Relativity Part 3: Relativity of length and subjective nature of perception



Now, that we have discussed relativity of time, we are aware that all measurable phenomenon that depends on the passage of time, must also therefore be relative. In this article, we are going to discuss the relative nature of yet another important entity- length. We had spoken how length of an object can be measured only because there is a set standard, to demonstrate what relativity means in a very broad sense. But, when it comes to pure physics and intuition, the theory falls short of being satisfactory, because we are simply not concerned with it while taking everyday measurements.

It is important to understand that relativity of distance work hand-in glove with the relativity of time and to maintain the speed of light as a constant which we’ll discuss later. We were satisfied with the relative nature of velocity even if there wasn’t any formal theory to support that claim, but such would not be the case when it comes to length .


Let us define what length/ distance is before discussing its relative nature:

A length between two points is the measure of how many times a standardized yardstick of fixed dimension has to be placed before reaching from one point to another.

This means that, if somehow, we were to change the very definition of this yardstick A, by chipping off a small part of the yardstick B with no trace of A’s existence, and then chip off every measurable entity in the same proportion we will be justified in saying that the yardstick hasn’t really shrunk, unless we have someone else outside our system to compare and contradict our statement. By this time, it is imperative to note that in physics, two frames of reference are different only by virtue of nature of their motion (i.e. two frames of motion are moving at uniform velocity k w.r.t each other)and not anything else.

Two different frames of reference are  different  only when their relative velocities are of any value other than zero. Thus, motion in general plays a critical role in asserting the subjective nature of objects existing in the same reality. It means that all objects with same velocity (w.r.t to the ground) or zero relative velocity (w.r.t. each other) are in a way synced with each other whether it is velocity or length we are measuring. As their relative velocities increase or decrease (any value other than zero), their very definition of quantities in the localized frame of reference changes (goes out of sync).



Through our discussion in the previous articles, we have understood that there is no such thing as absolute “truth” because there is no frame of reference that is “better” or of “higher importance”. In our everyday lives, we simply assign priorities to certain frames of reference over the other to suit ourselves. For e.g., when we say that the tree is still, we are essentially saying that it is still with respect to the motion of the earth that is still moving when compared to a fixed point in outer space. This fixed point in space, in turn could be in motion with respect to some point in higher dimensional reality we are not aware of and thus the absolute nature of its motion remains ambiguous. This theory can be applied to everything. Simply put, this argument of a body being both in motion w.r.t to one frame and not w.r.t to another can be extended indefinitely till we grow tired of it and are forced to choose something as standard through which everything else must be measured. This is what we do when we say that “a day is of 24 hours” and that “the tree is still” or that a “length of a certain train is 200 feet”.



It is hard to grasp how length itself can shrink with respect to a frame of reference, but it is accurate if we were to consider space and time interwoven into something called space-time and that they value are conserved inside this higher dimension reference frame. The interchangeable nature of space and time is evident when objects move at relativistic speeds(comparable to the speed of light).


Consider taking a train’s measurement while at rest. To do so, we mark points on the track corresponding to the train’s ends. To take the actual measurement, we have at our disposal, a yardstick Y of non-standard length which we will place progressively on the track starting from one point A to other B. Say, we start from the rear end of the train(A) and proceed to the front(B) and in doing so we obtain a number n (which is the no of times it had to be laid down on the track i.e. measurement).

Let us take this measurement again with the same yardstick Y, this time marking the points on train again at rest(L and M) instead of the track. The results obtained will be no different from one another, due to the fact that their points coincide with one another. Thus, measuring AB and LM would give us same results. This would not be the case if the train was moving.

Say, at t=0, we place the yardstick on L and observe the points A and L coincide. At t=T, we reach point M with our yardstick and measure M. This time, we find out that B and M are out of sync and the value of LM is no longer equal to n.

This is due to the passage of time that shifts M away from B, resulting in a small change. This change cannot be considered as an error, because by virtue of relative movement, their frames of reference can no longer be considered to be in sync. But what if, the person on ground is somehow able to take the measurements “simultaneously” of points? This again doesn’t guarantee that the length shall be same for another person sitting on the train, due to the previously discussed concept of “relativity of simultaneity”. Thus, one cannot say that a length of one meter, as measured from a moving frame of reference is same as when kept at rest. These changes are negligible for objects moving than speeds much slower than the speed of light. The factor by which relative velocity /motion measurable phenomenon is in its own frame of reference is known as the Lorentz factor given by:

\gamma = \frac{1}{\sqrt{1 - v^2/c^2}} = \frac{1}{\sqrt{1 - \beta^2}} = \frac{dt}{d\tau}

This factor noticeably alters measurable quantities such as time and length only at relativistic speeds(speeds comparable to the speed of light). As evident from the above equation, the denominator can be rounded off to 1 for all practical phenomenon visible to the human eye. In the next and final part of this series, we shall discus space-time, length contraction and time dilation in great detail. If you’ve found this article useful and would like to be updated on further concepts, don’t forget to like and subscribe to our Facebook page.

This is the last part of relativity series. Click these if you missed out the previous ones

Relativity Part 1: A brief introduction

Relativity Part 2 : Relativity of simultaneity and time.


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