Originally Posted by **BillKet** Show that if you add a total derivative to the Lagrangian density $\displaystyle L \to L + \partial_\mu X^\mu$, the energy momentum tensor changes as $\displaystyle T^{\mu\nu} \to T^{\mu\nu}+\partial_\alpha B^{\alpha\mu\nu}$ with$\displaystyle B^{\alpha\mu\nu}=-B^{\mu\alpha\nu}$. (Note: the Lagrangian can depend on higher order derivatives of the field)
The attempt at a solution:
So we have $\displaystyle T_{\mu\nu}=\frac{\partial L}{\partial(\partial_\mu \phi)}\partial_\nu \phi-g_{\mu\nu}L$, where $\displaystyle \phi$ is the field that the Lagrangian depends on. If we do the given change on the Lagrangian, the change in $\displaystyle T_{\mu\nu}$ would be $\displaystyle \frac{\partial (\partial_\alpha X^\alpha)}{\partial(\partial_\mu \phi)}\partial_\nu \phi-g_{\mu\nu}\partial_\alpha X^\alpha =\partial_\alpha \frac{\partial X^\alpha}{\partial(\partial_\mu \phi)}\partial_\nu \phi-g_{\mu\nu}\partial_\alpha X^\alpha$. From here I thought of using this: $\displaystyle g_{\mu\nu}\partial_\alpha X^\alpha=g_{\mu\nu}\partial_\alpha \phi \frac{\partial X^\alpha}{\partial \phi}$ But I don't really know what to do from here. Mainly I don't know how to get rid of that $\displaystyle g_{\mu\nu}$. Can someone help me? |

Since you are posting this in Particle Physics I presume we can take $\displaystyle g_{\mu \nu}$ to be the Minkowski metric.

For the last term on the RHS I don't see why you can't just use

$\displaystyle g_{\mu \nu} \partial _{\alpha } X^{\alpha } = \partial _{\alpha } \left ( g_{ \mu \nu } X^{\alpha } \right )$

and factor out the $\displaystyle \partial _{\alpha }$ to get

$\displaystyle \partial _{\alpha } \frac{ \partial X^{\alpha } }{\partial ( \partial _{\mu } \phi ) } \partial _{\nu } \phi - g_{\mu \nu } \partial _{\alpha }X^{\alpha } = \partial _{\alpha } \left ( \frac{ \partial X^{\alpha } }{\partial ( \partial _{\mu } \phi ) } \partial _{\nu }\phi - g_{\mu \nu } X^{\alpha } \right )$

so we get

$\displaystyle B^{\alpha }_{~~ \mu \nu } = \frac{ \partial X^{\alpha } }{\partial ( \partial _{\mu } \phi ) } \partial _{\nu } \phi - g_{\mu \nu } X^{\alpha }$

-Dan