# Show scalar field stress-energy tensor satisfies dominant energy condition

The stress- energy tensor for a minimally-coupled real massive scalar field is:

$$T_{\mu \nu}=\partial_{\mu} \phi \partial_{\nu} \phi-\frac{1}{2} g_{\mu \nu} \partial_{\alpha} \phi \partial^{\alpha} \phi-\frac{1}{2} g_{\mu \nu} m^{2} \phi^{2}$$

I want to show this satisfies the dominant energy condition. I've already shown it satisfies the weak energy condition ($$T_{\mu \nu} = U^{\mu}U^{\nu}$$). Now, I want to show that energy flux is timelike. I know this can be expressed as ($$J_{\nu} J^{\nu} \leq 0$$, where $$J^{\nu} = -U_{\mu}T^{\mu \nu}$$), and I am working in Minkowski space. I am unsure of how to proceed.

• As with this earlier question of yours, physics.stackexchange.com/questions/509635/… , I've added the homework-and-exercises tag. Again, please use this tag on questions of this type. – user4552 Oct 24 '19 at 2:51