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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 Shellfire said:Consider a spacetime, with a specific metric, sliced in spatial hypersurfaces. Take an observer located at a point of slide S_{0}. Are you telling me that you can perform a coordinate transformation such that the new time direction is contained in S_{0}(or has a vanishing normal component to S_{0})?

_{0}as well as the normal component.

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 T

_{0}which is the normal vector to S

_{0}will point in a different direction than the time vector T'

_{0}which is normal to S'

_{0}.