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.

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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 JustScience.net. (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

earth.jpg

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”.

Sagan

December 23, 2006

Unsurprisingly, I am late for Carl Sagan Blog-a-thon(See here for a list of posts) . It’s ten years since sagan died.

(Hmm.. Lately my blog has reduced to a series of posts in memorial of different anniversaries ! I promise my readers that I will try to come up with something else for the next post..)

Carl Sagan affected so many of us in so many different ways. I still remember reading Cosmos [Amazon] when I was younger and being awed by the wonder of it all. If you hadn’t noticed before, the quote just beneath this blog’s title comes from sagan.

Sagan is no more. But the spirit which drove him still survives in his wife Ann Druyan who is equally unflinching in supporting science and its popularisation. I could clerly recognise that spirit, for example, when I heard Druyan answer a question in the middle of a discussion about science and religion.

The greatest legacy of Sagan, in my eyes, is the realisation that a culture without science is as impoverished as a culture without art or music. Given how few people realise that fact (with my limited experience, I can safely assert that even most science students don’t realise it) world really needs more sagans today.

These are two videos of sagan I could get hold of at youtube.
Carl Sagan on Alexandria
Part 1

Part 2

You might also like to read this post – “Casting out the demons” by Jennifer Oullette.

Feeling the touch of science

November 27, 2006

Came across the following you-tube video by Neil deGrasse Tyson

Since, I’m not as eloquent as Tyson to describe what it feels like to be in science, I will fall back on one of my favourite poems by Tagore.

Thou hast made me endless, such is thy pleasure.

This frail vessel thou emptiest again and again,
and fillest it ever with fresh life.

This little flute of a reed thou hast carried over hills and dales,
and hast breathed through it melodies eternally new.

At the immortal touch of thy hands my little heart loses
its limits in joy and gives birth to utterance ineffable.

Thy infinite gifts come to me only on these very small hands of mine.
Ages pass, and still thou pourest, and still there is room to fill.

Rabindranath Tagore

Salam

November 21, 2006

(Crossposted to BlogPhysica )
salam.jpg

Ten years ago, on 21st of Novomber, 1996, Abdus Salam, who was among the co-founders of the Standard Model died at his home at Oxford.

For those who don’t know him, he was one of the recipient of the Nobel Prize in Physics(1979) “for their contributions to the theory of the unified weak and electromagnetic interaction between elementary particles, including, inter alia, the prediction of the weak neutral current” in short, for what physicists call the Glashow-Salam-Weinberg Model (which along with Quantum Chromodynamics and Classical General Relativity form the foundations on which all physics stands).

Was just reminded of him as I was doing some calculation with the Salam-Strathdee Superfield Formalism in (Supersymmetric extension of) Glashow-Salam-Weinberg Model. It is amusing how pursuit of science brings back names from far away…

As Salam himself mused before beginning his Nobel Lecture

Scientific thought and its creation is the common and shared heritage of mankind. In this respect, the history of science, like the history of all civilization, has gone through cycles. Perhaps I can illustrate this with an actual example.

Seven hundred and sixty years ago,a young Scotsman left his native glens to travel south to Toledo in Spain. His name was Michael, his goal to live and work at the Arab Universities of Toledo and Cordova, where the greatest of Jewish scholars, Moses bin Maimoun, had taught a generation before.

Michael reached Toledo in 1217 AD. Once in Toledo, Michael formed the ambitious project of introducing Aristotle to Latin Europe, translating not from the original Greek, which he did not know, but from the Arabic translation then taught in Spain. From Toledo, Michael travelled to Sicily, to the Court of Emperor Frederick II.

Visiting the medical school at Salerno, chartered by Frederick in 1231, Michael met the Danish physician, Henrik Harpestraeng – later to be-come Court Physician of King Erik Plovpenning. Henrik had come to Salerno to compose his treatise on blood-letting and surgery. Henrik’s sources were the medical canons of the great clinicians of Islam, Al-Razi and Avicenna, which only Michael the Scot could translate for him.

Toledo’s and Salerno’s schools, representing as they did the finest synthesis of Arabic, Greek, Latin and Hebrew scholarship, were some of the most memorable of international assays in scientific collaboration. To Toledo and Salerno came scholars not only from the rich countries of the East and the South, like Syria, Egypt, Iran and Afghanistan, but also from developing lands of the West and the North like Scotland and Scandinavia. Then, as now, there were obstacles to this international scientific concourse, with an economic and intellectual disparity between different parts of the world. Men like Michael the Scot or Henrik Harpestraeng were singularities. They did not represent any flourishing schools of research in their own countries. With all the best will in the world their teachers at Toledo and Salerno doubted the wisdom and value of training them for advanced scientific research. At least one of his masters counselled young Michael the Scot to go back to clipping sheep and to the weaving of woollen cloth.

In respect of this cycle of scientific disparity, perhaps I can be more quantitative. George Sarton, in his monumental five-volume History of Science chose to divide his story of achievement in sciences into ages, each age lasting half a century. With each half century he associated one central figure. Thus 450 BC – 400 BC Sarton calls the Age of Plato; this is followed by half centuries of Aristotle, of Euclid, of Archimedes and so on. From 600 AD to 650 AD is the Chinese half century of Hsiian Tsang, from 650 to 700 AD that of I-Ching, and then from 750 AD to 1100 AD – 350 years continuously – it is the unbroken succession of the Ages of Jabir,Khwarizmi, Razi, Masudi, Wafa, Biruni and Avicenna, and then Omar Khayam – Arabs, Turks, Afghans and Persians – men belonging to the culture of Islam. After 1100 appear the first Western names; Gerard of Cremona, Roger Bacon – but the honours are still shared with the names of Ibn-Rushd (Averroes), Moses Bin Maimoun, Tusi and Ibn-Nafi-the man who anticipated Harvey’s theory of circulation of blood. No Sarton has yet chronicled the history of scientific creativity among the pre-Spanish Mayas and Aztecs, with their invention of the zero, of the calendars of the ‘moon and Venus and of their diverse pharmacological discoveries, including quinine, but the outline of the story is the same – one of undoubted superiority to the Western contemporary correlates.

After 1350, however, the developing world loses out except for the occasional flash of scientific work, like that of Ulugh Beg – the grandson of Timurlane, in Samarkand in 1400 AD; or of Maharaja Jai Singh of Jaipur in 1720 – who corrected the serious errors of the then Western tables of eclipses of the sun and the moon by as much as six minutes of arc. As it was, Jai Singh’s techniques were surpassed soon after with the development of the telescope in Europe. As a contemporary Indian chronicler wrote: “With him on the funeral pyre, expired also all science in the East.” And this brings us to this century when the cycle begun by Michael the Scot turns full circle, and it is we in the developing world who turn to the
Westwards for science. As Al-Kindi wrote 1100 years ago: “It is fitting then for us not to be ashamed to acknowledge and to assimilate it from whatever source it comes to us. For him who scales the truth there is nothing of higher value than truth itself; it never cheapens nor abases him.”

And by the way, do read the whole thing – if not for anything else atleast for the Pauli Stories 🙂

…The hut also contained Professor Villars of MIT, who was visiting Pauli the same day in Zurich. I gave him my paper. He returned the next day with a message from the Oracle;
“Give my regards to my friend Salam and tell him to think of something better”. This was discouraging, but I was compensated by Pauli’s excessive kindness a few months later, when Mrs. Wu’s, Lederman’s and Telegdi’s experiments were announced showing that left-right symmetry was indeed violated and ideas similar to mine about chiral symmetry were expressed independently by Landau and Lee and Yang. I received Pauli’s first somewhat apologetic letter on 24 January 1957.

Thinking that Pauli’s spirit should by now be suitably crushed, I sent him two short notes I had written in the meantime. These contained suggestions to extend chiral symmetry to electrons and muons, assuming that their masses were a consequence of what has come to be known as dynamical spontaneous symmetry breaking. With chiral symmetry for electrons, muons and neutrinos, the only mesons that could mediate weak decays of the muons would have to carry spin one.

Reviving thus the notion of charged intermediate spin-one bosons, one could then postulate for these a type of gauge invariance which I called the “neutrino gauge”. Pauli’s reaction was swift and terrible. He wrote on 30th January 1957, then on 18 February and later on 11, 12 and 13 March: “I am reading (along the shores of Lake Zurich) in bright sunshine quietly your paper…”
“I am very much startled on the title of your paper ‘Universal Fermi interaction’ …For quite a while I have for myself the rule if a theoretician says universal it just means pure nonsense. This holds particularly in connection with the Fermi interaction, but otherwise too, and now you too, Brutus, my son, come with this word. …”….

Although he signed himself “With friendly regards”, Pauli had forgotten his earlier penitence. He was clearly and rightly on the warpath.

… I must admit I was taken aback by Pauli’s fierce prejudice against universalism – against what we would today call unification of basic forces – but I did not take this too seriously. I felt this was a legacy of the exasperation which Pauli had always felt at Einstein’s somewhat formalistic attempts at unifying gravity with electromagnetism – forces which in Pauli’s phrase “cannot be joined – for God hath rent them asunder”….

There is something more to Salam’s Legacy than Electroweak Unification. And of course, I’m thinking of the Abdus Salam International Centre for Theoretical Physics (ICTP) at Italy. And in a more subtle way, He also stands for a struggle – a struggle to provide the students from the third world(and in particular Pakistan) the joys of science…

To Quote Hoodhboy

In interacting with Salam, I could see that two strong passions governed his life. Physics research occupied him intensely; his mind would lock onto a problem making him oblivious to all else. He would engage only the most challenging and difficult problems of the field, problems that only the greatest can dare try. The elegance of his solutions were startling, as for example in his brilliant creation of what are called superfields. Without this powerful mathematical concept, physicists would have a very hard time to progress beyond a certain point in grappling with the basic laws of nature.

Salam’s other passion was Pakistan. I have never been able to understand why he was so dedicated to the country of his birth given that he was virtually ostracised there, being an Ahmadi. I can remember that when the members of the physics department at Quaid-i-Azam University sought to invite him for a lecture after he received the Nobel Prize, the idea was vetoed when the student arm of a vociferous religio-political party threatened to use violence if he came to the campus. In spite of this and much more, Salam was never embittered and he never gave up trying to do whatever he could for his country.

So on this day, let us wish that hundreds of years hence, let nobody speak of him the way Salam spoke of Jai Singh – let nobody say “With him .. expired also all science in ” Pakistan .

The title says it all ! And this is enough excuse for me to post some of my favourite photoes of Einstein. 😉

einstein_violin.jpg

einstein.jpg

Further, I came across a beautiful article titled Albert Einstein’s Theory of Relativity- In Words of Four Letters or Less which you should definitely read ! And then, here is an attempt to answer why Everyone Loves Einstein. Well, may be not everyone !

Einstein quite deservingly has become an icon for science in general and physics in particular. He was not infallible (His rejection of Quantum mechanics is an interesting example) and indeed by the very nature of science, no hero of science can be infallible . But, the fact remains that he saw farther than his colleagues and his benign influence still reverberates through many fields of theoretical physics – from cosmology of the very big to the quantum gravity of the very small. The theory of general relativity, The postulate of light quanta and The phenomenon of Bose-Einstein condensation – each of them is worth a nobel prize.

After celebrating the whole of the last year as the worldyear of physics, people have heard so much about Einstein that I have very little to add. Meanwhile, Astronomers are reporting that Jupiter has developed a new redspot

jup.jpg

On this very day, Americans are celebrating Pi day. If you are still scratching your head, March 14th is written as 3.14 in the American way of writing dates – No one can compete with mathematicians for inventing excuses for math 😉