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In the second book, Field Theory, of popular series of Theoretical Physics by Landau-Lifschitz are obtained following equations. After making linear coordinate transformation (prime=old+ipsilon) a new prime metric written in old coordinates

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My question is how (94.2) could be verified by direct trial as Landau said?

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  • $\begingroup$ Title is horrible "Killing Equations"?? $\endgroup$ – Wrichik Basu Jun 22 '17 at 16:50
  • $\begingroup$ Yes, because second equation in (94.2) is called Killing equation. That is equal to the last three terms of right hand side elements of the equation just before (94.2). $\endgroup$ – Constantin Jun 22 '17 at 16:54
  • $\begingroup$ These equations are used for obtaining energy-momentum tensor. In which book these equation are explained in detail? $\endgroup$ – Constantin Jun 22 '17 at 17:14
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Just look up the definition of the covariant derivative, and use the formula for the Christoffel symbols. The Landau and Lifshitz book is excellent, but it's probably not the best place to start when learning GR. Sean Carroll's textbook is an excellent place to start, and Wald's book is very thorough if you prefer a more mathematical approach. Misner is good if you're a fast reader and don't mind reading through a lot of somewhat heuristic explanations.

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    $\begingroup$ A little additional remark: Weinberg is also a great place to start. Cuts the crap and gets to the point without compromising rigor and not as sophisticated as Landau to make it a bit out-of-the-reach for a first-time learner of GR. $\endgroup$ – Dvij Mankad Jun 22 '17 at 20:37
  • $\begingroup$ I have used the definition of covariant derivative and Christoffel symbols. Only the second term on right-hand side with minus sign in equation before (94.2) is not derived. $\endgroup$ – Constantin Jun 23 '17 at 8:49
  • $\begingroup$ This indicates to me that there is a chance you are confused about how Christoffel symbols relate to the metric tensor. Are you familiar with the relationship? $\endgroup$ – TotallyRhombus Jun 23 '17 at 16:15
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Calculations of covariant derivative

I have carried out these calculations. It seems everything is ok. I am missing the key point here. How should it be continued to obtain in the proper form?

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