In the last article, we have read, what “relativity” means in a very broad sense and how properties of everyday objects can be measured, because a certain initial point or zero marker has been assigned. We have discussed some really abstract examples and have seen how this theory is intuitive (even obvious) to some.
QUANTITATIVE ANALYSIS OF RELATIVITY
To get even a clearer picture, we must introduce some mathematics to our discussion. Let us consider the system consisting of a man sitting on a tree, a moving train and a man standing on the rear end of the train.
Say, the train is moving with a velocity v and the man inside walking with m towards the front, with respect to train. We are going to measure the distances between two objects in distinct intervals of one seconds and not, in between, so as to prevent any kind of confusion. Say, for someone sitting on the tree, he observes the rear end of the train, the man inside and the tree coinciding with each other at t=0. True to the definition of velocity, after a distinct interval of one second(t=1), he observes the rear position of train get shifted by a distance v away from the tree.
If the above proposition is true, it must also be true that the man walking towards the front of the train, has been displaced by a length m with respect to the rear end of the train. Since the rear end of the train, itself has been shifted by a distance v, the total displacement of the man with respect to the tree, is now algebraic addition of the two displacements i.e. v +w. This is a very precise definition of relativity, that gives a clear picture.
Thus, even velocity and displacement depends upon the frame of reference. Had the man been moving in the opposite direction with the same velocity of the moving train v, then after one second, he wouldn’t have moved with the tree’s frame of reference, because his net displacement with respect to the tree would have been v – v = 0.
THE RELATIVE NATURE OF TIME
Now, that we have successfully, convinced ourselves how everything is relative and the critical role a frame of reference plays, in any kind of measurement, we are forced to analyze whether relativity is applicable to very fundamental aspect -time.
Before the discovery of relativity, time was considered as something absolute that binds everything else relative to it. It is important, to note that when we say time, we are essentially referring to its intrinsic value and not what is measured by ticking of the clock. Clock is a man-made object, and so it ticking is simply an object of reference. Another way of understanding why ticking of a clock might prove unreliable because, clock time t and intrinsic time T are not mutually inclusive. This is to say, that even if a person managed to stop his clock from ticking, he wouldn’t essentially be stopping intrinsic time from passing through his system.
He would still be late for work, by virtue of another working clock being present in his system. This is true, because, not all clocks have stopped working and their combined synchronicity shall be considered much more reliable than the one on his wrist. The office goer argues with his boss and rest of the world, that he has not been late because there’s some mechanical fault with the clock, but time itself has stopped working. The above statement is debatable simply because of the concept of frame of reference. Say, all the clocks in the world slowed down simultaneously by a factor of 0.5 and that only 12 clock hours pass between two consecutive sunrises (and sunset). It is by sheer observation and previously set standard (that two sunrises must span a length of 24 clock hours), we are able to detect the error and re-calibrate our clocks accordingly.
If we were to go even deeper and say, that time is much more than ticking of the clock, we might be thrown off-guard for a moment, if we consider the fact that we are capable of distinguishing a span of two hours versus a span of two minutes, with or without the aid of a clock. This again, is relativity. Consider, waiting in a laundry queue for 1 minute every day (regular day) for and getting used to it, until a day arrives when you’re forced to wait 100 minutes(slow day). There is no clock inside the laundry room and therefore, no way of mathematically determining that you’re late. In the above scenario, the feeling of being late can be considered relative, in the sense we have defined what “not being late” is. Say, if the speed at which the brain processes information (and also other senses responsible for perceiving the passage of time) shrinks by a factor of 100, on the slow day, then there is no way of determining that the day was, in fact, slow.
If say, for some reason the time between two consecutive sunrises were to increase, would be justified in saying that ticking of our clocks have slowed down?(because a day is now larger than 24 hours). It is imperative to understand, that as far as practical measurements are concerned, we as human beings prioritize certain frames of references over other. Certain frames of references commonly used are deemed as “absolute” because of the ease of measurement. (We are satisfied in saying that the ground is still, because for our purpose, we don’t want to bother ourselves with the fact that earth in itself, is in rotation and factor in its motion in relatively trivial calculations.)
In case of measuring time, the interval of consecutive sunrises proves much more reliable than clocks, and therefore is considered absolute. Thus, an hour is simply 1/24 of what is considered a day. The definition of day, in turns depends on the definition of something else. Thus, the definition of time is depends on the scope of system and the relative change in values. In the above example, our system is the solar system and everything outside is in simple terms, irrelevant to our discussion. In the above scenario, if all aids of measuring time and all functions of time, were to increase or decrease by the same factor, then there would be no way of telling that rate of time has changed, from inside the system. Thus, there is no such thing as intrinsic time . This shall help us understand “special theory of relativity.”
RELATIVITY OF SIMULTANEITY
The concept of simultaneity, if strictly associated with clock time, can be defined as the phenomena where multiple event occurs at the same time. But, we have seen as to how time is relative. This, definition therefore falls short of being sufficient.
WE shall therefore try and define it in its most natural form – subjective perception/intuition. Say person A and B both standing on a platform on the same exact point, hears two gunshots coming from opposite sides, given the two guns P and Q on their left and right and are equidistant from each other.
If the sounds from the two guns P and Q reaches A’s left and right ear, respectively such that no time has passed between the two sounds reaching his eardrums, he can consider the gunshots to be simultaneous. (We are assuming that A and B are capable of measuring simultaneity with utmost precision . )
Given, that that A and B are virtually indistinguishable and standing on the same exact point, they can both agree that the gunshots were simultaneous.
But if B was moving with a velocity on one direction, then the sound would reach one of his ears earlier than the other, and he could rightfully disagree as to whether or not, the gunshots were indeed, simultaneous. Thus, the most natural definition of simultaneity is relative to the frame.The following illustration shows that even if two lights are switched on simultaneous with respect to the ground, it is not so with a man moving towards one on a skateboard.
This means that in the example of train, we cannot say that the man who has traveled a distance w in one second(as judged from the train) might not necessarily have covered the distance v+w in the same one second(as judged from the tree). This theory further enforces the theory that flow of time in itself, is a relative quantity and depends on the body(its frame of reference), as discussed above.
This is part 2 of 3 part series: