SciWriter: Where/how did you come up with…
I got it from here:
I found this on the internet, by Eddie Current:
The Pythagorean theorem works in three dimensions, too. If you see a blimp in the sky, you can calculate the exact straight-line distance to the blimp by knowing its altitude above the ground (the value z), as well as how far east-west (x) and north-south (y) you'd have to go to get right under the blimp:
x2 + y2 + z2 = d2
where d is the distance to the blimp. I don't have a 3D diagram, but you can prove it for yourself with a little effort.
Now it gets interesting. This trick extends to four dimensions. Time is typically cited as the fourth dimension. Does the decidedly non-geometric idea of time work into the Pythagorean theorem? Incredibly, it does — but first, you have to convert the time measurement into a distance-like measurement. Then, the total distance you’re calculating is the spacetime distance in the bizarre four-dimensional world where east-west, north-south, up-down, and earlier-later mean the same thing, only in eight different directions. Represented by the letter s, spacetime distance (also known as a Minkowski interval) is determined by an amazing formula. Let’s break it down:
x2 + y2 + z2 – (ct)2 = s2
As before, x is the distance (for example) to the east, y is north, and z is up, but we’ve added a fourth term for time (t), which gets multiplied by a constant, c. Notice the minus sign before the term for time. When it comes to distance through spacetime, elapsed time counteracts spatial distance, and vice versa: If we travel a distance through space, and do it in a very short interval of time,* the distance traversed is effectively reduced. This is why a space traveler could reach stars across the galaxy within their lifetime if they got close enough to the speed of light. Time goes in the opposite “direction” of space!
That constant, represented by c? It’s the same c that represents the speed of light in equations such as E = mc2. What better number to convert units of time (seconds) into a distance-like measurement — after all, we know that for light, there are 186,000 miles per second. See what Einstein did there? The speed of light is more than just a speed; it’s a universal conversion factor that turns time into a distance-like measurement. By treating time as a negative and multiplying it by c, we can exchange time and space in our formulas as readily as nature exchanges them. That’s what special relativity is all about.
It’s as if the presence of mass causes “zero” to pull apart into the familiar ideas of spatial distance and temporal duration, like taffy. But since the universe is by definition everything there is, you have to be inside the universe to witness this incredible stretching apart of zero, to experience space and time as different things. If you were taking in the all-seeing “God’s-eye view” from a timeless, spaceless, massless perspective outside, you would see the same thing the speed-of-light traveler sees — nothing. To witness the action, you have to be inside the theater, in your seat.
Space and time cancel out to exactly zero for the universe as a whole. But that’s just one example of the zero-sum nature of the physical world. A few others:
• The kinetic energy of everything in the universe is exactly balanced by the gravitational potential energy of everything in the universe. The latter is expressed as a negative number, just as time is in the spacetime formula. A while back Alex Filippenko, who’s a familiar smiling face to science-TV geeks, co-wrote an essay about how this means the universe may have come from “nothing at all.” Like the pulling apart of space and time, kinetic energy and gravitational potential were also pulled apart in the Big Bang.
• For similar reasons, the net charge of the universe is generally believed to be zero, with the number of positively charged particles equaling the number of negative. (This is unproven.)
• Certain pairs of phenomena, like electricity and magnetism or mass and the curvature of space, are linked such that they seem to keep each other in check. The great physicist John Wheeler was fascinated by these “automatic” connections, pointing out how they are constrained together by zero sums, the way the ends of a see-saw are always the same total distance from horizontal. “That this principle should pervade physics, as it does,” he asked in 1986, “is that the only way that nature has to signal to us a construction without a plan, a blueprint for physics that is the very epitome of austerity?”
On the one hand, it’s surprising that quantities totaling zero show up again and again in nature. But on the other it makes sense, if the universe is a closed system incorporating everything there is. As a teen I remember being into the Taoist idea of Yin and Yang — I thought that in the final analysis, the universe as a whole couldn’t be anything but perfectly balanced. On a level deeper than I imagined, I may have been right.
* Slow speeds (which mean long elapsed times) cause the time part of the formula to overwhelm the space part, resulting in large spacetime distances. Spacetime distances only get small when you approach the speed of light, for example, covering 186,000 miles in 1.1 seconds — then the (negative) time part almost cancels the space part.