A Complex components of stress-energy tensor

[xpet]

Hi All,
I am evaluating the components of the stress-energy tensor for a (Klein-Gordon) complex scalar field. The ultimate aim is to use these in evolving the scalar field using the Klein-Gordon equations, coupled to Einstein's equations for evolving the geometric part. The tensor is given by
T_ab = ½(∂_aΦ* ∂_bΦ + ∂_aΦ ∂_bΦ*) - ½g_ab(g^cd ∂_cΦ* ∂_dΦ + V(|Φ|²))
All indices run from 0 to 3, and Φ* is the complex conjugate of Φ.
Now, evaluating the components of this stress-energy tensor, not all components the stress-energy tensor are real. In particular, when indices a and b are different. I just need to confirm if this observation is correct. The reason being that, part of the simulation code I will use seems to expect a stress-energy tensor with all real components.

Any help and insights will be appreciated.

Thank you

 

[PeterDonis]

evaluating the components of this stress-energy tensor, not all components the stress-energy tensor are real.

They should be. If they aren't, there's a mistake somewhere.

In particular, when indices a and b are different.

Note that the stress-energy tensor is symmetric, so $T_{ab} = T_{ba}$. The formula you wrote down should make that easy to prove; but it also should make it easy to see that this symmetry ensures that all of the components are real.

(Another way to see that all the components must be real is to work in local inertial coordinates, where the metric is the Minkowski metric and all of the terms with a and b being different vanish.)

Also note that, in a general curved spacetime, you should be using covariant derivatives, not partial derivatives. (But if you use local inertial coordinates, they are the same.)

 

[xpet]

Thanks for the reply. I am re-doing the calculations again for the components, to see if I get them to be real. Just one correction though: My point about different indices was meant to refer to c and d. Basically, the non-real contributions will come from #g^cd ∂_c Φ*∂_dΦ#
Yes, I'm aware of the symmetry of the tensor, and of the need for replacing the partial derivatives with covariant derivatives in curved spacetimes. I will get back with feedback.

 

[PeterDonis]

My point about different indices was meant to refer to c and d.

But in a local inertial frame, where the metric is Minkowski, there are no terms where c and d are different. And even in the general case, the metric is symmetric, so terms with different c and d occur in pairs with c and d swapped. If you work it out, you should find that those things imply that all of the stress-energy components are real.