General Expressions for Measurable Geometric and Kinematic Quantities in Curved Spacetime
General Relativity , Geometry , Astrophysics
General covariant expressions for measurable angles, distances, velocities, and accelerations are provided in terms of fundamental parameters that can be applied in any setup. The relativistic aberration of light relationship is presented in full generality, which is applicable to any orientation of observers and light rays. An expansion for the geometrical exponential map is established and used to form an expression for the physical distance between an observer and a nearby object within its extended local frame. Curvature effects on such distances, velocities, and accelerations are made explicit and appear in general tensorial form. The concepts of Fermi frames on timelike worldlines and the Fermi-Walker derivative are discussed in detail and used throughout; and in examining the meaning of relative stationarity between timelike observers, the Fermi-Walker derivative is established from first principles through physically meaningful consideration. A generalized type of Taylor expansion is provided for tensors of any rank in a covariant form. Expressions for the optically based angular diameter distance and luminosity distance are provided in general forms, and the reciprocity theorem is discussed and verified. Effect of the Weyl tensor component of curvature on volume deformation is demonstrated (contrary to the common claim in the opposite), and a generalized version of the geodesic deviation equation, applicable to extreme relative motion, is provided.