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 $B_B$ 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.

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