Let $(M,g)$ be spacetime and consider the space of solutions to the Klein-Gordon equation endowed with the inner product:
$$(\phi_1,\phi_2)=\int_\Sigma (\phi_1\nabla_\mu \phi_2^\ast-\phi_2^\ast\nabla_\mu \phi_1) n^\mu \sqrt{|\gamma|}d^{n-1}x$$$$(\phi_1,\phi_2)=i\int_\Sigma (\phi_1\nabla_\mu \phi_2^\ast-\phi_2^\ast\nabla_\mu \phi_1) n^\mu \sqrt{|\gamma|}d^{n-1}x$$
Let $\{\phi_i\}$ be a complete set of solutions on this space and $K$ a timelike Killing vector field on $(M,g)$.
If I understood well (and I might have not), in his book Spacetime and Geometry, Sean Carroll says that $\phi_i$ is a positive frequency solution to the KG equation if there is $\omega \in(0,\infty)$ such that
$$\mathscr{L}_K \phi_i=K\phi_i=-i\omega \phi_i,$$
and it is a negative-frequency solution if
$$\mathscr{L}_K \phi_i=K\phi_i=i\omega \phi_i.$$
Now why does one need a Killing field to judge this? Why this can't be done with an arbitrary timelike vector field?
The Lie derivative of a $C^\infty(M)$ function is just the directional derivative of the function along the integral lines of the field.
Why this condition says that a function is a positive or negative frequency, and why we need the Killing field?
Edit: I know a Killing field is a generator of isometries, so that the metric tensor is constant along its flow. Nevertheless, I really can't see why to define positive/negative frequency solutions we need a Killing field instead of just a timelike field. I don't see why any symmetry of spacetime should be involved.
Actually, a timelike future-directed vector field defines a family of observers and is a reference frame. I thought that this was enough to define positive/negative frequency: it is what this reference frame sees. I honestly don't know where the Killing part comes into play.