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## Main Question or Discussion Point

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?

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?