Energy-Momentum Tensor: Exploring Einstein-Hilbert Action

[lion8172]

I was wondering if someone could clarify something that I read in a book (Nakahara's book on Geometry, Topology, Physics). In the section on the Einstein-Hilbert action, the author defines the energy-momentum tensor as
\delta S_M = \frac{1}{2} \int T^{\mu \nu} \delta g_{\mu \nu} \sqrt{- g} d^4 x.
Shortly thereafter, he then writes that, "for example, T_{\mu \nu} of a real scalar field is given by
T_{\mu \nu} (x) = 2 \frac{1}{\sqrt{-g}} \frac{\delta}{\delta g^{\mu \nu} (x)} S_s = \cdots.
My question is, where did the integral over dx^4 go?

 

[samalkhaiat]

\delta S_M = \frac{1}{2} \int T^{\mu \nu} \delta g_{\mu \nu} \sqrt{- g} d^4 x.
Shortly thereafter, he then writes that, "for example, T_{\mu \nu} of a real scalar field is given by
T_{\mu \nu} (x) = 2 \frac{1}{\sqrt{-g}} \frac{\delta}{\delta g^{\mu \nu} (x)} S_s = \cdots.
My question is, where did the integral over dx^4 go?

The functional derivative of the actions is defined by

\delta S = \int d^{4}x \frac{\delta S}{\delta g^{ab}} \delta g^{ab}

regards

sam