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 oneforms, curvature twoforms, 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 oneforms i are given by
1 = cos d + sin sin d,
2 = sin d − cos sin d,
3 = d + cos d.
General Relativity Ph236bProblem Set 7Due: In class, March 6, 2007Preview:The frst fve problems are all exercises on diFerential ±orms. Problem 1 isjust an algebraic exercise. Problems 2 and 3 have you think about Maxwell’s equations inthe language o± diFerential ±orms. Problem 4 is a neat problem that has you work throughsome aspects o± gauge theories in higher dimensions. Problem 5 is a good exercise ±or you tounderstand 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 ±ormalismused to calculate the Riemann tensor ±or the Schwarzchild spacetime. Problem 6 is a cuteproblem (that really has nothing to do with GR) in Newtonian mechanics.1. (Carroll, problem 2.8)Leibnitz rule for the exterior derivative:Show that theaction o± the exterior derivative d on the wedge product o± a product o± ap±ormωandaq±ormηisd(ω∧η) = (dω)∧η+ (1)pω∧(dη).2. (Carroll, problem 2.9)Euclideanspace E&M:In Euclidean threespace, supposesF=qsinθdθ∧dφ.a. Evaluate ds F=sJ.b. What is the two±ormFequal to?c. What are the electric and magnetic felds equal to?d. EvaluteiVds F, whereVis a ball o± radiusRin a Euclidean threespace.3. (Carroll, problem 2.10)(1+1)d Maxwell’s equations:Consider Maxwell’s equation,dF= 0, ds F=sJ, in a 2dimensional spacetime. Explain why one o± the two setso± equations can be discarded. Show that the electromagnetic feld can be expressed interms o± a scalar feld. Write out the feld equations ±or this scalar feld in component±orm.4. (Carroll, problem 2.11)Extradimensional 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 theconnection one±orms, curvature two±orms, and hence the components o± the Riemanntensor ±or the Mixmaster universe. The metric is given bydS2=dt⊗dt+α2σ1⊗σ1+β2σ2⊗σ2+γ2σ3⊗σ3.
Hereα,β, andγare functions oftonly, and the oneformsσiare given byσ1= cosψdθ+ sinψsinθdφ,σ2= sinψdθcosψsinθdφ,σ3= dψ+ cosθdφ.6.Extradimensional Keplerian motion:The solution is supposed to show that if theUniverse had more than three spatial dimensions, then the orbit of the Earth aroundthe Sun would not be stable, and hence unable to support life. This argument has beengiven (perhaps tongue in cheek) as an anthropic argument for why the Universe has threespatial dimensions. In three spatial dimensions, the gravitational acceleration due to apoint mass (e.g., the Sun in the Solar System) is proportional to 1/r2, whereris thedistance from the point mass.a. Show that with this force law, circular orbits are stable to small perturbations.b. Withdextra spatial dimensions, the force law becomes proportional to 1/r2+d. Determine the condition ondfor circular orbits to be stable.
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