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.

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”.

fermibb.jpg
In mechanics, the concept of a pseudo-force is a useful one. It allows you to work in frames which are non-inertial and apply there the usual laws of Newton. But, one thing about pseudoforces which is rarely discussed, is the fact that pseudoforces tell you something about the geometry of the space around you.

One has to wait till one learns general relativity to recognise this fact. And even then, it is quite likely that this fact is missed in the rain of tensor indices and four vector algebra 🙂

But the relation between pseudoforces and geometry is a simple one, independent of relativity or gravity. And this relation between pseudoforces and geometry is better discussed first in the familiar Newtonian framework.

    Sidenote : Once you understand how pseudoforces affect geometry, the relation between geometry and gravity- the core idea of GR is self-evident. According to Einstein’s equivalence principle, gravitational forces are indistinguishable from pseudoforces and if pseudoforces affect geometry, gravitational forces should do so too !

    Well, the beauty of the equivalence principle is that by identifying gravitational forces with pseudoforces it connects a lot of seemingly unrelated facts into a beautiful theory. All this factsabout pseudoforces, geometry and gravity fall into place in General relativity which combines all of them by identifying gravitational forces with pseudoforces and making matter affect geometry via gravitational/pseudo-forces

Now, coming back – how do we see that pseudoforces affect geometry, and yeah why haven’t we seen it before when we learnt mechanics ?

I’ll answer the second question first. We haven’t seen it before, because the usual examples chosen for non-inertial frames – accelerating/rotating frames have such a familiar geometry that we tend to overlook the effect of pseudoforces. The geometry of space is same as the usual High-school geometry which applies to inertial frames , so there is not much new to talk about here. Once we realise this, the question how do we see the effect of pseudoforces on geometry has a straightforward answer : Think about more general non-inertial frames !

    Sidenote : By the way, if you take Einstein’s relativity into account, you would see that this is not true – you can prove that there is no non-inertial frame which has the same geometry as the highschool geometry. In particular, in accelerating and rotating frames we would see two additional geometric effects- gravitational redshift and Sagnac effect due to pseudoforces. That indeed is the reason why in relativity, you are forced to acknowledge the effect of pseudoforces on geometry, whether you like it or not.

Now, what do I mean by non-inertial frames more general than accelerating/rotating frames ? Consider, for example, what I call an “expanding” frame – a frame of observers who when seen from an inertial frame are expanding radially outwards from the origin.The frame is defined by the usual convention that according to these set of observers, they are all at rest with respect to each other. If you think about it a bit, you will see why this frame is different from the usual non-inertial frames .

In the case of accelerating/rotating frames, the corresponding observers are in a rigid-body like motion. In particular, this means that the distances between the observers does not change. But, for an expanding frame, the distance between the observers increases with time. It is very simple to convince oneself of this fact- just think of two diametrically opposite observers. The distance between them increases with time. And since distances are absolute in Newtonian framework, these observers will actually see the distance between them increasing, though they are at “rest” with respect to each other (by definition) !

I like this particular example very much, since it brings into a sharp focus our conception of what do we mean by being at rest . If you say that being at rest with respect to some frame means not moving, then these two observers are not moving in the “expanding” frame – they are at rest due to the very way the “expanding” frame has been defined. And the distance between the two observers can still increase, though neither of them is “moving” in this sense ! So, how do the expanding frame’s observers interpret this weird thing they observe ? There is only one way out – in that frame, the space itself is expanding ! Though they are at rest, the space is continuously “created” in between them resulting in the increase of distance.

But, what causes this “creation of space” in this non-inertial frame ? What is that entity which is responsible for the creation of space ? Well, that is easily answered. Remember this golden rule about pseudoforces- “Whatever weird is happening in non-inertial frame, can eventually be blamed on the pseudo forces” ! So, it is the pseudoforce fields that create space.

You can convince yourself by calculating the pseudoforces due to this expansion and by showing that the expanding observers can predict the rate of creation of space by measuring the pseudoforces in this frame. So here we have the first example of effect of pseudoforces on geometry- there are some pseudoforce fields which can create space and increase the distance between the two objects which are at rest. In fact, they can do the reverse thing too, they can “eat up” or “swallow” the space in between two objects at rest.

So in this sense, non-inertial frames are like the wonderlands of Lewis Caroll. If you’ve read “Alice in the Wonderland” you would remember this scene in which the red queen and Alice start running

The most curious part of the thing was, that the trees and the other things round them never changed their places at all : however fast they went, they never seemed to pass anything. …
Alice looked round her in great surprise. `Why, I do believe we’ve been under this tree the whole time! Everything’s just as it was!’
redqueen12.gif
`Of course it is,’ said the Queen, `what would you have it?’
`Well, in our country,’ said Alice, still panting a little, `you’d generally get to somewhere else — if you ran very fast for a long time, as we’ve been doing.’

`A slow sort of country!’ said the Queen. `Now, here, you see, it takes all the running you can do, to keep in the same place.

If you want to get somewhere else, you must run at least twice as fast as that!’

Expanding-frame is a bit like that. You’ve to run in this frame to be at the same distance from something at rest in this frame !

And once you realise this, you are on your way to understand the philosophy behind GR- that space is not a quite thing sitting silently allowing all other things to move. It is not a dead-silent stage for the other actors. The stage called space itself is an actor in the drama. It can be created and eaten up, squished and stretched by the pseudoforces.

In fact, there are only two more steps from here to GR. First, is the realisation that inertial frames need not exist- a spacetime where there is no frame in which pseudoforces vanish everywhere is said to be a “curved” spacetime. So, this realisation can be rephrased as the fact that the spacetime can be “curved”. This terminology comes from the fact that for example, a two-dimensional being over a sphere cannot establish a frame over the sphere in which all the pseudoforces vanish. So, this is one way of defining what do you mean by a curved space-time – you are in a curved spacetime if you cannot establish a single inertial frame everywhere.

The universe we live in is such a space. There are pseudoforce fields that extend all over the universe creating new space – this is what we mean by the expansion of the universe. But unlike the example before, this expansion cannot be gotten rid of. The pseudoforce fields cannot be transformed away everywhere by a change of frame since there are no global inertial frames. And in fact, you can use this to tell about the past of our universe- in the past when all this space was not yet created by the pseudoforce fields, the universe was small and hot, and this is what the BigBang theory is all about.

    And this should clear the confusing question “where did BigBang happen ?” – you see when bigbang happened there was only one place where it could happen – the rest of the space was simply not there !

    And then there are people who think everything in the universe expands – you know from our measuring rods to earth-sun distance to distance between the galaxies. Then they get confused about how we can see the expansion if everything expands. Well, I just tell them – think of this expanding frame and imagine what those non-inertial guys would see. Equivalence principle assures you that what we see in an expanding universe would be very similar.(The only difference is that the pseudoforces cannot be wished away bya frame choice- the efffect of pseudoforces on geometry are very similar.). It is immediately obvious that the measuring rods these people carry would not expand. True there will be a tendency to expand- there will be stresses developed on the rods due to the pseudoforces for example ! But, they won’t expand appreciably unless the pseudoforces are really strong .

Second, is of course the equivalence principle- There is no way to distinguish gravitational forces from the pseudoforces in a general spacetime. So, they are just two different names of the samething. This in turn, means that matter can produce pseudoforce fields which can then eat up space. How cool !

This gives rise to really cool objects called Black Holes ! What happens here is that the matter produces a pseudoforce field which eats up space at a large rate. In fact, it eats up space so fast that invariably any two things inside a blackhole are brought near each other. The rate of eating of space is so high that you can’t even run away ! Remember – there is a limit to how fast you can run since you cannot run faster than light. There is no escape from the interior of a blackhole because eventually the pseudoforces will eat up the space between any two objects and everything will be crushed together. Like the expanding universe case, these pseudoforces which create blackholes cannot be transformed away or gotten rid of by changing frames.

So, in GR you have this picture of space which is active and alive , with birth and death ! The space is like a deep sea on which all ships stand. When the sea is calm and serene, the explorers aboard can delude themselves that they are on the land – That there is a ‘ground’ beneath their feet, that their life is lived over a rock-steady stage. Oh explorers! how far would you delude yourselves ? When the sea comes to her full glory, When her waters drench your clothes, When the platform on which you stand begins to shake, would you still mainatain that you are on land ? Would you still dismiss its dynamical nature ? That, in a way, is the challenge of gravity to anyone who seeks to understand her.

P.S. : Do have a look at this article titled Albert Einstein’s Theory of Relativity- In Words of Four Letters or Less . It talks about the same topic but in a different(and probably better) way.

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 $v _1$ with respect to A.(Remember that A is moving with a velocity $v$ with respect to B already.) So, the vectors $\hat{e}_t''$ and $\hat{e}_x''$ of the third observer are related to $\hat{e}_t'$ and $\hat{e}_x'$ of the observer A by the equations

\begin{displaymath}\hat{e}_t'' = \gamma_1 \hat{e}_t' + \gamma_1 v_1 \hat{e}_x' \end{displaymath}
\begin{displaymath}\hat{e}_x'' = \gamma_1 \hat{e}_x' + \frac{\gamma_1}{v_1}\left(1-\frac{1}{\gamma_1^2}\right) \hat{e}_t' \end{displaymath}

We can substitute for $\hat{e}_t'$ and $\hat{e}_x'$ in terms of $\hat{e}_t$ and $\hat{e}_x$ to get

\begin{displaymath}\hat{e}_t'' = \gamma\gamma_1\left(1+\frac{v_1}{v}\left(1-\frac{1}{\gamma^2}\right)\right) \hat{e}_t + (\ldots)\ \hat{e}_x \end{displaymath}
\begin{displaymath}\hat{e}_x'' = \gamma\gamma_1\left(1+\frac{v}{v_1}\left(1-\frac{1}{\gamma_1^2}\right)\right) \hat{e}_x + (\ldots)\ \hat{e}_t \end{displaymath}

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

There is another way we can find how $\hat{e}_t''$ and $\hat{e}_x''$ of the third observer C are related to $\hat{e}_t$ and $\hat{e}_x$ of the observer B at rest. If the third observer C is moving with a velocity $v_2$ with respect to B, then we can directly write

\begin{displaymath}\hat{e}_t'' = \gamma_2 \hat{e}_t + \gamma_2 v_2 \hat{e}_x \end{displaymath}
\begin{displaymath}\hat{e}_x' = \gamma_2 \hat{e}_x + \frac{\gamma_2}{v_2}\left(1-\frac{1}{\gamma_2^2}\right) \hat{e}_t \end{displaymath}

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

\begin{displaymath}\frac{v_1}{v}\left(1-\frac{1}{\gamma^2}\right)=\frac{v}{v_1}\left(1-\frac{1}{\gamma_1^2}\right)\end{displaymath}

or

\begin{displaymath}\frac{1}{v^2}\left(1-\frac{1}{\gamma^2}\right)=\frac{1}{v_1^2}\left(1-\frac{1}{\gamma_1^2}\right)\end{displaymath}

Now, since we can choose $v$ and $v _1$ to be anything among the velocities that are physically possible,we conclude that for any velocity the following expression should hold.

\begin{displaymath}\frac{1}{v^2}\left(1-\frac{1}{\gamma^2}\right)= \ some\ constant\ =\lambda (say)\end{displaymath}

This implies that $\gamma$ should be related to $v$ as

\begin{displaymath}\gamma=\frac{1}{\sqrt{1-\lambda v^2}} \end{displaymath}

which is the condition on $\gamma$ if principle of relativity and our other “sensible? assumptions are true ! Now, to conform with notation used by everybody else, we will call $\frac{1}{\sqrt{\lambda}}$ as $c$ or we define $c$ such that $\lambda\equiv\frac{1}{c^2}$(Note that $c$ has the same units as speed).Until now, with all we know now, since $\lambda$ can be any real number, $c$ 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,

\begin{displaymath}\hat{e}_t' = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\left(\hat{e}_t + v \hat{e}_x \right) \end{displaymath}
\begin{displaymath}\hat{e}_x' = \frac{1}{\sqrt{1-\frac{v^2}{c^2}}}\left(\hat{e}_x + \frac{v}{c^2} \hat{e}_t \right) \end{displaymath}

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 $\hat{e}_t.\hat{e}_t=-c^2$, $\hat{e}_x.\hat{e}_t =0 $ then the above equations imply that $\hat{e}_t'.\hat{e}_t'=-c^2$ and $\hat{e}_x'.\hat{e}_t' =0 $ ! 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)

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 $\gamma$. What does it mean ? It means when one second is past, the moving clock shows a time $\frac{1}{\gamma}$ seconds. Note that if $\gamma$ 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 $\gamma$ ? Our normal intuition says that $\gamma$ 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,$\gamma$ 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 $\gamma$. 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 $\vec{v}=v\hat{e}_x$ along x direction is running slow by a factor of $\gamma$, 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 $\vec{v}$ 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 $\vec{v}=v\hat{e}_x$, or in the notation I introduced before, they are separated by $\hat{e}_t+v\hat{e}_x$. But what would A see ? In this one second, his clock would be showing $\frac{1}{\gamma}$ 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 $\frac{1}{\gamma}$ seconds. So, according to him, the two events are separated by $\frac{1}{\gamma}\hat{e}_t'$ 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 $\frac{1}{\gamma}$ seconds as seen by A or wait for one second and move $v$ 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

\begin{displaymath}\frac{1}{\gamma}\hat{e}_t' = \hat{e}_t + v \hat{e}_x \end{displaymath} (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).

\begin{displaymath}\frac{1}{\gamma'}\hat{e}_t = \hat{e}_t' + v' \hat{e}_x' \end{displaymath}

where $\gamma'$ is how much B’s clock has slowed down according to A and $v'$ 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 $\gamma$ and $v'$ to be functions of $v$. 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, $\gamma'$ and $v'$ should be related in the same way as $\gamma$ and $v$ are related,i.e, If $\gamma=f(v)$ then $\gamma'=f(v')$. Similarly, if $v'=g(v)$ then $v=g(v') $.

Two, let me assume that the function $\gamma$ depends only on the speed with which the clock is moving, but not on the direction along which the clock moves. In particular, assume $\gamma(-v)=\gamma(v)$. 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 $v'=-v$ . i.e, If A sees B move with a velocity $v$ then B sees A moving with a velocity $-v$. 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 $v'=-v$ and $\gamma'=\gamma(v')=\gamma(-v)=\gamma$. Substituting this into the previous equation, we get

\begin{displaymath}\frac{1}{\gamma}\hat{e}_t = \hat{e}_t' - v \hat{e}_x' \end{displaymath} (2)

Now,we can substitute Eqn.(% latex2html id marker 333 $\ref{et}$) in Eqn.(% latex2html id marker 335 $\ref{et'}$) and solve for $\hat{e}_x'$ to get

\begin{displaymath}\hat{e}_x' = \gamma \hat{e}_x + \frac{\gamma}{v}\left(1-\frac{1}{\gamma^2}\right) \hat{e}_t \end{displaymath} (3)

Which is quite a weird relation between $\hat{e}_x'$ , $\hat{e}_x$ and $\hat{e}_t$ which should hold if the moving clocks run slow and still don’t violate the principle of relativity…
(continued here )

1

This is a stronger statement than just saying two observers are equivalent. For example, If one observers x-axis is rotated with respect to the other at an angle, then these two observers 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


  1. 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 $\gamma$ turns out to be anything other than unity.They also don’t agree upon the velocity of a ball.

  2. 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.

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 $\hat{e}_x$. 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

\begin{displaymath}5\hat{e}_t = 2\hat{e}_t +3\hat{e}_t \end{displaymath}

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

\begin{displaymath}5\hat{e}_t = 3\hat{e}_t + 2\hat{e}_t \end{displaymath}

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 $5\hat{e}_t+10\hat{e}_x$ 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 $10\hat{e}_x+5\hat{e}_t$ which means you walk first and then wait. In fact, I can write

\begin{displaymath}10\hat{e}_x+5\hat{e}_t = 5\hat{e}_t+10\hat{e}_x = 2\hat{e}_t+6\hat{e}_x + 3\hat{e}_t+4\hat{e}_x \end{displaymath}

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…

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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 $B_B$ 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 )

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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.

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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.

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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.