克爾度規

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廣義相對論中,克爾度規或稱克爾真空Kerr vacuum),描述的一旋轉之質量龐大物體(例如:克爾黑洞)周遭的時空幾何。其為愛因斯坦場方程式精確解,故又稱克爾解。Lense和Thirring曾使用弱场近似方法得到过旋转轴对称球状物体度规的近似解,而其严格解是罗伊·克尔于1963年提出来的[1],但他并没有给出推导过程。1973年Schiffer等人给出了克尔度规的推导[2]

克尔度规的数学表示[编辑]

若以Boyer-Lindquist座標寫出克爾真空解,則為:

\mathrm{d}s^2 = -\left(1-\frac{2Mr}{\Sigma}\right)\mathrm{d}t^2 -\frac{4aMr\sin^2\theta}{\Sigma}\mathrm{d}t\mathrm{d}\phi +\frac{\Sigma}{\Delta}\mathrm{d}r^2
+\ \Sigma \mathrm{d}\theta^2 + \left(\Delta+\frac{2Mr(r^2+a^2)} {\Sigma}\right) \sin^2\theta \mathrm{d}\phi^2

其中

\Sigma=r^2+a^2\cos^2\theta,
\Delta=r^2-2Mr+a^2,
M是旋轉物體質量
a描述此黑洞的旋轉,與角動量J有關,關係式為a = J/M
所有的物理量採用幾何單位(geometrized units),亦即c=G=1。

當自轉參數(spin parameter)a(常稱作特定角動量specific angular momentum))其值為零,則無旋轉,度規退化成史瓦西度規a=M的例子對應到最大旋轉程度的質量物體。

注意到:

  • 一般而言,Boyer/Lindquist徑向座標(radial coordinate)r並「沒有」簡單而直接、如同徑向座標般的詮釋。
  • 「最大」旋轉程度指的是一黑洞可以存在的最大a值,而非旋轉質量物體可以具有的最大a值。


註釋[编辑]

  1. ^ Kerr, R.P., 1963, Physical Review Letters, 11, 237. NASA ADS doi:10.1103/PhysRevLett.11.237
  2. ^ Schiffer, M.M. et al., 1973, J. Math. Phys., 14, 52.

參考文獻[编辑]

  • Stephani, Hans; Kramer, Dietrich; MacCallum, Malcolm; Hoenselaers, Cornelius & Herlt, Eduard. Exact Solutions of Einstein's Field Equations. Cambridge: Cambridge University Press. 2003. ISBN 0-521-46136-7. 
  • O'Neill, Barrett. The Geometry of Kerr Black Holes. Wellesley, MA: A. K. Peters. 1995. ISBN 1-56881-019-9. 
  • D'Inverno, Ray. Introducing Einstein's Relativity. Oxford: Clarendon Press. 1992. ISBN 0-19-859686-3.  See chapter 19 for a readable introduction at the advanced undergraduate level.
  • Chandrasekhar, S. The Mathematical Theory of Black Holes. Oxford: Clarendon Press. 1992. ISBN 0-19-850370-9.  See chapters 6--10 for a very thorough study at the advanced graduate level.
  • Griffiths, J. B. Colliding Plane Waves in General Relativity. Oxford: Oxford University Press. 1991. ISBN 0-19-853209-1.  See chapter 13 for the Chandrasekhar/Ferrari CPW model.
  • Adler, Ronald; Bazin, Maurice & Schiffer, Menahem. Introduction to General Relativity Second Edition. New York: McGraw-Hill. 1975. ISBN 0-07-000423-4.  See chapter 7.
  • Perez, Alejandro; and Moreschi, Osvaldo M.. Characterizing exact solutions from asymptotic physical concepts. arXiv:Dec 2000 gr-qc/001210027 Dec 2000. 2000.  Characterization of three standard families of vacuum solutions as noted above.
  • Sotiriou, Thomas P.; and Apostolatos, Theocharis A. Corrections and Comments on the Multipole Moments of Axisymmetric Electrovacuum Spacetimes. Class. Quant. Grav. 2004, 21: 5727–5733. arXiv eprint Gives the relativistic multipole moments for the Ernst vacuums (plus the electrogmagnetic and gravitational relativistic multipole moments for the charged generalization).
  • Penrose R. In ed C. de Witt and J. Wheeler. Battelle Rencontres. W. A. Benjamin, New York. 1968. 222. 
  • "The Classical Theory of Fields", L.D. Landau, E.M. Lifshitz, Fourth revised English edition, Elsevier, Amsterdam ... London, New York ... Tokyo, 1975 (based on B. Carter, 1968).
  • B. Carter, Phys. Rev. Lett. 26, 331, 1971