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...
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...
Ok, so for the change in action we have: $$\frac{\delta S}{\delta y^\mu}=\int(\frac{\delta L}{\delta y^\mu}+\frac{\delta \partial_\alpha X^\alpha }{\delta y^\mu})$$ And the first term gives the old result. Is this ok so far?
Thank you for your reply. So I tried this: $$\frac{\delta(L+\partial_\mu X^\mu)}{\delta y^\nu}=\Big(\frac{\partial L}{\partial \phi}+\frac{\partial(\partial_\mu X^\mu)}{\partial \phi}-\partial_\mu\frac{\partial L}{\partial (\partial_\mu \phi)}-\partial_\mu\frac{\partial (\partial_\mu...
Homework Statement
Show that if you add a total derivative to the Lagrangian density ##L \to L + \partial_\mu X^\mu##, the energy momentum tensor changes as ##T^{\mu\nu} \to T^{\mu\nu}+\partial_\alpha B^{\alpha\mu\nu}## with ##B^{\alpha\mu\nu}=-B^{\mu\alpha\nu}##.
Homework Equations
The...