Tensors and Relativity: Assignment 6

Hints 6

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  • (2) This is straight out of section 6.5.1 of the lecture notes.

     

  • (3) First write down the standard transformation laws which take you from (x,y,z) to tex2html_wrap_inline324 . Next calculate the basis vectors tex2html_wrap_inline326 and then the metric from tex2html_wrap_inline328 . The metric for a 2- sphere, then follows by taking r=1, so dr=0.

     

  • (4) Here it is enough to show that the Christoffel symbols all vanish.

     

  • (5) The only non- zero component is tex2html_wrap_inline334 , so that's all you need to calculate.

     

  • (6) In both parts use a local inertial frame to prove the results.

     

  • (7) Calculate tex2html_wrap_inline336 , where tex2html_wrap_inline338 is an affine parameter along the geodesic.