Computing Spatial Distance in General Relativity

[eok20]

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 $$\gamma$$ in the spacetime (M,g). At every point $$\gamma(t)$$, I can find a coordinate neighborhood and vector fields (not arising from coordinates in general) $$T, X_1,X_2,X_3$$ defined in this neighborhood that are orthonormal with respect to g. Then I would somehow want to integrate the distribution determined by $$X_1,X_2,X_3$$, 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!

 

[djy]

To get a distance, you need to divide up space and time. For some observer, the logical way to do this is to send out geodesics that are orthogonal to the observer's world line at a given time. Assuming one of these geodesics intersects the spaceship's world line, then its distance to the observer is simply the length of the geodesic.

I think this is basically what you said.