But it does, it falls out of the geometric relationship between the three space dimensions and the time dimension. In a way, the speed of light can be thought of as the speed of time.

Think of the relationship between space and time as orthogonal, as a right angle -- which makes sense, since the time dimension is at right angles to the three space dimensions. If you move quickly through space, you can't also move quickly through time, and the relationship between your time and space velocities is just what you would expect for an orthogonal relationship:

t' = t √(1-v^2/c^2)

t = time at rest

t' = time at velocity v

c = speed of light

Einstein wrote the above in 1905, then his math teacher (Minkowski) read Einstein's paper, saw the above equation, and realized it meant time was a fourth dimension. Minkowski then famously said, "Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality."

In response, Einstein said, "Since the mathematicians have invaded the theory of relativity, I do not understand it myself anymore."

But from that point to 1915, Einstein acquired much more math knowledge and used it to write the General theory.