Homework 1

Grade weighting

10%

Due date

Deadline: Wednesday 6th February 2008 before 2:15pm.

Lateness penalty: 3% per day up to a total of two weeks.

Drop-dead deadline: 2:15pm on Wednesday 20th February. Assignments handed in more than two weeks after the due date, without a relevant valid medical certificate or equivalent gold-standard excuse, will not be marked.

PROBLEMS

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1. Zwiebach 3.9 and 3.10: gravitational field of point mass in one extra dimension [Relatively easy]
(3.9a) Find the gravitational potential of a point mass located at the origin in five noncompact dimensions.

(3.9b) Now put the fifth dimension on a circle of radius a and find the gravitational potential as an infinite series. (Hint: think method of images)

(3.9c) Show that, for distances much bigger than the compact dimension’s radius, r>>a, the four-dimensional form of the gravitational potential is recovered as an approximation.

(3.10a) Sum the series using the math identity in Zwiebach Q3.10 or your fave math reference.

(3.10b) Expand the analytic expression to find the leading correction at large radius to the four-dimensional answer. For what value of r/a is this correction a 1% effect?

(3.10c) Look in the opposite limit of small radius r<<a. What are the first two terms in the small-radius expansion of the five-dimensional potential? Is the leading term familiar?

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2. Zwiebach 5.5: relativistic point particle in EM field [easy]
Synopsis: In proper time gauge, add the term

to the geometric (kinetic) action for a free point particle,

and derive (from first principles) the equation of motion following from the combined action. Prove that this equation reproduces the Lorentz force law in 3-vector form familiar from undergraduate physics.

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3. Zwiebach 6.7: Open strings ending on Dp-branes of various dimensions [easy]

In a [flat] spacetime of dimension d+1, imagine a [lone] Dp-brane with p spatial worldvolume dimensions. Consider an open string ending on this Dp-brane. Let the p directions parallel to the Dp-brane worldvolume be labelled as the {x^i, i=1,…,p} and the orthogonal directions be {x^a, a=(p+1),…,d}.

(a) State the conditions satisfied by the curly-P_sigma. Treat separately the time {0}, {i}, and {a} components of curly-P_sigma.

(b) Prove that all boundary conditions (BCs) are satisfied for the case of the D0-brane (p=0).

(c) Prove that if the string ends on a D1-brane, the tangent to the string at the endpoint is orthogonal to the D1-brane, and the endpoint velocity is unconstrained.

(d) Prove that if the string ends on a Dp-brane with p>=2, either

(i) the string is orthogonal to the Dp-brane at the endpoint, and the endpoint velocity is unconstrained;

(ii) the string is not orthogonal to the Dp-brane at the endpoint, and the endpoint moves with the speed of light transversely to the string.

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4. Zwiebach 7.6: Planar open string motion and cusp formation. [Harder]
Consider a relativistic open string in planar motion in the {x,y} plane. Let the string endpoints be attached to (x,y)=(0,0) and (x,y)=(a,0) where a>0. Use the general formalism from class for solving classical open string motion, with the vector function F(u) and the quasi-periodicity condition F(u+2sigma_1) = F(u) + (2a,0).

Use the solution ansatz from Zwiebach 7.5,

and the fact that

a/sigma_1 = J_0(gamma),

where J_0 is the Bessel function of order zero and gamma may be assumed to satisfy

0 < gamma < pi/2.

Since J_0 is not a periodic function, the relationship between a, sigma_1 and gamma is unusual: open string motions corresponding to gamma and 2pi+gamma are not the same.

(a) Show that the instantaneous slope of the string is described by

where

Show that at ct=sigma_1/2 the string is horizontal.

(b) Prove that the instantaneous (transverse) velocity of the string satisfies

Note that at t=0 the string has zero velocity. Conclude that whenever gamma < pi/2 no point on the string ever reaches the speed of light. Moreover, show that for gamma =pi/2 the string midpoint sigma=sigma_1/2 acquires the speed of light when the string is horizontal.

(c) A tractable case is obtained for gamma=sqrt{2}(pi/2). Show that at ct=sigma_1/4 one point on the string reaches the speed of light. Examine the string at the slightly later time ct=sigma_1/3, show that there are two points that have the speed of light, and find the corresponding values of sigma. Analyze X-prime (partial_sigma X) as a function of sigma to show that the string has a cusp at each of these points. A cusp on a string is a point where the two outgoing string segments form zero angle. Equivalently, at a cusp the oriented tangent to the string reverses direction.

(d) Use your favourite mathematical software package to generate the picture of the string considered in (c) at various times (use numerical integration!). Assume that a=1 and verify that sigma_1 ~ 10.155. Show the string for ct=0, sigma_1/4 and sigma_1/3.

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