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In three dimentions the distance between two points is given by:
Where D is the distance, X is the distance in the X direction, etc.
So if a room is 12m long, 3m wide, and 4m tall, the distance between the most distant corners would be D = √(144 + 9 + 16) = √(169) = 13.
No matter how you spin the room, and no matter how you look at it D will stay the same. X, Y, and Z can change, but D won't.
But in order to specify an exact event you need 4 coordinates; 3, for the 3 dimentions of space, and 1 more for the time of the event (physicsists call this 3+1 dimentions).
If opposing football teams each make a touch down, most people will tell you that the events occured 100 yards apart (
= (100 yards)^2). But there's a lot of information missing from that, which raises some problems.1) The touch downs may have happened 100 yards apart, but not necesarily in the same game, or even the same year. You'd hope that the longer the time between events, the greater the space-time distance.
2) From the ball's point of view the distance between the touch downs was zero. The ball (being self centered) claims that the end zones came to it (as opposed to the ball moving to the end zones). The only difference between the events is time (D = 0).
These problems are easily solved using the "timespace interval":
(S^2 can be nagative?)
The interval is a way of measuring distance in both time and space that is invarient. Everyone, no matter how fast they move or in what direction, will always measure the same S.
Since time isn't really a distance (it's time) it is multiplied by C. Light travels at 300,000 km/s. So if you ask: "How far away is 5 seconds from now?", the answer is: 5x300,000 km = 1,500,000 km.
In general you never solve for S the way you might solve for D. Most people leave it squared.
This is a light cone diagram. The blue lines are light beams. Everything below the lines are things that can affect what's happening to you right now, and everything above the lines are things that can be affected by what you're doing.
The green line is something moving slower than light that can influence you: a breeze, a car, a letter, etc.
The red line is coming from something to far away to affect you. It would have to travel faster than light. If something happened yesterday on Alpha Centauri, nobody will even know for several years.
To figure out the interval between two points: square the distance in the X direction, then subtract the square of the distance in the T direction.
You'll notice that the shorttest lines are the lines closest to 45º.
Technically...