How would you define the present of an observer at a certain time? In Special Relativity that is trivial, just transform the coordinates with the Lorentz equations and determine the region with t' constant, that is going to be the present of the observer at its time t'. However in curved spaces this is not that easy to understand. I believe that one way of determining this region would be to consider the four-velocity of the observer at some time and determine the vector-space perpendicular to this four-vector. Then we can prolong all the vectors with geodesics or world-lines any length (I don't know the precise term in this case). The region defined could be called the "present" of the observer at that time.
This definition agrees with the case in Special Relativity. However, for the case of the FLRW metric, an observer at constant comovil coordinates would have a "present" which does not coincide to the hypersurface of constant cosmological time, which is the definition of the "present" Universe.
I would like to know if I can find more information about this "definition" or idea. And if it makes sense, for example, let a and b be events in this region, would the world-line that connects both events be contained in the region? If not, another observer in the region whose four-velocity is perpendicular to the region would define a different "present". Thank you in advance for your answer!