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Suppose a spaceship is leaving earth, moves close to the speed of light, passes regions of strong gravitational fields, etc., then the Einsteins Field Equations (EFE) should be able to predict the path of the spaceship with great precision. My question is how we specify the position of the spaceship relative to Earth (in curved spacetime). The EFE are differential equations, so that lenghts are obtained as integrals of ds along some path. So, what is the relativistically correct way to describe the position and movement of the spaceship relative to Earth ?