Is this a basis for the Bergman space?

The faster you travel through space, the slower you travel through time, and vice versa - Does this have a mathematical basis in physics?

• When someone asks why time slows down significantly for a particle travelling near c, you often hear the following explanation: Spacetime is a continuum. The faster you travel through space, the slower you travel to time; that's a property of this continuum. So for an object near c (travels fast through space), time slows down. For an object at rest, time moves at the fastest possible rate. Does this have a mathematical basis in special relativity, or is it just a nice way of simplifying the true physics?

• Answer:

  When you are moving away from a position and then back to the position, very fast, your trajectory in space time makes two legs of a triangle. A friend that stays home makes the third leg. In geometry, the length of two legs of a triangle is always longer than the third, it's a consequence of the pythagorean theorem. In relativity, the pythagorean theorem has a minus sign for time, and so the sum of the two legs is shorter than the third, when all three are mostly pointing in the time direction. The length of the time-pointing leg is the time passing along the leg as measured by a clock moving along the trajectory of the leg, by definition. To understand why the pythagorean theorem has a minus sign, you should know that for flat space, so that parallel lines are unique and parallel and described by linear equations, there are exactly three possibilities for a symmetrical space, the Galilean space of Newtonian mechanics, where space and time are separate, the Euclidean space of geometry, which reduces to Galilean space when all the slopes are small, and the Einstein-Minkowski space, with the minus sign in the pythagorean theorem, which is the spacetime of special relativity, which unlike Euclidean geometry contains special slope--- the slope of a light-ray trajectory in space-time. The Minkowski space also reduces to Galilean geometry for small slopes, in space-time, that means small speeds. To understand this, I gave a quick synthetic proof of the relativistic pythagorean theorem in my answer to this question on stackexchange: http://physics.stackexchange.com/questions/12435/einsteins-postulates-minkowski-space-in-laymans-terms .