hw6 - Physics 570 Homework No 6 due Monday 8 March 2010 1 A...

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Unformatted text preview: Physics 570 Homework No. 6 due Monday, 8 March, 2010 1. A Killing vector is a generator for symmetries of the metric; more precisely, if one “drags” the metric along the curves to which that vector is tangent one finds no change in the metric. Mathematically we describe this process as saying that a Killing vector, e ξ , is one that satisfies the following equations ξ μ ; ν + ξ ν ; μ = 0 , where of course the dependence on the metric is concealed within the (metric-compatible) con- nection used to determine the covariant derivatives. For static metrics, i.e., those with no dependence on time in the metric components, then ∂ t is clearly such a vector. For the spherically-symmetric metric we have been studying, with the gravitational potential, Φ = Φ( r ), please show that both ∂ ϕ and the following vector field are Killing vectors: e ξ ≡ cos ϕ∂ θ- cot θ sin ϕ∂ ϕ . Then compute their vector commutator and show that it, also, is a Killing vector. [Hint: Note that ∂ ϕ = δ μ ϕ e μ , where { e μ } 4 1 is a (coordinate) basis for tangent vectors. 2. Let F ∼ denote the electromagnetic 2-form, often referred to as the Faraday, as defined in the first portion of Eq. (6.15) of the handout on Connections. Using the definitions of the Hodge dual as given earlier in Section 6, please give a quite-detailed derivation of the form given in the latter half of that same equation for its Hodge dual,...
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This note was uploaded on 03/08/2010 for the course PHYSICS AN 570 taught by Professor Davids.king during the Spring '10 term at Caltech.

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hw6 - Physics 570 Homework No 6 due Monday 8 March 2010 1 A...

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