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…
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