# SE tensor for scalar field

## 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!

## Answers and Replies

Orodruin
Staff Emeritus
Science Advisor
Homework Helper
Gold Member
Your proposed Lagrangian term is not even Lorentz invariant ...

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...