MATH 423 January 2011 Examiner: Prof. A.E. Faraggi, Extension 43774. Time allowed: Two and a half hours Full marks can be obtained for complete answers to FIVE questions. Only the best FIVE answers will be counted. You may use a university approved pocket calculator and the constants: c = 3 10 10cm s ; h = 1.054 10 27 erg s; 8 cm3 G N = 6.674 10 m g s 2 e = 9.109 10 28 g; m p = 1.672 10 24 g; k = 1.380 10 16erg K ; m Planck = 2.17 10 5 g; m sun 2 10 33 g; Paper Code MATH 423 January 2011 Page 1 of 6 CONTINUED
1. Consider the infinitesimal line element, ds 2 = g µν dx µ dx ν = dt 2 dx 2. (a) Write the metric g µν and its inverse in an explicit in matrix form. (b) Find the set of independent transformations of the form [2 marks] t t + ǫa(t, x) x x + ǫb(t, x), where ǫ is an infinitesimal constant and the functions A and B have to be determined by the requirement that ds 2 is invariant. State what each transformation represents in space time. [18 marks] 2. The electromagnetic field strength tensor is given by: 0 E x E y E z E F µν = x 0 B z B y E y B z 0 B x E z B y B x 0 where E and B are the electric and magnetic fields respectively. We define the tensor T λµν by T λµν = λ F µν + µ F νλ + ν F λµ (a) Derive the source free Maxwell equations. Additionally the electromagnetic current four vector is given by j µ = (cρ, j 1, j 2, j 3 ) (b) Derive the Maxwell equations with sources. Paper Code MATH 423 January 2011 Page 2 of 6 CONTINUED
3. (a) The vacuum energy associated with the current acceleration of the universe is ρ vac = 7.7 10 27 kg/m 3. Derive a fundamental length scale, l vac associated with this vacuum energy in terms of ρ vac, the Planck constant h, and the speed of light c. Express the numerical value for l vac in µm where 1µm = 10 6 m. (b) The Standard Bohr radius is a 0 = h2 5.29 10 9 cm, and arises from me 2 the electric potential V = e2. What would be the gravitational Bohr radius if r the attraction force binding the electron to the proton was gravitational? 4. Consider the action S[x, e] = 1 2 edτ 1 ( ) dx µ 2 m 2, (1) e 2 dτ where τ is an arbitrary parameter and edτ is an invariant line element. (a) Perform variations of x µ and of e to obtain the equations of motion for x µ and e respectively. [6 marks] (b) For m 2 > 0, eliminate e by its equation of motion, and substitute the result back into (1). Show that you obtain the action for a free massive particle. (c) Instead of eliminating e by its equation of motion, you can set it to a constant value by reparameterisation. Derive the equation of motion and the contraint equation for the two cases m 2 > 0 and m 2 = 0. Paper Code MATH 423 January 2011 Page 3 of 6 CONTINUED
5. The Polyakov action is given by: S P = T d 2 σ hh αβ α X µ β X ν η µν, 2 where h = det(h αβ ). (a) For a 2 2 matrix show that ( ) ( ) a11 a A = 12 δa11 δa and δa = 12 a 21 a 22 δa 21 δa 22 δdeta = detatr(a 1 δa). [4 marks] (b) Hence, derive from the Polyakov action the world sheet energy momentum tensor T αβ, which is defined by T αβ = 2 T 1 h δs δh αβ [8 marks] (c) In the conformal gauge h αβ = η αβ, the energy momentum tensor takes the form T αβ = α X µ β X µ 1 2 η αβη γδ γ X µ δ X µ. Show that η αβ T αβ = 0 α T αβ = 0 [8 marks] Paper Code MATH 423 January 2011 Page 4 of 6 CONTINUED
6. The equations of motion for a relativistic string in the conformal gauge are ( 0 2 1)X 2 µ = 0 Solutions must satisfy the constraints 0 X µ 1 X µ = 0, 0 X µ 0 X µ + 1 X µ 1 X µ = 0 We consider closed strings with boundary conditions X µ (σ 0, σ 1 ) = X µ (σ 0, σ 1 + π) (a) Show that equations of motion and constraints are solved by X 0 = 2Rσ 0 X 1 = R cos(2σ 1 ) cos(2σ 0 ) X 2 = R sin(2σ 1 ) cos(2σ 0 ) X i = 0 for i > 2 Describe in words how the closed string moves. (b) Compute the length, the mass and the momentum of the string. (c) Show that the string is at rest at time X 0 = 0. Express the total energy in terms of its length at time X 0 = 0 and interpret the result. [6 marks] Paper Code MATH 423 January 2011 Page 5 of 6 CONTINUED
7. Consider the lightlike compactification, in which we identify events with position and time coordinates related by ( ) ( ) x x ct ct ( ) R + 2π R (2) (a) Rewrite this identification using light cone coordinates. [5 marks] (b) Consider coordinates (ct, x ) related to (ct, x) by a boost with veclocity parameter β. Express the identifications in terms of the primed coordinates. (c) Consider the family of indentifications given by ( ) ( ) x x ct ct + 2π ( R2 + R 2 s R ). (3) Show that there is a boosted frame S in which the identification (3) becomes a standard identification ( i.e. the space coordinate is identified but the time coordinate is not). [8 marks] Paper Code MATH 423 January 2011 Page 6 of 6 END