a1012 - (44 In the following K 1 and S 3 are generators as...

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ADVANCED DYNAMICS — PHY 4241/5227 HOME AND CLASS WORK – SET 12 (March 12, 2010) (42) Derive the Taylor expansions of the functions exp( x ), cosh( x ), cos( x ), sinh( x ) and sin( x ) about x = 0 from the equation f ( x ) = X n =0 f ( n ) (0) x n n ! by calculating f ( n ) (0) for each case explicitly. Do NOT use Euler’s formula. Due April 2 before class (5 points). (43) The matrix L is defined by L = l 0 0 l 0 1 l 0 2 l 0 3 l 1 0 l 1 1 l 1 2 l 1 3 l 2 0 l 2 1 l 2 2 l 2 3 l 3 0 l 3 1 l 3 2 l 3 3 1. Calculate - g L (2 points). 2. Write down the transpose matrix ˜ L (2 points). 3. Calculate ˜ L g (2 points). 4. Compare 1. and 2. to find the general form of L ( i.e. use ˜ L g = - g L , 4 points). 5. Obtain the same result by discussing the elements of the equation g αβ ˜ l γ β g γδ = - l α δ (4 extra points. Hint: Do the contractions first.) Due April 7 in class.
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Unformatted text preview: (44) In the following K 1 and S 3 are generators as deFned in class. 1. Calculate exp(-ζ K 1 ) and explain its physical meaning (3 points). 2. Calculate exp(-φ S 3 ) and explain its physical meaning (3 points). Due April 12 before class. (45) Consider the spaceship journey again and plot versus the proper time τ the following quantities (due April 5 before class, 10 points): 1. The time on earth at which the spaceship, seen from earth, is at its position at time τ . 2. The distance from earth as seen from earth. 3. The date of news received from earth, which is transmitted at the speed of light....
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