special relativity - Why do things slow down when you move faster, rather than speed up?

I've been trying to get to grips with SpaceTime.

As I understand it, we move at a set rate through spacetime. Any increase in our rate of travel through space results in a decrease in our rate of travel through time (via standard vector maths, pythagoras)

Where I get confused is the implication for our perception of time. Do we perceive time as the magnitude of our movement through spacetime, and the speed of someone stationary as the 'time' component of our vector through spacetime? Wouldn't this mean that, as a diagonal line is longer than a straight one of the same height, that things around us should slow down as we move more quickly?

Say this triangle is our vector in Spacetime. If $a$ is the 'time' component of spacetime, and $b$ is the 'space' component, does that mean that the $c$ component is of fixed magnitude? Do we perceive things around us moving at a rate of $a/c$? (ie, slower than they occur now)

I know other questions about spacetime have been asked, but I can't find one that explains this aspect.

Answer

You've hit on one of the fundamental "weirdnesses" of the Minkowski "metric". Its name "metric" is a bit misleading if you're a mathematician: it does not fulfil two out of the of the three axioms of the distance function defining a metric (in the topologist's words) space. There are null vectors, i.e. nonzero vectors $X$ for which $\langle X,\,X\rangle=0$ and it is NOT subadditive i.e. it does not fulfill the triangle identity. So the intuition you have from Euclidean (and general metric) space that the sum of the lengths of two of a triangle's sides must be longer than the third alone is WRONG!

The names "metric", "inner product" and "norm" in connexion with Minkowski spacetime reflects (1) the "structural" likeness that these operations as algebraic operations have to genuine metrics, inner products and norms (written on a page, they look a great deal like the genuine ones) and (2) the fact that many of the main theorems of Riemannian geometry also hold when we replace genuine inner products with nondegenerate ones (i.e. the matrix representing the two form is nonsingular) such as the Minkowski "inner product".

I recall many years ago reading Bert Mendelson in his wonderful introductory undergrad textbook "Topology" making the memorable and intuitive statement:

".... the [triangle] inequality $\mathrm{d}(x,\,z)\leq \mathrm{d}(x,\,y) + \mathrm{d}(y,\,z)$ may be thought of as asserting the transitivity of [the] closeness [relationship]; that is, if $x$ is close to $y$ and $y$ is close to $z$, then $z$ is close to $x$." (Chapter 2, Section 2, p 31 in my edition)

This wonderful statement - this transitivity - is precisely what is shattered in the twin paradox. The spacefaring twin at the halfway point ($y$) in his journey can be close to his beginning point ($x$) (he ages very little - the path has only short proper time), and at the end of his journey at ($z$) he can be near to his halfway point (again, he ages very little coming back), but ($z$) in spacetime is far, far away from ($x$): the Earthbound twin is now a decrepit, bent up old man.

PS if you study topology and they try to get you to read Munkres, I highly recommend Bert Mendelson's book to be read together with the heavier Munkres.