/sci/ - Science & Math

Is there a sign error in picrel? The stress-energy tensor is defined as:
[math]T_\mu{}^\nu = \frac{\partial \mathcal{L}}{\partial \left ( \partial_\nu \eta_\rho \right )} \partial_\mu \eta_\rho - \mathcal{L}\delta_\mu{}^\nu[/math]
Lowering the nu index gives
[math]T_{\mu \nu} = g_{\lambda \nu} \frac{\partial \mathcal{L}}{\partial \left ( \partial_\lambda \eta_\rho \right )} \partial_\mu \eta_\rho - g_{\mu \nu} \mathcal{L}[/math]
Now, the Lagrangian density that's given is that of the complex Klein-Gordon field:
[math]\mathcal{L} = c^2 \partial_\lambda \phi \partial^\lambda \bar{\phi}-\mu_0^2 c^2 \phi \bar{\phi}[/math]
My concern is with the last part of the equation that's multiplied by the metric:
[math]\cdots +c^2 \left ( \partial_\lambda \phi \partial^\lambda \bar{\phi}+\mu_0^2 c^2 \phi \bar{\phi} \right )g_{\mu \nu} [/math]
which should correspond to the
[math]- g_{\mu \nu} \mathcal{L}[/math]
part of the tensor. But should this part not be
[math]\cdots +c^2 \left ( -\partial_\lambda \phi \partial^\lambda \bar{\phi}+\mu_0^2 c^2 \phi \bar{\phi} \right ) g_{\mu \nu}[/math]
given that there's a minus sign in the Lagrangian density? Where does that minus sign go?