Energy-Momentum tensor components for complex Klein-Gorden field

[Dixanadu]

Hey guys,

So I have the stress energy tensor written as follows in my notes for the complex Klein-Gordon field:

$T^{\mu\nu}=(\partial^{\mu}\phi)^{\dagger}(\partial^{\nu}\phi)+(\partial^{\mu}\phi)(\partial^{\nu}\phi^{\dagger})-\mathcal{L}g^{\mu\nu}$

Then I have the next statement that $T^{0i}$ is given by

$T^{0i}=-[(\partial_{t}\phi)^{\dagger}\nabla^{i}\phi+(\partial_{t}\phi)\nabla^{i}\phi^{\dagger}]$

And I was wondering how this comes about. I can sort of see what's happened here. Obviously you replace mu and nu with 0 and i respectively to split the spatial and time derivatives. However I have a couple of questions:

1) is it true that $\partial^{t}\phi=-\partial_{t}\phi$?
2) why does the Lagrangian density at the end, $\mathcal{L}g^{\mu\nu}$ drop out?

Thanks a lot guys :)

 

[Dixanadu]

Okay maybe I know why the Lagrangian density drops out...is it cos $g^{\mu\nu}$ is diagonal and so $g^{0i}$, for i = 1, 2, 3, are off-diagonal elements which are 0?