How to define time in a time-dependent solution? If a spacetime has no timelike killing vector, how can we define "time" in such spacetime, in order to calculate the time evolution behaivor of some quantities in it?
 A: 
If a spacetime has no timelike killing vector, how can we define "time" in such spacetime, in order to calculate the time evolution behaivor of some quantities in it?

GR lets you use any coordinates you like. That means you can define any timelike coordinate you like. A coordinate system doesn't even need to have a timelike coordinate. For example, you could have two null coordinates and two spacelike coordinates. This is also true in a spacetime that does have a timelike Killing vector: nothing forces you to use a time coordinate that matches up with the Killing vector, although it would be the natural choice.
A: A good reference for such things is Wald's General Relativity. In that book, you'll find the following result:
Theorem. Let $(M,g)$ (a manifold with a metric) be globally hyperbolic. Then

*

*There exists a global time function, that is a map $t:M\to \mathbb{R}$ such that $-dt$ is future-directed and timelike;

*Surfaces of constant $t$ are Cauchy surfaces, and they all have the same topology $\Sigma$;

*The topology of $M$ is $\mathbb{R} \times \Sigma$.

This result essentially tells you that if your spacetime is globally hyperbolic, then you have a uniquely defined time evolution. This "global hyperbolicity" is required in several results in GR and intuitively it means, well, that you have a unique time evolution. More formally, we can define what follows:
Definition. Let $(M,g)$ be a time-orientable spacetime. A partial Cauchy surface is a hypersurface for which no two points are connected by a causal curve in $M$.
Intuitively a partial Cauchy surface is like a spacelike hypersurface, on which you can define some initial data for time evolution. Moreover,
Definition. If $\Sigma$ is a partial Cauchy surface then its future domain of dependence $D^+(\Sigma)$ is the set of points $p \in M$ such that every past-inextendible causal curve through $p$ intersects $\Sigma$. Similarly you can define the past domain of dependence $D^-(\Sigma)$. Then you can also define the domain of dependence $D(\Sigma)=D^-(\Sigma) \cup D^+(\Sigma)$.
Suppose that you have initial data defined on $\Sigma$. Then the domain of dependence $D(\Sigma)$ is intuitively the set of points on which you can propagate this initial data (say with some hyperbolic PDE). Finally,
Definition. A Cauchy surface $\Sigma$ is a partial Cauchy surface with $D(\Sigma)=M$. A spacetime with a Cauchy surface is called globally hyperbolic.
So a Cauchy surface is nothing but some hypersurface on which you can define initial data which can be propagated everywhere in your spacetime. Then the above theorem tells you a bit more about how such a spacetime will look like.
