Problems with a complex stress-energy tensor

1. Jun 25, 2011

Time Suspect

Hi , im 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)eiwt

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 Ttr /= Trt by a minus sign on the term of the partial derivates.

2. Jun 26, 2011

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 Ttr /= Trt. 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.

3. Jun 26, 2011

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, Tμν = 2δL/δgμν 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)/δgμν = 1/2 √(-g) gμν.

4. Jun 28, 2011

Time Suspect

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

Thanks a lot.