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Proper Time Vs Coordinate Time

Posted by Roaring Lovely on June 7, 2015 at 10:30am
Many people find Einstein's theory of relativity hard to understand. For instance, the concept of 'time dillation' or 'relativity of simultaneity'. Well, physicists say that it is because it goes against how we understand 'time' and 'simultaneity'. Right? Wrong! I will espouse the true reason why people can't grasp Einstein's theory. Read on.Earlier on, when you were 12 or so, they introduced you to the idea of speed as distance/time taken. But if you remember well, you might have notice that there is something missing in this notion. Infact, such definition is behind Zeno's paradox and one exellent way of solving Zeno's paradox is to say that speed is not distance/time!! Speed is not realistically analytic! Here it goes: we donnot need both a ruler and a clock to note the following things in the real world:1.)The tortoise is moving as fast as a cheetah2.)The cheetah is moving faster than the tortoise3.)The tortoise is moving faster than the cheetah4.)If a cheetah is moving faster than a tortoise, and then a leopard is moving faster than the cheetah, then it follows that the leopard is moving faster than the tortoise.The above 4 things is what we need to grasp the concept: speed. You could also just see directly a speeding thing and a slowly moving thing. Distance/time is just some racing between say a cheetah and something invisible as 'time' that is somehow running for unjustifyable reasons. It is enough to make a tortoise a standard and then talk of the speed of a cheetah, leopard, elephant etc/the speed of a tortoise. And so we see that logics and even physics do allow the speed of a tortoise to vary. Should the speed of the tortoise vary, then such is precisely the meaning of 'time dillation'. So the theory isn't hard at all to understand.So to understand the theory, I would advice you to completely remove the notion of 'time' from the equations (NOT from your mind) and replace it with 'unseen clock'. Then remove the notion of 'space' and replace it with 'unseen ruler'. So 'spacetime continum' becomes 'rulersclocks continum'. So to understand how time dillation can happen, just imerce a pendulum or an hourglass in water and watch it swing slowly. 'Time' does not dilate like a stomach filled with lots of beer, clocks do run slowly or faster under different conditions and the standard clock is just abitrarily by a human being. Conceptually, there is essentially nothing more mysterious about time dillation.4 DIMENSIONAL SPACTIME CONTINUMSo there are realy lots of clocks. I can use an hourglass, a pendulum, a song, bla bla bla. These donnot necesarily ran at the same rate. Then two otherwise identical pendulums etc donnot swing synchronously if one of them is inside the Einsteinian train. But I like using a tortoise as a clock. So it will mean it is moving along a straight line and not round and round or back and forth. So every time I talk of time, or I use symbol 't', visualise a tortoise moving along an arbitrary axis WITHIN 3D. It is a REAL, litteral tortoise. time dillation, t/T for instance would be two twin tortoises that are racing along two identical say 100m racing fild. So momentarily, t is pretending to be any other thing moving along space axis with a speed t/T given by measuring it with the clock T. If T is the tortoise carried by the Einsteinian train, then T is called proper time and t is called coordinate time.So we say t is behaving similar to the space axis in that in space axis, I often have some freedom to move in any inclination. I can move parralel to the train, or at an angle to it. If I am moving at an angle A from the train and then cover distance x, then the distace along the train do increment by distance X so that x/X is the cosine of angle A. Similarly, the freedom to choose the ratio t/T is very symetrical. But I chose various 'angles'=cos-1(t/T) by simply moving in an approriate speed. So it appears as though moving faster gives me the opportunity to move farther away from axis T in the same way I can move away from the axis that is along the train. But you should be able to see that t is not necesarily moving at any mysterious 'angle' with t at all it is simply moving faster or slowlier than T. Comparing t/T to a cosine of some angle is baseless. But did Minkowski got confused, was he trying to confuse people or was he trying to give some supernatural explanation as to why t is moving at different rate from T? Let me hope it is the latter.CURVED SPACETIMEThings become tough when we try to generalise relativity to accelerating bodies or those moving curvilinearly such as planets orbiting stars. As you may guess, t/T etc nolonger equals to a constant. So when we identify t/T with some cosine, t/T better be curved lines than straight lines.Another interesting case is seen by say a planet orbiting the sun. Remember that as we speed, lengths also contract in the direction of the travel. So if a planet revolves at a relativistic speed around a 'sun', the distance moved around would be shorter than espected because the distance from the sun to the planet is invariant. So when you use the formular for the circumfrence, c=2pr, it won't be correct! So it is quite easy to see that the only way for a generalised relativity to be correct is if spacetime is generaly curved (geometry is non Euclidian, it is Riemaniann).CLOSED UNIVERSEEinstein beleived that the universe is 'closed'. This is very hard to understand if you take the spacetime geometry quite literaly. However, it is a quite simple idea. Let us say that the pluto was moving at the speed of light around the sun. Then relativity tells us that the entire circumfrence of the orbit shrinks to zero!! The pluto is sort of omnipresent in its entire orbit! So if you were to move towards the pluto in eastwards direction, reaching pluto, you become omnipresent along the entire orbit and you may as well come back to earth from westward direction while still thinking you are heading straight away!! So by making the event horizon simply move circularly at c, it becomes impossible to cross the event horizon. The universe becomes finite and closed.
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