Four-momenta of fields in general curved spacetime

Question

I want to understand when and how can we define the four momentum of a matter field in a general curved spacetime.

To elaborate, I wish to know what conditions must the spacetime satisfy in order for 4-momentum of a matter field to be a valid concept. In such situations, what is the exact definition of 4-momentum. For simplicity, we may take the matter field to be a scalar field.

Comments

In my opinion, the four-momentum makes sense if it is conserved. I am afraid in "a general curved space-time" it is not conserved, so you must solve the equations at every time $t$ if you want to know its current value.