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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

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# A Complex components of stress-energy tensor

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