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# hw2 - Homework 2 Physics 776 Spring 2005 Due Thursday Feb...

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Homework 2, Physics 776, Spring 2005 Due Thursday Feb. 24, at the beginning of class. 1. If the metric components are independent of a particular coordinate x ˆ α in a particular coordinate system, then the corresponding vector ﬁeld ξ a = ( ˆ α ) a is called a Killing vector ﬁeld . Such a coordinate system is said to be “adapted” to ξ . Show that the coordinate invariant condition for a vector ﬁeld ξ a to be a Killing ﬁeld is ( a ξ b ) = 0 , (1) where the parentheses represent index symmetrization, and ξ b = g bc ξ c as usual.( Hint : Use the fact that a g bc = 0, and evaluate (1) in a coordinate system adapted to ξ a .) 2. The Kerr metric in Boyer-Lindquist coordinates is ds 2 = (1 - 2 Mr Σ ) dt 2 + 4 Mra sin 2 θ Σ dtdφ - Σ Δ dr 2 - Σ 2 - A sin 2 θ Σ 2 (2) with Σ = r 2 + a 2 cos 2 θ (3) Δ = r 2 - 2 Mr + a 2 (4) A = ( r 2 + a 2 ) 2 - a 2 Δ sin 2 θ. (5) (a) The angular velocity Ω H of the horizon is deﬁned by the condition that the Killing vector

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hw2 - Homework 2 Physics 776 Spring 2005 Due Thursday Feb...

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