This version is obsolete.
Reference: Disturbance Theory
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This is continuation of the examination of Einstein’s postulates underlying his theory of Relativity, specifically, how these postulates were translated into his mathematics.
Einstein’s 1905 paper: http://www.fourmilab.ch/etexts/einstein/specrel/www/#tex2html1
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I. KINEMATICAL PART – § 1. Definition of Simultaneity
“Let us take a system of co-ordinates in which the equations of Newtonian mechanics hold good. In order to render our presentation more precise and to distinguish this system of co-ordinates verbally from others which will be introduced hereafter, we call it the ‘stationary system’.”
Einstein takes up a system of coordinates in which equations of Newtonian mechanics hold well. These he calls inertial frames in which a body remains at rest or moves with constant linear velocity unless acted upon by forces. This property of a body is called inertia.
Inertia represents the resistance to change in motion of a body in space. When this resistance is overcome there is acceleration. We have assumed all along that space is completely permeable to matter. This is not so as evidenced by inertia.
Therefore, matter is ‘stationary’ relative to space when there is no acceleration. All inertial frames in “uniform motion” are actually stationary relative to space. This we identified earlier as the space reference frame (SRF).
“If a material point is at rest relatively to this system of co-ordinates, its position can be defined relatively thereto by the employment of rigid standards of measurement and the methods of Euclidean geometry, and can be expressed in Cartesian co-ordinates.”
A particle is essentially a disturbance propagating through space. This particle of disturbance has a configuration. As the complexity of this configuration increases, the inertia of the particle also increases, and its speed of propagation decreases. A light particle has the simplest configuration and its speed of propagation is ‘c’. An electron is a particle of complex configuration, whose speed is less than 1% of the speed of light. A neutron is a still more complex particle whose speed is thousand times still less.
Einstein’s “material point” refers to a matter particle that has a configuration more complex than that of a neutron. In its most complex configuration a matter particle shall have a speed that is infinitesimal compared to ‘c’. Euclidean geometry and Cartesian co-ordinates apply only to this extreme case of a matter particle. They do not apply to light particles.
All motion considered by Einstein is in reference to matter. This we identified earlier as the material reference frame (MRF). MRF represents a limiting case of a more general SRF that addresses a much wider range of particle configurations.
“If we wish to describe the motion of a material point, we give the values of its co-ordinates as functions of the time. Now we must bear carefully in mind that a mathematical description of this kind has no physical meaning unless we are quite clear as to what we understand by ‘time.’ We have to take into account that all our judgments in which time plays a part are always judgments of simultaneous events. If, for instance, I say, ‘That train arrives here at 7 o’clock,’ I mean something like this: ‘The pointing of the small hand of my watch to 7 and the arrival of the train are simultaneous events’.”
The natural speed of propagation in space then depends on the complexity of configuration of a disturbance as particle. This we perceive as motion that is balanced by the inertia of the particle. Any change in this balance is perceived as acceleration. Acceleration implies presence of force.
Motion is described by the property of TIME. Time essentially describes the sequence of change. A change is referred back to the previous step in the sequence. Thus, time lies in the continuity of a sequence, and it is unique to the configuration of that sequence.
To compare two time sequences in terms of simultaneity they must have comparable configurations. This is reflected in comparability in terms of inertia of the particles. The property of time shall then be a function of inertia. The “time” that we are used to is tied with the material level of inertia. In other words, our experience of time depends on the inertial characteristic of MRF (material reference frame).
The “time” associated with light shall depend on the configuration of the light particle or its inertia. To consider simultaneity of time for matter and light particles, their relative inertia shall have to be taken into account.
“It might appear possible to overcome all the difficulties attending the definition of ‘time’ by substituting ‘the position of the small hand of my watch’ for ‘time.’ And in fact such a definition is satisfactory when we are concerned with defining a time exclusively for the place where the watch is located; but it is no longer satisfactory when we have to connect in time series of events occurring at different places, or—what comes to the same thing—to evaluate the times of events occurring at places remote from the watch.”
The “time characteristics” of particles of different inertia shall be measurable from a “particle” that has no inertia. Such a particle may be postulated as “undisturbed space”. We can then assess the “simultaneity” of two particles by determining their “time characteristics” in terms of their inertia.
The complexity of configuration, and thus the inertia of a particle may be measured in terms of “disturbance levels” as described earlier in The Disturbance Theory. On this scale the disturbance level of zero is a frequency of 1. The disturbance level of 77.6 represents a neutron. All higher disturbance levels represent matter. Earth has a disturbance level of about 235.
At the disturbance levels of matter the wavelength, period and speed become infinitesimal; and the sinusoidal variations in time and space become imperceptible. Time and space then acquire an appearance of constancy that does not exist at electrodynamic and quantum levels.
“We might, of course, content ourselves with time values determined by an observer stationed together with the watch at the origin of the co-ordinates, and co-ordinating the corresponding positions of the hands with light signals, given out by every event to be timed, and reaching him through empty space. But this co-ordination has the disadvantage that it is not independent of the standpoint of the observer with the watch or clock, as we know from experience. We arrive at a much more practical determination along the following line of thought.”
The idea of observer basically represents the characteristics of the reference frame that is being used to interpret motion. The time measured by Einstein’s clocks follows the inertial characteristics of matter. To combine the velocity of light with material velocity would be equivalent to assuming light to have same inertial characteristics as matter. Any mathematics that combines the velocity of light with material velocity using simple addition or subtraction shall lead to erroneous results. It would be like adding a penny to a dollar and calling it two coins of same magnitude.
Unfortunately, Einstein’s mathematics does just that in the rest of this section. We shall skip this mathematics and focus on those aspects of Einstein’s theory that make correct predictions of physical phenomena. Hopefully, a closer look at such aspects will provide better insight into Einstein’s thinking.
[To be continued…]
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