Einsteins Miraculous Year, Five Papers That Changed the Face of Physics

PAPER 3 where √y =t x =0 V 2 − v2 hence η = a√ V y V 2 − v2 and ζ = a√ V z V 2 − v2 If we substitute for x its value, we obtain τ =ϕ v β t − v x V2 ξ = ϕ v β x − vt η=ϕvy ζ =ϕ v z where β= 1 v 2 V 1− and ϕ is an as yet unknown function of v. If no assumptions are made regarding the initial position of the moving system and the zero point of τ, then a constant must be added to the right-hand sides of these equations. Now we have to prove that, measured in the moving sys- tem, every light ray propagates with the velocity V , if it does so, as we have assumed, in the rest system; for we have not yet proved that the principle of the constancy of the velocity of light is compatible with the relativity principle. Suppose that at time t = τ = 0 a spherical wave is emitted from the coordinate origin, which at that time is common to 134

ELECTRODYNAMICS OF MOVING BODIES both systems, and that this wave propagates in the system K with the velocity V . Hence, if x y z is a point reached by this wave, we have x2 + y2 + z2 = V 2t2 We transform this equation using our transformation equations and, after a simple calculation, obtain ξ2 + η2 + ζ2 = V 2τ2 Thus, our wave is also a spherical wave with prop- agation velocity V when it is observed in the moving system. This proves that our two fundamental principles are compatible. 3 The transformation equations we have derived also con- tain an unknown function ϕ of v, which we now wish to determine. To this end we introduce a third coordinate system K , which, relative to the system k, is in parallel-translational motion, parallel to the axis , 4 such that its origin moves along the -axis with velocity −v. Let all three coordinate origins coincide at time t = 0, and let the time t of system K equal zero at t = x = y = z = 0. We denote the coordi- nates measured in the system K by x y z and, by twofold application of our transformation equations, we get t = ϕ −v β −v τ + v ξ = ϕ v ϕ −v t V2 x = ϕ −v β −v ξ + vτ = ϕ v ϕ −v x y = ϕ −v η = ϕ v ϕ −v y z = ϕ −v ζ = ϕ v ϕ −v z 135