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# a1014 - (1 point 3 Calculate T αβ A T S αβ Due April 19...

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ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 14 (April 8, 2010) (51) An electron and a positron, each with mass equal to 0 . 511 MeV , annihilate at rest into two photons. Chose the rest frame of the positronium, such that one of the photons moves in positive x = x 1 direction. 1. For each photon ±nd the momentum 4-vector. 2. Find the 4-momentum in a frame that moves at a velocity ˆ x with respect to the rest frame of the positronium. 3. Suppose that the annihilation took place 10 9 years ago in a galaxy that is receding from us at β = 4 / 5. What is the energy of the photon that we observe? Due April 21 before class (10 points). (52) Let T αβ be a rank two tensor. 1. Write down T αβ S , the symmetric part of T αβ . Due April 19 in class (1 point). 2. Write down T αβ A , the antisymmetric part of T αβ . Due April 19 in class
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Unformatted text preview: (1 point). 3. Calculate T αβ A T S αβ . Due April 19 in class (1 point). 4. Derive the continuity equation from the antisymmetry of F αβ . Due April 19 in class (1 point). 5. Show ∂ α * F αβ = 0, where * F αβ = 1 2 ² αβγδ F γδ is the dual tensor. Here ² αβγδ is the completely antisymmetric Levi-Cevita tensors: +1 for ( α, β, γ, δ ) and even permutation of (0 , 1 , 2 , 3),-1 for ( α, β, γ, δ ) and even permutation of (0 , 1 , 2 , 3). Due April 19 in class (2 points). (53) Under a gauge transformation the vector potential transform according to A α → A α = A α + ∂ α Λ . Calculate the corresponding transformation of the electromagnetic ±eld tensor F αβ → F αβ . Due April 21 before class (4 points)....
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