Coordinate Systems
In Einstein’s model, he defines one type of coordinate system. He then applies the postulates and equations to this single type of system. When this coordinate system is moving at velocity v, Einstein concludes that everything within this coordinate system must behave according the same laws of physics.
Extending the Coordinate Systems
Consider two 18-wheeled trucks. On the back of one truck is an enclosed trailer. On the back of the second truck is an animal cage, with the bars just large enough to keep the animals from escaping. The cage and the trailers will be generally referred to as the enclosure. The length of each enclosure is x’.
Cats
At the rear of each enclosure is a cat. The sole purpose of this cat is to walk at velocity j from the rear of the enclosure to the front, turn around, and then return to the rear, completing one “cycle.” At the end of each cycle, the cats repeat their behavior.
When both trucks move forward at velocity v, will both cats return to the rear of the cage at the same time? Ignoring resistance from the air hitting the cat in the cage, the answer is yes. Both cats will return at the same time, and will have traveled the same distance 2x’.
Birds
At the rear of each enclosure, out of reach of the cat, is a bird. The sole purpose of the bird is to fly at velocity w from the rear of the enclosure to the front, turn around, and return to the rear, completing one “cycle.” At the end of each cycle, each bird repeat its behavior.
When both truck move forward at velocity w, will both birds return to the rear of the cage at the same time? In this case the answer is no. The bird in the trailer will return first because the air that it is traveling through is completely enclosed and is also traveling forward. However, the air in the cage is not enclosed. Although the bird is in the cage, it might as well be outside of the cage when it comes to flying.
Both birds perform the same behavior; they fly from the rear of the cage, to the front, and again to the rear. Yet, when their respective enclosure is in motion, the bird in the cage flies further and takes longer to complete the journey.
Complete and Incomplete Coordinate Systems
The difference in behaviors for the two birds requires a revised definition of a coordinate system. This model creates two; a Complete Coordinate System, that explains the behavior of the bird in the back of the trailer, and an Incomplete Coordinate System, that explains the behavior of the bird in the back of the animal cage.
A Complete Coordinate System is one in which the phenomena under observation (or its associated travel medium) is contained within the coordinate system and moves with it at velocity v.
An Incomplete Coordinate System is one in which the phenomena under observation (or its associated travel medium) is not completely contained within the coordinate system and does not move with it at velocity v.
The trailer containing the bird traveling through the enclosed body of air is an example of a Complete Coordinate System, while the cage containing the bird traveling through the unenclosed body of air is an example of an Incomplete Coordinate System. Both enclosures, with respect to the cats traveling on their respective floors, are examples Complete Coordinate Systems.
