Regular and Conformal Regular Cores for Static and Rotating Solutions

Physics Letters B 730 (2014) 95–98

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Physics Letters B

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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 solutions 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 solutions [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 equations 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 components 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 possible 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 . In this case the components of the SET diverge on erties of both the normal (Section 3) and conformal (Section 4) the ring ρ2 = 0unless f ∝ r4 as r → 0, resulting in (1 − F ) ∝ r2 as rotating solutions that can be constructed using the unique simple r → 0, which corresponds to the (anti) de Sitter case and to regu- solution Ψ available to us, which is given by (8).Fromnowon,we lar black holes. In fact, most of regular black holes derived so far shall use the prime notation to denote derivatives of functions. have de Sitter-like behavior near r = 0 [17,19,20]. From the third equality in Eq. (14), one sees that the tangen- 3. Physical properties of the model-independent normal interior tial pressures, (pθ , pφ ), are generally nonequal and are equal only core: G = F if p2 = 0 or/and if a = 0 (the static case). Hence, in the general rotating case, the tensor T μν has four different eigenvalues repre- The constraints G = F and K = r2 + p2 yield H = K ,sowedeal senting thus a totally imperfect fluid. withanormalfluidsincelima→0 Ψ = H [Eq. (8)]. The invariants R It is straightforward to verify the validity of the continuity μναβ −6 −12 μ and Rμναβ R are proportional to ρ and ρ , respectively, equation: (u );μ = 0, where the semicolon denotes covariant 2 2 2 2 2 μν with ρ ≡ K + a y = H + a y . Thus, the static and rotating so- derivative. The conservation equation, T ;ν = 0, is consistent with 2 μ ν lutions (3) are regular if H(r) is never zero (p = 0), which is u ;ν u = 0 which shows that the motion of the fluid elements is the case for wormholes and some type of regular phantom black not geodesic. This is attributable to the nonvanishing of the r- and holes [15,21].IfH = r2 (p2 = 0), then the rotating solution (3) θ-components of the pressure gradient. may have a ring singularity in the plane θ = π/2(y = 0) at r = 0 The purpose of constructing rotating and nonrotating solutions (more details are given in [22]). As we shall see below, there are with negative pressure components, as might be the case in (13) μναβ cases where the numerators of R and Rμναβ R also vanish for to (14), is, as was made clear in [18], two-fold, in that, following a M. Azreg-Aïnou / Physics Letters B 730 (2014) 95–98 97 suggestion by Sakharov and Gliner [23,24], (1) the core of collaps- Notice that the Kerr solution (q = 0) and the rotating IΛF ing matter, with high matter density, should have a cosmological- one are derived from each other on performing the substitution type equation of state =−p, (2) the problem of the ring singu- 2M ↔ Λr3/3, so that most of the Kerr solution properties, where larity, which characterizes Kerr-type solutions, could be addressed no derivations with respect to r are performed, are easily car- if the interior of the hole is fitted with an imperfect fluid of the ried over into the rotating IΛF properties. For instance, the static type derived above. Fitting the interior of the hole with a de Sit- limit, which is the 2-surface on which the timelike Killing vector μ ter fluid is one possible solution to the ring singularity [18,19]. t = (1, 0, 0, 0) becomes√ null, corresponds to gtt(rst,θ)= 0lead- 2 = + + 2 2 Another possibility is to consider a regular core or a conformal ing to 2Λrst 3 9 12Λa cos θ. Thus, observers can remain regular one as we shall see in the case G = F (Section 4). static only for r < rst. Similarly, the cosmological horizon, which sets a limit for stationary√ observers, corresponds to Λ(rch) = 0 2 = + + 2 3.1. Rotating imperfect Λ-fluid—de Sitter rotating solution leading to 2Λrch 3 9 12Λa . Hence, the static limit is en- closed by the cosmological horizon and intersects it only at the Instances of application of (3) to re-derive the Kerr–Newman poles θ = 0orθ = π (in contrast with the Kerr solution where the solution from the Schwarzschild solution and to generate a rotating static limit encloses the event horizon). imperfect Λ-fluid (IΛF) from the de Sitter solution are straightfor- The four-velocity of the fluid elements may be expressed, in ward. To derive the Kerr–Newman solution, we take F = G = 1 − terms of the timelike tμ and spacelike φμ = (0, 0, 0, 1) Killing 2 2 2 μ μ μ 2 2 2 2m/r + q /r and H = r , the solution is then given by (10) with vectors, as u = N(t + Ωφ ),withN = (r + a )/ ρ Λ and = − 2 = 2 + 2 − + 2 2 = 2 + 2 2 = a r2 + a2 is the differentiable ( = constant) angular veloc- 2 fKN 2Mr q , KN r a 2Mr q , ρKN r a cos θ Ω /( ) Ω 2 2 2 2 2 ity of the fluid. Since the norm of the vector tμ + Ωφμ,1/N2,is and ΣKN = (r + a ) − a KN sin θ. Consider the de Sitter solution positive only for Λ > 0, which corresponds to the region r < rch, the fluid elements follow timelike world lines only for r < rch.As − 2 = − 2 2 − − 2 1 2 r → r , Ω approaches the limit a/(r2 + a2) that is the lowest dsΛ 1 Λr /3 dt 1 Λr /3 dr ch ch angular velocity of the fluid elements which we take as the angu- − r2 dθ 2 + sin2 θ dφ2 (15) = 2 + 2 lar velocity of the cosmological horizon: Ωch a/(rch a ).Atthe cosmological horizon, tμ + Ωφμ becomes null and tangent to the where F = G = 1 − Λr2/3 and H = r2.Themetricds2 of the rotat- Λ horizon’s null generators, so that the fluid elements are dragged = 4 = 2 + 2 − 4 ing IΛFisgivenby(10) with 2 fΛ Λr /3, Λ r a Λr /3, with the angular velocity Ω . 2 = 2 + 2 2 = 2 + 2 2 − 2 2 ch ρΛ r a cos θ and ΣΛ (r a ) a Λ sin θ.Except from a short description made in [25], the rotating IΛF has never 4. Physical properties of the conformal interior core: G = F been discussed deeply in the scientific literature. The compo- 4 4 2 2 nents of the SET are = Λr /ρ , pr =−, pθ = pφ =−Λr (r + Λ In this case H = K = r2 + p2,unlessp2 = 0, leading to 2a2 cos2 θ)/ρ4 .Thelimita → 0 leads to de Sitter solution where Λ lim → Ψ = H.WithΨ = K + a2 y2 [Eq. (8)], the conformal ro- the fluid is perfect with = Λ and p = p = p =−Λ. a 0 r θ φ tating solution ds2 is again given by (10) to (11) and the basis The rotating IΛF is only manifestly singular on the ring ρ2 = 0 c Λ (u, e , e , e ) by (12) but this time ρ2 ≡ K + a2 y2 = H + a2 y2.The [(θ, r) = (π/2, 0) or (y, r) = (0, 0)]. In fact, the curvature and r θ φ components of the SET are different due to the non-covariance of Kretchmann scalars the field equations under conformal transformations [26].TheSET 2 2 4Λr related to dsc is only partly proportional to that related to metric R =− , 2 r2 + a2 y2 dsn and includes terms involving first and second order derivatives of the conformal factor (K + a2 y2)/(H + a2 y2), which are the μναβ Rμναβ R residual terms in the transformed Einstein tensor. Finally, the SET 2 8Λ2r4(r8 + 4a2 y2r6 + 11a4 y4r4 − 2a6 y6r2 + 6a8 y8) related to dsc takes the form = (16) 3(r2 + a2 y2)6 p2[6 f − r2 − p2 − a2(2 − cos2 θ)] 2(rf − f ) = + , do not diverge in the limit (y, r) → (0, 0). Despite the fact that ρ6 ρ4 the limits do not exist, we can show that they do not diverge. 2 2 + 2 + 2 − Let C : r = ah(y) and h(0) = 0 be a smooth path through the 2p (r p a 2 f ) pr =− − , (18) point (y, r) = (0, 0) in the yr plane. We choose a path that reaches ρ6 (y, r) = (0, 0) obliquely or horizontally but not vertically, that is, 2 2 2 2 2(r + a cos θ)f p + 2rf f we assume that h (0) is finite [for paths that may reach (y, r) = p =− + − , θ 6 4 2 (0, 0) vertically we choose a smooth path y = g(r)/a and g(0) = 0 ρ ρ ρ C 2 where g (0) remains finite]. On , the limits of the two scalars as 2a2 p2 sin θ → p = p + (19) y 0read φ θ ρ6 4Λh (0)2 − which is finite and defined everywhere if p2 = 0. If p2 = 0, the , 1 + h (0)2 SET if finite, but undefined on the ring ρ2 = 0, if f ∝ r4 as r → 0 8Λ2h (0)4[6 − 2h (0)2 + 11h (0)4 + 4h (0)6 + h (0)8] ((anti) de Sitter behavior for F = G). The curvature scalar , (17) 3[1 + h (0)2]6 2{p2[r2 + p2 + a2(2 − cos2 θ)]−2p2 f } 2 f R = − (20) which are nonexisting [for h (0) depends on the path] but they re- ρ6 ρ2 main finite. Thus, the rotating IΛF is regular everywhere, however, the components of the SET are undefined on the ring ρ2 = 0. Paths is also finite for all p2. The Kretchmann scalar is certainly finite of the form: y = g(r)/a and g(0) = 0, where g (0) remains finite, everywhere for all p2. Conclusions made earlier concerning the μν lead to the same conclusion. The other scalar, Rμν R , behaves in continuity and conservation equations apply to the present case the same way as the curvature and Kretchmann scalars. of the conformal fluid. 98 M. Azreg-Aïnou / Physics Letters B 730 (2014) 95–98

4.1. Examples of static and rotating conformal imperfect fluids custom to issues pertaining to regular cores [18,27,28].These violations worsen in the rotating case as was concluded in [18]. Consider a static regular black hole or a wormhole of the form (2) where G = F are finite at r = 0 and H(r) = r2 + q2.In References the (t, u,θ,φ) coordinates, where u is the new radial coordinate, G(u) = G(r(u)), F (u) = G(u)/r (u)2 and H(u) = r(u)2 + q2.Since [1] W. Van Stockum, Proc. R. Soc. Edinb. 57 (1937) 135. we want K (u) = u2 + p2 [Eq. (8)], we have to solve the differential [2] R.P. Kerr, Phys. Rev. Lett. 11 (1963) 237. =[ 2 + 2] 2 + 2 [3] E.T. Newman, A.I. Janis, J. Math. Phys. 6 (1965) 915. equation: dr/du r(u) q /(u p ), yielding [4] B. Carter, Commun. Math. Phys. 10 (1968) 280. [5] F.J. Ernst, Phys. Rev. 167 (1968) 1175. r(u) = q tan (q/p) arctan(u/p) (21) [6] M. Gürses, F. Gürsey, J. Math. Phys. 16 (1975) 2385. [7] W.B. Bonnor, J. Phys. A, Math. 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