Just watched a documentary about galaxies. They mentioned something called Dark Energy that acts opposite to Dark Matter. Apparently Dark Energy repulsive powers will eventually overcome the attractive powers of Dark Matter and cause galaxies to move apart faster than the speed of light. This reminded me of something I've never really been able to fully get my head around.
Suppose you are standing on planet A, you can see two other planets, X and Y in opposite directions (so if you drew a straight line from X to Y it would pass through a):
X and Y start traveling away from you in opposite directions at 100kph. In the frame of reference of planet A each are traveling at 100kph in opposite directions, you might represent this as a vector of 100kph and -100kph in the X-Axis or X(100,0,0) and Y(-100,0,0). If one goes to the X frame of reference then we see A moving away at 100kph and Y at 200kph. Easy enough to model with a long straight road, two cars and a few of those speed guns the police use, maybe Top Gear will give it a go some day.
Speed (or velocity) is just distance traveled over time or increase in distance between two points (one designated as fixed in the frame of reference). So when we say that X and Y are traveling away from A at 100kph were actually saying that in the A frame of references the distance between X and A and between Y and A is increasing by 100km for every hour that elapses. In the X frame of reference The distance between X and A is increasing at 100km each hour and the distance between X and Y is increasing by 200km each hour.
Now suppose we increase the speed, in the A frame of reference, to 5.395x10^8kph. Back in the X frame of reference A is now whizzing away at 5.395X10^8kph and Y at 1.079x10^9kph. Thing is that if we increase the speed, in the A frame of reference, whilst in the X frame of reference the speed of A will increase the speed of Y will remain at 1.079x10^9kph because that is the speed of light in free space (c) and is the top speed matter can attain.
Now say we increase the speed in the the A frame of reference to c and measure the distance between our 3 planets (we'll call that T0) then measure the distance again an hour later (T1). Say at T0 the distance between A and both X and Y is 1.079x10^9km so the distance between X and Y will be 2.158x10^km:
After one hour the distance, still in the A frame of reference, from A to X would now be 2.158x10^9, same as from A to Y, whilst the distance from X to Y would be 4.316x10^9km as both gave been traveling away from A at 1.079x10^9kph:
In the X frame of reference, however, whilst A has been traveling away at 1.079x10^9 Y has also been traveling away at only 1.079x10^9kph, not at 2.158x0^9kph, as that is the maximum speed of matter. Therefore whilst the distance from X to A has increased to 2.158x10^9km to distance fro X to Y in the X frame of reference will be 3.237x10^9km:
That's what confuses me.