克爾度規

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的精確解，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，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。

克尔度规的数学表示

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。

参考文献

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