Misconceptions about Time Dilation and Length Contraction

The common perception of the theory of Relativity is that moving bodies suffer from dilation of their time and contraction of their length, and as a result of time dilation, their processes slowdown and their lifespans increase. There are innumerable particles around us, which suffer from these distortions for us, but we don’t, although we are moving too, with respect to them, with all speeds and in all directions.

Does it not mean that we have positioned ourselves, going against the very theory of Relativity, as the privileged stationary frame where no such distortions occur? This question needs to be answered first, before casting our assumptions on the particles we claim to analyze.

The article discusses as to how these occurrences, emerging in specific contexts of Relativity, have incorrectly taken the form of a compulsory attribute of moving bodies/particles.

Introduction:

Time dilation and Length Contraction are the two terms that came to be widely used in modern physics. Numerous attempts — supposedly successful — have been made to explain many physical phenomena, and also to conceptualize/support many theories in modern physics.

The routine usage of the terms has, however, led to a disconnect from their parent theory, and a lot many practitioners of physics are found to be harbouring misconceptions about these occurrences.

The Two Together Make All Speeds Invariant!

Time dilation means more time in the stationary frame, or less time in the moving frame. Similarly, length contraction also means less length in the moving frame. The two together are in the right direction, from the viewpoint of conformity to Lorentz Transformation Condition, and any view of these being of opposite signs would contradict the Lorentz Transformation Condition. However, the prevailing perception, of both happening by the same factor √(1 − 𝑣2/𝑐2) in the moving frame, has grave consequences which are not taken note of. If the distance as well as the time in the moving frame change by the same factor, their ratio would be free from any changes, and therefore, all speeds of the stationary frame would remain unchanged in the moving frame. Thus a property, solely reserved for light, gets applicable to all moving bodies. So, nothing could be more obviously wrong than this. Even then, unfortunately, the perception has been ruling innumerous minds.

Distort for Whom?

If the current perception professes that the time dilation and length contraction happen within the moving bodies themselves, a natural question arises as to which benchmark these arise for. The universal impression is that these happen with respect to Earth by default, without realising that these are only observed from the Earth’s frame, and do not actually happen within the bodies. Because a body is moving with respect to not only Earth but also other innumerable bodies. Thus it would have infinite number of velocities, with respect to the infinite number of particles around us, and all pairs are equivalent from Relativity point of view.

Does a moving body distort in space and time in infinite ways for the infinite number of benchmarks available? The question leads us to the correct answer i.e. no. The distortion is always one, with respect to only the observer judging it, and other benchmarks (or relatively moving bodies) do not matter. So, we reach the following conclusion.

Time dilation and length contraction are only the effects of Relativity, observed from other frames, and do not occur within the frame holding the time/clock or the length.

Under the conditions, it would be interesting, as well as educating, to recall the parent theory that gave birth to these phenomena, and more importantly, to examine under what conditions and to what extent these arose.

Parent Relations:

All would agree that the interpretations arose from the theory of Special Relativity, and their value is worked out from the Lorentz transformation given below.

Here, all the terms are familiar. Remember that the non-primed frame (say, K) is stationary and the primed frame (say, K’) is moving with a velocity of 𝑣 with respect to the stationary frame, in the (+)ve 𝑥-direction. So, the non-primed parameters 𝑥 and 𝑡 are for the stationary frame, where these objects of observation i.e. 𝑥 and 𝑡 are placed. These objects are observed from the moving (primed) frame, with the corresponding results as 𝑥′ and 𝑡′ respectively.

This setup of relations has to be adhered to strictly under all conditions, for any correct interpretation.

For instance, when one has to consider the moving frame as stationary, and the stationary one as moving in the opposite direction, the same scheme has still to be followed. Now, 𝑥′ and 𝑡′ become the objects of observation, and 𝑥 and 𝑡 become the results. Therefore, the non-primed and the primed parameters have to swap their places, and 𝑣 gets replaced by −𝑣.

With these changes, the above relations take the following form.

Now, the relations (3) and (4) are valid, in the two frames, for any length (Δ𝑥 or Δ𝑥′) and the time interval at its ends (Δ𝑡 or Δ𝑡′ respectively).

It must, however, be noted that a given length and its associated time interval are inseparably tied together in a frame, for the purpose of working out transformations in other frames.

Time Dilation:

The time dilation is stated to be happening in moving frames by a factor equal to √(1 − 𝑣2/𝑐2). The origin of the moving frame is considered to be the same as the moving body. Thus the object of observation, i.e. the time interval, is in the moving frame, and we judge the results in Earth’s (stationary) frame. As a result, the second of relations (4) becomes the appropriate one for this case.

Since the clock in the moving frame is constantly positioned at the frame’s origin, the distance gap between any two ticks of the clock is zero i.e. Δ𝑥′ = 0. On substituting it in the second of relations (4), one gets

Δ𝑡 =Δ𝑡′/√(1 −𝑣2/𝑐2)

This is the familiar relation for time dilation.

However, we must not lose sight of the fact that it is not the time (Δ𝑡′) between the two ticks of the moving clock that has changed but it is the time (Δ𝑡), as judged in the Earth’s frame, which has become larger (by Lorentz Factor).

The rate of time for the moving clock always remains the same, and the moving clock, being stationary to itself, always gives the proper time to itself. Thus no change can occur in the rate of processes, or lifespan, in moving particles/bodies themselves.

Further, it may be interesting to note that the time dilation changes to time compression by the same factor, if the start and end of the time interval correspond to locations separated by a distance (Δ𝑥′) equal to −𝑣Δ𝑡′. This also leads to Δ𝑥 = 0 from the first of relations (4), meaning a situation where the two times correspond to the same location in the observer’s (stationary) frame.

To summarize, we got time dilation with the condition Δ𝑥′ = 0, and got time compression with the condition Δ𝑥 = 0.

Thus the conclusion emerging out is as follows.

Time dilation is an effect of Relativity, with its value depending on the locations corresponding to time being observed, in frames other than the one holding the time, and it never occurs in the holding frame itself.

Length Contraction:

The moving objects are said to contract in length, in the direction of their motion, by a factor √(1 − 𝑣2/𝑐2). Let us examine this too, with respect to the parent relations.

Since the (object’s) length is moving, and we judge the results in Earth’s (stationary) frame, the first of relations (4) becomes the appropriate one for this case.

It may be recalled that in the previous case of time dilation, the co-parameter, distance, was kept the same at both the ends (ticks) of the time interval. Similarly, in this case too, if we keep the same value of the co-parameter, time, at both of its ends, we get similar results i.e. expansion.

That is, however, a rude shock, as we are getting just the opposite of what we profess.

So, to save the situation, we have to compulsorily resort to other favourable setups. A little application of mind would reveal that instead of choosing Δ𝑡′ = 0, if we chose Δ𝑡′ = − (𝑣/𝑐2) Δ𝑥′, the desired result is obtained, as follows.

Δ𝑥 = Δ𝑥′√(1 −𝑣2/𝑐2)

It may further be noted that the sought relation Δ𝑡′ = − (𝑣/𝑐2) Δ𝑥′ also leads to Δ𝑡 = 0 from the second of relations (4), meaning the time corresponding to the two ends of the length should be the same in the observer’s (stationary) frame.

To summarize, we got length elongation with the condition Δ𝑡′ = 0, and length contraction with the condition Δ𝑡 = 0.

It must again be noted here that these effects are noticed only in the other frame, and never occur in the frame where the observed length is lying.

So, the conclusion emerges as follows.

Length contraction is an effect of Relativity, with its value depending on the time corresponding to the ends of the length being observed, in frames other than the one holding the length, and it never occurs in the holding frame itself.

Conclusion:

It has been shown above from the parent relations, which serve as the basis for proclamations such as time dilation and length contraction, that these are nothing more than effects of Relativity, observed in other relatively moving frames, and do never occur in the frame holding the time/clock or the length.

Further, even the effects observed in other frames, do not always happen in the same ratio (by Lorentz factor or its inverse), but depend a lot on the values of the co-parameter selected at the two ends of the (length or time) interval.