Homework Solutions (12) - Ph 236 Homework 9 Due Monday 1 Higher symmetries[12 points In class we considered spacetimes with Killing elds with the

Homework Solutions (12) - Ph 236 Homework 9 Due Monday 1...

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Ph 236 – Homework 9 Due: Monday, January 23, 2012 1. Higher symmetries.[12 points]In class we considered spacetimes with Killing fields with the standard commutation relations of rotation[Ji,Jj] =-ǫijkJk,(1)with two-dimensional orbitsS(P). Cases where the orbits are higher-dimensional tend to be either trivialdue to the high degree of symmetry, or very weird. Here we will consider a trivial example.Consider the vectors in standard Minkowski spacetime given byHi=12(Ji+iKi)(2)(see Notes XVI,§IIIA for definitions ofJiandKiin Minkowski).(a) Show that[Hi,Hj] =-ǫijkHk.(3)Thus the three fields described commute like rotations.(b) Show that at a generic pointP, the three vectorsH1(P),H2(P), andH3(P) are linearly independent.Conclude that the orbit has dimension higher than 2.(c) What symmetries do the Killing fieldsHicorrespond to? Construct an alternative (non-Minkowski)metric that has theHias Killing fields.2. Spherical polytropes.[24 points]Consider a star with a polytropic equation of state,

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