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A Derivation of the Kerr Metric by Ellipsoid Coordinate Transformation

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A Derivation of the Kerr Metric by Ellipsoid Coordinate Transformation Vol. 12(11), pp. 130-136, 22 June, 2017 DOI: 10.5897/IJPS2017.4605 Article Number: AEE776964987 International Journal of Physical ISSN 1992 - 1950 Copyright ©2017 Sciences Author(s) retain the copyright of this article http://www.academicjournals.org/IJPS Full Length Research Paper A derivation of the Kerr metric by ellipsoid coordinate transformation Yu-ching Chou M. D. Health101 Clinic, Taipei, Taiwan. Received 3 February, 2017; Accepted 20 June, 2017 Einstein's general relativistic field equation is a nonlinear partial differential equation that lacks an easy way to obtain exact solutions. The most famous of which are Schwarzschild and Kerr's black hole solutions. Kerr metric has astrophysical meaning because most cosmic celestial bodies rotate. Kerr metric is even harder than Schwarzschild metric to be derived directly due to off-diagonal term of metric tensor. In this paper, a derivation of Kerr metric was obtained by ellipsoid coordinate transformation which causes elimination of large amount of tedious derivation. This derivation is not only physically enlightening, but also further deducing some characteristics of the rotating black hole. Key words: General relativity, Schwarzschild metric, Kerr metric, ellipsoid coordinate transformation, exact solutions. INTRODUCTION The theory of general relativity proposed by Albert Nearly 50 years later, the fixed axis symmetric rotating Einstein in 1915 was one of the greatest advances in black hole was solved in 1963 by Kerr (Kerr, 1963). Some modern physics. It describes the distribution of matter to of these exact solutions have been used to explain topics determine the space-time curvature, and the curvature related to the gravity in cosmology, such as Mercury's determines how the matter moves. Einstein's field precession of the perihelion, gravitational lens, black equation is very simple and elegant, but based on the hole, expansion of the universe and gravitational waves. fact that the Einstein' field equation is a set of nonlinear Today, many solving methods of Einstein field equations differential equations, it has proved difficult to find the are proposed. For example: Pensose-Newman's method exact analytic solution. The exact solution has physical (Penrose and Rindler, 1984) or Bäcklund transformations meanings only in some simplified assumptions, the most (Kramer et al., 1981). Despite their great success in famous of which is Schwarzschild and Kerr's black hole dealing with the Einstein equation, these methods are solution, and Friedman's solution to cosmology. One year technically complex and expert-oriented. after Einstein published his equation, the spherical The Kerr solution is important in astrophysics because symmetry, static vacuum solution with center singularity most cosmic celestial bodies are rotating and are rarely was found by Schwarzschild (Schwarzschild, 1916). completely at rest. Traditionally, the general method of E-mail: [email protected]. Tel: (+886) 0933174117. Author(s) agree that this article remain permanently open access under the terms of the Creative Commons Attribution License 4.0 International License Chou 131 the Kerr solution can be found here, The Mathematical Schwarzschild metric, one can obtain the components: Theory of Black Holes, by the classical works of Chandrasekhar (1983). However, the calculation is so 2푀 2푀 −1 lengthy and complicated that the College or Institute 푔 = 1 − , 푔 = − (1 − ) , 00 푟 11 푟 students also find it difficult to understand. Recent 푔 = −푟2 , 푔 = −푟2 푠푖푛2 휃 literature review showed that it is possible to obtain Kerr 22 33 metric through the oblate spheroidal coordinate’s transformation (Enderlein, 1997). This encourages me to Which can also be presented as: look for a more concise way to solve the vacuum solution −1 of Einstein's field equation. 2푀 2푀 푔 = 1 − , 푔 = − (1 − ) , The motivation of this derivation simply came from the 푡푡 푟 푟푟 푟 2 2 2 use of relatively simple way of solving Schwarzschild 푔휃휃 = −푟 , 푔휙휙 = −푟 푠푖푛 휃 (5) metric to derive the Kerr metric, which can make more students to be interested in physics for the general Differences of metric tensor 푔휇휈 between the relativity of the exact solution to self-deduction. Schwarzschild metric in Equation 4 and Minkowski In this paper, a more enlightening way to find this space-time in Equation 2 are only in time-time terms (푔 ) solution was introduced, not only simple and elegant, but 푡푡 and radial-radial terms (푔 ). also further deriving some of the physical characteristics 푟푟 Kerr metric is the second exact solution of Einstein field of the rotating black hole. equation, which can be used to describe space-time geometry in the vacuum area near a rotational, axial- symmetric heavenly body (Kerr, 1963). It is a generalized SCHWARZSCHILD AND KERR SOLUTIONS form of Schwarzschild metric. Kerr metric in Boyer- Lindquist coordinate system can be expressed in The exact solution of Einstein field equation is usually Equation 6: expressed in metric. For example, Minkowski space-time is a four-dimension coordinates combining three- 2 2 dimensional Euclidean space and one-dimension time 2푀푟 4푀푟푎 sin 휃 휌 푑푠2 = (1 − ) 푑푡2 + 푑푡푑휙 − 푑푟2 can be expressed in Cartesian form as shown in 휌2 휌2 ∆ 2푀푟푎2 sin2 휃 Equation 1. In all physical quality, we adopt c = G = 1. −휌2푑휃2 − (푟2 + 푎2 + ) sin2 휃푑휙2 (6) 휌2 푑푠2 = 푑푡2 − 푑푥2 − 푑푦2 − 푑푧2 (1) Where 휌2 ≡ 푟2 + 푎2 푐표푠2 휃 and ∆ ≡ 푟2 − 2푀푟 + 푎2 and in polar coordinate form in Equation 2: M is the mass of the rotational material, 푎 is the spin 푑푠2 = 푑푡2 − 푑푟2 − 푟2푑휃2 − 푟2 sin2 휃푑휙2 (2) parameter or specific angular momentum and is related to the angular momentum 퐽 by 푎 = 퐽 /푀. Schwarzschild employed a non-rotational sphere- symmetric object with polar coordinate in Equation 2 with By examining the components of metric tensor 푔 in two variables from functions ν(r), λ(r), which is shown in 휇휈 Equation 3: Equation 6, one can obtain: 2 2 2ν(r) 2 2휆(푟) 2 2 2 2 2 2 2푀푟 휌 푑푠 = 푒 푑푡 − 푒 푑푟 − 푟 푑휃 − 푟 푠푖푛 휃푑휙 (3) 2 푔00 = 1 − 2 , 푔11 = − , 푔22 = −휌 , 휌 ∆ 2 In order to solve the Einstein field equation, 2푀푟푎 푠푖푛 휃 푔03 = 푔30 = 2 Schwarzschild used a vacuum condition, let 푅휇휈 = 0, to 휌 2 2 calculate Ricci tensor from Equation 3, and got the first 2 2 2푀푟푎 푠푛 휃 2 푔33 = −(푟 + 푎 + ) 푠푖푛 휃 (7) exact solution of the Einstein field equation, 휌2 Schwarzschild metric, which is shown in Equation 4 (Schwarzschild, 1916): Comparison of the components of Schwarzschild metric Equation 4 with Kerr metric Equation 6: −1 2 2푀 2 2푀 2 2 푑푠 = (1 − ) 푑푡 − (1 − ) 푑푟 − 푑훺 푟 푟 Both 푔03(푔푡휙) 푎푛푑 푔30(푔휙푡) off-diagonal terms in Kerr 푑훺2 = 푟2푑휃2 + 푟2 푠푖푛2 휃푑휙2 (4) metric are not present in Schwarzschild metric, apparently due to rotation. If the rotation parameter However, Schwarzschild metric cannot be used to 푎 = 0, these two terms vanish. describe rotation, axial-symmetry and charged heavenly 푔00푔11 = 푔푡푡푔푟푟 = −1 in Schwarzschild metric, but not bodies. From the examination of the metric tensor 푔휇휈 in in Kerr metric when spin parameter 푎 = 0, Kerr metric 132 Int. J. Phys. Sci. 2 2 2 2 2 turns into Schwarzschild metric and therefore is a 푑푠 = 퐺′00푑푇 + 퐺′11푑푟 + 퐺′22푑휃 + 퐺′33푑∅ (11) generalized form of Schwarzschild metric. To eliminate the off-diagonal term: TRANSFORMATION OF ELLIPSOID SYMMETRIC 푑푇 ≡ 푑푡 − 푝푑휑, 푑∅ ≡ 푑휑 − 푞푑푡 (12) ORTHOGONAL COORDINATE to obtain To derive Kerr metric, if we start from the initial assumptions, we must introduce 푔00, 푔11, 푔22, 푔03, 푔33 퐺′00퐺′11 = −1 (13) (five variables) all are function of (r, θ), and finally we will get monster-like complex equations. Apparently, due to By comparing the coefficient, Equations 14 to 18 were the off-diagonal term, Kerr metric cannot be solved by the obtained. spherical symmetry method used in Schwarzschild 2 metric. ′ −2푀푟푎 sin 휃 퐺′00푝 + 퐺 33푞 = 2 (14) Different from the derivation methods used in classical 휌 works of Chandrasekhar (1983), the author used the 2푀푟 퐺′ + 퐺′ 푞2 = 1 − (15) changes in coordinate of Kerr metric into ellipsoid 00 33 휌2 symmetry firstly to get a simplified form, and then used 2푀푟푎2 sin2 휃 Schwarzschild's method to solve Kerr metric. First of all, 퐺′ 푝2 + 퐺′ = − (푟2 + 푎2 + ) sin2 휃 (16) the following ellipsoid coordinate changes were apply to 00 33 휌2 Equation 1 (Landau and Lifshitz, 1987): 퐺′ = −휌2 (17) 22 1 푥 → (푟2 + 푎2)2 푠푖푛 휃 푐표푠 휙, 휌2 퐺′ = − (18) 1 11 ∆ 2 2 2 푦 → (푟 + 푎 ) 푠푖푛 휃 푠푖푛 휙, 푧 → 푟 푐표푠 휃, By solving six variables 퐺′00, 퐺′11, 퐺′22, 퐺′33, 푝, 푞 in the six 푡 → 푡 (8) dependent Equations 13 to 18, the results shown in Equation (19) were obtained: Where a is the coordinate transformation parameter. The metric under the new coordinate system becomes 푝 = ±푎 sin2 휃 , take the positve result Equation 9: 푎 푞 = ± , take the positive result 휌2 푟2 + 푎2 푑푠2 = 푑푡2 − 푑푟2 − 휌2푑휃2 − (푟2 + 푎2) sin2 휃푑휙2 (9) 푟2+푎2 ∆ 휌2 퐺′ = , 퐺′ = − , 퐺′ = −휌2 Equation 9 has physics significance, which represents 00 휌2 11 ∆ 22 (푟2+푎2)2 sin2 휃 the coordinate with ellipsoid symmetry in vacuum, it can 퐺′ = − (19) also be obtained by assigning mass M = 0 to the Kerr 33 휌2 metric in Equation 6. Due to the fact that most of the celestial bodies, stars and galaxy for instance, are Put them into Equation 8 and obtain Equation 20: ellipsoid symmetric, Bijan started from this vacuum ∆ 휌2 ellipsoid coordinate and derived Schwarzschild-like 푑푠2 = (푑푡 − 푎 sin2 휃 푑휑)2 − 푑푟2 − 휌2푑휃2 solution for ellipsoidal celestial objects following Equation 휌2 ∆ 10 (Bijan, 2011): (푟2+푎2)2 sin2 휃 푎 2 − 2 (푑휑 − 2 2 푑푡) (20) 휌 푟 +푎 2푀 2푀 −1 휌2 푑푠2 = (1 − ) 푑푡2 − (1 − ) 푑푟2 Equation 20 can be found in the literature and also 푟 푟 푟2 + 푎2 2 2 2 2 2 2 textbook by O’Neil.
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