How does one go about in finding a variation of a tensor quantity say, with respect to the variation in the metric tensor? I've gone through calculus of variations but I can't figure out how one obtains the result of a variation of, say the Ricci tensor with respect to variation in the metric tensor? Is there some formula that tells us what to do or is the concept behind it something else?
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$\begingroup$ Do you know how to vary a (classical) Lagrangian with respect to the coordinate $q$? $\endgroup$– Kyle KanosCommented Oct 10, 2015 at 11:39
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$\begingroup$ Nope. We learned the shortest time problems along with the Euler Lagrange equations for different situations in mechanics. The variation part only appeared when we derived the Euler Lagrange equations and I didn't understand how the variation part worked. $\endgroup$– nihalCommented Oct 10, 2015 at 12:41
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$\begingroup$ I would go back to understanding the classical case before diving into the general relativistic case then. $\endgroup$– Kyle KanosCommented Oct 10, 2015 at 12:55
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