hw2soln - Homework 2 William D. Linch III March 2, 2005 1....

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Homework 2 William D. Linch III March 2, 2005 1. We work in a coordinate system adapted to ξ and compute α ξ β = ( α ξ γ ) g γβ = ( ξ γ + Γ γ δα ξ δ ) g γβ . (1) Since the coordinates { x α } are adapted to ξ , the components ξ γ = δ γ ˆ α for some coordinate direction ˆ α . In particular, ξ γ 0 and 2 α ξ β = g γ± ( g ±δ,α + g ±α,δ - g δα,± ) g γβ ξ δ = (2 g δ [ β,α ] + g αβ,δ ) ξ δ , (2) It follows that ( α ξ β ) ∂g αβ ∂x ˆ δ ξ ˆ δ . (3) Since g αβ does not depend on x ˆ α this vanishes. Since 0 is a tensor, we find in general that the Killing condition is ( a ξ b ) = 0. 2.a Orthogonality of χ and φ at the horizon, H , gives 0 = χ · φ | H = 2 Mr + a sin 2 θ Σ | - Ω H A | sin 2 θ Σ | Ω H = 2 Mr + a A | . (4) Now, at the horizon Δ | = 0 which implies 2 Mr + / ( r 2 + + a 2 ) = 1 and A | = ( r 2 + + a 2 ) 2 . This gives Ω H = a r 2 + + a 2 = a 2 Mr + . (5) 2.b Let ˜
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hw2soln - Homework 2 William D. Linch III March 2, 2005 1....

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