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In this page there are calculations of variation of Einstein-Hilbert action.

I see variations of terms like this:

$\delta {R^{\rho }}_{{\sigma \mu \nu }}$

where the term is not a functional, and

$\frac {\delta {\mathcal {L}}_{{\mathrm {M}}}}{\delta g^{{\mu \nu }}}$

where we have a functional derivative of a term that is not also a functional.

What is the exact meaning of those expressions?

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It is a notational short hand. (See, e.g. Appendix E in Wald's General Relativity).

Given a function $\psi$ and a one-parameter family of functions $\psi_{\lambda}$ with $\psi_0 = \psi$, the notation $\delta \psi$ refers to the short hand $\delta\psi := \frac{d}{d\lambda} \psi_\lambda \Big|_{\lambda = 0}$.

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  • $\begingroup$ Thank you very much. What $\frac {\delta {\mathcal {L}}_{{\mathrm {M}}}}{\delta g^{{\mu \nu }}}$ stands for? If you want, you could write this short hand in wikipedia. $\endgroup$
    – asv
    Mar 20, 2018 at 16:09
  • $\begingroup$ Formally, it is the "partial derivative of $\mathcal{L}_M$ with respect to $g^{\mu\nu}$". However, there can be some shenanigans with integration by parts involved. So you should think that given $\mathcal{L}_M$ that can depend on a one-parameter family of inverse metrics $g^{\mu\nu}_{(\lambda)}$ the term $\delta \mathcal{L}_M / \delta g^{\mu\nu}$ is the object that makes the statement $$ \frac{d}{d\lambda} \left(\int \mathcal{L}_M \right) |_{\lambda = 0} = \int \frac{\delta \mathcal{L}_M}{\delta g^{\mu\nu}} \delta g^{\mu\nu} $$ hold for all perturbations $\delta g^{\mu\nu}$ $\endgroup$ Mar 20, 2018 at 17:44
  • $\begingroup$ Example in the simplest case: if your functional is given by the density $\mathcal{L}_M = f \triangle \phi$, where $\phi$ is the unknown function and $f$ is given, then $\delta \mathcal{L}_M / \delta \phi = \triangle f$ provided we only look at compactly supported perturbations. $\endgroup$ Mar 20, 2018 at 17:47
  • $\begingroup$ Thanks, can you indicate some free resource (math rigorous) in internet on this argument? $\endgroup$
    – asv
    Mar 21, 2018 at 14:48

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