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General Relativity Ph236b Problem Set 7 Due: In class, March 6, 2007 Preview: The rst ve problems are all exercises on dierential forms. Problem 1 is...

Tetrad formalism for the mixmaster universe:
Calculate the connection one-forms, curvature two-forms, and hence the components of the Riemann tensor for the Mixmaster universe. The metric is given by
ds2 = −dt
dt + 21

1 + 22

2 +
23

3.
Here , , and
are functions of t only, and the one-forms i are given by
1 = cos d + sin sin d,
2 = sin d − cos sin d,
3 = d + cos d.
General Relativity Ph236b Problem Set 7 Due: In class, March 6, 2007 Preview: The frst fve problems are all exercises on diFerential ±orms. Problem 1 is just an algebraic exercise. Problems 2 and 3 have you think about Maxwell’s equations in the language o± diFerential ±orms. Problem 4 is a neat problem that has you work through some aspects o± gauge theories in higher dimensions. Problem 5 is a good exercise ±or you to understand how to calculate the Riemann tensor using the tetrad ±ormalism. As a warmup, you may want to read through Ch. 6.1 on Wald’s book to see, as an example, this ±ormalism used to calculate the Riemann tensor ±or the Schwarzchild spacetime. Problem 6 is a cute problem (that really has nothing to do with GR) in Newtonian mechanics. 1. (Carroll, problem 2.8) Leibnitz rule for the exterior derivative: Show that the action o± the exterior derivative d on the wedge product o± a product o± a p -±orm ω and a q -±orm η is d( ω η ) = (d ω ) η + ( - 1) p ω (d η ) . 2. (Carroll, problem 2.9) Euclidean-space E&M: In Euclidean three-space, suppose sF = q sin θ d θ d φ . a. Evaluate d s F = sJ . b. What is the two-±orm F equal to? c. What are the electric and magnetic felds equal to? d. Evalute i V d s F , where V is a ball o± radius R in a Euclidean three-space. 3. (Carroll, problem 2.10) (1+1)-d Maxwell’s equations: Consider Maxwell’s equation, d F = 0, d s F = sJ , in a 2-dimensional spacetime. Explain why one o± the two sets o± equations can be discarded. Show that the electromagnetic feld can be expressed in terms o± a scalar feld. Write out the feld equations ±or this scalar feld in component ±orm. 4. (Carroll, problem 2.11) Extra-dimensional gauge theories: This is problem 2.11 ±rom Carroll’s book. As you’ll see, its a long problem, and I’m too lazy to type it in. You, however, should not be too lazy to solve it; its a good one. 5. (Carroll, problem J.2) Tetrad formalism for the mixmaster universe: Calculate the connection one-±orms, curvature two-±orms, and hence the components o± the Riemann tensor ±or the Mixmaster universe. The metric is given by dS 2 = - d t d t + α 2 σ 1 σ 1 + β 2 σ 2 σ 2 + γ 2 σ 3 σ 3 .
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Here α , β , and γ are functions of t only, and the one-forms σ i are given by σ 1 = cos ψ d θ + sin ψ sin θ d φ, σ 2 = sin ψ d θ - cos ψ sin θ d φ, σ 3 = d ψ + cos θ d φ. 6. Extra-dimensional Keplerian motion: The solution is supposed to show that if the Universe had more than three spatial dimensions, then the orbit of the Earth around the Sun would not be stable, and hence unable to support life. This argument has been given (perhaps tongue in cheek) as an anthropic argument for why the Universe has three spatial dimensions. In three spatial dimensions, the gravitational acceleration due to a point mass (e.g., the Sun in the Solar System) is proportional to 1 /r 2 , where r is the distance from the point mass. a. Show that with this force law, circular orbits are stable to small perturbations. b. With d extra spatial dimensions, the force law becomes proportional to 1 /r 2+ d . De- termine the condition on d for circular orbits to be stable.
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