comelli D, Nesti F and Pilo L, Weak Massive Gravity, Phys. Rev. D 87 , 124021 (2013).\n- [5] P. Li, X.Z. Li, and P. Xi, Analytical expression for a class of spherically symmetric solutions in Lorentz-breaking massive gravity, Classical Quantum Gravity 33 , 115004 (2016).\n- [6] C. de Rham, G. Gabadadze, and A. J. Tolley, Resummation of Massive Gravity, Phys. Rev. Lett 106 , 231101 (2011).\n- [7] C. de Rham, Massive Gravity, Living Rev. Relativity 17 , 7 (2014).\n- [8] T. Baker, E. Bellini, P. G. Ferreira, M. Lagos, J. Noller and I. Sawicki, Departures from the Friedmann-Lemaitre-Robertston-Walker Cosmological Model in an Inhomogeneous Universe: A Numerical Examination Phys. Rev. Lett. 116 , 251301 (2016).\n- [9] A. Nicolis, R. Rattazzi and E.Trincherini, The Galileon as a local modification of gravity, Phys. Rev. D 79 , 064036 (2009).\n- [10] C. de Rham, G.Gabadadze, L. Heisenberg and D. Pirtskhalava, Cosmic Acceleration and the Helicity-0 Graviton, Phys. Rev. D 83 , 103516 (2011).\n- [11] T. M. Nieuwenhuizen, Exact Schwarzschild-de Sitter black holes in a family of massive gravity models, Phys. Rev. D 84 , 024038 (2011).\n- [12] Y. F. Cai, D. A. Easson, C. Gao, E. N. Saridakis, Charged black holes in nonlinear massive gravity, Phys. Rev. D 87 , 064001 (2013).\n- [13] L. Berezhiani, G. Chkareuli, C. de Rham, G. Gabadadze, and A. J. Tolley, On Black Holes in Massive Gravity, Phys. Rev. D 85 , 044024 (2012).\n- [14] Ping Li, Xin-zhou Li, and Ping Xi, Black hole solutions in de Rham-Gabadadze-Tolley massive gravity, Phys. Rev. D 93 , 064040 (2016).\n- [15] Ping Li, Xin-Zhou Li, and Xiang-Hua Zhai, Vaidya solution and its generalization in de RhamGabadadze-Tolley massive gravity, Phys. Rev. D 94 , 124022 (2016).\n- [16] Ping Li, Yang Huang, Chao-Jun Feng, and Xin-Zhou Li, Superradiant instabilities for a charged black hole in de Rham-Gabadadze-Tolley theory, Phys. Rev. D 102 , 024063 (2020).\n- [17] Rong-Gen Cai, Ya-Peng Hu, Qi-Yuan Pan, and Yun-Long Zhang, Thermodynamics of Black Holes in Massive Gravity, Phys. Rev. D 91 , 024032 (2015).\n- [18] Allan Adams, Daniel A. Roberts, and Omid Saremi, Hawking-Page transition in holographic massive gravity, Phys. Rev. D 91 , 046003 (2015).\n- [19] Jianfei Xu, Li-Ming Cao, and Ya-Peng Hu, P-V criticality in the extended phase space of black holes in massive gravity, Phys. Rev. D 91 , 124033 (2015)  \n- [20] Li-Ming Cao, Yuxuan Peng, and Yun-Long Zhang, de Rham-Gabadadze-Tolley massive gravity with degenerate reference metrics, Phys. Rev. D 93 ,124015 (2016).\n- [21] Hongsheng Zhang, and Xin-Zhou Li, Ghost free massive gravity with singular reference metrics, Phys. Rev. D 93 , 124039 (2016).\n- [22] Ya-Peng Hu, Xin-Meng Wu, and Hongsheng Zhang, Generalized Vaidya Solutions and MisnerSharp mass for n -dimensional massive gravity, Phys. Rev. D 95 , 084002 (2017).\n- [23] Ghadir Jafari, M. R. Setare, and Hamid R. Bakhtiarizadeh, Static spherically symmetric black holes of de Rham-Gabadadze-Tolley massive gravity in arbitrary dimensions, Phys. Lett. B 773 (2017) 395-400.\n- [24] Ping Li, Yong-Qiang Liu and Xianghua Zhai, Nonrotating and rotating black hole solutions in f(R)=R-2 α √ R model , International Journal of Modern Physics D 31 (2022) 2250123.\n- [25] Ping Li, Yongqiang Liu, Siwei Xu, Jianghe Yang, Xianghua Zhai, Analytical generalization of Kerr-Ads and KN-Ads solutions in f ( R ) gravity (in Chinese). Sci Sin-Phys Mech Astron . https://doi.org/10.1360/SSPMA-2024-0106.\n- [26] Ping Li, Yong-qiang Liu, Jiang-he Yang, Siwei Xu, and Xiang-hua Zhai, Analytical calculation of Kerr and Kerr-Ads black holes in f ( R ) theory, Eur. Phys. J. C , (2024) 84:704.\n- [27] E. T. Newman, E. Couch, K. Chinnapared, A. Exton, A. Prakash et al, Metric of a Rotating, Charged Mass, J. Math. Phys. 6 , 918 (1965).\n- [28] G. C. Debney, R. P. Kerr, and A. Schild, Solutions of the Einstein and Einstein-Maxwell Equations, J. Math. Phys. 10 , 1842 (1969).\n- [29] B. Carter, Republication of: Black hole equilibrium states, Gen Relativ. Gravit. 41 (2009) 12, 2873-2938.\n- [30] S. Chandrasekhar, The Mathematical Theory of Black Holes , Clarendon Press (Oxford) (1983) P67-P69.'}