# Is the Energy Momentum Tensor for Scalar Fields Always Symmetric?

• BillKet
In summary, the conversation discusses the proof that the energy momentum tensor is always symmetric when the Lagrangian only depends on scalar fields. The equation for the energy momentum tensor is provided, and it is stated that the second term is already symmetric. The discussion then moves on to the first term and addresses the issue of Lorentz invariance. The speaker is unsure how to proceed in the general case and asks for clarification on what terms are allowed while respecting Lorentz invariance.
BillKet

## Homework Statement

Show that if the Lagrangian only depends on scalar fields ##\phi##, the energy momentum tensor is always symmetric: ##T_{\mu\nu}=T_{\nu\mu}##

## Homework Equations

##T_{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu\phi)}\partial_\nu\phi-g_{\mu\nu}L##

## The Attempt at a Solution

So the second therm in the SE tensor is symmetric so we need to prove that the first term is, too. But I am really not sure how to proceed for a general Lagrangian. For example, if we have a term like ##(\partial_\mu \phi)^2\partial_\nu \phi##, that wouldn't by symmetric. Am I reading this the wrong way? Thank you!

Your proposed Lagrangian term is not even Lorentz invariant ...

Orodruin said:
Your proposed Lagrangian term is not even Lorentz invariant ...
Oh sorry for that, I didn't mean that term is alone, it was just an example. It could be ##(\partial_\mu \phi)^2\partial_\nu \phi J^\nu## where I have an external source. My point was, what terms am I allowed to use (respecting Lorentz invariance) in the general case? I just don't know where to start from in solving this...

## 1. What is the SE tensor for scalar field?

The SE tensor for scalar field, also known as the Einstein tensor, is a mathematical quantity used in general relativity to describe the curvature of spacetime caused by matter and energy. It is a symmetric tensor that combines the Ricci tensor and the scalar curvature to describe the curvature of a four-dimensional spacetime.

## 2. How is the SE tensor for scalar field calculated?

The SE tensor for scalar field is calculated by taking the Ricci tensor and subtracting one-fourth of the scalar curvature from it. This calculation is based on Einstein's field equations, which relate the curvature of spacetime to the distribution of matter and energy within it.

## 3. What is the significance of the SE tensor for scalar field?

The SE tensor for scalar field is significant because it is a crucial component of Einstein's field equations and is used to describe the gravitational effects of matter and energy. It allows us to understand how the curvature of spacetime is influenced by the distribution of matter and energy, and plays a key role in our understanding of general relativity.

## 4. Can the SE tensor for scalar field be used to solve equations?

The SE tensor for scalar field itself cannot be used to solve equations, but it is an essential element in solving Einstein's field equations. These equations are highly complex and are used to describe the behavior of gravity on a large scale, such as the motion of planets and galaxies.

## 5. How does the SE tensor for scalar field relate to the energy-momentum tensor?

The SE tensor for scalar field and the energy-momentum tensor are closely related, as both are used to describe the distribution of matter and energy in spacetime. The energy-momentum tensor represents the flow of energy and momentum in a given region of spacetime, while the SE tensor for scalar field describes the curvature of spacetime caused by that matter and energy. Together, they help us understand the gravitational effects of matter and energy in the universe.

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