Back in 1921, Theodor Kaluza published his idea of a very tiny curled up fifth dimension, which I will refer to as the "w" dimension. In a space consisting solely of the x and w dimensions, space is cylindrical, and a ray traveling at the speed of light along x looks like a light wave, while the same ray traveling in the w direction looks like a particle. Changing the direction the ray travels produces all the distortions of length and time we recognize as the Lorentz transformations. Einstein was intrigued by the possibilities of such a curled-up dimension, but the idea never really caught on in his lifetime. I'm trying to think through the mathematics of Kaluza-space, and need some mathematical help. I've worked out the formula for a ray traveling at the speed of light through an xw space where the "w" dimension gets larger as one travels along the x axis. (In such a case, xw forms a cone rather than a cylinder.) If the cone is long and skinny enough (i.e., about a light year from the vertex to a diameter much smaller than an electron) the formula for the observable motion of a ray is indistinguishable from d=1/2 a*t^2 (for small elapsed time "t") and from v=c (for very large t). I'd like to compare my formula for displacement as a function of time with what the basic theory of relativity would predict, but it turns out to be hard to find the formula for the motion of a particle subjected to a constant force like that of gravity. I am NOT looking for the formula for a rocket that "accelerates" at a constant rate (within its own frame of reference), but for the path of a rock thrown down an "infinitely deep well" which just happens to have a physically impossible constant gravitational pull all the way down. Does anybody happen to know this formula off hand? If not, can someone derive it for me?