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.)

A frame is different from a co-ordinate system. I can’t repeat it enough, so I will say it again – A frame is different from a co-ordinate system.

The confusion between a frame and a co-ordinate system is unfortunately quite common among physicists[1]. 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[2] 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.[3]

That I hope settles the confusion[4].

Endnotes :

[1] 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.

[2] 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”.

[3] 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.

[4] 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.

Sometime ago, Micheal Fisher had come to TIFR. And he gave a public lecture about critical phenomena. It was a great lecture and the presentation was quite impressive.

However, it was supposed to be a public lecture and a lot of non-physicists were there in the audience. Fisher did try his best to take them along, but, I suspect most of them were lost [1]. It is, in fact, quite unfortunate. I think the kind of things he was talking about, are the kind of things everybody should know about.

As Cosma Shalizi writes in his notebook on statistical mechanics, “Statistical Mechanics (and Condensed Matter) [are the] first mathematical, natural science of emergent properties”. But, sadly enough, as he adds

If a non-scientist wants to learn about some large and important part of science, say planetary astronomy or genetics, there are usually a handful of reliable, uncontroversial, well-written, non-technical books about it to be found in the stores and libraries, which will convey at least something of the field’s history, problems, results and methods. By this point there must be dozens of good popular books written on evolution, particle physics, cosmology, relativity and quantum mechanics, notwithstanding that the last two are about as abstract and abstruse as science gets. There are even excellent popularizations of mathematics, in a continuous tradition from E. T. Bell (if not before). ….

A few months ago, when I was trying to explain some parts of my research to my father, I realized I was assuming he knew what statistical mechanics was, and something about how it worked, when in fact he did not. My first thought was to pass on some popular work about statistical mechanics (it’s only fair; he did it to me constantly when I was younger). A great many thoughts later I realized I could not think of a single one which didn’t stake out some very peculiar philosophical position, or did more than just blab about the second law, never mind something as good as Einstein for Beginners or The First Three Minutes or Does God Play Dice? Granted that relativity and particles and chaos are sexy, and statistical mechanics is not, it’s peculiar that there’s nothing. Stat. mech. is, after all, one of the essential theories of current physics, actually used by chemists and biologists and materials scientists, etc., the part of physics most directly applicable to daily life (you could illustrate the core of it with a coffee cup, and the whole with a kitchen), and bound up with deep puzzles about why time goes the way it does. This cries out for a remedy.(italics in the original)

And the deeper you go in condensed matter, popular science books become rarer to find. Whenever I try to talk about condensed matter physics to somebody back home, I am irrevocably drawn into talking about its “usefulness” vs what makes it scientifically interesting. Peter Armitage indeed has a point when he wrote

As a field we can be justifiably proud to have discovered the physics that led to the transistor, NMR, superconducting electronics etc etc. But this boon has also been a curse. It has made us lazy and has stifled our capacity to think creatively about outreach in areas where we don’t have the crutch of technological promise to fall back on.

This is a luxury our cosmology colleagues don’t have. They feel passionately about their research and they have to (get to?) convey that passion to the public (with predictably good results). We feel passionately about our research, but then feel compelled to tell boring stories about this or that new technology we might develop (which predictably elicits yawns and perhaps only a mental note to take advantage of said technology when it is available in Ipod form). We do this because we are bred and raised to think that technological promise is a somehow more legitimate motivation to the outside public than genuine fundamental scientific interest. It doesn’t have to be this way.

Due to our tremendous technological successes there is also the feeling then that at some level ALL our work should touch on technology. This is the easy strategy, but ultimately it hasn’t been good for the health of the field. This is because, for many of us, technology isn’t our passion and it shows…..

The reality is that many of us in CMP don’t have the inclination or interest to ‘make’ anything at all. For instance, we may pursue novel states of matter at low temperature and consider the concept of emergence and the appearance of collective effects to be just as fundamental and irreducible as anything in string theory. We should promote what excites us in the manner that it excites us….

Meanwhile, I just got hold of a book from TIFR library titled “Constitutions of matter” by Martin.H.Krieger[Amazon][American Scientist Review] [Journal of Chemical Education Review]. It is really not a popular science book, but if you like the kind of mathematical physics that statistical mechanics throws up, you might like it.

[1] Some of my colleagues were of the same opinion. In fact, I opined that biology students might have got something about scaling whereas my colleagues were a bit more pessimistic.

What is a frame ?

February 5, 2007

I was looking around for something to post for (I’ll come around to plasma physics as promised sometime later this week ) .

Jennifer (of Cocktail Party Physics) makes a list of “the Top Ten Things About Physics We Wish Everyone Knew” – Lo and behold, I find one of my favorite topics to ramble about –

3. Frames of reference. Yet another bit of jargon so common to scientists, they forget that the phrase might not hold any real meaning for John/Jane Q. Public, even though it’s a fairly simple concept. It’s still necessary to define the term. Chad touched on this in his post on forces, but it’s central enough that it bears repeating. For instance, it’s tough for a non scientist to grasp why scientists occasionally argue about centrifugal versus centripetal force without a solid grasp of frames of reference. It’s just as critical when considering the differences, physics-wise, between linear and rotational motion, and to understanding why Einstein’s theory of special relativity was such a revolutionary advance….

The fact that this is my favorite topic is not a secret, of course. But, I’ll try a slightly different tune this time – I will assume that you’ven’t read any of the links in the previous line.

A frame of reference is basically a convention that is very useful in physics. Before going into what it actually is, let us look at a simpler but a related concept.

Consider the surface of the earth . On the earth, we find it very useful to name a specific direction as “North”. You can goto any place on earth(except the poles) and you have a reference direction which all of us have agreed to call as “North”. Similarly, we call a specific direction as “East”.

Now, why is this a useful thing ? It is useful because it gives a way for people to communicate with each other. Consider, for example, a person on a plane flying over the place marked P in the figure below. Now, if we want to tell the pilot to goto the place marked Q , one of the easiest ways to communicate the instruction is to ask him to go say 5 kilometres towards East.[1]

Let me invent a shorthand and give an instruction – “Fly 5 E


Similarly, if you have a person at the place marked R , to goto Q , you just have to tell him to go 5 kilometres towards East and then 4 kilometres towards North. So, now the instruction is “Fly 5 E + 4 N“. So far so good.

Now, imagine that I intend to meet this pilot at the place Q some 10 hours after now. So, including the travel time, I want him to wait for 10 hours and be at Q after 10 hours. Now, how do I say that ?

Let’s assume the time taken for travel is very small. So, basically, I tell the pilot to fly 5 E + 4 N and then wait for 10 hours. Let us combine the two instructions into one and send him a single line instruction “Fly 5 E + 4 N + 10 T ” – of course, T represents waiting or “flying in time”.[2]

Now, the question is this – Is my instruction unambiguous ? At the first sight, it does seem to be . But, I will insist that it is not !

To understand why, consider this possibility – say the pilot goes to the place Q and he is very tired after the journey. Since he has ten more hours, he decides to have a good sleep. So, he boards a good train going towards the place S and goes to sleep. He wakes up after ten hours to find himself at S. Having faithfully followed my instructions, he is angry that I am not there !

You might be saying – ” Come on, this is cheating. He didn’t just wait. He also traveled some more distance !” But, the pilot can insist that no-he didn’t go anywhere, that it was the stations which moved towards him as he slept. This might sound very philistine, but, technically, he is right !

What is mere waiting for one person can actually be waiting plus some additional motion for a second person, provided the first person is moving as seen by the second person. So, in a sense, what the first person calls waiting is actually what second person sees as waiting with flying.

So, the moral of the story is that it is not enough if I just say “Wait for ten hours”. It is like saying “Fly for five kilometres”. If I tell you “Fly for five kilometres”, you should ask me back – “Along which direction ? ” . Similarly, if I say “Wait for ten hours”, you should ask me according to whose definition of “waiting” – you see like “North” and “East” we also have to define a “way of waiting” so that our instructions are unambiguous.

So, you might be wondering, what has all this got to do with frame of reference ? The answer is simple – A frame of reference 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.

The point about frames of reference is that one way of definition is as good as any other – sky is not going to fall if tomorrow everybody starts calling East as North and North as East. But, I am saying that and much more – heavens are not going to fall even if all of us change our convention of what it means to say that we are just “waiting” .

So, that in short, is what a frame of reference is . I’ve not addressed the other things that Jennifer mentioned – “centrifugal versus centripetal force”, “linear and rotational motion” and things like inertial and non-inertial frames of reference. I’ll probably take it up later in some other post.

[1] Actually I am cheating you. If you take the given figure to be a representation of earth, the distance shown would be about a thousand kilometres. If I had actually shown 5 km on that figure, it would be so small that you would have a hard time seeing what I’ve drawn.

[2] Of course, I am just repeating what I had already told before . The things I put in bold are basically vectors, and T is what physicists like to call a “Time-like vector”.