HW-571-2 - K 3 from K 00 to K 000 moving with velocity u...

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1 Phys 571 , Fall 2010 Assignment #2 Due: 9/8/2010 1. Express the components of 4-acceleration w μ as a function of the velocity ~v and acceleration ~w . Find w 2 μ in terms of ~w and ~w × ~v . Useful identity: ( ~w × ~v ) 2 = w 2 v 2 - ( ~w · ~v ) 2 . 2. A bundle of light rays forms a solid angle d Ω = sin θ dθ dφ in a certain reference frame. How does this bundle look like in another reference frame? (In other words, find d Ω 0 ). What is the total solid angle R d Ω in each frame? Hint: first derive the transformation laws of azimuthal φ and polar θ angles using the transformation law of velocity ~v . 3. Consider three successive Lorentz transformations: 1) from K to K 0 moving with velocity V = V ˆ x with respect to K ; 2) from K 0 to K 00 moving with velocity v = v ˆ y 0 with respect to
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Unformatted text preview: K ; 3) from K 00 to K 000 moving with velocity u with respect to K 00 . (a) With the help of the formulas derived in the Home Assignment #1 (problem 3) show that the frame K 000 is at rest with respect to K if u equals to the relativistic sum of-v and-V (as expected!). Prove, that u =-V p 1-v 2 /c 2 ˆ x-v ˆ y . Hint: first find the relation t and t 000 . (b) Although K 000 is stationary with respect to K their axes do not coincide, i.e. frame K 000 is rotated with respect to K . This is known as Thomas precession . Calculate the Thomas precession angle φ . (c) Study limits v,V << c and v,V → c ....
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