Problems with a complex stress-energy tensor

[Time Suspect]

Hi , I am working with the following stress-energy tensor:

T_\mu\nu=\partial_\mu\phi\partial_\nu\phi^* - g^\mu\nu(\nabla^\mu\phi\nabla_\mu\phi^* - m²\phi\phi^*)

Where \phi is a complex scalar of the form:
\phi(r,t) = \psi(r)e^iwt

that obeys the Klein-gordon equation and \phi^* is the complex conjugate, and g the metric tensor.

My problem is that i think this tensor is not symmetric as T_tr /= T_rt by a minus sign on the term of the partial derivates.

Thanks a lot for reading.

 

[Time Suspect]

I think my problem wasn't really clear, this tensor is supposed to be a stress energy tensor and therefore it should be symmetric but I'm finding that it isn't T_tr /= T_rt. I'm not sure if I'm differentiating wrong the complex scalar field or I think originally instead of partial derivatives they were covariant derivatives but since it is a scalar quantity I thought it wouldn't matter.

Hope this gives a better idea of the problem at hand.

Thanks a lot.

 

[Bill_K]

So why don't you just symmetrize it.

Or better yet, forget the "canonical" stress-energy tensor and use the correct and much simpler definition, T^μν = 2δL/δg_μν whose calculation typically involves no taking of derivatives and is guaranteed to come out symmetrical. All you have to remember is that the Lagrangian density contains a factor √(-g) which must also be varied, and δ√(-g)/δg_μν = 1/2 √(-g) g^μν.

 

[Time Suspect]

Thanks Bill_K, with that and the Klein-Gordon Lagrangian I found a Tensor which is indeed symmetric.

Thanks a lot.