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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(∂

_{a}Φ* ∂

_{b}Φ + ∂

_{a}Φ ∂

_{b}Φ*) - ½g

_{ab}(g

^{cd}∂

_{c}Φ* ∂

_{d}Φ + 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