Relation between Time and Velocity in Special Relativity

The theory of special relativity explains that time and length can be changed depending on the observer's speed. A few months ago, I have struggled with instantly coming out with the answers to easy science questions in special relativity - I had to calculate the change of time everytime I confronted that kind of problem. So I was wondering if there would be any easy and straightforward way of remembering the results.

 

A few months later, I read a comic book titled <Quantum>. In the comic, there was a page showing how increase of velocity in space leads to the decrease of speed(?) in time. Its explanation, which involved dividing the area of a circle into two parts - speed in space & speed in time - and showing how the area of the two sections change. It was a fairly straightforward explanation to me, and I accepted it. But suddenly, a few days ago, I wondered if the area representation is mathematically correct, and if it isn't, what was?

 

I decided to figure it out myself. Since there is no such thing as the 'speed' fo 'time', I defined the 'time velocity' as τ(tau), along with 'c-velocity'- a slightly different meaning of velocity - as σ(sigma).

By the law of time dilation, Δt'= γΔt, thus τ= Δt/Δt'. For example, when v=0, τ=1 and γ=1, progress of time is 1, so time flows the same as proper time. But when v→c, τ→0 and γ→∞. In that situation, Δt'→ ∞*Δt. This means that a second observed by the moving observer needs to be repeated infinitly to be equal to a second measured by a stationary observer. In other words, time slows down as v nears c, and τ decreases. The larger the value of τ, the faster(nearer to proper time) time flows. Of σ, the progress in space, its value is proportionate to the observer's velocity. With these two new variables, a simple formula can be derived.

 

An incredible result that the squares of progress of time & progress in space is always 1 is derived! For an even simpler look, draw it on a unit circle: 

You can easily picture the relation between speed and time by simply moving point X on the unit circle. When velocity increases(σ increases), τ, the progress of time, decreases and time slows down.