ИСТИНА

We consider two most popular definitions of velocities of remote objects in General Relativity. First definition is the velocity as derivative of proper distance to the object with respect to the proper time of the observer. This definition is widely used in cosmology. Another definition is based on parallel transport of the initial emitter’s 4-velocity along some curve in 4-dimensional space-time to the observer. We consider advantages and counterintuitive properties of both definitions in detail. We derive the generalization of the formula connecting these two velocity definitions in Friedmann-Lemaître- Robertson-Walker (FLRW) metric found by Chodorowski (M. Chodorowski, The kinematic component of the cosmological redshift. MNRAS, 413, 585 (2011)). We show that it is valid for any synchronous spherically symmetric metrics. Then we extend this formula to non-zero peculiar velocities of the emitter.