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PHYS195 NotesChapter12

PHYS195 NotesChapter12 - Chapter 12 The nature of...

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Chapter 12 The nature of space-time 12.1 The problem of coordinates The basic problem of physics is to track in space and time the development of elements of a system. This requires that we have some method to com- municate where and when something took place. In a three dimensional space the place is a set of three numbers; for instance, in a room you could use how far along the floor in a direction along one wall, how far along another wall, and how far up towards the ceiling. The time comes from a clock. This seems so obvious that we generally do not even think about it but, like all the things that we do, this is a subtle operation and we should understand what it is that we are doing when we make a coordinate system. In fact, the realization, that the establishment of the coordinate system is arbitrary is the key to understanding General Relativity. That will come later, Chapter 15. First, lets talk about places. The idea is to label the places. Think of a large parking lot, say at Disney Land. What you need is a unique label for every place. This could be done simply by going around and labeling spots on the lot with the name of a Disney character. This though is not an e ffi cient way to label places. It is a unique label for each place which is how we started but there are many better ways to proceed. For one thing, this labeling scheme does not provide a guide for movement. If you are at Donald Duck, you do not know how far or in what direction to go to get to Goofy, the labels are not an ordered set. You could fix this by say ordering the characters alphabetically. This system is nice in that it provides a guide to how to move, it does not indicate how far. It is also not extendable. Another reason that it won’t work because you are really in a two di- 267
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268 CHAPTER 12. THE NATURE OF SPACE-TIME mensional space and you are really only using one sequence of labels. Of course, you could wrap the ordered set of labels so that they would still cover the parking lot but this does not help you to know how far to go when you want to move between labeled points and compounds the extendability problem. Thus there are two problems. First, you need a distance. You can use the length that we discussed in Section 2.3.1. In the present case, this means that we define length from how far light travels in a given time. Secondly, what happens with the idea of extension. What happens when you add to the lot? You have to relabel everything. You can still cover the lot with labels but it is not convenient. By the way, this fact that you can cover a two dimensional space with a wrapped one dimensional label is also a simple proof of the size of the spaces are the same and thus that, although it might appear that a two dimensional infinite space seems bigger than an infinite one dimensional, there are as many points on the plane as there are on a line. Thus since you want to extend in a direction that is not along the direction of the chosen sequence, you can improve things quite a
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