## New Year !

### December 31, 2005

A New Year Begins !

It is an excuse to look back on what change a single revolution of our little blue planet can bring.

Let the time unfold !

## To Frame My Frame…(Frame II : Einstein) -4. The end of the argument

### December 8, 2005

In fact, we are almost at the end of our arguments. To complete the argument, consider now a third observer C moving with a velocity with respect to A.(Remember that A is moving with a velocity with respect to B already.) So, the vectors and of the third observer are related to and of the observer A by the equations

We can substitute for and in terms of and to get

where we have lazily omitted the terms which we don’t need for further argument 😉

There is another way we can find how and of the third observer C are related to and of the observer B at rest. If the third observer C is moving with a velocity with respect to B, then we can directly write

The only feature of interest to us is the fact that the coefficient of in the first equation is same as the coefficient of in the second equation. Looking back at the equations in the last paragraph, we conclude that this can be true only if

or

Now, since we can choose and to be anything among the velocities that are physically possible,we conclude that for any velocity the following expression should hold.

This implies that should be related to as

which is the condition on if principle of relativity and our other “sensible? assumptions are true ! Now, to conform with notation used by everybody else, we will call as or we define such that (Note that has the same units as speed).Until now, with all we know now, since can be any real number, can be any real or any purely imaginary number.Now of course we will be very much ineinterested knowing its value !

One way is to directly measure the behaviour of moving clocks, and it can be done. But, there are other ways which are widely used to determine it. And we will return to this thing later.Anyway, we can now write down with all due glory,

And in these equations lies the essence of relativity !I’ll just conclude for now with an interesting observation by Minkowski. Say, we take the notation of a time-like vector seriously. AtlAt leastnoenoughriously to imagine *dot product* of time-like vectors with the other vectors. Then, what Minkowski said was basically this- If we assume , then the above equations imply that and ! which essentially means that time-like vectors are more like vectors than we imagined and the above transformations are a “kind? of rotation in which the “angle? between vectors are preserved ! This is among the most valuable insights that relativity offers- time is very much like one another direction in space, only slightly different !

“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.?HERMANN MINKOWSKI

(To be continued)

## To Frame My Frame…(Frame II : Einstein) -3. The Trick

### December 8, 2005

Now, we are ready to ask what does relativity say about the question – Can moving clocks run slow ? Let us start by assuming that a moving clock runs slower by a factor . What does it mean ? It means when one second is past, the moving clock shows a time seconds. Note that if turns out to be less than one, then the moving clock would be actually running fast !

The question is of course, what is the value of ? Our normal intuition says that should be equal to one(moving clocks run at the same rate as clocks at rest), if relativity is true. But as we have seen, this problem might not be as simple as it appears. But one thing is clear, cannot be arbitrary- moving clocks can run slow, but the fact that they should not be able to find out whether they are moving should place some condition on . The question is what is that condition ?

To answer this,Let us start by writing down this question in terms of the funny-looking time-like vectors. Assume a clock A moving with a velocity along x direction is running slow by a factor of , and another clock B at rest with respect to the absolute space. Consider again two explosions which occur one after another. We also assume they happen in such a way that A sees the first explosion then moves with the velocity for a second along x (as seen by B) and then sees the second explosion.

So, according to B,these two explosions are separated by a time duration of one second and a net displacement , or in the notation I introduced before, they are separated by . But what would A see ? In this one second, his clock would be showing seconds. And according to him, he did not move at all. So, he would say that, the only thing he did for going from one explosion to another is just to wait for seconds. So, according to him, the two events are separated by where we have put a prime to show that it is waiting as seen by a moving observer.

So, now we want to write down an expression to convey the fact that to go from one explosion to another you can either wait for seconds as seen by A or wait for one second and move metres as seen by B.Since, these two ways of going from one explosion to another are equivalent and or merely two different ways of going from one explosion to another, we write

(1) |

If the principle of relativity is right, equivalently we can treat A to be at rest, imagine two *other* explosions now seen by B and repeat the whole argument above.Or, we can just interchange all the primed things(measured by A) with all the un-primed things(measured by B).

where is how much B’s clock has slowed down according to A and is the velocity of B as seen by A.

Now, to proceed we have to make some further assumptions. These are assumptions precisely because I don’t know how to prove them ! Probably, the only way is to show you that the results that come out of these assumptions agree with whatever we see in the real world. But, even without that, I hope, you would agree with me that these assumptions are ’sensible’.

One, let me **assume and **** to be functions of .** What I am saying is this – If you give me the velocity of a moving clock A with respect to clock B at rest, that is sufficient to determine a) how much does A is slow as seen by B and b) the velocity of B as seen by A. Note that, by principle of relativity, and should be related in the same way as and are related,i.e, If then . Similarly, if then .

Two, let me * assume that the function depends only on the speed with which the clock is moving, but not on the direction along which the clock moves. In particular, assume *. Well, you will hear people say, of course, that should be true because of what is called as the principle of isotropy. (This is basically the idea that there is no way the Newtonian absolute space distinguishes between two observers

**rotated**with respect to each other by a constant angle. In that sense, the principle of isotropy is completely analogous to the principle of relativity.) But, we have to be a bit more careful in going from the principle of isotropy to the statement that clocks moving at different directions with the same speed run at the same rate.

We have already seen, for example, that principle of relativity alone (which asserts the equivalence of two observers moving with respect to each other by a uniform velocity) DOESNOT imply that the rate of a moving clock should be same as a clock at rest. Similarly, principle of isotropy alone DOESNOT imply that the rate of a clock moving towards the left should be same as one that is moving towards the right with the same speed. So, you have to add some more assumption than just the principle of isotropy.And the necessary assumption is basically this- Any two observers rotated with respect to each other should agree on the time intervals. ^{1}

Now, let me come to my third assumption. Let me **assume that **** **** **. i.e, If A sees B move with a velocity then B sees A moving with a velocity . In fact, this in my opinion is the most non-trivial assumption among the three. I have some ideas on how to replace it with a weaker assumption, and if you have any idea you are of course welcome to mail me ! But, to discuss these ideas might take us a bit too far from the topic under discussion. So, just take this as one of the assumptions – again sensible,but non-trivial.

Now we are ready. We have and . Substituting this into the previous equation, we get

(2) |

Now,we can substitute Eqn.() in Eqn.() and solve for to get

(3) |

Which is quite a weird relation between , and which should hold if the moving clocks run slow and still don’t violate the principle of relativity…

(continued here )

^{1}

**would not**agree upon the x-component of the velocity of the particle. And, this disagreement to the question is there despite the fact that both the observers are equivalent !Given this fact, it’s quite non-trivial to assume that they do agree upon the time intervals.It’s a

*sensible*thing to assume. But, we should acknowledge the fact that it is a new assumption over and above just the principle of isotropy. The non-trivial nature of this assumption is clear if you consider these facts

- Two observers moving at uniform velocity with respect to each other are equivalent(The principle of relativity). But, they don’t agree on the time intervals if turns out to be anything other than unity.They also don’t agree upon the velocity of a ball.

- Two observers rotated with respect to each other by an angle are equivalent(The principle of isotropy). And they do agree upon the time intervals according to the assumption above. But, they don’t agree upon the components of the velocity of a ball.

If at all anything, this just shows that we have to be careful about arguments which go like “Because the two observers are equivalent, they should agree on blah blah.” In fact, if you read about Reichenbach’s arguments on relativity, you would see that there is a way of formulating relativity in which two observers rotated with respect to each other DON’T agree on the time intervals, whereas the principle of isotropy is well and fine ! But, these are really micro-details and I repeat- We would get into it only if we have to.

## To Frame My Frame…(Frame II : Einstein) – 2. A time-like what ?

### December 8, 2005

Well, I wanted to talk about something called a time-like vector. It’s really a simple thing, actually. To understand what I am talking about, just think of what a vector is. If you remember, a High-school vector is defined as something which has a magnitude and direction. It’s an arrow extending from one point to another. For example, consider a unit vector . It’s just an arrow of unit length. I think of it as some arrow one metre long on which you can walk starting from the tail to the head. Roughly these vectors give you a direction to walk and they also come with a distance that is to be walked.

So, what is a time-like vector ? well, it is **not** something which tells you how much to walk, rather it is something which tells you how much to wait ! For example, imagine that there are two explosions which happen one after another. If I say there are two explosions separated by , I mean this- Say you see the first explosion. Now to see the second explosion, what you have to do is to stand there and *wait* for five seconds. Now I can write

By which I mean, waiting for five seconds is same as waiting for two seconds first, and then waiting for three seconds after that. Of course I can write

which means I am saying waiting for five seconds is same as waiting for three seconds first and then waiting for two seconds after that. The advantage is of course the fact that I can save a lot of effort if I just write it down as an equation rather than writing one long sentence.

Now, I want to combine waiting and walking. If I say, for example the two explosions are separated by that means, after seeing the first explosion, you have to wait for five seconds and in the same time walk ten metres to see the second explosion. Of course, this is same as which means you walk first and then wait. In fact, I can write

where the last form means you can walk and then wait, walk again and then wait. You will see the second explosion anyway.

You get the idea, right .Now, this is a very neat way of giving instructions on how much you should walk and wait. The whole point is this- it is not important how you walk and how you wait. To be able to see the second explosion, what is important is that in total you should wait some amount of time and the net displacement should be some fixed thing.

If I have one hour free between two classes, I can either go to my room wait for an hour and then come back to the class, or go to library wait for half an hour, go to the room wait for another half an hour and come back. It doesn’t matter how I do it, provided I am just in time for the next class. And that’s all we need to know about time-like vectors for now ! Okay, now we are ready to address the mystery of the moving clocks…

(continued here )

## To Frame My Frame…(Frame II : Einstein) – 1. Moving Clocks

### December 8, 2005

I sometimes ask myself how it came about that I was the one to develop the theory of relativity. The reason, I think, is that a normal adult never stops to think about problems of space and time. These are things which he has thought about as a child.ALBERT EINSTEIN

It all started with a simple question- Can an ideal clock moving at a uniform velocity with respect to the absolute space run slower than a similar clock at rest ? It seemed such a simple question that everyone thought they had an answer- NO,if we assume that the principle of relativity is right.

Remember what is called the principle of relativity (or what I would like to call the great conspiracy!) is basically this – there is no way for an observer moving at a uniform velocity with respect to the absolute space to find out whether he is moving.Or we can make it a more stronger statement: Newton’s absolute space is such a thing that it doesnot distinguish between two observers moving at a uniform velocity with respect to one another. In that sense, being at rest with the absolute space is same as moving at a uniform velocity with respect to it. ^{1}

The argument seems quite straightforward- Assume that moving clock call it A, runs slower than the clock B at rest with respect to the absolute space.Then, just by comparing the rate of two clocks A can find out that he is moving. But, according to principle of relativity he can’t find out.That means a moving clock can’t run slow.

This was such a neat argument which almost everybody(including physicists and philosophers) would have immediately accepted a hundred years ago. It’s a fact that among all those who thought, wrote and philosophised about Newton’s concept of absolute space and time before 1890’s, nobody thought that a moving clock running slow can be consistent with principle of relativity. But, about a century ago, to our awe and amusement,the above argument was shattered into pieces !

As Poincare,Einstein and Lorentz pointed out, the above reasoning is naive enough to be misleading. And THAT signalled the beginning of modern relativity as we know it. The Newtonian concept of an ? Absolute, true, and mathematical time, of itself, and from its own nature, flowing equably without relation to anything external? was showing its first cracks..

So you might ask, what is wrong with above reasoning ? The point is that this argument is meaningless as it stands.The easiest way to see why the argument is junk is to rephrase the question and the answer in terms of something we know. Take for example, a ball that is at rest with respect to the absolute space along with clock B. And let us ask, ? Can the speed of this ball according to A be more than the speed of the ball according to B? ?

Let us translate the previous argument to this problem – word by word.?Assume that the ball’s speed according to A is more than the ball’s speed according to B.Then, just by comparing the two speeds A can find out that he is moving. But, according to principle of relativity he can’t find out.That means the speed must be the same.? But this “argument? is a total crap as anyone can see. B will see the ball at rest, and hence the ball’s speed according to B is zero. Whereas A will see the ball moving, and hence will assign a non-zero speed to the ball. And that’s all there is to it.

So where did we go wrong ? It is in “comparing the two speeds?. Yes, It’s correct to say A will see B measure the ball’s speed. And A will see B’s apparatus register a value of zero.And A will also see his apparatus register a non-zero speed. O.K., so what ? Why should A compare his value with the value B registers ? According to him, B’s apparatus is being fooled because it is moving along with the ball and *A has no necessity to grant any significance to a speed measured by a moving apparatus*.^{2}

Now, the only thing left is to translate back the above paragraph to our original problem- again word by word. Yes, It’s correct to say A will see B measure the rate of A’s clock. And A will see B’s apparatus showing that A’s clock is slower.And A will also see his apparatus showing that his clock is completely normal. O.K., so what ? Why should A compare the rate of his clock according to him with the rate B registers ? According to him, B’s apparatus is being fooled because it is moving and *A has no necessity to grant any significance to the rate of a clock measured by a moving apparatus*.^{3}

In fact, the two observers disagree on the answers to so many questions- Who is moving ? who is at rest ? what is the energy of the particle so on and so forth. So, what exactly is the problem in supposing that they disagree on the answer to the question “which clock is slower ?? too. Why does it sound mysterious to add one more question to the long list of questions A and B disagree on ?

So, we see that the question whether moving clocks can run slow is more subtle than it appears to be.To answer this question, we have to carefully consider how the moving apparatus are “fooled? as seen by an observer at rest.And we have to find out whether you can find a way the moving observer can be fooled such that One-He cannot find out whether he is moving with a uniform velocity and Two – His clock would be running slow as he is moving.

This is quite a tricky thing, moving clocks should run slow but still, they should not know that they are running slow ! In fact, it seems such a trick is impossible to pull off. But, Einstein , in his 1905 work precisely pulled off this trick ! Before going into this trick, it helps to know something called a time-like vector….

(continued here )

.

^{1}- This fact is usually put in terms of inertial frames. But, I am not using it here, because the very definition of what constitutes a frame is an involved question in relativity. And we will get into it, when we have to.
.

^{2}- I am not saying that A can’t use a moving apparatus to measure the ball’s speed. But, if ever he uses it then he should not take the “zero” registered by such a moving apparatus as such. He HAS to CORRECT for the fact that the apparatus was moving and hence the value measured by it is not the “true” speed.
.

^{3}- I am not saying that A can’t use a moving apparatus to measure the rate of a clock. But, if ever he uses it then he should not take the rate registered by such a moving apparatus as such. He HAS to CORRECT for the fact that the apparatus was moving and hence the value measured by it is not the “true” rate.

## To Frame My Frame…(Frame I : Newton) – 3. The Scholium

### December 7, 2005

Among the beginning sections of Newton’s *Principia* is a wonderful section titled *Scholium*. In it, Newton with his natural grandeur wrote about absolute space and absolute time and presented an argument for absolute space and absolute time that has ever since puzzled physicists and philosophers alike.

You may be wondering why I am spending so much time on Newton when my declared intention is to talk about GR, the theory by Einstein.

That’s because I believe that the thrill of the roller coaster descent that relativity is, is lost to those who had not yet heard about(or who are not yet convinced about) the absolute space and the absolute time of Newton.

So, Here we go. The scholium starts off with the assertion that there is an absolute space and time and “true” motion is the motion with respect to this absolute space and time. This, Newton said, should be distinguished from “relative” or “apparent” motion where one body moves in relation to another. Similarly, the “true” time is just the absolute time. Whereas relative or apparent time is the time measured by a certain clock or some repeating motion like the motion of the sun, for example. This, Newton insisted, is NOT the absolute time since all days need not be equal in duration.

Indeed, Newton wrote,

I. Absolute, true, and mathematical time, of itself, and from its own nature, flows equably without relation to anything external, and by another name is called duration: relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by the means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year.

II. Absolute space, in its own nature, without relation to anything external, remains always similar and immovable. Relative space is some movable dimension or measure of the absolute spaces; which our senses determine by its position to bodies; and which is commonly taken for immovable space;…

Now, all this is O.K. But, still, the question remains- How do you detect absolute,immovable and whatever space ? Indeed what is the necessity of assuming the existence of such a thing if all we can observe is one thing moving with respect to another. If there is no way whatsoever to tell whether you are moving absolutely or not, what’s the point anyway ?

The point is,(and as far as I know, Newton was the first to realise this) that the argument given above is WRONG ! It is NOT true that absolute motions are unobservable. Some absolute motions do have observable effects. Of course, I am talking about cases where we observe a pseudo-force.For example, if you are sitting in a moving bus, and the driver applies a break, you feel a jolt. Now, if I ask why you felt a jolt, you would say – “well, I felt a jolt because the bus came to halt !”. But, notice, as far as you are concerned, you are sitting in the same seat as before. So, with respect to the bus, you are at rest. So, whether you feel a jolt or not cannot be determined by your position relative to the bus,

So, there should be something out there, relative to which if you reduce your speed, you will feel a jolt. That something, Newton triumphantly concluded, is the absolute space ! This indeed is among the most strange things in this world. Consider these facts – Number one, If you move with a constant speed along a straight line with respect to the absolute space, there is no way you can find it out. Number two, If, however you CHANGE your speed or the direction or both, it is very straightforward to tell whether that “change” is an absolute one or a relative one. It is really weird that the jolt I feel is determined by whether I am moving with respect to something out there rather than my motion with respect to the nearby bus ! But, such is the world, and physics is interesting precisely because it’s a weird weird world we live in ! But, as you might have very well guessed, physics has a lot more things to say about this mystery.

I took jolt as an example and once you get the idea, it’s not difficult to notice how nature allows you to find out your “true” motion.

The Foucault pendulum is among the interesting things you can think of in this context. In the year 1851, the French physicist

Jean Bernard Léon Foucault demonstrated the earth’s rotation with respect to the absolute space through a long and heavy pendulum that bears his name. The basic idea is that, if you allow such a pendulum to oscillate the plane of oscillation keeps on rotating, thus betraying the fact that earth is rotating with respect to the absolute space !

O.K,in one sentence Newton’s argument is this-**The absolute space is such a thing that if you accelerate with respect to it, then you feel a force.** And as is easily demonstrated, such a thing does exist.

This argument for existence of absolute space stood the test of many centuries. Not everyone was happy with it, but it worked. And theories of physics achieve immortality not through a scripture but through the fact they work where they are supposed to. And, Newton’s theory worked not because it was written down by Newton, but because Newton could rightly decipher the rules that nature played by. Newton was gone, days passed, days became years, years became centuries.

Until one fine day, Einstein stared at the definition Newton gave **The absolute space is such a thing that if you accelerate with respect to it, then you feel a force.** And muttered- well, that looks like the definition of a gravitational field …! Ahem ! The birth of general relativity !

” So the principles which are set forth in this treatise will, when taken up by thoughtful minds, lead to many another more remarkable result; and it is to be believed that it will be so on account of the nobility of the subject, which is superior to any other in nature.”

GALILEO GALILEI

(Continued here)

### Bibliography

1

Thanks to Eigenspace and its editors Shanth and Akash for giving me this opportunity and thanks to all those juniors (especially Karan and Charu) who inspired this article. I truly owe my understanding in relativity(whatever little that is there) to discussions with my juniors!

2

The whole text of Newton’s scholium can be found here.

## To Frame My Frame… (Frame I – Newton) – 2.Genesis

### December 6, 2005

Newton was a clever man.That is actually an understatement for the man who discovered the laws of dynamics and gravitation, dabbled his hands in optics and alchemy(!), co invented calculus and played a pivotal role in the birth of modern science ! But, above all he was clever.

And this clever man happened to believe in absolute space. According to him, there is an absolute space and there is an absolute time. All our rulers and clocks are just our way of measuring it.Good clocks are that which measure the absolute time accurately. Bad clocks are those which do not show this time. Good rulers are those which measure the absolute distances accurately. And bad rulers are those which do not show the correct distance. This is how we started understanding nature. And Newton is arguably among the best explorers that we humans ever produced and sent out to understand nature.

Almost everyone would agree to this way of stating how we usually look at the world. Well, not everyone, even from the start there were some philosophers who did not like this idea at all, but then philosophers complain about each and everything, don’t they ? Well, if we can live without giving a damn about “proof of our existence”, we can surely live with absolute space and time not giving a damn about the philosophical objections. Can’t we ?

Talking of Philosophy, I think philosophy in physics is like a spice to our food. You can eat food without it in principle, but it is less tasty that way. And what is the right amount of spice depends on your taste and habit.

So, It is good to have an idea of the philosophical problems. So what is the problem with absolute space and time ? Whereas this whole thing seems okay, there is one weird result that people noticed early on. That is what I would call the first great conspiracy of nature, and this result was a tiny flare from which the flame of General Relativity was born.

Well, and that mystery was basically this – consider someone who is moving with respect to this absolute space in absolute time with a **uniform** velocity (again defined through the absolute space and time). It seemed that nature provided NO way for him to know that he is moving with respect to the so called absolute space as the so called absolute time passed.

And even now, about 400 years after Galileo, the degree to which the nature has gone to hide this information from us is simply amazing. Nature will do anything, and when I say anything I mean **anything** to prevent you from knowing whether you are moving with a uniform velocity or whether you are at rest. After so many years, we have got so used to this conspiracy foiling our attempts that modern physics elevates it into one of its fundamental principles. Like the one ring of LOTR, The principle of relativity is the one rule born to rule all rules of nature !

Now, here is where the problems start. Once you stare at the mystery of this conspiracy for sometime, you might come to think – Physics is after all an observational science. And if you cannot detect the absolute space then probably it is simply not there. If Newton refused to talk about unseen angels moving the planets, why did he so willingly talk about the absolute space and absolute time ?

Well, you should never forget what I told you in the beginning – Whatever Newton was, he was not stupid…

(continued here)