International Journal of the Physical Sciences Vol. 6(1), pp. 93-97, 4 January, 2011 Available online at http://www.academicjournals.org/IJPS DOI: 10.5897/IJPS10.594 ISSN 1992 - 1950 ©2011 Academic Journals
Full Length Research Paper
Approximate Kerr–like interior and exterior solutions for a very slowly rotating star of perfect fluid
Bijan Nikouravan
1Department of Physics and Astrophysics, Islamic Azad University (IAU) - Varamin Pishva Branch, Iran, 2Department of Physics, University of Malaya, 50603 Kuala Lumpur, Malaysia. E-mail: [email protected], [email protected]. Tel: (+60) 0123-851513
Accepted 30 December, 2010.
Approximate Kerr- like interior and exterior solutions for a very slowly rotating star of perfect fluid is presented here which is valid for a very slowly rotating and hence very slightly oblate star of perfect fluid rotating with a very small constant angular velocity. Corresponding approximate exterior metric represents approximate Kerr exterior metric for a perfect fluid. The aim here is to obtain Kerr-like approximate exterior and interior solutions for a very slowly rotating star of perfect fluid simply following standard procedure used to obtain Schwarzschild solutions.
Key words: Slowly rotating oblate star, approximate Kerr- like metric, exterior interior solution.
INTRODUCTION
The first solution of Einstein field equation for a single equations Einstein (1916). spherical and non-rotating mass has been derived by 1 Schwarzschild (1916). The next solution for a rotating R i − g i R = −8π G T i (1) star has been derived by Kerr (1965). However the j 2 j j Schwarzschild metric is very nice, but it is describing only for, non-rotating and spherical objects. By Schwarzschild The quantities in the right-hand side of the above metric it is possible to match the Schwarzschild vacuum equations vanish where no matter is present. In this exterior to a Schwarzschild fluid interior and it is more purpose, we start by describing the situation (Landau and general for static spherically symmetric perfect fluid Lifshitz, 1987) for a very slow rotating and hence very solutions. But the problem of finding a rotating fluid slight oblate star in the following spheroid coordinate interior which can be matched to Kerr exterior or to any system. Here we supposed r >> a and c = 1 (velocity of asymptotically flat vacuum exterior solution has proven light). very difficult and it is a major unsolved problem in general 2 2 2/1 2 2 2/1 relativity (Wiltshire, 2003; Lorenzo, 2004). x=(r +a ) SinθCosϕ y=(r +a ) SinθSinϕ z=rCosθ t=t (2) Wahlquist (1968) considered this problem and described a solution with a new metric. Bradley et al. In this surfaces r = constant, are oblate ellipsoids of (2000) contradicted Wahliquist’s metric. Here we are rotation. In these coordinate systems, the rotating flat- trying to derive an approximate Kerr-like interior and space metric assumes the following form (Nikouravan, exterior solution for a very slowly rotating star and slightly 2001). oblate of perfect fluid with a different line element. As we know most of stars are not exactly in spherical r2 + a2Cos 2θ ds 2 = − dr 2 −(r2 + a2Cos 2θ)dθ 2 −(r 2 + a2 )Sin 2θ dϕ2 − (3) form but rather slightly oblate in shape. This essentially r2 + a2 involves obtaining the interior solution to Einstein’s 2ω (r2 + a2 ) Sin 2θ dϕ dt +[]1−ω2 (r 2 +a2 )Sin 2θ dt 2
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Therefore in the presence of matter, the metric (3) takes The Non-zero Christoffel symbols of second kind are given as the form: follows:
2 2 2 −λ −λ 2 2 2 2 1 λ′ ra Sin θ 1 λ& a Sin 2θ 1 re ∆ 1 re ∆Sin θ 2 λ r +a Cos θ 2 2 2 2 2 2 2 2 2 ds =−e dr −(r +a Cos θ)dθ −(r +a )Sin θ dϕ − (4) Γ11 = + 2 , Γ12 = − 2 , Γ22 =− 2 , Γ33 =− 2 r2 +a2 2 ρ ∆ 2 2ρ ρ ρ 2ω (r2 +a2) Sin 2θ dϕdt +eυ []1−ω2(r2 +a2)Sin 2θ dt 2 −λ 2 ′ υ−λ λ 2 λ 2 r 1 rωe ∆Sin θ 1 υ e ∆ 2 λ&e a e Sin θ 2 Γ34 =− 2 , Γ44 =− 2 , Γ11 =− + 2 , Γ12 = 2 , Here supposed λ = λ(r,θ) and υ =υ(r,θ) not only ρ 2ρ 2∆ 2ρ ∆ ρ depends on r but also depends on θ , and is a constant a2 Sin 2θ ∆Sin 2θ ω∆Sin 2θ υeυ r Γ2 =− , Γ2 =− , Γ2 =− , Γ2 = & , Γ2 = , angular velocity. 22 2 33 2 34 2 44 2 11 2ρ 2ρ 2ρ 2ρ ∆ The assumption used here enables us to study the behavior of the object not only in the radial direction but ωr ωυ′ ωυ′ υ′ υ Γ3 =− + , Γ3 =−Cotgθ, Γ3 =−ωCotgθ+ , Γ4 = , Γ2 = & also along the angle θ simultaneously. However, if the 14 ∆ 2 23 24 2 14 2 24 2 basis of this metric is in the spherical coordinate (9) system (r,θ,ϕ) , but basically the metric (4) explains the shape of any stars and/or galaxies in more complete form For the calculation of all Christoffel symbols of first and second kind so that the spherical form is only a part of this coordinate and also the Ricci tensors which are very tedious, it is possible to system. Otherwise, instead of working in terms of use the software Mathematica (Nikouravan, 2009). Here a prime ∂λ (ξ,η,ϕ) with difficulty in calculation we can transform the denotes derivative with respect to r , (λ ′ = ) and the dot ∂r coordinate system in term of (r,θ,ϕ) as an easier way. ∂λ denotes derivative respect to t (λ& = ) . Then Rij 's are given ∂t as follows: APPROXIMATE KERR-LIKE INTERIOR SOLUTION
To arrive at the solution let us start with the metric (4). When we ′′ ′2 ′′ ′ λ 2 υ υ λυ λ e λ& λ&υ& 2 3 4 assume that the star is rotating very slowly and hence ω being R11 = + − − + 2 λ&&+ + +λ&Cotgθ+[smalltermsa ,a or a ] 2 4 4 4 2r 2 2 2 very small, the terms involving ω in the metric (4) may be neglected as compared with the other higher order terms. It can 2 2 −λ −λ (υ′−λ) λ&& υ&& (λ& +υ& ) 2 3 4 2 R22 =−1+e +re + + + +[]smalltermsa ,a or a even be shown that the terms involving ω in Rij ’s are smaller by 2 2 2 4 many orders of magnitude in comparison with other terms at the (υ′−λ) (λ&+υ) surface of the Sun or planets. Therefore, it is justifiable to neglect 2 1 −λ −λ & 2 R33 =Sin θ − +e +re + Cotgθ the terms involving ω compared to other higher order terms. The 2 2 metric (4) therefore assumes the form: υ′′ υ′2 λυ′′ υ′ eυ υ2 λ&υ 2 2 2 (ν−λ) & & 2 λ r + a Cos θ 2 2 2 2 2 2 2 2 2 (5) R =−e + − + − υ+ + +υCotgθ ds = − e dr − (r + a Cos θ )dθ − (r + a )Sin θ dϕ + 44 2 && & r 2 + a 2 2 4 4 r 2r 2 2 2ω (r 2 + a 2 ) Sin 2θ dϕ dt + eυ dt 2 2 −λ −λ (υ′−λ) (λ&+υ&) 2 2 2 2 2 2 R34 =R43 =−ωSin θ −1+e +re + Cotgθ Let us write ρ ≡ r + a Cos θ and ∆ ≡ r + a . Then the 2 2 above metric can be written as, 1 υ′ υ& 2 R12 =R21 =−(λ&+ν&) + + 2 λ ρ 2 2 2 2 2 2 ν 2 (6) 2r 4 2 ds = − e dr − ρ dθ − ∆ Sin θ dϕ − 2ω ∆ Sin θ dϕ dt + e dt ∆ (10)
Therefore, gij are given by, 2 2 3 4 In Rij 's , we can neglect the terms containing ω , a , a , a
2 2 λ as they are very small. Similar way the terms involving & , − e 0 0 0 λ υ& 2 (7) 0 − ρ 0 0 and λ&υ can also be neglected. g ij = 2 2 & 0 0 − ∆ Sin θ − ω ∆ Sin θ For zeroth approximation we may even neglect terms 2 υ 0 0 − ω ∆ Sin θ e involving λ& and υ& . Terms involving λ& and υ& are not zero but are and, very small so that we can approximate λ& ≈ 0 and υ& ≈ 0 . This is rather stringent approximation but to obtain zeroth order solution we make this approximation. Nevertheless, we do consider that the star λ +υ 4 2 (8) g = g ij = −e ρ Sin θ rotates extremely slow and hence have a very small bulge. Under
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i And ρ being the constant density of an oblate sphere of equatorial this approximation scheme, we get R j from Rij and obtained Einstein’s field equations as follows: radius r1 , ε the oblateness and M is the mass of the oblate sphere contained within the equatorial radius . r1 ( M = M ( r1 )) υ ′ 1 1 e − λ + − = 8πρ 3M 2 2 Taking ρ = for a slightly oblate sphere and expressing r r r 3 4π r1 2 − λ υ ′′ υ ′ λ ′υ ′ λ ′ υ ′ V in terms of M and ε at the equator r = r1 , we get the e + − − + = 8πρ 2 4 4 2 r 2 r potential with a negative sign as,
λ ′ 1 1 M ε e − λ − + = 8πρ V = − 1 + ()1 − 3 Cos 2θ (17) r r 2 r 2 r 5 (11) 1
These are same as those in Schwarzschild static interior solution To determine B(θ ) and consequently A(θ ) , we use the well but with a difference that while integrating them the constants of known result, that is, approximation, integration A and B turn out to be the function of θ and not just pure constants as they are in the Schwarzschild static case. This is g 44 = 1 + 2V (18) because, although in stringent approximation λ& ≈ 0 and υ& ≈ 0 , where, is the internal Newtonian potential of a rotating and but λ and υ are functions of θ . The relationship between A(θ ) V slightly oblate sphere at , as given by Equation (15) and get and B(θ ) is, r1 B(θ ) to be, 2 r1 (12) A (θ ) = B (θ ) 1 − 2 2 R r ε 1 − 1 1 + ( 1() − 3 Cos 2θ ) 1 R 2 5 (19) B (θ ) = Where, 2 r 2 1 1 − 2 3 R R 2 = (13) 8πρ 15 ω 2 Note that ε = hence, even for a small ω , ρ may be 16 and r1 is the equatorial radius of the spheroid. One can therefore π ρ arrive at the metric for a very slowly rotating and very slightly oblate 2 such that the ratio ω has small but significant value and hence star of perfect fluid of constant density ρ and rotating with a ρ constant velocity ω to be, may not be neglected and hence in this situation ε will have a small but still significant value. In this situation, (14) along with (12) and (19), represents the gravitational field of a slowly rotating and 2 2 2 2 1 r +a Cos θ 2 2 2 2 2 2 2 2 2 slightly oblate sphere. Corresponding exterior metric is, ds =− dr −(r +a Cos θ)dθ −(r +a )Sin θ dϕ + r2 r2 +a2 (14) 1− 2 R 2 2 2 2 1 r +a Cos θ 2 2 2 2 2 2 2 2 2 2 ds =− dr −(r +a Cos θ) dθ −(r +a )Sin θdϕ + 2M r2 a2 1− + 1 r 2ω (r2 +a2) Sin 2θ dϕdt +A(θ)−B(θ) dt 2 2 2 r r ε 2 1− 1 ( 1 )1 1 3Cos 2 1 1 2 − 2 + ()− θ 2 2 R 1 R 5 r 2 r 2 2 2 2 1 2 2ω (r +a ) Sin θdϕdt + 31− 2 − 1− 2 dt 4 r2 R R 1 1− 2 where relation between A(θ ) and B(θ ) is given by Equation (12). R Now it remains to determine B(θ ) . Ramsey (1961) and MacMillan (20)
(1958) have given the guidelines to find the expression for a Newtonian potential for a rotating oblate sphere to be, RELATION WITH KERR EXTERIOR METRIC
3 2 2 M r ω r ε M 2 1 1 V = + 1 1 − Cos θ − (15) As a << r , the terms involving , 1 3 2 2 2 2 2 r r 2 r1 3 ( r + a ) ( r + a Cos θ ) (r 2 + a 2 Cos 2θ ) (r 2 + a 2 ) where, , , can be expanded in 2 2 2 2 2 (r + a ) (r + a Cos θ ) 16 2 2 πρε (16) a ω = powers of . The Equation (14) for inside the body and with 15 2 r
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a 4 RESULTS AND DISCUSSION neglecting terms and other higher order terms, in the 4 r following form, By comparing approximate metrics (Equations 24 and 2aJ 2 2 22), we get =ω(r + a ) . In the same level of r 1 a2Sin 2θ ds 2 =− 1− dr 2 −(r2 +a2Cos2θ)dθ2 −(r2 +a2)Sin 2θdϕ2 + 2 aJ 2 2 approximation, this condition becomes, . This r r 3 = ω 1− 1 r R2 2 a 2 condition is approximately valid because = ω , r ε 2 3 1 2 1 1 r 1−( 2 )1+ ()1−3Cos θ 1 R 5 2 2 2 2 2 2 2 r1 r 2 where J = aM . The discussion here shows that in 2ω(r +a ) Sin θdϕdt + 31− 2 −1− 2 dt 4 r2 R R 1 approximation, the exterior metric (Equation 22) to the 1− 2 R interior metric (Equation 21) obtained in this paper, really (21) corresponds in approximation to the approximate exterior Kerr metric (Equation 24) of exterior metric (Equation 23). The above metric, for outside of the body assumes in the following Thus we can conclude that the metrics (Equation 14), form, along with Equations (12) and (19), represents Kerr interior metric. 2 2 1 a Sin θ ds 2 =− 1− dr 2 −(r2 +a2Cos 2θ)dθ2 −(r2 +a2)Sin 2θdϕ2 + 2M 2 1− r r CONCLUSION 2 M ε 2 1 2 1− 1+ ()1−3Cos θ 1 2 2 1 r1 5 2M 2M 2 2 2 2 + 31− −1− dt +2ω (r +a ) Sin θdϕdt The aim of this work was to obtain Kerr-like interior and 4 2M r1 r exterior solutions for a slowly rotating star or galaxy with 1− r1 small in angular velocity. This approximate solution has (22) been obtained by the standard procedure which is used in getting the standard Schwarzschild solutions. The Moreover the Kerr exterior metric, in Boyer and Lindquist advantage of the procedure is that the exterior Kerr coordinates system is given by, Landau and Lifshitz (1987), solution has been obtained without going into detail, the procedure adopted in obtaining the actual exterior Kerr 2 2 2 2 solution. The objective of the work has been achieved 2 r +a Cos θ 2 2 2 2 2 2 2 2Mra 2 2 ds =− dr −(r +a Cos θ)dθ −(r +a + )Sin θdϕ + successfully. r2 −2Mr +a2 r2 +a2Cos2θ
4 2 2 Mra 2 Mr 2 2 2 2 Sin θdϕdt +1− 2 2 2 dt r +a Cos θ r +a Cos θ REFERENCES (23) Bradley M, Fodor G, Marklund M, Perjés Z (2000). The Wahlquist metric cannot describe an isolated rotating body. Quantum Grav., 17: 351- In approximation as, a << r , then the term 1 2 2 2 359. (r + a Cos θ ) Einstein A (1916). On the general theory of relativity, Annals of Physics, 2 49: 769–822. can be expanded in powers of a . By neglecting terms 2 Kerr RP (1965). Gravitational Collapse and Space-Time Singularities, r Phys. Rev. Lett., 14(3): 57-59. a 4 Landau LD, Lifshitz EM (1987). The classical Theory of Relativity. containing and higher order terms and similarly Pergamon Press, Volume 2. 4 r Lorenzo I (2004). Some comments on a recently derived approximated solution of the Einstein Equations for a spinning body with negligible approximating the term 1 , we can express the 2 2 mass. Gen. Relat. Gravit., 36(9): 1987-2003. (r − 2 Mr + a ) MacMillan ED (1958). Theoretical mechanics, the theory of the Kerr exterior metric (Equation 23) in the following form, potential, Dover Publications. Nikouravan B (2001). Gravitational field of an Ellipsoidal star in General Relativity and its Various Applications. PhD dissertation, University of 2 2 2 1 a Sin θ 2 2 2 2 2 2 2 2 2 Mumbai, Mumbai, India. ds =− 1− dr −(r +a Cos θ)dθ −(r +a )Sin θdϕ + 2M r2 Nikouravan B (2009). Calculation of Ricci Tensors by Mathematica, 1− V5.1. IJPS, 4(12): 818–823. r Ramsey AS (1961). An introduction to the theory of Newtonian 2 attraction. Cambridge Press, pp. 176-178. 4Ma 2 2M 2 + Sin θ dϕdt +1− dt Schwarzschild K (1916). On the Gravitational Field of a Point-Mass, r r According to Einstein's Theory, (English translation). Abraham (24) Zelmanov J., 1: 10-19. Wahlquist HD (1968). Interior Solution for a Finite Rotating Body of
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