Axisymmetric Stationary Spacetimes of Constant Scalar Curvature in Four Dimensions Rosikhuna F. Assafari,a,Bobby E. Gunara,,b 1,Hasanuddin,c, and Abednego Wiliardy,d arxiv:1606.02805v1 [gr-qc] 9 Jun 2016 Indonesian Center for Theoretical and Mathematical Physics ICTMP and Theoretical Physics Laboratory Theoretical High Energy Physics and Instrumentation Research Group, Faculty of Mathematics and Natural Sciences, Institut Teknologi Bandung Jl. 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 rosikhuna@yahoo.co.id, b bobby@fi.itb.ac.id, c hasanuddin@physics.untan.ac.id, d abednego.wiliardy@gmail.com Abstract In this paper we construct a special class of four dimensional axissymmetric stationary spacetimes whose Ricci scalar is constant in the Boyer-Lindquist coordinates. The first step is to construct Einstein metric by solving a modified Ernst equation for nonzero cosmological constant. Then, we modify the previous result by adding two additional functions to the metric to obtain a more general metric of constant scalar curvature which are not Einstein. 1 INTRODUCTION The aim of this paper is to construct a special class of four dimensional axissymmetric stationary spacetimes of constant scalar curvature in the Boyer-Lindquist coordinates. First, we discuss the construction of Einstein spacetimes or known to be Kerr-Einstein spacetimes with nonzero cosmological constant by solving a 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 spaces of constant scalar curvature which are not Einstein by modifying the previous result, namely, we add two additional functions to the metric to have a more 1 Corresponding author. 1
general form of the metric but it has the structure of polynomial, namely, it is a quartic polynomial. The structure of the paper can be mentioned as follows. In section 2 we give a quick review on axissymmetric stationary spacetimes. Then, we begin our construction of Kerr-Einstein spacetimes in section 3. We discuss the construction of axissymmetric stationary spacetimes of constant scalar curvature in section 4. 2 Axissymmetric Stationary Spacetimes: A Quick Review Suppose we have a metric in the general form ds 2 = g µν xdx µ dx ν, 1 defined on a four dimensional spacetimes M 4, where x µ parametrizes a local chart on M 4 and µ,ν = 0,... Then, we simplify the case as follows. In a stationary axisymmetric spacetime, the time coordinate t and the azimuthal angle ϕ are considered to be x 0 and x 1 respectively. A stationary axisymmetric metric is invariant under simultaneous transformations t t and ϕ ϕ which yields g 02 = g 03 = g 12 = g 13 = 0, 2 and moreover, all non-zero metric components depend only on x 2 r and x 3. The latter condition implies g 23 = 0 and the metric 1 can be simplified into [2, 3] ds 2 = e 2ν dt 2 +e 2ψ dϕ ωdt 2 +e 2µ2 dr 2 +e 2µ3 d 2, 3 where ν,ψ,ω,µ 2,µ 3 νr,,ψr,,ωr,,µ 2 r,,µ 3 r,. In the following we list the non-zero components of Christoffel symbol related to the metric 3: Γ 0 02 = ν, 2 1 2 ωω, 2 e2ψ ν, Γ 0 03 = ν, 3 1 2 ωω, 3 e2ψ ν, Γ 0 12 = 1 2 ω, 2 e2ψ ν, Γ 0 13 = 1 2 ω, 3 e2ψ ν, Γ 1 20 = ωψ, 2 ν, 2 1 2 ω, 21+ω 2 e 2ψ ν, Γ 1 12 = ψ, 2 + 1 2 ωω, 2 e 2ψ v, Γ 1 13 = ψ, 3 + 1 2 ωω, 3 e 2ψ ν, Γ 2 00 = ν, 2 e 2v µ2 ωω, 2 +ωψ, 2 e 2ψ µ2, 4 1 Γ 2 01 = 2 ω, 2 +ωψ, 2 e 2ψ µ2, Γ 2 11 = ψ, 2 e 2ψ µ2, 2
Γ 2 22 = µ 2, 2, Γ 2 23 = µ 2, 3, Γ 2 33 = µ 3, 2 e 2µ2 µ3, Γ 3 00 = ν, 3 e 2ν µ3 ωω, 3 +ωψ, 3 e 2ψ µ3, 1 Γ 3 01 = 2 ω, 3 +ωψ, 3 e 2ψ µ3, Γ 3 11 = ψ, 3 e 2ψ µ3, Γ 3 22 = µ 2, 3 e 2µ2 µ3, Γ 3 23 = µ 3, 2, Γ 3 33 = µ 3, 3, and the non-zero components of Ricci tensor: R 00 = e 2ν µ2 ν, 2, 2 +v, 2 ψ +ν µ 2 +µ 3, 2 1 2 ω2, 2 e 2ψ ν +e 2ν µ3 ν, 3, 3 +ν, 3 ψ +ν +µ 2 µ 3, 3 1 2 ω2, 3 e 2ψ ν ωe 2ψ µ2 ω, 2, 2 +ω, 2 3ψ ν µ 2 +µ 3, 2 ωe 2ψ µ3 ω, 3, 3 +ω, 3 3ψ ν +µ 2 µ 3, 3 ω 2 e 2ψ µ2 ψ, 2, 2 +ψ, 2 ψ +ν µ 2 +µ 3, 2 + 1 2 ω2, 2 e 2ψ ν ω 2 e 2ψ µ3 ψ, 3, 3 +ψ, 3 ψ +ν +µ 2 µ 3, 3 + 1 2 ω2, 3 e 2ψ ν R 01 = 1 2 e2ψ µ2 ω, 2, 2 +ω, 2 3ψ ν µ 2 +µ 3, 2 + 1 2 e2ψ µ3 ω, 3, 3 +ω, 3 3ψ ν +µ 2 µ 3, 3 +ωe 2ψ µ2 ψ, 2, 2 +ψ, 2 ψ +ν µ 2 +µ 3, 2 + 1 2 ω2, 2 e 2ψ ν +ωe 2ψ µ3 ψ, 3, 3 +ψ, 3 ψ +ν +µ 2 µ 3, 3 + 1 2 ω2, 3 e 2ψ ν R 11 = e 2ψ µ2 ψ, 2, 2 +ψ, 2 ψ +ν µ 2 +µ 3, 2 + 1 2 ω2, 2 e 2ψ ν e 2ψ µ3 ψ, 3, 3 +ψ, 3 ψ +ν +µ 2 µ 3, 3 + 1 2 ω2, 3 e 2ψ ν R 22 = ψ, 2, 2 +ψ, 2 ψ µ 2, 2 ν, 2, 2 +v, 2 ν µ 2, 2 e 2µ2 µ3 µ 2 +µ 2 ψ +ν +µ 2 µ 3, 3 µ 3 +µ 3 µ 3 µ 2, 2 + 1 2 ω2, 2 e 2ψ v, R 23 = ψ, 2, 3 +ψ, 2 ψ µ 2, 3 ν, 2, 3 +ν, 2 v µ 2, 3 +µ 3 ψ ν, 3 + 1 2 ω, 2ω, 3 e 2ψ ν, R 33 = ψ, 3, 3 +ψ, 3 ψ µ 3, 3 ν, 3, 3 +ν, 3 v µ 3, 3 e 2µ3 µ2 µ 3 +µ 3 ψ +ν µ 2 +µ 3, 3 µ 2 +µ 2 µ 2 µ 3, 3 + 1 2 ω2, 3 e 2ψ ν.,,,5 3
Then, Ricci scalar can be obtained as R = 2e 2µ2 ψ +ψ ψ µ 2 +µ 3 +ψ ν, 2 +ν +ν ν µ 2 +µ 3 +µ 3 +µ 3 µ 3 µ 2, 2 1 4 ω2 e 2ψ ν +2e 2µ3 ψ +ψ ψ +µ 2 µ 3, 3 +ψ ν +ν +ν ν +µ 2 µ 3, 3 +µ 2 +µ 2 µ 2 µ 3 1 4 ω2 e 2ψ ν, 6 where we have defined f,µ f x µ, f,µ,ν 2 f x µ x ν. 7 3 EINSTEIN SPACETIMES In this section, we construct a class of axissymmetric spacetimes satisfying Einstein condition R µν = Λg µν, 8 with Λ is named cosmological constant, yielding the following coupled nonlinear equations: e 3ψ ν µ 2+µ 3 ω + e 3ψ ν+µ2 µ3 ω = 0, 9 ψ +ν ψ +ν µ 2 ψ +ν µ 3 +ψ ψ +ν ν = 1 2 e2ψ ν ω ω, e µ3 µ2 e β + e µ2 µ3 e β e β µ2+µ3 ψ ν e + β+µ2 µ3 ψ ν 10 = 2Λe β+µ2+µ3, 11 = e3ψ ν e µ3 µ2 ω 2 +e µ2 µ3 ω 2, 4e µ3 µ2 β, 2 µ 3 +ψ ν 4e µ2 µ3 β µ 2 +ψ ν = 2e β [ e µ 3 µ 2 e β, 2 where we have defined 12 + e µ2 µ3 e β e 2ψ ν e µ3 µ2 ω 2 e µ2 µ3 ω 2, 13 β ψ +ν. 14 This class of solutions is called Kerr-anti de Sitter solutions. ] 3.1 The Functions µ 2 and µ 3 First of all, we simply take e µ2 as e µ2 = r2 +a 2 cos 2 1 2 0 1 2 r, 15 4
where 0 r 0 r r and a is a constant related to the angular momentum of a black hole [3]. Next, we assume that the function e 2µ3 µ2 and e 2β are separable as e 2µ3 µ2 = 0 sin 2 r, 0 e 2β = 0 r 0, 16 with 0 0. Thus, 11 can be cast into the form [ ] [ ] 01 2 r 01 2 r + 1 01 2 01 2 = 2Λ r 2 +a 2 cos 2. 17 sin sin Employing the variable separation method, we then obtain 0 r = Λ 3 r4 +c 1 r 2 +c 2 r +c 3, 0 = Λ 3 a2 cos 4 c 1 cos 2 c 4 cos +c 5, 18 where c i i = 1,...,5, are real constant. To make a contact with [1], one has to set c i to be such that we have c 1 = Λ 3 a2, c 2 = 2M, c 3 = a 2, c 4 = 0, c 5 = 1, 19 0 r = Λ 3 r2 r 2 +a 2 +r 2 2Mr+a 2, 1+ Λ 3 a2 cos 2 0 = sin 2. 20 3.2 The Functions ω, ν, ψ and Ernst Equation To obtain the explicit form of ω,ν,ψ, we have to transform 9 and 12 into a so called Ernst equation with nonzero Λ using 20. This can be structured as follows. First, we introduce a pair of functions Φ,Ψ via Φ = e 2ψ ν 0 ω,p, Φ,p = e 2ψ ν 0 r ω, 21 Ψ e ψ ν 01 2 r 01 2, 5
where p cos. Then, 9 and 12 can be cast into [ ] Ψ 0 r Φ + 0 Φ,p,p = 2 0 r Ψ Φ +2 0 Ψ,pΦ,p, 22 [ ] Ψ 0 r Ψ + 0 Ψ,p,p = 0 [ r Ψ 2 Φ 2] 0 [ Ψ,p 2 Φ,p 2], respectively. Defining a complex function Z Ψ+iΦ, 22 can be rewritten in Ernst form [ ] ReZ 0 r Z + 0 Z,p,p = 0 r Z 2 + 0 Z,p 2. 23 Note that one could obtain another solution of 23, say Z Ψ + i Φ by a conjugate transformation Ψ = 01 2 r 01 2 χ, Φ = 0 χ 2 ω, 24 Φ = 0 r χ 2 ω, where In the latter basis, we find χ ω e ν ψ e 2ν ψ ω, 2 ω e 2ν ψ ω. 25 2 Ψ = 0 r a 2 0 r 2 +a 2 cos 2, 2aM cos Φ = r 2 +a 2 cos 2 + 2Λ 3 After some computation, we conclude that [1] e 2ψ = e 2ν = arcos. 26 r 2 +a 2 2 0 0 r a 2 sin 4 r 2 +a 2 cos 2, r 2 +a 2 cos 2 0 0 r r 2 +a 2 2 0 0 r a 2 sin 4, 27 ω = ar2 +a 2 0 a sin 2 0 r r 2 +a 2 2 0 0 r a 2 sin 4. 6
4 SPACETIMES OF CONSTANT RICCI SCALAR In this section we extend the previous results to the case of spaces of constant Ricci 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 in 16, 20, and 27 by r = 0 r +fr, = 0 +h. 29 Then, inserting these modified functions mentioned above to 28, we simply have kr 2 +a 2 cos 2 = r + 1 [ ], 30 sin sin which gives [ k 4Λr 2 +a 2 cos 2 = 2 f 1 2 The solution of 31 is given by ] f 1 2 + 2 sin fr = 1 12 k 4Λr4 + 1 2 C 1r 2 +C 2 r+c 3, [ h 1 2 sin ] h 1 2. 31 h = 1 12 k 4Λa2 cos 4 1 2 C 1cos 2 C 4 cos +C 5, 32 where C i, i = 1,...,5, are real constant. It is worth mentioning some remarks as follows. First, the functions fr and g have the same structure as in 18, namely they are quartic polynomials with respect to r and cos, respectively. Second, the constant Λ here is no longer the cosmological constant. Finally, Einstein spacetimes can be obtained by setting k = 4Λ and C 3 = a 2 C 5 with other 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 4 endowed with metric ds 2 = e 2ν dt 2 e 2ψ dϕ ωdt 2 e 2µ2 dr 2 e 2µ3 d 2, 33 7
satisfying e 2µ3 µ2 = r sin 2, e 2β = r, e 2ψ = r 2 +a 2 2 r a 2 sin 4 r 2 +a 2 cos 2, 34 e 2ν = r 2 +a 2 cos 2 r r 2 +a 2 2 r a 2 sin 4, ω = ar2 +a 2 a sin 2 r r 2 +a 2 2 r a 2 sin 4, where r and are given by 29. Then, there exist a familiy of spacetimes of constant scalar curvature with r = 1 3 Λr2 a 2 +r 2 2Mr+a 2 k 12 r4 + 1 2 C 1r 2 +C 2 r +C 3, 35 = cos 2 + Λ 3 a2 cos 2 k 12 a2 cos 4 1 2 C 1cos 2 C 4 cos +C 5 +1, where C i, i = 1,...,5, are real constant. The metric 33 becomes Einstein if k = 4Λ and C 3 = a 2 C 5 for all C i. The norm of Riemann tensor for the case at hand in general has the form R µναβ 384r 4 R µναβ = r 2 +a 2 cos 2 6 2a 2 C 4 cos a 2 C 5 +rc 2 2M+C 3 + a 2 C 5 +rac 4 +C 2 2M+C 3 C 5 a 2 +r ac 4 +C 2 2M+C 3 192r 2 + r 2 +a 2 cos 2 5 a 2 C 4 cos 3a 2 C 5 4C 2 r 3C 3 +8Mr 5rC 2 2M C 3 a 2 C 5 2 C3 a 2 2 C 5 3r 2 ac 4 +C 2 2MaC 4 +C 2 2M 8 + r 2 +a 2 cos 2 4 6a 2 C 4 cos a 2 C 5 +3rC 2 2M+C 3 +30rC 2 2M C 3 a 2 C 5 +7 C3 a 2 2 C 5 +27r 2 ac 4 +C 2 2M ac 4 +C 2 2M 12 r 2 +a 2 cos 2 3 ac 4 +C 2 2MaC 4 +C 2 2M+ k2 6 36 8
which might have a negative value as observed in [4] for Kerr-Newman metric with 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 = π/2 with radius a. Generally, the metric described in Theorem 1 may not be related to Einstein general relativity since our method described above does not use the notion of energy-momentum tensor. To make a contact with general relativity, we could simply set some constants, for example, namely C 1 = C 2 = C 4 = C 5 = 0, C 3 = q 2 +g 2, 37 with k = 4Λ where q and g are electric and magnetic charges, respectively. This setup gives the Kerr-Newman-Einstein metric describing a dyonic rotating black hole with nonzero cosmological constant [5]. A SPACETIME CONVENTION In this section we collect some spacetime quantities which are useful for the analysis in the paper. Christoffel symbol: Γ λ µν = 1 2 gρσ µ g ρν + ν g ρµ ρ g µν. 38 Riemann curvature tensor: R ρ µνσ = σγ ρ µν ν Γ ρ µσ +Γ λ µνγ ρ λσ Γ λ µσγ ρ λν. 39 Ricci tensor: R µν = R ρ µρv = ργ ρ µν ν Γ ρ µρ +Γ λ µνγ ρ λρ Γ λ µργ ρ λν. 40 Ricci scalar: R = g µν R µν. 41 Acknowledgments The work in this paper is supported by Riset KK ITB 2014-2016 and Riset Desentralisasi DIKTI-ITB 2014-2016. 9
References [1] B. Carter, Black holes equilibrium states, in Proceedings, Ecole d Et de Physique 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 Gravitational Field, Proc. Roy. Soc. Lond. A 358, 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, Astrophys. J. 535, 350 2000 [astro-ph/9912320]. [5] M. R. Setare and M. B. Altaie, The Cardy-Verlinde formula and entropy of topological Kerr-Newman black holes in de Sitter spaces, Eur. Phys. J. C 30, 273 2003 [hep-th/0304072]. 10