I am wondering if, in general relativity, there is a way to make sense of a statement such as, "the spaceship is 100km from me." In special relativity, we could define this (as long as I am an inertial observer) by choosing global coordinates (t,x1,x2,x3) corresponding to my notions of time and space, and then restricting the metric to the hypersurface t = 0. Then I have a Riemannian metric which will give me the distance to any point with t = 0. Now, in general relativity I am represented by a curve [tex]\gamma[/tex] in the spacetime (M,g). At every point [tex]\gamma(t)[/tex], I can find a coordinate neighborhood and vector fields (not arising from coordinates in general) [tex]T, X_1,X_2,X_3[/tex] defined in this neighborhood that are orthonormal with respect to g. Then I would somehow want to integrate the distribution determined by [tex]X_1,X_2,X_3[/tex], giving a submanifold to which the pullback of g is definite. Then I would have a Riemanian metric and I can define the distance to anything that is in this submanifold. The problem with this is that I can only define the distance to something in this submanifold, and that is even only if I can integrate the distribution. Is this the right way to think about this? I feel like I may be missing something obvious... Thanks!