Lorentz Transformations

in the special theory of relativity, transformations of the coordinates and time of an event on changing from one inertial frame of reference to another. These transformations were derived in 1904 by H. A. Lorentz as transformations for which the equations of classical microscopic electrodynamics (the Lorentz-Maxwell equations) retain their form. In 1905, A. Einstein derived them by proceeding from two postulates that formed the basis of the special theory of relativity: the equivalence of all inertial frames of reference and the independence of the velocity of light in a vacuum from the motion of the source of light.

Let us examine a particular case of two inertial reference frames Σ and Σ’ with axes x and x’ lying on the same straight line and with the other correspondingly parallel axes (y and y’, z and z’). If the system Σ’ moves with respect to Σ at a constant velocity v in the direction of the x-axis, then the Lorentz transformations for the transition from Σ to Σ’ have the form

where c is the velocity of light in a vacuum (the coordinates with primes refer to system Σ’, and the coordinates without primes, to Σ).

The Lorentz transformations lead to a number of important consequences, such as the law of addition of velocities in the theory of relativity and the dependence of the linear dimensions of bodies and time intervals on the selected frame of reference. For rates of motion that are small in comparison with the velocity of light (v « c), the Lorentz transformations become Galilean transformations, which are valid in classical Newtonian mechanics.