- #1

- 1,344

- 32

## Homework Statement

The energy-momentum tensor ##T^{\mu\nu}## of the Klein-Gordon Lagrangian ##\mathcal{L}_{KG} = \frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2}## is given by

$$T^{\mu\nu}~=~\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}.$$

Show that ##\partial_{\mu}T^{\mu\nu}=0##.

## Homework Equations

## The Attempt at a Solution

$$\partial_{\mu}T^{\mu\nu} \\

=\partial_{\mu}[\partial^{\mu}\phi\partial^{\nu}\phi-\eta^{\mu\nu}\mathcal{L}_{KG}]\\

=\partial_{\mu}(\partial^{\mu}\phi\partial^{\nu}\phi-\partial^{\nu}\mathcal{L}_{KG})\\

=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)-\partial^{\nu}(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi-\frac{1}{2}m^{2}\phi^{2})\\

=(\partial_{\mu}\partial^{\mu}\phi)(\partial^{\nu}\phi)-\partial^{\nu}(\frac{1}{2}\partial_{\mu}\phi\partial^{\mu}\phi)+m^{2}\phi(\partial^{\nu}\phi)$$

Where do I go from here? I know I need to use the Klein-Gordon equation, but using the KG equation cancels the first and third terms and leaves the second term.