= OK, and the red line representing the world line of a particle in motion has the equation w = x/β = xc/v. A bit of algebraic manipulation yields The expression involving the square root symbol appears very frequently in relativity, and one over the expression is called the Lorentz factor, denoted by the Greek letter gamma : If v is greater than or equal to c, the expression for becomes physically meaningless, implying that c is the maximum possible speed in nature. For any v greater than zero, the Lorentz factor will be greater than one, although the shape of the curve is such that for low speeds, the Lorentz factor is extremely close to one. In Fig. 3-3b, segments OA and OK represent equal spacetime intervals. Length contraction is represented by the ratio OB/OK. The invariant hyperbola has the equation , where k = OK, and the edges of the blue band representing the world lines of the endpoints of a rod in motion have slope 1/β = c/v. Event A has coordinates (x, w) = (γk, γβk). Since the tangent line through A and B has the equation w = (x − OB)/β, we have γβk = (γk − OB)/β and Lorentz transformations The Galilean transformations and their consequent commonsense law of addition of velocities work well in our ordinary low-speed world of planes, cars and balls. Beginning in the mid-1800s, however, sensitive scientific instrumentation began finding anomalies that did not fit well with the ordinary addition of velocities. Lorentz transformations are used to transform the coordinates of an event from one frame to another in special relativity. The Lorentz factor appears in the Lorentz transformations: The inverse Lorentz transformations are: When v ≪ c and x is small enough, the v2/c2 and vx/c2 terms approach zero, and the Lorentz transformations approximate to the Galilean transformations. etc., most often really mean etc. Although for brevity the Lorentz transformation equations are written without deltas, x means Δx, etc. We are, in general, always concerned with the space and time differences between events. Calling one set of transformations the normal Lorentz transformations and the other the inverse transformations is misleading, since there is no intrinsic difference between the frames. Different authors call one or the other set of transformations the "inverse" set. The forwards and inverse transformations are trivially related to each other, since the S frame can only