No, I don't think that that is possible. What is possible is that the new time direction will include a non-vanishing component of S0 as well as the normal component.hellfire said:Consider a spacetime, with a specific metric, sliced in spatial hypersurfaces. Take an observer located at a point of slide S0. Are you telling me that you can perform a coordinate transformation such that the new time direction is contained in S0 (or has a vanishing normal component to S0)?
You will wind up with a new coordinate system with a different notion of simultaniety, and different space-like hypersurfaces S'0. In other words, at the same point in the two coordinate systems, the time vector T0 which is the normal vector to S0 will point in a different direction than the time vector T'0 which is normal to S'0.