Frames and Co-ordinates
February 7, 2007
Few days ago, I tried to answer the question – What is a frame ? As we saw, it is not a very difficult concept. But, let me just add something I forgot to mention the previous time. (Actually, I’m worrying about some other problem today – so it’ll be a short post.)
The confusion between a frame and a co-ordinate system is unfortunately quite common among physicists. A frame, as I have already explained, is basically a convention which decides ad hoc – what is “North”, what is “East”, What is “up” and what kind of “waiting” is a standard at each place and for all times.
A co-ordinate system, in contrast, is a set of numbers given to each place at a particular time. For example, I can say give a set of numbers (1,2,3,4) to the place I was born as it was in the time I was born. Or give a set of numbers (5,6,8,0) to the place I was sitting in as it was when I was typing the previous full-stop. And if I can give such a set of numbers to each and every place, as it was/is/will be at every instant of time, then I am supposed to have “established” a co-ordinate system.
That I hope settles the confusion.
 Even in a cartan-conscious book like the one titled “Gravitation” (Misner,Thorne and Wheeler), it is not unusual to see a paragraph in which they use these terms interchangeably. And I think this confusion is a de-facto standard in engineering and physics outside general relativity.
 It is an interesting question to ask – how many such numbers do you require to cover every place at every instant ? Experience tells us that we need at least four numbers. This is what we mean when we say we live in a “four-dimensional space-time”.
 Quite often, it happens that it is neither necessary nor possible to “establish” such a system. In that case, we tone down our ambitions and worry only about some places as they were/are/will be during some instants. Such a thing can be called a local co-ordinate system.
 Stated like that, you might wonder why people confuse between these two words. The point is this – often co-ordinate system is used to construct a frame. The trick goes something like this . To give a standard way of waiting, you go about as follows.
a) Take the set of numbers associated with the place where you start waiting as it was at the instant you started waiting.
b) Similarly, take the set of numbers associated with the place where you end up after waiting as it was at the instant you finished waiting.
c) Now, choose the kind of waiting which keeps the first three numbers the same between a) and b) and declare that kind of waiting to be standard.
Similarly, by fixing the third number instead of the fourth, you can define “North”. And you can do the same thing for defining “East” and “Up” using the rest of the two numbers. Such a frame defined using co-ordinates is said to be a co-ordinate frame.