8 251 Homework 6 Corrected 3 17 071 B Zwiebach Spring 2007 Due Tuesday March 20 1 15 points Three dimensional motion of closed strings and cusps We considered in lecture the closed string motion described by v with u ct v ct t 1 F u G X 2 1 Here 1 where 1 E T0 and E is the energy of the string We showed that u 2 G v 2 1 F u 1 F u F v 1 G v and G 2 3 v can be described as two independent closed Equations 2 and 3 imply that F u and G parameterized paths on the surface of a unit two sphere We assumed that the paths intersect at u u0 and v v0 v0 F u0 G 4 The quantities u0 and v0 de ne a time t0 and position 0 We showed that at t t0 the point u0 0 on the string moves with the speed of light in the direction of F a We choose a coordinate system so that the cusp generated by 4 appears at the origin u0 G v0 0 Use the Taylor expansions of F u and G v around u0 and v0 to prove F that for near 0 t0 T 0 2 R 0 3 X 5 are given by where the vectors T and R 1 v0 T F u0 G 4 1 F u0 G v0 R 12 6 Assume that the intersection of the paths on the two sphere indicated in equation 4 u0 nor G v0 is regular the paths are not parallel at the intersection and neither F does not vanishes Explain why T is non zero and orthogonal to F u0 In general R vanish but it may under special conditions b One can use equation 5 to show that the cusp opens up along the direction of the vector For this align the positive y axis T and is contained in the plane spanned by T and R lies on the x y plane and demonstrate that near along T position the x axis so that R 2 3 the cusp y x In what plane does the velocity of the cusp lie 1 Problem 1 d was revised 1 u and G v given by c Consider the functions F u 1 sin 2 u cos 2 u 0 F 2 1 1 v 1 sin 4 v 0 cos 4 v G 4 1 1 7 Verify that the conditions in 2 and 3 are satis ed For the cusp at t 0 give its direction the plane it lies on and its velocity Draw a sketch d Show that the motion of the closed string has period 1 4c How many cusps are formed during a period Hint recall that you found in Problem 7 3 an example of a situation in which the string returns to its original position in less time than the function F ct takes to repeat itself In fact any free closed string when viewed in its rest frame will return to its original position in time 1 2c where 1 is the period of the functions F and G 2 5 points Gravitational lensing by a cosmic string A cosmic string produces a conical de cit angle An observer is a distance d from the cosmic string and a quasar is a distance from the cosmic string The position of the quasar is such that the observer sees a double image separated by an angle Calculate in terms of d and in the approximation that is small 3 10 points Problem 7 5 4 5 points Problem 8 1 5 10 points Problem 8 3 6 10 points Problem 8 5 I recommend Problem 8 2 as good practice to reinforce concepts 2