� � 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 1 X(t, σ)= F(u)+ G(v) , with u = ct + σ, v = ct − σ. (1)2 Here σ ∼ σ + σ1,where σ1 = E/T0 and E is the energy of the string. We showed that |F�(u)|2 = |G�(v)|2 =1 , (2) F�(u + σ1)= F�(u) and G�(v + σ1)= G�(v) . (3) Equations (2) and (3) imply that F�(u) and G�(v) can be described as two independent closed parameterized paths on the surface of a unit two-sphere. We assumed that the paths intersect at u = u0 and v = v0 F�(u0)= G�(v0) . (4) The quantities u0 and v0 define a time t0 and position σ0. We showed that at t = t0,the point σ = σ0 on the string moves with the speed of light in the direction of F�(u0). (a) We choose a coordinate system so that the cusp generated by (4) appears at the origin: F(u0)+ G(v0)= 0. Use the Taylor expansions of F(u) and G(v) around u0 and v0 to prove that for σ near σ0, X(t0,σ)= T(σ − σ0)2 + R(σ − σ0)3 + ... , (5) where the vectors Tand Rare given by 1� � 1 � � T= F��(u0)+ G��(v0) , R= F���(u0) − G���(v0) . (6)4 12 Assume that the intersection of the paths on the two-sphere indicated in equation (4) is regular: the paths are not parallel at the intersection and neither F��(u0) nor G��(v0) vanishes. Explain why Tis non-zero and orthogonal to F�(u0). In general Rdoes not 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 Tand is contained in the plane spanned by Tand R. For this, align the positive y axis along T, position the x axis so that Rlies on the (x, y) plane, and demonstrate that near the cusp y ∼ x2/3 . In what plane does the velocity of the cusp lie? Problem 1(d) was revised. 1 1� � � � (c) Consider the functions F(u) and G(v) given by F(u)= σ1 sin 2πu , − cos 2πu, 0 , G(v)= σ1 sin 4πv , 0 , − cos 4πv . (7)2π σ1 σ1 4π σ1 σ1 Verify that the conditions in (2) and (3) are satisfied. For the cusp at t = σ =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 σ1/(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 + σ) takes to repeat itself. In fact, any free closed string, when viewed in its rest frame, will return to its original position in time σ1/(2c),where σ1 is the period of the functions F� and G�.) 2. (5 points) Gravitational lensing by a cosmic string A cosmic string produces a conical deficit 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 δφ.Calculate δφ 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.