The spacetime of special relativity, formally called Minkowski space, has a different way of calculating the distance between points.
In a bit more detail: Take a space with 4 space dimensions, which I'll label with x, y, z, w. Also take Minkowski spacetime with 3 space and 1 time dimension and coordintes x, y, z, t. If I pick two points in either space, then ds denotes the distance between them. dx, dy, dz, dt or dw denote the difference in the values of that coordinate between the two points.
Then distance in the purely spatial space is calculated by: ds^2 = dw^2 + dx^2 + dy^2 + dz^2
And distance in Minkowski space is calculated by: ds^2 = -dt^2 + dx^2 + dy^2 + dz^2
The "weird" geometry comes down to that minus sign.
But I still don't see how traveling in the spatial deminsions can make the spacetime distance between two points SHORTER.
I'm not sure I get the question but in SG's example the "all spatial" universe is not ours.
We move through a 4 dimensional spacetime and distance is calculated as shown by the x, y, z and T example. That is the correct method of calculating distance through our spacetime and produces the odd results. At least that is my understanding (again, waiting for correction).
But I'd love to see an actual calculation because I can't show it :( in mathematical detail. (and I tried ) :(
It's the Pythagorean theorem. x, the spatial separation is the hypotenuse of a right triangle. s, the spacelike spacetime interval; and, t, the time interval make up the adjacents. The hypotenuse is always the longest side.
Edited by lyx2no, : Meant time interval, not timelike spacetime interval. Got carried a way with parallel sentence structure.
Edited by lyx2no, : Typos, confusing ones.
Everyone deserves a neatly dug grave. It is the timing that's in dispute.
If nobody minds I'll do the calculation for a twin that stays on Earth and one that goes to the moon and back. All calculations are done in the twin who remains on Earth's frame. I'm also going to use ds^2 = dt^2 - dx^2 - dy^2 - dz^2, which is equivalent to ds^2 = -dt^2 + dx^2 + dy^2 + dz^2, but easier to use in this case.
Basic set up: First of all I have to get the problem of units out of the way.
Since I'm computing a spacetime distance I'm going to have to use the same units for all quantities. Distances in space are measured using meters. So let's take a meter as our measurement. However humans use seconds for measuring distances in time, in order to compute the spacetime distance I'll have to convert a second into meters.
Well basically a second is (roughly) 300,000,000 meters in the temporal direction.
Now I'll ignore y and z. So I'll be using ds^2 = dt^2 - dx^2.
First of all the starting location of both twins will be labelled as: (0 ; 0). That is t=0, x=0.
Stationary twin: If one twin sits where they are for four seconds they end up with coordinates: (1,200,000,000 ; 0) or t=1,200,000,000 and x=0.
Now I'll compute the spacetime distance. Since dx=0 (no difference or change in spatial coordinate) we just have ds^2 = dt^2.
Moving twin: The moon is roughly 384,000,000 meters from Earth. The second twin starts at Earth and travels to the moon in two seconds. So they start at (0 ; 0) and end up at (600,000,000 ; 384,000,000). The spatial difference is dx = 384,000,000 - 0 = 384,000,000 Similarly, dt = 600,000,000. dx^2 = 147,456,000,000,000,000 dt^2 = 360,000,000,000,000,000
So far as I can understand it, the travelling twin is not affected by his velocity, but by his acceleration. One of the twins is accelerated, the other isn't, and so far as I understand it, that's the difference between them.
Now I'd like to make a couple of points.
First, Einstein's ideas have been borne out by meticulous experiments: his ideas may sound crazy, but they're right.
Second, I'm a mathematician. Of the two of us, I am much, much closer to being able to understand the math of general relativity. Nonetheless, I can't be bothered. There is so much more interesting stuff to learn and do that I shall live and die without bothering to learn what it was Einstein was trying to tell us, and on my deathbed I won't be saying "Damn, I should have spent more time learning about General Relativity", I'll be saying "Damn, I should have spent more time having sex".
This is kind of a serious point. The people who study physics know that Einstein was right, and they know why Einstein was right. Unless we're going to try to become physicists, and learn the math, we might as well shrug our shoulders and say "yeah, it's weird, but apparently it's true".
However, relativity theory has to transform between the different frames of reference and that produces different numerical values for a "tick" as you transform space and time variables. Neither clock (on earth or in the GPS satellites) are changed. But the calculations to compare them to one another (in whichever reference frame you pick) changes the numbers attached.
Thanks for the explanation, it made perfect sense.
All great truths begin as blasphemies
I smoke pot. If this bothers anyone, I suggest you look around at the world in which we live and shut your mouth.