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Intro to Differential Geometry and General Relativity - S. Warner Episode 9

Tham khảo tài liệu 'intro to differential geometry and general relativity - s. warner episode 9', kỹ thuật - công nghệ, cơ khí - chế tạo máy phục vụ nhu cầu học tập, nghiên cứu và làm việc hiệu quả | g Rabcd e Rabec d Rabde c 0 Since gj k 0 see Exercise Set 8 we can slip the gbc into the derivative getting Rad e Rae d Ra de c 0 Contracting again gives gữd -Rad e Rae d RaCde c 0 -R e Rded Rdcdec 0 -Re Rded Rcec 0. Combining terms and switching the order now gives RÌ e b - ĨR e 0 Rb ib - 0 Multiplying this by gae we now get Rab b - 2 gabR b 0 Rab is symmetric or Gabb 0 where we make the following definition Einstein Tensor G ab Rib 1 gabR Einstein s field equation for a vacuum states that Gab 0 as we shall see later. . Example 10.7 Take the 2-sphere of radius r with polar coordinates where we saw that 81 g r2sin2f 0 0 r2 The coordinates of the covariant curvature tensor are given by Rabcd 2 gbc ad gbd ac gad bc gac bd a c jbd a d jbc Let us calculate Rgýgý. Note when we use Greek letters we are referring to specific terms so there is no summation when the indices repeat So a c 0 and b d Ộ. Incidentally this is the same as RỌIIỌII by the last exercise below. The only non-vanishing second derivative of g is geew 2r2 cos2ộ - sin20X giving 1 2a 2ja 2 gộd dộ gTT ee g8ộ ộd gee ộộ r sin T cos t . The only non-vanishing first derivative of g is gee T 2r2sin T cos f giving TLTjbd j 0 since b d T eliminates the second term two of these indices need to be e in order for the term not to vanish. T-j rj r 1 2 cos T Í n 2- 2_2 radrjbc vej 4 1 sinf J -2r sin T cos T -r cos f Combining all these terms gives n _ 2 .2 2 .2 2 Reộ0ộ r sin T - cos f r cos T r2sin2f. We now calculate Ra Ree g Re.f0.f sin2f 82 and Rộộ - g Rộdộe _ sin2 3ự _ 1 - sin20 All other terms vanish since g is diagonal and R is antisymmetric. This gives R - gabRab - geeRee g R - 2 2 sin20 2 - J r sin Ộ r r .2 Summary of Some Properties of Curvature Etc. rabc Rabcd R abcd R abcd Rab - R -ab R - gabR Ra b Rab - gágbÍR I Gab - Rab 2 gabR - r a Rabdc R bacd R cdab ầbi - gijRa.ibi I ba ab ab - gaiR h ib - gaCgbdRabcd abc cba Rabcd Rabdc Note that a b and c d always go together R t - R Exercise Set 10 1. Derive the .