What's the variation of a product of two metrics?

Asked Aug 28 '18 at 18:30

Active Aug 28 '18 at 18:43

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I was trying variate an action in General Relativity, and I come to the next calculus:

$\delta(g^{\alpha\beta}g^{\mu\nu})$

And I did:

$\delta(g^{\alpha\beta}g^{\mu\nu})=g^{\alpha\beta}\delta g^{\mu\nu}+g^{\mu\nu}\delta g^{\alpha\beta}$

I don´t know how to follow, what's the next step?

If $\delta=\partial/\partial g^{\mu\nu}$

The fist term would be just $g^{\alpha \beta}$? And what about the second term?

edited Aug 28 '18 at 18:43

Qmechanic

asked Aug 28 '18 at 18:30

It may be a question of language, but what do you mean by "variate an action"? What is the desired outcome - it looks like you are attempting a partial differential? – Floris Aug 28 '18 at 18:37
Related: https://physics.stackexchange.com/q/149066/2451 and links therein. – Qmechanic Aug 28 '18 at 18:47

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