be moving forwards or reverse with respect to . So inverting the equations simply entails switching the primed and unprimed variables and replacing v with −v. Example: Terence and Stella are at an Earth-to-Mars space race. Terence is an official at the starting line, while Stella is a participant. At time , Stella's spaceship accelerates instantaneously to a speed of 0.5 c. The distance from Earth to Mars is 300 light-seconds (about ). Terence observes Stella crossing the finish-line clock at . But Stella observes the time on her ship chronometer to be as she passes the finish line, and she calculates the distance between the starting and finish lines, as measured in her frame, to be 259.81 light-seconds (about ). 1). Deriving the Lorentz transformations There have been many dozens of derivations of the Lorentz transformations since Einstein's original work in 1905, each with its particular focus. Although Einstein's derivation was based on the invariance of the speed of light, there are other physical principles that may serve as starting points. Ultimately, these alternative starting points can be considered different expressions of the underlying principle of locality, which states that the influence that one particle exerts on another can not be transmitted instantaneously. The derivation given here and illustrated in Fig. 3-5 is based on one presented by Bais and makes use of previous results from the Relativistic Composition of Velocities, Time Dilation, and Length Contraction sections. Event P has coordinates (w, x) in the black "rest system" and coordinates in the red frame that is moving with velocity parameter . To determine and in terms of w and x (or the other way around) it is easier at first to derive the inverse Lorentz transformation. There can be no such thing as length expansion/contraction in the transverse directions. y must equal y and must equal z, otherwise whether a fast moving 1 m ball could fit through a 1 m circular hole would depend on the observer. The first postulate of relativity states that all inertial frames are equivalent, and transverse expansion/contraction would violate this law. From the drawing, w = a + b and From previous results using similar triangles, we know that . Because of time dilation, Substituting equation (4) into yields . Length contraction and similar triangles give us and Substituting the expressions for s, a, r and b into the equations in Step 2