An approximate global solution of Einstein's equations for differentially rotating compact body
aa r X i v : . [ g r- q c ] J un An approximate global solution of Einstein’sequations for a differentially rotating compactbody
A. Molina , and E. Ruiz Dep. de F´ısica Qu`antica i Astrof´ısica Institut de Ci`encies del Cosmos (ICCUB)Universitat de Barcelona , Mart´ı Franqu`es 1, 08028 Barcelona, Spain Instituto Universitario de F´ısica Fundamental y Matem´aticas , Universidad de Salamanca , Plaza de la Merced s/n, 37008 Salamanca, Spain .October 21, 2018
Abstract
We obtain an approximate global stationary and axisymmetricsolution of Einstein’s equations which can be thought of as a sim-ple star model: a self-gravitating perfect fluid ball with a differen-tial rotation motion pattern. Using the post-Minkowskian formalism(weak-field approximation) and considering rotation as a perturba-tion (slow-rotation approximation), we find approximate interior andexterior (asymptotically flat) solutions to this problem in harmoniccoordinates. Interior and exterior solutions are matched, in the sensedescribed by Lichnerowicz, on the surface of zero pressure, to obtaina global solution. The resulting metric depends on four arbitrary con-stants: mass density; rotational velocity at r = 0; a parameter thataccounts for the change in rotational velocity through the star; andthe star radius in the non-rotation limit. The mass, angular momen-tum, quadrupole moment and other constants of the exterior metricare determined in terms of these four parameters.PACS number(s) 04.40.Nr, 04.20.Jb Introduction
One of the regrettable facts about General Relativity is that, up to now,despite the numerous exact solutions and modern methods to generate them,there has been no exact solution describing a rotating stellar model, i.e.,a space-time corresponding to an isolated self-gravitating rotating fluid inequilibrium, other than the rotating disc of dust described by Neugebauer andMeinel in [1] and its generalization for counter-rotating discs [2]. Althoughthese infinitesimally thin disc solutions are useful models for galaxies andaccretion discs, they are far removed from a description of spheroidal sources,which are the most common astrophysical objects.Stellar models are built by matching an interior space-time describingthe source and the exterior space-time that encloses it. A candidate interiorsolution should correspond to a stationary axisymmetric perfect fluid withoutextra symmetries and admit a zero pressure surface.To our knowledge, for a long time the only candidates have been theWahlquist metric [3] and the generalization of this solution [4], both fora rigidly rotating perfect fluid with the equation of state: µ + 3 p = Ct. and the family of differentially rotating solutions with the equation of state: µ = p + Ct. which have good properties [5, 6]; for instance, they verify theenergy conditions, zero pressure surface, finite body and regular symmetryaxis; but they have a small problem: there are two singular points at thenorth and south poles. This could be avoided, and the work of Haggag [7]moved in this direction, but again, that solution has a singularity: a Diracdelta in the equatorial plane.Numerical relativity predicts stationary toroidal sources[8] that can be ob-tained by starting from a spheroidal topology for a sufficiently strong degreeof differential rotation; however, in the case of rigid rotation, these cannotbe attained[9].Added to the difficulty in finding suitable interiors, there are those aris-ing from the matching to the asymptotically flat exterior. For stellar models,it is an overdetermined problem[10], so in general we cannot find an exte-rior that matches a given interior. This seems to be the case for Wahlquist,where the derivations of the impossibility of matching it with an asymptoti-cally flat exterior come from analysis of the shape of its surface and involveapproximations.Within the field of approximations, one has to choose between accuracyand closeness to the real physical problem that numerical methods provide,and the density of information and greater flexibility for theoretical workthat analytic perturbation theory offers.We would like to build an approximate solution of the Einstein equations2hich describes the gravitational field inside a ball of perfect fluid differen-tially rotating, and to match it, on the zero pressure surface, to an asymp-totically flat approximate solution of the vacuum Einstein equations. Theknown exact solutions for this problem are either not physical [3, 4] or theyhave some point of singularity [5, 6, 7].Although the metrics analysed fulfil the Einstein equations for differen-tially rotating perfect fluids, none of them fulfils the set of requirements tobecome a physically relevant solution. The search for exact solutions for sta-tionary axisymmetric gravitational fields coupled with differentially rotatingperfect fluids remains open.In some previous papers [11, 12, 13, 14], we studied this problem for rigidrotation; and now, we will use the same approximation scheme to study adifferentially rotating perfect fluid.The scheme we proposed consists of a slow rotation approximation on apost-Minkowskian algorithm. We introduce two dimensionless parameters.One, λ , measures the strength of the gravitational field, the other, Ω whichwas a constant, measures the deformation of the matching surface due tofluid rotation.If there is no rotation (Ω = 0), we are faced with the post-Minkowskianperturbation to the Newtonian gravitational field of a spherically symmetricmass distribution. Meanwhile, Newtonian deformation of the source due torotation is included in first-order λ terms, up to some order in the rotationparameter. In this paper, with differential rotation, the parameter used isthe rotation around the symmetry axis: Ω .In the Newtonian formulation, a barotropic equation of state for the fluidimplies that differential rotation can only depend on the cylindrical coordi-nate ρ as a consequence of the Poincar´e-Wavre theorem.In the barotropic case, the integrability conditions of the relativistic Eulerequation imply that Φ = Ψ u ϕ must be a function of Ω, φ (Ω). The choice ofthis function will determine the rotational model.A linear law has been used for some numerical results [16, 17]. We willmake this easy choice for our analytical model.In this paper, we keep terms of order less than or equal to Ω and λ / ;that is, we have gone beyond a simple linear analysis but not so far as tocompute strong non-linear effects. The solution we are looking for is a stationary, axisymmetric and asymptoti-cally flat space-time that admits a global system of spherical-like coordinates3 t, r, θ, ϕ } .Our coordinates are adapted to the space-time symmetry, ξ = ∂ t and ζ = ∂ ϕ , which are respectively the time-like and space-like Killing vectors;so that the metric components do not depend on coordinates t and ϕ , andthe coordinates { r, θ } parametrize two-dimensional surfaces orthogonal tothe orbits of the symmetry group. Then we have: g = γ tt ω t ⊗ ω t + γ tϕ ( ω t ⊗ ω ϕ + ω ϕ ⊗ ω t ) + γ ϕϕ ω ϕ ⊗ ω ϕ + γ rr ω r ⊗ ω r + γ rθ ( ω r ⊗ ω θ + ω θ ⊗ ω r ) + γ θθ ω θ ⊗ ω θ , (1)where ω t = dt , ω r = dr , ω θ = r dθ , ω ϕ = r sin θ dϕ is the Euclidean or-thonormal co-basis associated with these coordinates.Furthermore, coordinates { t, x = r sin θ cos ϕ, y = r sin θ sin ϕ, z = cos θ } associated with the spherical-like coordinates are harmonic and the metricin these coordinates tends to the Minkowski metric in standard Cartesiancoordinates for large values of the coordinate r .We assume that the source of the gravitational field is a perfect fluid, T = ( µ + p ) u ⊗ u + p g (2)whose density and pressure p are functions of the r and θ coordinates. More-over, we assume the fluid has no convective motion, so its velocity u lies onthe plane spanned by the two Killing vectors, u = ψ ( ξ + ω ζ ) , (3)where ψ ≡ h − (cid:16) γ tt + 2 ω γ tϕ r sin θ + ω γ ϕϕ r sin θ (cid:17)i − (4)is a normalization factor, i.e., u α u α = − ∂ a p = ( µ + p ) ( ∂ a ln ψ − Φ ∂ a ω ) ( a, b, . . . = r , θ ) . (5)where Φ ≡ ψu ϕ = − γ tϕ r sin θ + ωγ ϕϕ r sin θγ tt + 2 ωγ tϕ r sin θ + ω γ ϕϕ r sin θ (6)and ω is a function of r and θ .If we consider a barotropic fluid, µ ( p ) then the integrability conditions for(5) are satisfied if and only if Φ is a function of ω only, we call this function φ ( ω ) to distinguish it from the function defined in (6) which depends on themetric functions. 4herefore the solution of equations (5) is implicitly defined by the equa-tion Z p dp ′ µ ( p ′ ) + p ′ = ln ψ − χ ≡ ln ζ (7)where χ ( ω ) ≡ Z ω φ ( ω ′ ) dω ′ and ζ ≡ ψe − χ (8)Since p must be a function of ζ , it will play the same role as ψ played in therigid rotation problem, i.e., it will determine the surfaces p = constant. Forinstance, the p = 0 surface can implicitly be defined as: p = 0 ⇐⇒ ζ = ζ Σ , (9)where ζ Σ is an arbitrary constant.Equation (7) and (9) play an important role in our scheme. We use themto derive approximate expressions for the pressure and the matching surfacein a coherent way with the expansion for the metric we propose below.Given an equation of state (EoS), we can integrate the left-hand side of(7) and even obtain explicit expressions for the pressure and density. Forinstance, a linear equation of state, µ + (1 − n ) p = µ , gives: p = µ n ζζ Σ ! n − ! and µ = µ n ( n − ζζ Σ ! n + 1 ! (10)whereas a polytropic EoS, p = aµ /n , leads to: p = 1 a n ζζ Σ ! n +1 − n +1 and µ = 1 a n ζζ Σ ! n +1 − n (11)As we said above, the function Φ can be written from (6) in terms of themetric components or, taking into account the integrability condition, as afunction of only ω and φ ( ω ); by identifying both expressions we arrive at anequation which relates ω to the metric functions. Solving that equation for ω we will provide us with ω , as a function of the metric and of the coordinates r and θ .To continue, we need to make some assumptions concerning the function φ ( ω ). The simplest hypothesis is to choose a linear function: φ = ω − ωα (12)This choice was already made [16],[17] within a numerical approach to dif-ferentially rotating polytropes[18]. 5ow we can build up the function χ : χ = Z φdω = − ( ω − ω ) α where we have chosen the constant of integration in such a way that ζ = ψ at ω = ω , thereby recovering previous results in rigid rotation. As in our previous work [12, 13, 14, 15] on the rigid rotation problem, herewe introduce a post-Minkowskian parameter, λ , and a dimensionless rotationparameter, Ω = λ − / ωr , where r is the radius of the source in the non-rotation limit. Then we can rewrite (12) as:Φ = λ / Ω − Ω r α . (13)Moreover, we assume the following expansion of the metric components, (see[12]) (we will not use labels to distinguish between exterior or interior metricswhenever it can be clearly understood to which of them we are referring). γ tt ≈ − λf tt , γ tϕ ≈ λ / Ω f tϕ , γ ϕϕ ≈ λf ϕϕ ,γ rr ≈ λf rr , γ rθ ≈ λf rθ , γ θθ ≈ λf θθ . (14)We can determine Ω by solving the equation φ ( ω ) = Φ, the expression (6)up to order λ reads: Φ ≈ λ / Ω r sin( θ ) r + O ( λ / ) (15)which means that, to the lowest order in λ , Ω will only depend on the cylin-drical coordinate ρ = r sin( θ ), then from (13) and (15) we obtain:Ω ≈ Ω αρ + λF where function F , written in terms of the interior metric coefficients (14), is: F ≈ − α Ω r ρ
11 + αρ f tϕ + ρr f tt + f ϕϕ αρ + ρ r Ω (1 + αρ ) ! (16)Since we would like to follow the approximation scheme used in [12], weneed to assume that the constant α must be proportional to Ω because it6ultiplies a second-order Legendre polynomial. Furthermore, as we want allour constants to be dimensionless, we redefine α as: α → Ω r α the expression for the angular velocity to the lowest order in λ and up to Ω is: Ω ≈ Ω (cid:16) − α Ω sin θη + α Ω sin θη (cid:17) , (17)substituting this approximate expression for Ω, we obtain the following ap-proximate expressions for ψ and χ , and omitting terms higher than Ω , weobtain: χ ≈ − λα Ω ρ r and e − χ ≈ λα Ω ρ r (18)and: ψ ≈ λ f tt + Ω ρ r − α Ω ρ r ! (19)and finally, ζ : ζ ≈ λ f tt + Ω ρ r − α Ω ρ r ! ≡ λ (cid:18) f tt +13 Ω η (1 − P ) − α
105 Ω η (7 − P + 3 P ) (cid:19) (20)where η ≡ r/r , and P l are the Legendre polynomials of cos( θ ). This approx-imate function is used to determine approximate expressions for the pressure,density and matching surface up to the order λ and Ω , and consequentlythe energy–momentum tensor up to order λ / in our expansion parameterand Ω . As in our previous papers [12, 13, 14, 15], here we use the post-Minkowskianapproximation scheme. g αβ = η αβ + λh αβ (21)In those references, the resulting equations and notation are explained .7 .1 Linear exterior solution The inhomogeneous part of the linear exterior equations is zero, i.e.: t αβ = 0 , N αβ = H α = 0and the equations to solve are: △ h αβ = 0 ,∂ k ( h kµ − h η kµ ) = 0 . (22)We are going to assume equatorial symmetry and the same dependence ofthe metric on the expansion parameters as in our previous work [12, 13]( M n ∝ Ω n , J n ∝ Ω n ). So, the exterior metric up to order λ and Ω can bewritten in terms of the spherical harmonic tensors as: h ≈ λ X l =0 , , Ω l M l η l +1 ( T l + D l ) + 2 λ / X l =1 , , Ω l J l η l +1 Z l + λ X l =0 , , Ω l A l η l +3 E l +2 + X l =2 , Ω l B l η l +1 F l , (23)where T n ≡ P n (cos θ ) ω t ⊗ ω t ( n ≥ , D n ≡ P n (cos θ ) δ ij dx i ⊗ dx j ( n ≥ , Z n ≡ P n (cos θ ) ( ω t ⊗ ω ϕ + ω ϕ ⊗ ω t ) ( n ≥ , (24)are spherical harmonic tensors and: E n ≡ n ( n − H n + ( n − H n − H n ( n ≥ , F n ≡ n (2 n − D n − n ( n + 1) H n −
12 ( H n + H n ) ( n ≥
1) (25)are two suitable combinations of D n , and these other three spherical har-monic tensors: H n ≡ P n (cos θ ) ( δ ij − e i e j ) dx i ⊗ dx j ( n ≥ , H n ≡ P n (cos θ ) ( k i e j + k j e i ) dx i ⊗ dx j ( n ≥ , H n ≡ P n (cos θ ) ( k i k j − m i m j ) dx i ⊗ dx j ( n ≥
2) (26) The definitions and notation used in this paper are the same as in [13], but not thosein the former work [12] i , e i and m i stand for Euclidean unit vectors of standard cylindrical coor-dinates, dρ = k i dx i , dz = e i dx i , ρ dϕ = m i dx i = ω ϕ (this is the set ofspherical harmonic tensors we use to write covariant tensors of rank-2 in thispaper); M n and J n are the multi-pole moments in Thorne [19] or, exceptfor a constant, those in Geroch-Hansen [20, 21], A n and B n are other con-stants which, unlike M n and J n , are not intrinsic, (gauge constants) but theyare needed to solve the Lichnerowicz matching problem. E has sphericalsymmetry (therefore it must be included in the spherical symmetric linearsolution in addition to the mass monopole term). E ij ≡ H ij + H ij − H ij = δ ij − n i n j , (27)where n i is the Euclidean unit radial vector of standard spherical coordinates.Let us remark that in this approximation, order M , J and A can bepolynomials of second order in Ω ; M , J , A and B can be linear functionsof Ω ; and M , J and B are pure numbers.To obtain the exterior metric, we must add the Minkowski part to theexterior solution for h (23), i.e.: g ext ≈ − T + D + h . (28) We will now find the interior solution for a fluid linear EoS. To this end, weneed an approximate expression of the energy–momentum tensor of the fluid.First of all, let us notice that the density µ = 3 λ/ (4 πr ) is a quantity oforder λ . Thus, taking into account (8) and (10), it is easy to check that thepressure is of order λ ; so, to this order of approximation in λ , the densityconstant EoS and the linear one give the same energy momentum tensor.Therefore, the energy–momentum tensor (2) contributes to the right-handside of the Einstein equations by means of:8 π t ≈ λr ( T + D ) + 6 λ / Ω r η Z , (29)if the terms of order equal to or higher than λ are disregarded. Now we needto substitute Ω for its approximate expression in term of Ω at the lowestorder in λ and up to order Ω given in (17); and, also as we did in (20),we substitute sin θ and sin θ for their expressions in terms of the Legendrepolynomials P l (cos θ ); and finally we split the term containing the tensor Z into the corresponding spherical harmonic tensors Z , Z , Z .9et us consider the following system of linear differential equations thatcorresponds to the linear post–Minkowskian approximation: △ h αβ = − πt αβ ,∂ k ( h kµ − h η kµ ) = 0 , (30)where t is given by (29). A particular solution for the inhomogeneous partwhich is regular at the origin of the coordinate system r = 0 is: h inh = − λη ( T + D ) − λ / (cid:20) η Z + α Ω η (cid:18) Z − Z (cid:19) +43 α Ω η (cid:18) − Z + 111 Z − Z (cid:19)(cid:21) (31)Now, for the homogeneous part up to order λ and Ω , the regular solutionat the origin with equatorial symmetry is: h hom ≈ λ X l =0 , , m l Ω l η l ( T l + D l ) + X l =0 , , a l Ω l η l E ∗ l + λ X l =0 , b l +2 Ω l +20 η l F ∗ l + λ / X n =1 , , j l Ω l η l Z l (32)We have also introduced two new sets of spherical harmonic tensors: E ∗ ≡ D , F ∗ ≡ H E ∗ l ≡ l + 16 ((4 l + 6) D l − l H l ) −
12 ( H l + H l ) ( l ≥ , F ∗ l ≡
12 ( l + 1)( l + 2) H l − ( l + 2) H l − H l ( l ≥ , (33)which seem to be well suited to the interior problem, indeed better than thosewe used to write the linear exterior metric.As before, in this approximation order, m , j , a and b are pure numbers; m , j , a and b are linear functions of Ω ; and m , j and a are quadraticfunctions of Ω .Finally, adding this homogeneous part to the inhomogeneous part, andto the Minkowski part, we obtain an approximate expression for the interiormetric up to the order λ / and Ω : g int ≈ − T + D + h hom + h inh . (34)10 .3 Matching surface and energy–momentum tensor If we assume that the metric components are continuous on the matchingsurface, then we can use their exterior expressions given by (14) to make (9)into a true equation for this surface. So we can search for a parametric formof the matching surface up to zero order in λ and up to order Ω by makingthe following assumption: r ≈ r (cid:16) σ Ω P (cos θ ) + σ Ω P (cos θ ) (cid:17) . (35)where σ and σ must be determined from the equation ζ = constant . Notethat as for the previous constants in this approximation level, σ is a constantbut σ could be a linear function of Ω . The function f tt can be read from(23), i.e.: f tt = 2 X n =0 , , Ω n M n η n +1 P n (cos θ )A simple but lengthy calculation leads to: σ = 1 M (cid:18) M − (cid:19) + 4Ω M (9 M − M + 6 αM −
2) (36) σ = 6 + 35 M M − M − M − αM M (37)We can also obtain a similar expression for ζ Σ (9) in terms of the exteriorconstants: ζ Σ ≈ λ M + 13 Ω − Ω M − M + 4 αM M ! . (38) λ Let us recall the matching conditions we are using in this paper: the metriccomponents and their first derivatives have to be continuous through thehyper-surface of zero pressure (matching surface). Imposing these conditionson metrics (28) and (34) on the surface given by (35) and bearing in mind theorder of approximation we are concerned with, a straightforward calculationleads to the following values for the multipole moments: M = 1 + 512 Ω , M = −
12 + 17 Ω (cid:18) α − (cid:19) , M = 17 (cid:18) − α (cid:19) ,J = 25 + Ω (cid:18) − α (cid:19) + Ω − α − α ! , = −
17 + 4 α
315 + Ω − α
735 + 16 α ! ,J = 542 − α − α A n and B n are zero. For the interior metric constantswe obtain: m = 3 + 512 Ω , m = − (cid:18) α − (cid:19) , m = − α j = 2 + Ω (cid:18) − α (cid:19) + Ω − α − α ! , j = − α α ) ,j = −
27 + 4 α
35 + Ω − α
735 + 8 α ! (40)and also all the gauge constants a n and b n are zero up to this order. The aboveexpressions include the rigid rotation case, which is arrived at by making α = 0. Furthermore, the resulting expressions enlarge the results given in[12],[13] since they include corrections in Ω at the first order in λ . In this paper we have extended our analytical approach to stationary ax-isymmetric global solutions for a barotropic rigidly rotating perfect fluid to adifferentially rotating perfect fluid. The integrability condition of the Eulerequation for barotropic fluids demands that the function Φ ≡ Ψ u ϕ must bea function of the differential rotation ω .We would like to comment on the function φ ( ω ). Following proposals ofother authors, [16],[17], we have chosen a linear function and it seems to bewell suited to our approximation scheme if a simple assumption concerningthe dependence of the constant α on Ω is made. Nevertheless, our schemecan also be compatible with a more general choice of this function.Let us consider the following choice: φ ( ω ) = ω − ωα (cid:16) β ( ω − ω ) + · · · (cid:17) (41)In our scheme, even terms in ( ω − ω ) must be omitted since under a changein the sign of the rotation γ tϕ → − γ tϕ , u ϕ → − u ϕ , ψ → ψ → − Φ.Finally, introducing Ω and Ω in (41): φ (Ω) = λ / r α (Ω − Ω) β λr (Ω − Ω) + O ( λ ) ! (42)Therefore, corrections to the linear function can be ignored unless we wantto evaluate Ω to the first order in λ ; that is, they change the function F in(16) but do not modify the metric up to order λ . However, this choice of φ ( ω ) is compatible with our approximation scheme. Nevertheless, it will leadto a slightly different second-order metric, also including a new constant, β .The choice of this function as a lineal function permits us to integrate theEinstein equations to each order, obtaining a series in two parameters: thepost-Minkowskian parameter, λ , and the rotation on the symmetry axis, Ω .With this choice, a new parameter appears: α . When this parameter iszero, we obtain our previous results for the rigid rotation problem.In this paper, we have obtained the global metric keeping terms of orderless than or equal to Ω and λ / ; that is, we have gone beyond a simple linearanalysis but no so far as to compute strong non-linear effects. However, sincethe algorithm is implemented by an algebraic computational programme,our results can easily be enhanced, if so desired, by going farther in theapproximation scheme.In the Newtonian formulation, a barotropic fluid implies that differentialrotation can only depend on the cylindrical coordinate ρ , as a consequenceof the Poincar´e-Wavre theorem. One interesting result is that we obtain thisconclusion in the post-Minkowskian perturbation at the first order, i.e.:Ω ≈ f ( ρ ) + λf ( ρ, z ) Acknowledgements
AM gratefully acknowledge the
Universidad de Salamanca for the warm hos-pitality which facilitated this collaboration. We also want to thank our friendJ.M.M. Senovilla for comments that have allowed us to improve the paper.Financial support to the authors for this work was provided by the Spanishpeople via the government award FIS2015-65140-P (MINECO/FEDER).13 eferences [1] Neugebauer, G. and Meinel, R. “General relativistic gravitational fieldof a rigidly rotating disk of dust: Solution in terms of ultraelliptic func-tions”,
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