immediately yield The above equations are alternate expressions for the t and x equations of the inverse Lorentz transformation, as can be seen by substituting ct for w, for , and v/c for β. From the inverse transformation, the equations of the forwards transformation can be derived by solving for and . Linearity of the Lorentz transformations The Lorentz transformations have a mathematical property called linearity, since and are obtained as linear combinations of x and t, with no higher powers involved. The linearity of the transformation reflects a fundamental property of spacetime that was tacitly assumed in the derivation, namely, that the properties of inertial frames of reference are independent of location and time. In the absence of gravity, spacetime looks the same everywhere. All inertial observers will agree on what constitutes accelerating and non-accelerating motion. Any one observer can use her own measurements of space and time, but there is nothing absolute about them. Another observer's conventions will do just as well. A result of linearity is that if two Lorentz transformations are applied sequentially, the result is also a Lorentz transformation. Example: Terence observes Stella speeding away from him at 0.500 c, and he can use the Lorentz transformations with to relate Stella's measurements to his own. Stella, in her frame, observes Ursula traveling away from her at 0.250 c, and she can use the Lorentz transformations with to relate Ursula's measurements with her own. Because of the linearity of the transformations and the relativistic composition of velocities, Terence can use the Lorentz transformations with to relate Ursula's measurements with his own. Doppler effect The Doppler effect is the change in frequency or wavelength of a wave for a receiver and source in relative motion. For simplicity, we consider here two basic scenarios: (1) The motions of the source and/or receiver are exactly along the line connecting them (longitudinal Doppler effect), and (2) the motions are at right angles to the said line (transverse Doppler effect). We are ignoring scenarios where they move along intermediate angles. Longitudinal Doppler effect The classical Doppler analysis deals with waves that are propagating in a medium, such as sound waves or water ripples, and which are transmitted between sources and receivers that are moving towards or away from each other. The analysis of such waves depends on whether the source, the receiver, or both are moving relative to the medium. Given