Basic Special Relativity deals essentially with the question of how observers describe physical events, that is, phenomena that occur at a given place in space and at a given time. In order to describe events, an observer must be capable of specifying their positions in space and the times at which they occur. For this purpose, he must carry with him devices to measure distances (such as a ruler or metric tape), directions (such as a theodolite), and times (such as a clock). An observer who is thus equipped and is in uniform motion (no acceleration) is called an inertial observer. One may think of an inertial observer as somebody traveling in a spaceship carrying devices capable of measuring positions and times, and moving without accelerating or slowing down. The expression inertial referential is used to refer generically to such a system.
Special Relativity is based essentially on two postulates which were enunciated by Albert Einstein and can be stated as follows:
The fact that the speed of a light pulse is the same for two observers moving with respect to each other is incompatible with the familiar additive law for velocities and implies that the adoption of Einstein's two postulates will require a profound reevaluation of the concepts of space and time.
Observers that are in motion with respect to each other will, in general, give different descriptions of the same event and the development of Relativity starts by establishing the relations between these descriptions. For this purpose, it is necessary to first specify the kinematical relation between the two observers. The more standard procedure consists in using the relative velocity, that is, the velocity of one of the observers with respect to the other. However, as pointed out by Hermann Bondi, the use of some other related quantity might turn out to be more convenient, as well as perhaps conceptually more sound.
Specifically, Bondi proposed to employ the ratio between the interval of emission of light pulses by the first observer and the corresponding interval of reception of the pulses by the second observer. Bondi adopted the symbol k to refer to this adimensional quantity and the development of relativistic kinematics based on its use came to be known under the name of k-calculus.
Since all light pulses move at the same speed, the k factor will obviously be equal to one if the two observers are at rest with respect to each other. But if the second observer is moving away from the first, the k factor will be larger than one and its value will constitute a measure of how fast the second observer is moving away. Correspondingly, if the second observer is moving toward the first, the k factor will be less than one and its value will indicate how fast the second observer is closing in. This is nothing but the familiar Doppler effect, which is well-known to occur, not only for light but also for other wave signals such as sound.
As mentioned above, the use of the k factor permits to distinguish the phase of motion in which the observers move toward each other from the subsequent phase in which they move apart. This is not the case if relative velocity is used instead; this quantity will be positive or negative, depending merely on convention, but will maintain the same value during the whole motion of the observers.
The main advantage of using the k factor is that, as a consequence of the second postulate of Relativity, it can be straightforwardly seen to satisfy a multiplicative combination law. That is, if one considers a third inertial observer, the k factor relating this third observer to the first one will simply be the product of the k factor relating the second observer to the first, by the k factor relating the third observer to the second one. As a result, the mathematical formulae of Relativity generally take a simpler form when expressed in terms of k factors than when expressed in terms of relative velocities. In particular, the ubiquitous square roots that plague standard formulae are not present if the k factor, rather than the relative velocity, is employed.
From a conceptual point of view, developing basic relativity in terms of k factors is attractive, since it involves only performing local time measurements and invoking the second postulate.
Here is the link to the text “Concepts of Special Relativity” in PDF format. This text, written by the author of this page, employs Bondi's k factor to develop basic Special Relativity. Chapter 1 introduces some essential concepts, such as events, observers, referentials, space-time, and also a graphic representation, known as Minkowski diagrams, used throughout the text to help the reader to visualize the situations and processes considered. Chapter 2 states and discusses the fundamental principles of Special Relativity. Chapter 3 is devoted to the definition of the k factor and the discussion of its basic properties. The relationship between k factor and relative velocity is established in Chapter 4. The multiplicative combination law of k factors is analyzed in Chapter 7.
A numerical illustration of the Bondi factor, with a comparison of the situation in which the observers are getting closer to the situation in which they are moving away, can be found in Appendix A.1.
An animation software written in the Java language is available. The definition of Bondi's k factor is illustrated by the second animation proposed.
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