The Energy-Momentum Tensor

• agostino981
In summary, the Einstein Field Equation has two versions of the stress-energy tensor: the symmetrical energy-momentum tensor and the canonical energy-momentum tensor. They are not equivalent, but for scalar fields, they are identical. In QFT, the stress-energy tensor is not unique and can be modified to match the stress-energy tensor used in General Relativity.
agostino981
I am a bit confused here.

In the Einstein Field Equation, there is a tensor called stress-energy tensor in wikipedia and energy-momentum tensor in some books or papers which is $$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L} \sqrt{-g})}{\delta g_{\mu\nu}}$$

Is it equivalent to the energy-momentum tensor I came across in QFT?

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial( \partial_\mu\phi_{a})}\partial^\nu\phi_a -g^{\mu\nu}\mathcal{L}$$

agostino981 said:
I am a bit confused here.

In the Einstein Field Equation, there is a tensor called stress-energy tensor in wikipedia and energy-momentum tensor in some books or papers which is $$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L} \sqrt{-g})}{\delta g_{\mu\nu}}$$
This is the general definition of the SYMMETRICAL energy-momentum tensor.

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial( \partial_\mu\phi_{a})}\partial^\nu\phi_a -g^{\mu\nu}\mathcal{L}$$

This is the CANONICAL energy-momentum tensor. For scalar fields, the two are identical. For other fields they differ by a total divergence. They are equivalent in the sense that both leads to the same energy-momentum 4-vector
$$P^{ \mu } = \int d^{ 3 } x T^{ 0 \mu } ( x )$$

agostino981 said:
I am a bit confused here.

In the Einstein Field Equation, there is a tensor called stress-energy tensor in wikipedia and energy-momentum tensor in some books or papers which is $$T_{\mu\nu}=\frac{2}{\sqrt{-g}}\frac{\delta(\mathcal{L} \sqrt{-g})}{\delta g_{\mu\nu}}$$

Is it equivalent to the energy-momentum tensor I came across in QFT?

$$T^{\mu\nu}=\frac{\partial\mathcal{L}}{\partial( \partial_\mu\phi_{a})}\partial^\nu\phi_a -g^{\mu\nu}\mathcal{L}$$

They are not, in general, the same. However, in QFT, the stress-energy tensor is not unique, because you can add additional terms to it that have no effect on conservation laws. There is a procedure for tweaking the canonical stress-energy tensor to get a modified tensor, the Belinfante–Rosenfeld stress–energy tensor, that (according to Wikipedia, at least) agrees with the Hilbert stress-energy tensor used by General Relativity:
http://en.wikipedia.org/wiki/Belinfante–Rosenfeld_stress–energy_tensor

Wald has a good discussion of this, and shows that the first form arises naturalliy from formulating GR as a Lagrangian theory.

Thanks! That clears things up.

1. What is the Energy-Momentum Tensor?

The Energy-Momentum Tensor is a mathematical representation of the distribution of energy and momentum in a physical system. It is a crucial concept in the theory of relativity and is used to describe the flow of energy and momentum in space and time.

2. How is the Energy-Momentum Tensor calculated?

The Energy-Momentum Tensor is calculated using a mathematical formula that takes into account the density of energy and momentum at each point in space and time. It is derived from the stress-energy tensor, which includes the contributions of both matter and non-matter sources.

3. What is the significance of the Energy-Momentum Tensor?

The Energy-Momentum Tensor is significant because it allows us to understand the behavior of energy and momentum in physical systems, particularly in the context of relativity. It also helps us make predictions about the behavior of matter and energy in extreme conditions, such as near black holes or during the early stages of the universe.

4. How is the Energy-Momentum Tensor related to conservation laws?

The Energy-Momentum Tensor is directly related to the conservation laws of energy and momentum. This means that the total energy and momentum in a closed system remain constant over time, even as they may be redistributed within the system. The Energy-Momentum Tensor allows us to track and calculate these changes.

5. What are some practical applications of the Energy-Momentum Tensor?

The Energy-Momentum Tensor has many practical applications, such as in the study of astrophysics, where it is used to understand the behavior of stars and galaxies. It is also used in high-energy physics to study subatomic particles and their interactions. Additionally, it has applications in engineering, particularly in the design of advanced propulsion systems and energy storage technologies.

• Special and General Relativity
Replies
1
Views
431
• Special and General Relativity
Replies
2
Views
1K
• Special and General Relativity
Replies
2
Views
715
• Special and General Relativity
Replies
4
Views
895
• Special and General Relativity
Replies
4
Views
495
• Special and General Relativity
Replies
22
Views
2K
• Special and General Relativity
Replies
3
Views
401
• Special and General Relativity
Replies
8
Views
408
• Special and General Relativity
Replies
4
Views
753
• Special and General Relativity
Replies
11
Views
1K