Relativistic speed
Relativistic speed refers to speed at which relativistic effects become significant to the desired accuracy of measurement of the phenomenon being observed. Relativistic effects are those discrepancies between values calculated by models considering and not considering relativity. Related words are velocity, rapidity, and celerity which is proper velocity. Speed is a scalar, being the magnitude of the velocity vector which in relativity is the four-velocity and in three-dimension Euclidean space a three-velocity. Speed is empirically measured as average speed, although current devices in common use can estimate speed over very small intervals and closely approximate instantaneous speed. Non-relativistic discrepancies include cosine error which occurs in speed detection devices when only one scalar component of the three-velocity is measured and the Doppler effect which may affect observations of wavelength and frequency.Relativistic effects are highly non-linear and for everyday purposes are insignificant because the Newtonian model closely approximates the relativity model. In special relativity the Lorentz factor is a measure of time dilation, length contraction and relativistic mass increase of a moving object.
Velocity-addition formula
To find the relativistic transformation law, it is useful to introduce the four-velocities , which is the motion of the ship away from the shore, as measured from the shore, and which is the motion of the fly away from the ship, as measured from the ship. The four-velocity is defined to be a four-vector with relativistic length equal to , future-directed and tangent to the world line of the object in spacetime. Here, corresponds to the time component and to the component of the ship's velocity as seen from the shore. It is convenient to take the -axis to be the direction of motion of the ship away from the shore, and the -axis so that the plane is the plane spanned by the motion of the ship and the fly. This results in several components of the velocities being zero; .