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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 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 solutions. Kerr metric has astrophysical meaning because most cosmic celestial bodies rotate. Kerr metric is even harder than to be derived directly due to off-diagonal term of . 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 .

Key words: , 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 in cosmology, such as 's determines how the matter moves. Einstein's field precession of the perihelion, , 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 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

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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 can be expressed in The exact solution of Einstein field equation is usually Equation 6: expressed in metric. For example, -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 of the rotational material, 푎 is the spin 푑푠2 = 푑푡2 − 푑푟2 − 푟2푑휃2 − 푟2 sin2 휃푑휙2 (2) parameter or specific angular and is related to the 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) 푔 = 1 − , 푔 = − , 푔 = −휌2 , 00 휌2 11 ∆ 22 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 푠𝑖푛 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: 2푀 2푀 −1 푑푠2 = (1 − ) 푑푡2 − (1 − ) 푑푟2 − 푑훺2 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): 2 퐺′22 = −휌 (17) 1 푥 → (푟2 + 푎2)2 푠푖푛 휃 푐표푠 휙, 휌2 1 퐺′11 = − (18) 2 2 ∆ 푦 → (푟 + 푎 )2 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, 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 ∆

(푟2+푎2)2 sin2 휃 푎 2 10 (Bijan, 2011): − (푑휑 − 푑푡) (20) 휌2 푟2+푎2

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. It is also called Kerr metric with Boyer- −휌 푑휃 − (푟 + 푎 ) sin 휃푑휑 (10) Lindquist in orthonormal frame (O'Neil, 1995). There is no off-diagonal terms, and 푔00푔11 = −1 after the coordinate Equation 10 morphs into the Schwarzschild’s solution in transformation. Equation 4 when the coordinate transformation parameter a = 0 and therefore Equation 10 is also a generalization of Schwarzschild’s solution. CALCULATING THE RICCI TENSOR In order to eliminate the difference between Kerr metric and Schwarzschild metric described earlier, we can From pervious discussion, Equation 9 can be recognized assume to rewrite the Kerr metric in the following as the coordinate under the ellipsoid symmetry in coordinates: vacuum. Therefore, when the mass M approached 0, Chou 133

Kerr metric Equation 20 will also be transformed into 3 3 2푟 푟 훤13 = 훤31 = − 2 (32) Equation 21, which equals Equation 9. The differences of ℎ 휌 metric tensor components are in time-time and radial- 1 -2휆 radial terms, just the same as between Schwarzschild 훤22 = −푟푒 (33) metric (Equation 4) and Minkowski space-time (Equation 3 3 ℎ 2). 푑푇 and 푑∅ defined in Equation 22 are ellipsoid 훤32 = 훤23 = 푐표푡 휃 ( 2) (34) coordinate transformation functions. 휌

2 2 2 2 2 2 2 2 1 -2휆 2 2ℎ ℎ 2 푟 +푎 2 휌 2 2 2 (푟 +푎 ) sin 휃 2 훤 = −푟푒 푠푖푛 휃 ( − ) (35) 푑푠 = 푑푇 − 푑푟 − 휌 푑휃 − 푑∅ (21) 33 휌2 휌4 휌2 푟2+푎2 휌2

2 2 2 ℎ 푠𝑖푛 휃 푐표푠 휃 푑푇 ≡ 푑푡 − 푎 sin 휃 푑휑 훤22 = − (36) 푎 휌2 푑∅ ≡ 푑휑 − 푑푡 (22) 푟2+푎2 ℎ3 훤2 = − 푠푖푛 휃 푐표푠 휃 ( ) (37) In this paper, Schwarzschild method was used to solve 33 휌6 Kerr metric starting from Equations 21 to 22 by introducing two new functions 푒2ν(푟,휃), 푒2휆(푟,휃): The calculation of tensor can be derived by the following (Equation 38), and the results are listed 푑푠2 = in Equations 39 to 50: (푟2+푎2)2 sin2 휃 푒2ν(푟,휃)푑푇2 − 푒2휆(푟,휃)푑푟2 − 휌2푑휃2 − 푑∅2 (23) 휌2 휌 휌 휌 휌 휆 휌 휆 푅훼훽 = 푅 = 휕휌훤 − 휕훽훤 + 훤 훤 − 훤 훤휌훼 (38) 훼휌훽 훽훼 휌훼 휌휆 훽훼 훽휆 2 Define the parameters 휌 and 푕 in Equation (24): 0 0 0 0 휆 0 휆 푅101 = 휕0훤11 − 휕1훤01 + 훤0휆훤11 − 훤1휆훤01 2 2 휌2 ≡ 푟2 + 푎2 푐표푠2 휃 = 휕1휈 휕1휆 − (휕1휈) − 휕1 휈 (39) 푕 ≡ 푟2 + 푎2 (24) 0 0 0 0 휆 0 휆 -2휆 푅202 = 휕0훤22 − 휕2훤02 + 훤0휆훤22 − 훤2휆훤02 = −푟푒 휕1휈 (40) Metric tensor in the matrix form shown in Equations 25 to 0 0 0 0 휆 0 휆 26: 푅303 = 휕0훤33 − 휕3훤03 + 훤0휆훤33 − 훤3휆훤03 2 -2휆 2 2ℎ ℎ = −푟푒 푠푖푛 휃 ( − ) 휕1휈 (41) 푒2ν(푟,휃) 0 0 0 휌2 휌4 2휆(푟,휃) 0 −푒 0 0 1 1 1 1 휆 1 휆 2 푅 = 휕 훤 − 휕 훤 + 훤 훤 − 훤 훤 푔휇휈 = 0 0 −휌 0 (25) 212 1 22 2 12 1휆 22 2휆 12 푟2 ℎ2푠𝑖푛2 휃 = 푒-2휆 (푟휕 휆 − 1 + ) (42) 1 휌2 0 0 0 − 2 ( 휌 )

푅1 = 휕 훤1 − 휕 훤1 + 훤1 훤휆 − 훤1 훤휆 푒−2ν(푟,휃) 0 0 0 313 1 33 3 13 1휆 13 3휆 13 2ℎ ℎ2 −2휆(푟,휃) = 푟푒-2휆푠푖푛2휃 ( − ) (휕 휆) (43) 0 −푒 0 0 휌2 휌4 1 휇휈 −2 푔 = 0 0 −휌 0 (26) 휌2 2 2 2 2 휆 2 휆 푅323 = 휕2훤33 − 휕3훤23 + 훤 훤33 − 훤 훤23 0 0 0 − 2 2 2휆 3휆 ( ℎ 푠𝑖푛 휃) ℎ4 5푟2−4휌2 푟2ℎ ℎ = 푠푖푛2휃 * ( ) − (2 − ) 푒-2휆+ (44) 휌8 ℎ 휌4 휌2 Chrostoffel symbols can be obtained by the following steps in Equation 27: 푅1 = 푔11푔 푅0 = 푒2(ν-휆)[−휕 휈 휕 휆 + (휕 휈)2 + 휕2휈 ] (45) 010 00 101 1 1 1 1 훼 1 훼훽 훤휇휈 = 푔 (휕휇푔휈훽 + 휕휈푔훽휇 − 휕훽푔휇휈) (27) 2 22 0 2(ν-휆) 푟 2 푅020 = 푔 푔00푅202 = 푒 휕1휈 (46) 휌2 Non-zero Chrostoffel symbols are listed in Equation 28 to 2푟 푟 37: 푅3 = 푔33푔 푅0 = 푒2(ν-휆) ( − ) 휕 휈 (47) 030 00 303 ℎ 휌2 1

훤1 = 푒2(ν-휆)휕 휈 (28) 1 푟2 00 1 푅2 = 푔22푔 푅1 = (푟휕 휆 + − 1) (48) 121 11 212 휌2 1 휌2 1 훤 = 휕1휆 (29) 11 2 2 3 33 1 2 1 2푟 2푟 푅131 = 푔 푔11푅313 = ( − 2) (푟휕1휆 + 2 − ) (49) 훤0 = 훤0 = 휕 휈 (30) ℎ 휌 휌 ℎ 10 01 1

ℎ2 5푟2−4휌2 푟2 ℎ 2 2 푟 3 33 2 -2휆 훤 = 훤 = (31) 푅232 = 푔 푔22푅323 = * 4 ( ) − (2 − 2) 푒 + (50) 12 21 휌2 휌 ℎ ℎ 휌 134 Int. J. Phys. Sci.

2 2휈 2푀 푟 −2푀푟 푙푖푚푒 = 1 − = 2 (62) 푅휇휈 can be calculated by the Equations 51 to 54: 푎→0 푟 푟

1 2 3 One could also demand the other limit condition of flat 푅00 = 푅010 + 푅020 + 푅030 2푟 space-time, where the mass approaches zero (M → 0) in = 푒2(ν-휆) *−휕 휈 휕 휆 + (휕 휈)2 + 휕2휈 + 휕 휈+ (51) 1 1 1 1 ℎ 1 the Equation 18, which could be represented as in Equation 63: 0 2 3 푅11 = 푅101 + 푅121 + 푅131 2푟 2 2 2 2 2 2 2휈 푟 +푎 푟 +푎 = 휕1휈 휕1휆 − (휕1휈) − 휕1 휈 + 휕1휆 (52) 푙푖푚푒 = = (63) ℎ 휌2 푟2+푎2 푐표푠2 휃 푀→0 0 1 3 푅22 = 푅202 + 푅212 + 푅232 Deduced from the above conditions in Equations 62 to 2푟2 2푟2 ℎ2 5푟2−4휌2 = 푒-2휆 (푟(휕 휆 − 휕 휈) − 1 + − ) + ( ) (53) 63, the equations of Ricci tensor could be solved as in 1 1 휌2 ℎ 휌4 ℎ Equation 64: 0 1 2 푅33 = 푅303 + 푅313 + 푅323 2 2 −1 2ℎ ℎ2 푟2 ℎ2 5푟2−4휌2 2ℎ ℎ2 2ν 푟 −2푀푟+푎 = 푠푖푛2휃 ( − ) [푒-2휆 (푟(휕 휆 − 휕 휈) − ) + ( ) ( − ) ](54) 푒 = 휌2 휌4 1 1 휌2 휌4 ℎ 휌2 휌4 푟2+푎2 푐표푠2 휃 푟2+푎2 푐표푠2 휃 푒2휆 = (64) 푟2−2푀푟+푎2 FINDING A SOLUTION OF THE VACCUM EINSTEIN FIELD EQUATIONS Finally, the Kerr metric was gotten as shown in Equation 65:

To solve vacuum Einstein's field equations, first, the Ricci 2 2 2 2 푟 −2푀푟+푎 2 2 휌 2 tensor was set to zero, which means: 푅휇휈 = 0, 푅 = 0, in 푑푠 = (푑푡 − 푎 sin 휃 푑휑) − 푑푟 휌2 푟2−2푀푟+푎2 the empty space, θ is approximately constant. Then (푟2+푎2)2 sin2 휃 푎 2 −휌2푑휃2 − (푑휑 − 푑푡) (65) combine with R00 and R11 to get Equation 55, and solve 휌2 푟2+푎2 the equation, Equations 56 to 58 were obtained:

2푟 푒-2(ν-휆)R + R = (휕 휈 + 휕 휆) = 0 (55) 00 11 ℎ 1 1 DISCUSSION

휕1휈 + 휕1휆 = 휕1(휈 + 휆) = 0 (56) It is proved that Kerr metric equation (Equation 65) can be obtained by combining the ellipsoid coordinate 휈 = −휆 + 푐, 휈(푟, ) = −휆(푟, ) + 푐 (57) transformation and the assumptions listed in Equations 21 to 23 following these steps: transforming the Euclidian four-dimension space-time in Equation 1 to vacuum ν -휆 푒 = 푒 (58) Minkowski space-time with ellipsoid symmetry in Equation 9; transforming from (푡, 푟, 휃, 휙) to (푇, 푟, 휃, ∅) To solve this partial differential equation, one has to under the new coordinate system to eliminate the major remember that when the angular momentum approaches difference in metric tensor components between Kerr zero (a → 0), the Kerr metric Equation 6 turns into metric and Schwarzschild metric, and the product of 푑푡푑휙 Schwarzschild metric (Equation 4). Then Equation 59 to and 푔00푔11 becomes -1; solving vacuum Einstein’s 61 were obtained: equation by using Schwarzschild method from Equation 23; applying limit method to calculate Ricci curvature 2 -2휆 2 푙푖푚푅33 = 푠푖푛 휃[푒 (푟(휕1휆 − 휕1휈) − 1) + 1] = 푠푖푛 휃푅22 (59) tensor; and finally deducting Kerr metric. 푎→0 Table 1 shows the list of the metric tensor components discussed in previous sections, including the Minkowski -2휆 푙푖푚푅22 = 푒 (푟(휕1휆 − 휕1휈) − 1) + 1 푎→0 space-time, the Schwarzschild solution, empty ellipsoid, a Schwarzschild-like ellipsoid solution and the Kerr solution. = 푒2ν(– 2푟 ∂ ν − 1) + 1 = 0 (60) The Minkowski space-time and the Schwarzschild solution 1 have spherical symmetry, and the others have ellipsoid

퐶 symmetry. 푒2ν = 1 + , 푙푒푡 퐶 = −2푀 (61) 푟 Further, some of characteristics with deeper physics meaning of ellipsoid symmetry, Kerr metric and rotating So under the limit condition of angular momentum black hole can be obtained from this new coordinate approaching zero (푎 → 0), the equations could be solved function d푇, 푑∅. Remember when a approaches to zero as shown in Equation 62: (푎 → 0), d푇, 푑∅ degenerate to d푡, 푑휙. Chou 135

Table 1. Metric tensor components and symmetry.

Metric tensor ( ) ( ) Symmetry and state Minkowski 1 -1 −푟2 −푟2푠푖푛2휃 Spherical, empty 2 2 푟 −2푀 푟 2 2 2 Schwarzschild − −푟 −푟 푠푖푛 휃 Spherical, static, mass 푟2 푟2−2푀 2 2 2 2 2 2 2 푟 +푎 휌 2 (푟 +푎 ) 푠𝑖푛 휃 Ellipsoid − −휌 − Ellipsoid, empty 휌2 푟2+푎2 휌2 2 푟2 휌2 Schwarzschild-like 푟 −2푀 − −휌2 −(푟2 + 푎2)푠푖푛2휃 Ellipsoid, static, mass 푟2 푟2−2푀 푟2+푎2 2 2 2 2 2 2 2 푟 −2푀+푎 휌 2 (푟 +푎 ) 푠𝑖푛 휃 Kerr 2 − −휌 − 2 Ellipsoid, axisymmetric, mass 휌 푟2 − 2푀 + 푎2 휌

Ellipsoid symmetry and Kerr metric Equation 67. As the mass approaches zero 푀 → 0, Equation 68 degenerates into Equation 67. While metric with spherical symmetry in vacuum has the following expression: In order to describe frame-dragging, Kerr metric can be

2 2 2 2 2 re-written as Equation 69: −푟 푑휃 − 푟 sin 휃푑휙 (66) 2 2 2 2 2 푑푠 = 푔푡푡푑푡 + 2푔푡휙푑푡푑휑 + 푔푟푟푑푟 + 푔휃휃푑휃 + 푔휙휙푑휑 And metric of ellipsoid symmetric in vacuum has the 2 푎 𝑔2 𝑔 following expression in Equation 67, where and = (푔 − 푡휙 ) 푑푡2 + 푔 푑푟2 + 푔 푑휃2 + 푔 (푑휑 + 푡휙 ) (69) 푟2+푎2 푡푡 𝑔 푟푟 휃휃 휙휙 𝑔 2 휙휙 휙휙 푎 푠푖푛 휃 term can be seen in multiply and divide combination: The definition of angular momentum (Ω) in frame-

dragging: −휌2푑휃2 − (푟2 + 푎2) sin2 휃푑휙2

2 2 +풂 2 −휌 푑휃 − ( ) (풂퐬퐢퐧 ) 푑휙 (67) 2푀푟푎 sin2 휃 풂 𝑔 2 푡휙 휌 Ω = − = 2푀푟푎2 sin2 휃 2 2 𝑔휙휙 (푟2+푎2+ ) sin2 휃 Terms of 푑휃 , 푑휙 in Kerr metric is shown in Equation 68, 휌2 푎 2푀푟푎 2푀푟 where and 푎 푠푖푛2 휃 term can also be seen in linear (70) 푟2+푎2 = 2 2 2 2 2 = +풂 휌 (푟 +푎 )+2푀푟푎 sin 휃 휌2( )+2푀푟(풂 퐬퐢퐧 ) combination: 풂

2푀푟푎2 푠𝑖푛2 휃 풂 −휌2푑휃2 − (푟2 + 푎2 + ) 푠푖푛2 휃푑휙2 So, we see both the 푎 sin2 휃 and term in Ω, which 휌2 +풂 +풂 2푀푟풂 𝒔풊풏 means d푇, 푑∅ would have some relation with frame- −휌2푑휃2 − ( + )(풂𝒔풊풏 ) 푑휙2 (68) 풂 휌2 dragging angular momentum.

The 푎 in Equation 67 represents a parameter in the ellipsoid symmetric coordinate transformation, however, a Black hole angular in Kerr metric Equation (Equation 68) represents a spin parameter, which is proportional to angular momentum. Its close relationship with the black hole angular velocity Both 푎’푠 are equivalent in mathematic perspective and (Ω퐻) can be easily identified by examining 푑∅ term in used to transform the space-time into ellipsoid symmetry Equation 71. with a rotational symmetric z-axis. As 푎 → 0, both Equation (67) and Equation (68) degenerate into spherical 푎 푑∅ = 푑휙 − 푑푡 symmetry equation (Equation 66). 푟2 + 푎2 푎 Ω퐻 = 2 2 푟+ + 푎 Frame-dragging angular momentum 2 2 푓푟표푚 Δ = 0, 푠표푙푣푒 푟± = 푀 ± √푀 − 푎 (71)

In physics, a spinning heavenly body with a non-zero Base on this derivation, in the future, we will further study mass will generate a frame-dragging phenomenon along whether the method mentioned in this paper can be the equator’s direction, which has been proven by Gravity extended to other more general cases. For example, Probe B experiment (Everitt et al., 2011). Therefore, an suppose we start with three functions 2푀푟푎2 sin2 휃 extra term in Kerr metric is found in Equation 2ν(푟,휃) −2ν(푟,휃) 2휆(푟,휃) 2휇(푟,휃) 휌2 푒 , 푒 , 푒 , 푒 as shown in Equation 68 as compared to the vacuum ellipsoid symmetry in (72): 136 Int. J. Phys. Sci.

푑푠2 = 푒2ν(푟,휃)푑푇2 − 푒−2ν(푟,휃)푑푟2 − 푒2휆(푟,휃)푑휃2 − 푒2휇(푟,휃)푑∅2 (72) ACKNOWLEDGMENTS

Besides, as 푑푇, 푑∅ is shown to be related to ellipsoid The author thanks Ruby Lin, Dr. Simon Lin of Academia symmetry, frame-dragging angular momentum, and black Sinica, Prof. Anthozy Zee, and a wonderful NTU Open hole angular velocity, which are all rotation parameters, it Course Ware taught by Prof. Yeong-Chuan Kao, and also deserves further study if this method could be extended indebted to Shanrong, for editing the English. to solve the other axial-symmetry exact solutions of vacuum Einstein's field equation. REFERENCES

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