8 251 Test B Zwiebach Wednesday March 16 2005 Only personal 2 page notes allowed Test duration 60 minutes Problem 1 10 points Boundary conditions for open strings Consider two static D2 branes in four dimensional spacetime ct x1 x2 x3 The rst one is at x3 0 The second one is parallel to the rst and is located at x3 a 0 Sketch the branes Consider open strings with 0 1 that stretch from the second brane 0 to the rst brane 1 State the boundary conditions Free or Dirichlet with value for the string coordinates X t list the eight conditions 0 1 2 3 and 0 1 Problem 2 10 points Spaces constructed by identi cations Give a simple fundamental domain F and describe the resulting space M for each of the following single identi cations acting on the complex plane z x iy a z z i b z 2z Problem 3 15 points Variation of an action Consider the Chern Simons action for three dimensional electromagnetism S dt d2 x A0 F12 A1 F20 A2 F01 Recall that the eld strength F A A Find the equation of motion resulting from the variation of the gauge eld component A0 as usual ignore boundary terms The equation of motion can be written fully in terms of eld strengths Problem 4 10 points How heavy is a cosmic string A nearby relativistic cosmic string of tension T0 produces a cylindrical gravitational lens in which two images of a single faraway source would be separate by an angle 8 GT0 1 This formula is given in units where c and are set equal to one the angle is measured in radians and G 6 7 10 11 m3 kg s2 is Newton s constant c 3 108 m s 1 06 10 34 kg m2 s a Complete 1 by adding whatever factors of c and or are needed b A string produces the plausible value of 0 5 arc seconds degree 60 arc minutes arcminute 60 arc seconds What is the linear mass density of such string in kg m 1 Problem 5 20 points Angular momentum of a rotating open string An open string of length and energy E rotates rigidly with angular velocity Recall that 2 c and 2 TE0 a Introduce a radial length r along the string and let dr denote a small piece of string a distance r from the center What is the magnitude dp of the relativistic momentum carried by this small piece of string What is the magnitude dJ of the angular momentum carried by this small piece of string Both answers should be in terms of T0 r dr and constants b Use integration to calculate the total angular momentum carried by the rotating string Give your answer in terms of the energy E of the string and the string tension T0 1 2 Useful integral 0 x1 dxx2 4 Problem 6 25 points Momentum of closed strings For a free closed string we have t 1 F u G v X 2 with u ct v ct 1 a Demonstrate that the periodicity condition 1 1 E T0 relates the lack of v periodicity of F u to the lack of periodicity of G b We now write F u f u u v and G v g v are constant vectors where f and g are strictly periodic functions with period 1 and and How does the result in a relate and t in terms of f u g v Plug back in 1 to nd X ct and possibly T20 X Calculate the total c The momentum density per unit carried of the string is P c t momentum p carried by the string in terms of the vector and other constants Problem 7 10 points You learned that a closed string stretched along a circle and released with zero initial velocity will contract to zero size at some later time Consider a closed string that is stretched along an ellipse and is released with zero initial velocity Will it contract to zero size If yes why if not why not A complete answer requires a precise justi cation but in fact no calculation 2