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Regular and Conformal Regular Cores for Static and Rotating Solutions Physics Letters B 730 (2014) 95–98 Contents lists available at ScienceDirect Physics Letters B www.elsevier.com/locate/physletb Regular and conformal regular cores for static and rotating solutions Mustapha Azreg-Aïnou Baskent ¸ University, Department of Mathematics, Ba˘glıca Campus, Ankara, Turkey article info abstract Article history: Using a new metric for generating rotating solutions, we derive in a general fashion the solution of an Received 10 January 2014 imperfect fluid and that of its conformal homolog. We discuss the conditions that the stress–energy Accepted 16 January 2014 tensors and invariant scalars be regular. On classical physical grounds, it is stressed that conformal fluids Available online 23 January 2014 used as cores for static or rotating solutions are exempt from any malicious behavior in that they are Editor: M. Cveticˇ finite and defined everywhere. © 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. μν = μ ν − μ ν + μ ν + μ ν 1. Introduction pleteness relation, g u u (e2 e2 e3 e3 e4 e4 ),leadsto T μν = ( + p)uμuν − pgμν . The quest for rotating solutions has always been a fastidious Given a static spherically symmetric solution to the field equa- task. It took more than two decades to discover the rotating solu- tions in spherical coordinates: tion of Van Stockum [1] and more than forty years to derive that of 2 2 2 dr 2 2 2 Kerr [2] since the foundation of General Relativity in 1916. Several ds = G(r) dt − − H(r) dθ + sin θ dφ (2) F (r) partial methods have been put forward to construct rotating so- lutions [1–15] but no general method seems to be available. This we generate a stationary rotating solution, the metric of which, work is no exception and presents a novel partial method for gen- written in Boyer–Lindquist (B–L) coordinates, we postulate to be erating rotating solutions from static ones. However, the method of the form will allow us (1) to generate rotating solutions without appeal- G(FH+ a2 cos2 θ)Ψ Ψ ds2 = √ √ dt2 − dr2 ing to linear approximations [16] and (2) to apply the matching ( FH+ a2 G cos2 θ)2 FH+ a2 methods [17–19] to regular black hole cores as well as to worm- √ √ F GH − FGH hole cores [15,20,21]. The excellent paper by Lemos and Zanchin + 2a sin2 θ √ √ Ψ dt dφ offers an up-to-date classification of the existing matching meth- + 2 2 2 ( FH a G cos θ) ods, discusses the types of regular black holes derived so far and presents new electrically charged solutions with a regular de Sitter − Ψ dθ 2 − Ψ sin2 θ 1 core [19]. The present method reduces the task of finding a ro- √ √ tating solution to that of finding a two-variable function that is a 2 F GH − FGH + a2G cos2 θ + a2 sin2 θ √ √ dφ2, (3) solution to two second order partial differential equations. + 2 2 2 μ μ ( FH a G cos θ) We work with R νρσ =−∂σ Γ νρ +··· (μ = 1 → 4) and a met- by solving the field equations for Ψ(r,θ), which depends also on ric gμν with signature (+, −, −, −). We make all necessary con- the rotating parameter a. More on the derivation and general- ventions such that the field equations take the form Gμν = Tμν . ization of (3) will be given elsewhere [22]. For fluids undergo- We consider a fluid without heat flux, the stress–energy tensor ing only a rotational motion about a fixed axis (the z axis here), (SET) of which admits the decomposition Trθ ≡ 0leadingtoGrθ = 0, which is one of the very two equa- tions to solve to obtain Ψ(r,θ).Fromnowon,weusethefollow- μν = μ ν + μ ν + μ ν + μ ν T u u p2e2 e2 p3e3 e3 p4e4 e4 (1) ing conventions and notation: μ : 1 ↔ t, 2 ↔ r, 3 ↔ θ,4 ↔ φ and (u, e2, e3, e4) = (u, er , eθ , eφ). where is the mass density and (p1, p2, p3) are the compo- nents of the pressure. We have preferred the notation uμ, instead μ 2. The solutions of e1 , which is the four-velocity of the fluid. The four-vectors are μ = μ =− = mutually perpendicular and normalized: u uμ 1, ei eiμ 1 To ease the calculations, we use the algebraic coordinate y 2 2 2 (i = 2 → 4). If the fluid is perfect, p2 = p3 = p4 ≡ p, then the com- cos θ and replace dθ by dy /(1 − y ) in (3).Forthesakeof http://dx.doi.org/10.1016/j.physletb.2014.01.041 0370-2693/© 2014 The Author. Published by Elsevier B.V. This is an open access article under the CC BY license (http://creativecommons.org/licenses/by/3.0/). Funded by SCOAP3. 96 M. Azreg-Aïnou / Physics Letters B 730 (2014) 95–98 subsequent applications (to regular black holes and wormholes), r = 0 and θ = π/2 to the same order, leading to a ring-singularity √we will√ assume H = r2 unless otherwise specified. Setting K (r) ≡ free solution (3). When this is the case, the components of the SET ≡ FH/ G and using an indexical notation for derivatives: Ψ,ry2 as well as the two invariants remain finite, but undefined, on the 2 2 2 = ∂ Ψ/∂r∂ y , K,r ≡ ∂ K /∂r, etc., the equation Grθ = 0yields ring ρ 0. Setting 2 f (r) ≡ K − FH, (r) ≡ FH+ a2 and Σ ≡ (K + a2)2 − + 2 2 2 − = 2 2 a2 sin2 ,thesolution(3) reduces to K a y (3Ψ,r Ψ,y2 2ΨΨ,ry2 ) 3a K,r Ψ . (4) θ This hyperbolic partial differential equation may possess different 2 2 2 2 f 2 ρ 2 4af sin θ solutions, but a simple class of solutions is manifestly of the form ds = 1 − dt − dr + dt dφ n 2 2 Ψ(r, y) = g(K + a2 y2) where g(z) is solution to ρ ρ Σ sin2 θ 2 − 2 2 + 2 = − ρ2 dθ 2 − dφ2 (10) 2z gg,zz 3z g,z 3g 0(5)ρ2 2 2 where z = K (r) + a y . A general solution depending on two con- 2 2 2 2 sin θ 2 2 stants is derived setting A(z) = g /g and leads to Ψgen = c2z/(z + = dt − a sin θ dφ − a dt − K + a dφ 2 ρ2 ρ2 c1) . However, this solution does not exhaust the set of all possi- ble solutions of the form g(z) to (5) which, being nonlinear, admits 2 ρ 2 2 2 ∝ n − dr − ρ dθ . (11) other more interesting power-law solutions g(z) z leading to 2 2 2 2 −3 Ψ1 = K (r) + a y or Ψ2 = K (r) + a y (6) We fix the basis (u, er , eθ , eφ) by = = √ where Ψ2 is included in Ψgen taking c1 0 and c2 1. A consis- + 2 = μ = (K a , 0, 0,a) μ = ( 0, 1, 0, 0) tency check of the field equations Gμν Tμν and the form of Tμν u , er , [Eq. (1)] yields the partial differential equation ρ2 ρ2 2 2 2 2 μ (0, 0, 1, 0) μ (a sin θ,0, 0, 1) Ψ K,r + K (2 − K,rr) − a y (2 + K,rr) e = , e =− . (12) θ 2 φ 2 + + 2 2 2 − = ρ ρ sin θ K a y 4y Ψ,y2 K,r Ψ,r 0, (7) The components of the SET are expressed in terms of G as: = 2 2 μν which is solved by Ψ1 (but not by Ψ2) provided K = r + p where μ ν rr θθ μ ν u u Gμν , pr =−g Grr, pθ =−g Gθθ, pφ = e e Gμν .Wefind: p2 is real. We have thus found a simple common solution to both φ φ Eqs. (4) and (7) given by 2(rf − f ) − p2 2p2(3 f − a2 sin2 θ) = + , (13) Ψ = r2 + p2 + a2 y2. (8) ρ4 ρ6 2 We do not know the set of all possible solutions to Eqs. (4) 2p f p =− − , p =−p − , and (7), however, we can still distinguish two families of rotating r ρ6 θ r ρ2 solutions. Depending on G(r), F (r) and H(r), a rotating solution 2 2 2 given by (3) is called a normal fluid, Ψ , if the static solution (2) is 2p a sin θ n pφ = pθ + . (14) recovered from the rotating one in the limit a → 0: This implies ρ6 lim → Ψ = H. Otherwise the rotating solution is called a con- a 0 Thus, for wormholes and some type of regular phantom black formal fluid, Ψ .GivenG(r), F (r) and H(r), the normal ds2 and c n holes [15,21] where always ρ2 > 0(H never vanishes), the compo- conformal ds2 fluids are conformally related c nents of the SET are finite in the static and rotating cases. Eqs. (13) 2 2 and (14) will be used in [22] to derive the rotating counterpart ds = (Ψc/Ψn) ds . (9) c n of the stable exotic dust Ellis wormhole emerged in a source-free = 2 = 2 = 2 Now, since lima→0 Ψc H (by definition) and lima→0 dsn dsstat radial electric or magnetic field [29].IfH r , corresponding to 2 = 2 2 [Eq. (2)], this implies that lima→0 dsc dsstat, and thus lima→0 dsc regular as well as singular black holes, the above expressions re- 2 =− = − 4 = is a new static metric conformal to dsstat. duce to those derived in [6,18]: pr 2(rf f )/ρ , pθ 2 For the remaining part of this work, we shall explore the prop- pφ = − f /ρ .
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