Axisymmetric Stationary Spacetimes of Constant Scalar Curvature in Four Dimensions
Rosikhuna F. Assafari, Bobby E. Gunara, Hasanuddin, Abednego Wiliardy
aa r X i v : . [ g r- q c ] J un Axisymmetric Stationary Spacetimes of ConstantScalar Curvature in Four Dimensions
Rosikhuna F. Assafari ♯,a ) , Bobby E. Gunara § ,♯,b ) 1 , Hasanuddin ‡ ,c , and Abednego Wiliardy ♯,d ) § Indonesian Center for Theoretical and Mathematical Physics (ICTMP) and ♯ Theoretical Physics LaboratoryTheoretical High Energy Physics and Instrumentation Research Group,Faculty of Mathematics and Natural Sciences,Institut Teknologi BandungJl. Ganesha no. 10 Bandung, Indonesia, 40132 and ‡ Department of Physics,Faculty of Mathematics and Natural Science,Tanjungpura University,Jl. Prof. Dr. H. Hadari Nawawi, Pontianak, Indonesia, 78124 a ) [email protected], b ) bobby@fi.itb.ac.id, c ) [email protected], d ) [email protected] Abstract
In this paper we construct a special class of four dimensional axis-symmetric stationary spacetimes whose Ricci scalar is constant in theBoyer-Lindquist coordinates. The first step is to construct Einstein met-ric by solving a modified Ernst equation for nonzero cosmological constant.Then, we modify the previous result by adding two additional functionsto the metric to obtain a more general metric of constant scalar curvaturewhich are not Einstein.
The aim of this paper is to construct a special class of four dimensional axissym-metric stationary spacetimes of constant scalar curvature in the Boyer-Lindquistcoordinates. First, we discuss the construction of Einstein spacetimes (or knownto be Kerr-Einstein spacetimes) with nonzero cosmological constant by solvinga modified Ernst equation for nonzero cosmological constant. In other words,we re-derive the Carter’s result in [1]. Then, we proceed to construct the spacesof constant scalar curvature which are not Einstein by modifying the previousresult, namely, we add two additional functions to the metric to have a more Corresponding author.
Suppose we have a metric in the general form ds = g µν ( x ) dx µ dx ν , (1)defined on a four dimensional spacetimes M , where x µ parametrizes a localchart on M and µ, ν = 0 , ....,
3. Then, we simplify the case as follows. Ina stationary axisymmetric spacetime, the time coordinate t and the azimuthalangle ϕ are considered to be x and x respectively. A stationary axisymmetricmetric is invariant under simultaneous transformations t → − t and ϕ → − ϕ which yields g = g = g = g = 0 , (2)and moreover, all non-zero metric components depend only on x ≡ r and x ≡ θ . The latter condition implies g = 0 and the metric (1) can be simplifiedinto [2, 3] ds = − e ν dt + e ψ ( dϕ − ωdt ) + e µ dr + e µ dθ , (3)where ( ν, ψ, ω, µ , µ ) ≡ ( ν ( r, θ ) , ψ ( r, θ ) , ω ( r, θ ) , µ ( r, θ ) , µ ( r, θ )). In the follow-ing we list the non-zero components of Christoffel symbol related to the metric(3): Γ = ν, − ωω, e ψ − ν ) , Γ = ν, − ωω, e ψ − ν ) , Γ = 12 ω, e ψ − ν ) , Γ = 12 ω, e ψ − ν ) , Γ = − ω ( ψ, − ν, ) − ω, (1 + ω e ψ − ν ) ) , Γ = ψ, + 12 ωω, e ψ − v ) , Γ = ψ, + 12 ωω, e ψ − ν ) , Γ = ν, e v − µ ) − ω ( ω, + ωψ, ) e ψ − µ ) , (4)Γ = (cid:18) ω, + ωψ, (cid:19) e ψ − µ ) , Γ = − ψ, e ψ − µ ) , = µ , , Γ = µ , , Γ = − µ , e µ − µ ) , Γ = ν, e ν − µ ) − ω ( ω, + ωψ, ) e ψ − µ ) , Γ = (cid:18) ω, + ωψ, (cid:19) e ψ − µ ) , Γ = − ψ, e ψ − µ ) , Γ = − µ , e µ − µ ) , Γ = µ , , Γ = µ , , and the non-zero components of Ricci tensor: R = e ν − µ ) (cid:18) ν, , + v, ( ψ + ν − µ + µ ) , − ω , e ψ − ν ) (cid:19) + e ν − µ ) (cid:18) ν, , + ν, ( ψ + ν + µ − µ ) , − ω , e ψ − ν ) (cid:19) − ωe ψ − µ ) ( ω, , + ω, (3 ψ − ν − µ + µ ) , ) − ωe ψ − µ ) ( ω, , + ω, (3 ψ − ν + µ − µ ) , ) − ω e ψ − µ ) (cid:18) ψ, , + ψ, ( ψ + ν − µ + µ ) , + 12 ω , e ψ − ν ) (cid:19) − ω e ψ − µ ) (cid:18) ψ, , + ψ, ( ψ + ν + µ − µ ) , + 12 ω , e ψ − ν ) (cid:19) ,R = 12 e ψ − µ ) ( ω, , + ω, (3 ψ − ν − µ + µ ) , )+ 12 e ψ − µ ) ( ω, , + ω, (3 ψ − ν + µ − µ ) , )+ ωe ψ − µ ) (cid:18) ψ, , + ψ, ( ψ + ν − µ + µ ) , + 12 ω , e ψ − ν ) (cid:19) + ωe ψ − µ ) (cid:18) ψ, , + ψ, ( ψ + ν + µ − µ ) , + 12 ω , e ψ − ν ) (cid:19) , (5) R = − e ψ − µ ) (cid:18) ψ, , + ψ, ( ψ + ν − µ + µ ) , + 12 ω , e ψ − ν ) (cid:19) − e ψ − µ ) (cid:18) ψ, , + ψ, ( ψ + ν + µ − µ ) , + 12 ω , e ψ − ν ) (cid:19) ,R = − ( ψ, , + ψ, ( ψ − µ ) , − ( ν, , + v, ( ν − µ ) , − e µ − µ ) ( µ , , + µ , ( ψ + ν + µ − µ ) , ) − ( µ , , + µ , ( µ − µ ) , ) + 12 ω , e ψ − v ) ,R = − ( ψ, , + ψ, ( ψ − µ ) , − ( ν, , + ν, ( v − µ ) , + µ , ( ψ − ν ) , + 12 ω, ω, e ψ − ν ) ,R = − ( ψ, , + ψ, ( ψ − µ ) , − ( ν, , + ν, ( v − µ ) , − e µ − µ ) ( µ , , + µ , ( ψ + ν − µ + µ ) , ) − ( µ , , + µ , ( µ − µ ) , ) + 12 ω , e ψ − ν ) . − R = 2 e − µ (cid:16) ψ , , + ψ , ( ψ − µ + µ ) , + ψ , ν, + ν , , + ν , ( ν − µ + µ ) , + µ , , + µ , ( µ − µ ) , − ω , e ψ − ν ) (cid:17) +2 e − µ (cid:16) ψ , , + ψ , ( ψ + µ − µ ) , + ψ , ν , + ν , , + ν , ( ν + µ − µ ) , + µ , , + µ , ( µ − µ ) , − ω , e ψ − ν ) (cid:17) , (6)where we have defined f ,µ ≡ ∂f∂x µ , f ,µ,ν ≡ ∂ f∂x µ ∂x ν . (7) In this section, we construct a class of axissymmetric spacetimes satisfying Ein-stein condition R µν = Λ g µν , (8)with Λ is named cosmological constant, yielding the following coupled nonlinearequations: (cid:0) e ψ − ν − µ + µ ω , (cid:1) , + (cid:0) e ψ − ν + µ − µ ω , (cid:1) , = 0 , (9)( ψ + ν ) , , − ( ψ + ν ) , µ , − ( ψ + ν ) , µ , + ψ , ψ , + ν , ν , = 12 e ψ − ν ) ω , ω , , (10) (cid:16) e µ − µ (cid:0) e β (cid:1) , (cid:17) , + (cid:16) e µ − µ (cid:0) e β (cid:1) , (cid:17) , = − e β + µ + µ , (11) (cid:16) e β − µ + µ ( ψ − ν ) , (cid:17) , + (cid:16) e β + µ − µ ( ψ − ν ) , (cid:17) , = − e ψ − ν (cid:0) e µ − µ ω , + e µ − µ ω , (cid:1) , (12)4 e µ − µ ( β, µ , + ψ , ν , ) − e µ − µ ( β , µ , + ψ , ν , ) = 2 e − β (cid:20)(cid:0) e µ − µ (cid:0) e β (cid:1) , (cid:1) , + (cid:16) e µ − µ (cid:0) e β (cid:1) , (cid:17) , (cid:21) − e ψ − ν ) (cid:0) e µ − µ ω , − e µ − µ ω , (cid:1) , (13)where we have defined β ≡ ψ + ν . (14)This class of solutions is called Kerr-(anti) de Sitter solutions. µ and µ First of all, we simply take e µ as e µ = ( r + a cos θ ) ∆ (0) r , (15)4here ∆ (0) r ≡ ∆ (0) r ( r ) and a is a constant related to the angular momentumof a black hole [3]. Next, we assume that the function e µ − µ ) and e β areseparable as e µ − µ ) = ∆ (0) r sin θ ∆ (0) θ ,e β = ∆ (0) r ∆ (0) θ , (16)with ∆ (0) θ ≡ ∆ (0) θ ( θ ). Thus, (11) can be cast into the form (cid:20) ∆ (0) r (cid:16) ∆ (0) r (cid:17) , (cid:21) , + 1sin θ " ∆ (0) θ sin θ (cid:16) ∆ (0) θ (cid:17) , , = − (cid:0) r + a cos θ (cid:1) . (17)Employing the variable separation method, we then obtain∆ (0) r = − Λ3 r + c r + c r + c , ∆ (0) θ = − Λ3 a cos θ − c cos θ − c cos θ + c , (18)where c i i = 1 , ..., , are real constant. To make a contact with [1], one has toset c i to be c = − Λ3 a , c = − M , c = a ,c = 0 , c = 1 , (19)such that we have ∆ (0) r = − Λ3 r (cid:0) r + a (cid:1) + r − M r + a , ∆ (0) θ = (cid:18) a cos θ (cid:19) sin θ . (20) ( ω, ν, ψ ) and Ernst Equation To obtain the explicit form of ( ω, ν, ψ ), we have to transform (9) and (12) intoa so called Ernst equation with nonzero Λ using (20). This can be structuredas follows.First, we introduce a pair of functions (Φ , Ψ) viaΦ , = e ψ − ν ) ∆ (0) θ ω ,p , Φ ,p = − e ψ − ν ) ∆ (0) r ω , , (21)Ψ ≡ e ψ − ν ∆ (0) r ∆ (0) θ , p ≡ cos θ . Then, (9) and (12) can be cast intoΨ h (∆ (0) r Φ , ) , + (∆ (0) θ Φ ,p ) ,p i = 2∆ (0) r Ψ , Φ , + 2∆ (0) θ Ψ ,p Φ ,p , (22)Ψ h (∆ (0) r Ψ , ) , + (∆ (0) θ Ψ ,p ) ,p i = ∆ (0) r (cid:2) (Ψ , ) − (Φ , ) (cid:3) − ∆ (0) θ (cid:2) (Ψ ,p ) − (Φ ,p ) (cid:3) , respectively. Defining a complex function Z ≡ Ψ + iΦ, (22) can be rewritten inErnst formR eZ h (∆ (0) r Z , ) , + (∆ (0) θ Z ,p ) ,p i = ∆ (0) r ( Z , ) + ∆ (0) θ ( Z ,p ) . (23)Note that one could obtain another solution of (23), say ˜ Z ≡ ˜Ψ + i ˜Φ by aconjugate transformation ˜Ψ = ∆ (0) r ∆ (0) θ ˜ χ , ˜Φ , = ∆ (0) θ ˜ χ ˜ ω , , (24)˜Φ , = − ∆ (0) r ˜ χ ˜ ω , , where ˜ χ ≡ e ν − ψ e ν − ψ ) − ω , ˜ ω ≡ ωe ν − ψ ) − ω . (25)In the latter basis, we find˜Ψ = ∆ (0) r − a ∆ (0) θ r + a cos θ , ˜Φ = 2 aM cos θr + a cos θ + 2Λ3 ar cos θ . (26)After some computation, we conclude that [1] e ψ = (cid:0) r + a (cid:1) ∆ (0) θ − ∆ (0) r a sin θr + a cos θ ,e ν = ( r + a cos θ )∆ (0) θ ∆ (0) r ( r + a ) ∆ (0) θ − ∆ (0) r a sin θ , (27) ω = a ( r + a )∆ (0) θ − a sin θ )∆ (0) r ( r + a ) ∆ (0) θ − ∆ (0) r a sin θ . SPACETIMES OF CONSTANT RICCI SCALAR
In this section we extend the previous results to the case of spaces of constantRicci scalar, namely R = g µν R µν = k , (28)where k is a constant. To have an explicit solution, we simply replace ∆ (0) r and∆ (0) θ in (16), (20), and (27) by∆ r = ∆ (0) r + f ( r ) , ∆ θ = ∆ (0) θ + h ( θ ) . (29)Then, inserting these modified functions mentioned above to (28), we simplyhave − k ( r + a cos θ ) = (∆ r ) , , + 1sin θ (cid:20) (∆ θ ) , sin θ (cid:21) , , (30)which gives − ( k − r + a cos θ ) = 2 (cid:20) f (cid:16) f (cid:17) , (cid:21) , + 2sin θ " h sin θ (cid:16) h (cid:17) , , . (31)The solution of (31) is given by f ( r ) = −
112 ( k − r + 12 C r + C r + C ,h ( θ ) = −
112 ( k − a cos θ − C cos θ − C cos θ + C , (32)where C i , i = 1 , ...,
5, are real constant. It is worth mentioning some remarksas follows. First, the functions f ( r ) and g ( θ ) have the same structure as in (18),namely they are quartic polynomials with respect to r and cos θ , respectively.Second, the constant Λ here is no longer the cosmological constant. Finally,Einstein spacetimes can be obtained by setting k = 4Λ and C = a C withother C i i=1, 2, 4, are free constants.Now we can state our main result as follows. Theorem 1
Suppose we have an axissymmetric spacetime M endowed withmetric ds = e ν dt − e ψ ( dϕ − ωdt ) − e µ dr − e µ dθ , (33)7 atisfying e µ − µ ) = ∆ r sin θ ∆ θ ,e β = ∆ r ∆ θ ,e ψ = (cid:0) r + a (cid:1) ∆ θ − ∆ r a sin θr + a cos θ , (34) e ν = ( r + a cos θ )∆ θ ∆ r ( r + a ) ∆ θ − ∆ r a sin θ ,ω = a ( r + a )∆ θ − a sin θ ∆ r ( r + a ) ∆ θ − ∆ r a sin θ , where ∆ r and ∆ θ are given by (29). Then, there exist a familiy of spacetimesof constant scalar curvature with ∆ r = −
13 Λ r a + r − M r + a − k r + 12 C r + C r + C , (35)∆ θ = − cos θ + Λ3 a cos θ − k a cos θ − C cos θ − C cos θ + C + 1 , where C i , i = 1 , ..., , are real constant. The metric (33) becomes Einstein if k = 4Λ and C = a C for all C i . The norm of Riemann tensor for the case at hand in general has the form R µναβ R µναβ = 384 r ( r + a cos θ ) a C cos θ (cid:16) − a C + r ( C − M ) + C (cid:17) + (cid:16) − a C + r ( aC + C − M ) + C (cid:17)(cid:16) − C a + r ( − aC + C − M ) + C (cid:17)! + 192 r ( r + a cos θ ) a C cos θ (cid:0) a C − C r − C + 8 M r (cid:1) − r ( C − M ) (cid:0) C − a C (cid:1) − (cid:0) C − a C (cid:1) − r ( − aC + C − M )( aC + C − M ) ! + 8( r + a cos θ ) a C cos θ (cid:16) − a C + 3 r ( C − M ) + C (cid:17) +30 r ( C − M ) (cid:0) C − a C (cid:1) + 7 (cid:0) C − a C (cid:1) +27 r ( aC + C − M )( − aC + C − M ) ! − r + a cos θ ) ( − aC + C − M )( aC + C − M ) + k k = 0. This is so because the spacetime metric is indefinite. The norm(36) shows that the spacetime has a real ring singularity at r = 0 and θ = π/ a .Generally, the metric described in Theorem 1 may not be related to Einsteingeneral relativity since our method described above does not use the notion ofenergy-momentum tensor. To make a contact with general relativity, we couldsimply set some constants, for example, namely C = C = C = C = 0 , C = q + g , (37)with k = 4Λ where q and g are electric and magnetic charges, respectively. Thissetup gives the Kerr-Newman-Einstein metric describing a dyonic rotating blackhole with nonzero cosmological constant [5]. A SPACETIME CONVENTION
In this section we collect some spacetime quantities which are useful for theanalysis in the paper.Christoffel symbol:Γ λµν = 12 g ρσ ( ∂ µ g ρν + ∂ ν g ρµ − ∂ ρ g µν ) . (38)Riemann curvature tensor: − R ρµνσ = ∂ σ Γ ρµν − ∂ ν Γ ρµσ + Γ λµν Γ ρλσ − Γ λµσ Γ ρλν . (39)Ricci tensor: R µν = R ρµρv = ∂ ρ Γ ρµν − ∂ ν Γ ρµρ + Γ λµν Γ ρλρ − Γ λµρ Γ ρλν . (40)Ricci scalar: R = g µν R µν . (41) Acknowledgments
The work in this paper is supported by Riset KK ITB 2014-2016 and RisetDesentralisasi DIKTI-ITB 2014-2016. 9 eferences [1] B. Carter, “Black holes equilibrium states,” in “Proceedings, Ecole d’Et dePhysique Thorique: Les Astres Occlus : Les Houches, ed. by B. DeWitt,C. M. DeWitt, (Gordon and Breach, New York, 1973).[2] S. Chandrasekhar, “The Kerr Metric and Stationary Axis-Symmetric Grav-itational Field,” Proc. Roy. Soc. Lond. A , 405 (1978).[3] For a review see for example:S. Chandrasekhar, “The mathematical theory of black holes,” OXFORD,UK: CLARENDON (1985) 646 p and references therein.[4] R. C. Henry, “Kretschmann scalar for a Kerr-Newman black hole,” Astro-phys. J. , 350 (2000) [astro-ph/9912320].[5] M. R. Setare and M. B. Altaie, “The Cardy-Verlinde formula and entropyof topological Kerr-Newman black holes in de Sitter spaces,” Eur. Phys. J.C30