Finally, if c = omega1 x r, and -c = omega2 x r, divide both sides of both equations by r (distance = light travel time) Then multiply them together and for a simple case, omega1=omega2: 1/t^2 = omega^2 1/t = +/- omega And we finally have the identity involving time Minkowski was missing. Notice that the speed of light isn't even involved unless we know that photons of sufficient energy can combine in this way to form matter. Time is always positive (one direction) no matter which direction omega (or light) propagates. The rest can be taken care of using the results of the Lorentz transformations. Be careful how you treat time dilations. They are just as valid for circular paths as they are for linear ones. In this way, you understand that time has two components. One is energy-like, and one is matter-like, but in both cases, space is simply light travel time and has no inertia until or unless it is bound, and by that we mean 'spinning'. Time literally is inertia. Triple sizing your soft drinks and French fries probably won't help extend your life by means of time dilation, unless you eat enough to become a black hole, that is. Yeah, that would work. Don't you just love physics that has real, practical applications? Minkowski didn't.