REMEMBER

Here are some things to remember before applying special relativity relations to get consistency in the results.

1. Identify Clearly Objects of Transformation and Results of Transformation

It is commonly noticed that while working out transformation of distance and time from the relativity relations, the objects of Transformation and results of transformation are mistakenly interchanged and also, so are the motion status of the two frames (stationary and moving). This obviously leads to incorrect conclusions spreading confusion among the practitioners of relativity.

The transformation relations are based on assumption that x and t of the stationary frame are the objects of transformation and x' and t' are the corresponding results when observed from the moving frame. So, application of the relations would be correct only if a correct bifurcation is made between what is being measured and what represents the results.

Further, it must be ensured that the frame carrying the object of transformation is designated as stationary and the one observing the results is treated as moving, no matter what the given setup is. Consequently, the transformation results i.e., contraction, elongation/expansion, dilation etc. should be referred from the moving frame only.

An example of such mistakes may be seen in my Observations 1.4.1.1, sr. no.2 of chapter 1 of the instant book.

2. Freeze Locations or Time for Two Events in Stationary Frame Only

Common time or location is often selected for two events to demonstrate phenomena like length contraction and time dilation respectively. Another set of confusions arise due to varying selection of frame - sometimes stationary and sometimes moving - while freezing one of these parameters.

It may be noted that contraction/expansion of distance/length or time is dependent on not only motion, as is generally believed and stated, but also on relative values of distance and time of the event. For example, while observing transformation of length, when the results are observed (from the moving frame) in a snapshot, one finds contraction but when the object (length) is measured (in stationary frame) in a snapshot, the reverse i.e. elongation is observed (from the moving frame). Similarly, for time, when the results are measured (from the moving frame) from a location, contraction is observed but when the object (time) is measured (in stationary frame) at a location, the reverse i.e. expansion is observed (from the moving frame).  It may be kept in mind that two locations with the same time or two times at the same location cannot exist together in both the frames; these get transformed to two locations with two times (one at each location) in the other frame.

Examples of varying selection of frame while freezing of time and location are the famous ‘contraction of electrons’ and ‘increased lifetime of muons’. Although both the statements refer from the laboratory’s frame, which is considered moving, the former is result of freezing time in the moving frame (photograph in laboratory) and the latter is of freezing location in the stationary frame (of muon).

This leads to misconception that for a given frame, length contraction and increase in time are two simultaneous phenomena, which is contrary to the fact that the two cannot exist together in a frame according to the Lorentz Transformation Condition.

The following exercises, showing reversal of results on change of frame to freeze distance or time, would demonstrate it further.

Let us assume that the two ends of an electron under observation are suitably flagged and the stationary distance between the flags is known. Further, the electron moves with a relativistic speed and locations of the flags are measured in the laboratory, though the flags would appear at different times. The length in laboratory frame is worked out by taking difference of the locations. Now that the times at the ends are the same in stationary (electron’s) frame, we would be observing elongation, not contraction, in the same ratio i.e. 1 ∕ √1-v²/c².

Another example may be seen in my Observations 1.4.1.1, sr. no.2 of chapter 1 of the instant book.

Further, the same conclusions emerge out in my Observations 6.1 of chapter 6 on Minkowski’s exercise on electron contraction in motion in his 1908 paper on space time.

Similarly, in case of muons, let us assume that two muons with identical lifespans are produced at different locations in such a way that a moving observer is able to record the production time of the first muon and the decay time of the second one at the same location in his frame. Here, when the locations for the two times are the same in moving (result measuring) frame, the results would be reversed i.e. contraction, not expansion, of muon’s lifespan would be observed and in the same ratio i.e. √1-v²/c².

Another example may be seen in my Observations 1.4.1.2, sr. no.2 of chapter 1 of the instant book.

In view of the above, it is necessary to establish a convention to always select the same frame, preferably the one considered stationary, for freezing of distance or time for two events, whenever a necessity arises.