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

 

[Entropia]

Thank you for your question. The energy-momentum tensor is a very important concept in general relativity and it can be defined in various ways, depending on the context. In the context of the Einstein-Hilbert action, the energy-momentum tensor is defined as the functional derivative of the action with respect to the metric tensor. This means that it is a mathematical object that describes how the action changes when the metric tensor is varied.

In the first equation that you provided, the energy-momentum tensor is defined as a functional that takes in the variation of the metric tensor, \delta g_{\mu \nu}, and outputs the variation of the action, \delta S_M. The integral over d^4x is still present in this definition, it is just not explicitly written out. This is because the energy-momentum tensor is a function of the spacetime coordinates, x, and the integral over d^4x is implied in the notation.

In the second equation, the author is providing an example of the energy-momentum tensor for a real scalar field. Here, the energy-momentum tensor is defined as a function of the metric tensor, g_{\mu \nu}, and its variation, \delta g_{\mu \nu}. In this case, the integral over d^4x is not explicitly written out because it is being evaluated at a specific point, x. This is why the notation is slightly different, with a (x) subscript on the energy-momentum tensor.

I hope this clarifies your question. The energy-momentum tensor is a complex and important concept in general relativity, and it is often defined and used in different ways depending on the context. It is always important to pay attention to the notation and make sure you understand the definitions being used.