   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 first one is at x3 = 0. The second one is parallel to the first and is located at x3 = a> 0. Sketch the branes. Consider open strings with σ ∈ [0,σ1] that stretch from the second brane (σ = 0) to the first brane (σ = σ1). State the boundary conditions (Free or Dirichlet, with value) for the string coordinates Xµ(t, σ )∗(list the eight conditions – µ =0, 1, 2, 3 and σ =0,σ1).∗ Problem 2. (10 points) Spaces constructed by identifications. Give a (simple!) fundamental domain F and describe the resulting space M for each of the following (single) identifications 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 A0F12 + A1F20 + A2F01 . Recall that the field strength Fµν = ∂µAν − ∂ν Aµ. Find the equation of motion resulting from the variation of the gauge field component A0 (as usual, ignore boundary terms). The equation of motion can be written fully in terms of field 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 δ =8πGT0 . (1) This formula is given in units where c and � are set equal to one, the angle δ is measured in radians, and G 6.7 × 10−11 m3/kg s2 is Newton’s constant (c =3 × 108m/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 δ =0.5 arc-seconds (degree = 60 arc-minutes, arc-minute = 60 arc-seconds). What is the linear mass density of such string in kg/m? 1Problem 5. (20 points) Angular momentum of a rotating open string. An open string of length  and energy E rotates rigidly with angular velocity ω. Recall that ω 2  = c and  = 2 E .π T0 (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 ω, 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. Useful integral:  01 √x12−dxx2 = π 4 . Problem 6. (25 points) Momentum of closed strings. For a free closed string we have 1  X(t, σ)= F(u)+ G(v) , with u = ct + σ, v = ct − σ. (1)2 (a) Demonstrate that the periodicity condition σ ∼ σ + σ1 (σ1 = E/T0) relates the lack of periodicity of F(u) to the lack of periodicity of G(v). (b) We now write F(u)= f(u)+  and g(v)+  αu, G(v)=  βv, where fand g are strictly periodic functions with period σ1 and α and βare constant vectors. How does the result in (a) relate α and β? Plug back in (1) to find X(t, σ) in terms of f(u),g(v),α, ct, and possibly σ. (c) The momentum density (per unit σ) carried of the string is Pτ = Tc20 ∂∂tX. Calculate the total momentum p carried by the string in terms of the vector α 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 justification but, in fact, no calculation.