# Energy-Momentum Tensor: Exploring Einstein-Hilbert Action

• lion8172
In summary, the author defined the energy-momentum tensor as an integral of the functional derivative of the actions over dx^4 in the section on the Einstein-Hilbert action. However, in a later example, the integral over dx^4 is not shown. The question is, where did the integral over dx^4 go?
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?

lion8172 said:
$$\delta S_M = \frac{1}{2} \int T^{\mu \nu} \delta g_{\mu \nu} \sqrt{- g} d^4 x.$$
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

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.

## 1. What is the Energy-Momentum Tensor?

The Energy-Momentum Tensor is a mathematical quantity used in Einstein's field equations of General Relativity. It is a symmetric tensor that describes the distribution of energy and momentum in a given space-time. In simpler terms, it tells us how much energy and momentum is present at a specific point in space and time.

## 2. How is the Energy-Momentum Tensor calculated?

The Energy-Momentum Tensor is calculated using the Einstein-Hilbert action, which is a mathematical expression that describes the curvature of space-time. It involves integrating the Ricci scalar (a measure of curvature) over the entire space-time. The resulting equation gives us the values for the components of the Energy-Momentum Tensor.

## 3. What is the significance of the Energy-Momentum Tensor in General Relativity?

The Energy-Momentum Tensor is a crucial component in Einstein's field equations of General Relativity. It relates the curvature of space-time to the distribution of energy and momentum, allowing us to understand how matter and energy affect the geometry of the universe. It also plays a key role in predicting the behavior of gravitational fields and the motion of objects in space.

## 4. Can the Energy-Momentum Tensor be used to describe all forms of energy?

No, the Energy-Momentum Tensor is specific to the theory of General Relativity and cannot be used to describe all forms of energy. It is primarily used to describe the energy and momentum of matter, but it does not account for other forms of energy like dark energy or dark matter.

## 5. How does the Energy-Momentum Tensor relate to conservation laws?

The Energy-Momentum Tensor is closely related to the conservation laws of energy and momentum. In General Relativity, the Einstein field equations state that the curvature of space-time is determined by the distribution of energy and momentum. This means that the Energy-Momentum Tensor is conserved, and any changes in its values must be balanced by corresponding changes in the curvature of space-time.

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