Category Archives: Physics

Euclidean and Non-Euclidean Continuum

Reference: Einstein’s 1920 Book

This paper presents Part II, Chapter 7 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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Euclidean and Non-Euclidean Continuum

The surface of a marble table is spread out in front of me. I can get from any one point on this table to any other point by passing continuously from one point to a “neighbouring” one, and repeating this process a (large) number of times, or, in other words, by going from point to point without executing jumps.” I am sure the reader will appreciate with sufficient clearness, what I mean here by “neighbouring” and by “jumps” (if he is not too pedantic). We express this property of the surface by describing the latter as a continuum.

Space is basically a continuum. Every dimension varies continuously in it.

Let us now imagine that a large number of little rods of equal length have been made, their lengths being small compared with the dimensions of the marble slab. When I say they are of equal length, I mean that one can be laid on any other without the ends overlapping. We next lay four of these little rods on the marble slab so that they constitute a quadrilateral figure (a square), the diagonals of which are equally long. To ensure the equality of the diagonals, we make use of a little testing-rod. To this square we add similar ones, each of which has one rod in common with the first. We proceed in like manner with each of these squares until finally the whole marble slab is laid out with squares. The arrangement is such that each side of a square belongs to two squares and each corner to four squares.

The surface of the table may be divided into rows and columns of unit squares.

It is a veritable wonder that we can carry out this business without getting into the greatest difficulties. We only need to think of the following. If at any moment three squares meet at a corner, then two sides of the fourth square are already laid, and as a consequence, the arrangement of the remaining two sides of the square is already completely determined. But I am now no longer able to adjust the quadrilateral so that its diagonals may be equal. If they are equal of their own accord, then this is an especial favour of the marble slab and of the little rods about which I can only be thankfully surprised. We must needs experience many such surprises if the construction is to be successful.

The Euclidean continuum assumes certain conditions for the above to happen.

If everything has really gone smoothly, then I say that the points of the marble slab constitute a Euclidean continuum with respect to the little rod, which has been used as a “distance” (line-interval). By choosing one corner of a square as “origin,” I can characterize every other corner of a square with reference to this origin by means of two numbers. I only need state how many rods I must pass over when, starting from the origin, I proceed towards the “right” and then “upwards,” in order to arrive at the corner of the square under consideration. These two numbers are then the “Cartesian co-ordinates” of this corner with reference to the “Cartesian co-ordinate system” which is determined by the arrangement of little rods.

The Cartesian coordinate system is based on the assumptions of the Euclidean continuum.

By making use of the following modification of this abstract experiment, we recognise that there must also be cases in which the experiment would be unsuccessful. We shall suppose that the rods “expand” by an amount proportional to the increase of temperature. We heat the central part of the marble slab, but not the periphery, in which case two of our little rods can still be brought into coincidence at every position on the table. But our construction of squares must necessarily come into disorder during the heating, because the little rods on the central region of the table expand, whereas those on the outer part do not.

Such squares shall distort if the units of length change, as they do in a field. So we shall not be able to use the Cartesian coordinate system.

With reference to our little rods—defined as unit lengths—the marble slab is no longer a Euclidean continuum, and we are also no longer in the position of defining Cartesian co-ordinates directly with their aid, since the above construction can no longer be carried out. But since there are other things which are not influenced in a similar manner to the little rods (or perhaps not at all) by the temperature of the table, it is possible quite naturally to maintain the point of view that the marble slab is a “Euclidean continuum.” This can be done in a satisfactory manner by making a more subtle stipulation about the measurement or the comparison of lengths.

We may be able to use the math based on Euclidean system, if we add the dimension of inertia. Changes in length and time units can be adjusted by changes in inertia.

But if rods of every kind (i.e. of every material) were to behave in the same way as regards the influence of temperature when they are on the variably heated marble slab, and if we had no other means of detecting the effect of temperature than the geometrical behaviour of our rods in experiments analogous to the one described above, then our best plan would be to assign the distance one to two points on the slab, provided that the ends of one of our rods could be made to coincide with these two points; for how else should we define the distance without our proceeding being in the highest measure grossly arbitrary? The method of Cartesian co-ordinates must then be discarded, and replaced by another which does not assume the validity of Euclidean geometry for rigid bodies. [Mathematicians have been confronted with our problem in the following form. If we are given a surface (e.g. an ellipsoid) in Euclidean three-dimensional space, then there exists for this surface a two-dimensional geometry, just as much as for a plane surface.] The reader will notice that the situation depicted here corresponds to the one brought about by the general postulate of relativity (Section XXIII).

Einstein is exploring the math that will geometrically take care of the field problem.

NOTE:—Gauss undertook the task of treating this two-dimensional geometry from first principles, without making use of the fact that the surface belongs to a Euclidean continuum of three dimensions. If we imagine constructions to be made with rigid rods in the surface (similar to that above with the marble slab), we should find that different laws hold for these from those resulting on the basis of Euclidean plane geometry. The surface is not a Euclidean continuum with respect to the rods, and we cannot define Cartesian co-ordinates in the surface. Gauss indicated the principles according to which we can treat the geometrical relationships in the surface, and thus pointed out the way to the method of Riemann of treating multi-dimensional, non-Euclidean continua. Thus it is that mathematicians long ago solved the formal problems to which we are led by the general postulate of relativity.

Einstein’s approach is that of using the math which applies to non-Euclidean geometry. But there is another approach possible that uses the dimension of inertia. This approach requires a definition of inertia as a form of density of substance that is tied to duration.

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FINAL COMMENTS

The problem of field presents the problem of varying space and time characteristics. Einstein had to figure out how to approach it mathematically. That math appears to be quite complex.

Another mathematical approach is possible by associating the changes in space and time characteristics to changes in inertia of substance. Inertia may be defined as the density of force as a substance. As the density of force increases, the length contracts and time gains endurance. This increasing endurance can be seen as decreasing velocity.

By working out a relationship between inertia and velocity, the problem of field may be solved more simply.

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Einstein 1920: Heuristic Value of Relativity

Reference: Einstein’s 1920 Book

This paper presents Part 1, Chapter 14 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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The Heuristic Value of the Theory of Relativity

Our train of thought in the foregoing pages can be epitomised in the following manner. Experience has led to the conviction that, on the one hand, the principle of relativity holds true, and that on the other hand the velocity of transmission of light in vacuo has to be considered equal to a constant c. By uniting these two postulates we obtained the law of transformation for the rectangular co-ordinates x, y, z and the time t of the events which constitute the processes of nature. In this connection we did not obtain the Galilei transformation, but, differing from classical mechanics, the Lorentz transformation.

The Principle of Relativity works in conjunction with the velocity of light simply because light is serving as a “point of zero inertia” for all inertial systems in the material domain, which is characterized by the rigidity of the rectangular co-ordinates x, y, z and t. The point of zero inertia makes it possible to account for subtle change in inertia in the material domain, and their effect on velocity. This effect was not accounted for by the Galileian transformation.

The law of transmission of light, the acceptance of which is justified by our actual knowledge, played an important part in this process of thought. Once in possession of the Lorentz transformation, however, we can combine this with the principle of relativity, and sum up the theory thus:

Every general law of nature must be so constituted that it is transformed into a law of exactly the same form when, instead of the space- time variables x, y, z, t of the original co-ordinate system K, we introduce new space-time variables x’, y’, z’, t’ of a co-ordinate system K’. In this connection the relation between the ordinary and the accented magnitudes is given by the Lorentz transformation. Or, in brief: General laws of nature are co-variant with respect to Lorentz transformations.

The x, y, z, t co-ordinates are a function of inertia. The Lorentz transformations provide this function for the material domain only in an approximate but fairly accurate manner. In other words, general laws of nature are co-variant with respect to Lorentz transformations in the material dimension only.

The law of transmission of light is incomplete until a relationship is found between light’s velocity and its inertia (quantization).

This is a definite mathematical condition that the theory of relativity demands of a natural law, and in virtue of this, the theory becomes a valuable heuristic aid in the search for general laws of nature. If a general law of nature were to be found which did not satisfy this condition, then at least one of the two fundamental assumptions of the theory would have been disproved. Let us now examine what general results the latter theory has hitherto evinced.

Inertia may be defined by finding the relationship of the x, y, z, t co-ordinates with mass.

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FINAL COMMENTS

Lorentz transformation is more accurate than the Galileian transformation because it takes into account the changes in velocity due to subtle changes in inertia in the material domain. These changes are accounted for because taking the large velocity of light as constant is equivalent to treating light as a point of zero inertia. This allows subtle changes in inertia to be converted into equivalent changes in velocity.

The Principle of Relativity is simply accounting the variation in inertia through the rectangular co-ordinates x, y, z and t. This relationship needs to be understood better and related with mass.

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Einstein 1920: Experiment of Fizeau

Reference: Einstein’s 1920 Book

This paper presents Part 1, Chapter 13 from the book RELATIVITY: THE SPECIAL AND GENERAL THEORY by A. EINSTEIN. The contents are from the original publication of this book by Henry Holt and Company, New York (1920).

The paragraphs of the original material (in black) are accompanied by brief comments (in color) based on the present understanding.  Feedback on these comments is appreciated.

The heading below is linked to the original materials.

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Theorem of the Addition of Velocities. The Experiment of Fizeau

Now in practice we can move clocks and measuring-rods only with velocities that are small compared with the velocity of light; hence we shall hardly be able to compare the results of the previous section directly with the reality. But, on the other hand, these results must strike you as being very singular, and for that reason I shall now draw another conclusion from the theory, one which can easily be derived from the foregoing considerations, and which has been most elegantly confirmed by experiment.

Velocities in the material domain are small compared with the velocity of light because of high inertia. High velocities exist in the radiation domain only where the inertia is much lower. Velocities closer to the velocity of light cannot be expected in the material domain.

In Section 6 we derived the theorem of the addition of velocities in one direction in the form which also results from the hypotheses of classical mechanics. This theorem can also be deduced readily from the Galilei transformation (Section 11). In place of the man walking inside the carriage, we introduce a point moving relatively to the co-ordinate system K’in accordance with the equation

x’ = wt’.

By means of the first and fourth equations of the Galilei transformation we can express x’and t’in terms of xand t, and we then obtain

x = (v + w)t.

This equation expresses nothing else than the law of motion of the point with reference to the system K(of the man with reference to the embankment). We denote this velocity by the symbol W, and we then obtain, as in Section 6,

W = v + w…………………………………. (A)

But we can carry out this consideration just as well on the basis of the theory of relativity. In the equation

x’ = wt’

we must then express x’ and t’ in terms of xand t, making use of the first and fourth equations of the Lorentz transformation. Instead of the equation (A) we then obtain the equation

Equation (B) is valid only when c is infinite or very large compared to the material velocity v that is being added to a radiation velocity w.

which corresponds to the theorem of addition for velocities in one direction according to the theory of relativity. The question now arises as to which of these two theorems is the better in accord with experience. On this point we are enlightened by a most important experiment which the brilliant physicist Fizeau performed more than half a century ago, and which has been repeated since then by some of the best experimental physicists, so that there can be no doubt about its result. The experiment is concerned with the following question. Light travels in a motionless liquid with a particular velocity w. How quickly does it travel in the direction of the arrow in the tube T(see the accompanying diagram, Fig. 3) when the liquid above mentioned is flowing through the tube with a velocity v?

In accordance with the principle of relativity we shall certainly have to take for granted that the propagation of light always takes place with the same velocity w with respect to the liquid, whether the latter is in motion with reference to other bodies or not. The velocity of light relative to the liquid and the velocity of the latter relative to the tube are thus known, and we require the velocity of light relative to the tube.

It is clear that we have the problem of Section 6 again before us. The tube plays the part of the railway embankment or of the co-ordinate system K, the liquid plays the part of the carriage or of the co-ordinate system K’, and finally, the light plays the part of the man walking along the carriage, or of the moving point in the present section. If we denote the velocity of the light relative to the tube by W, then this is given by the equation (A) or (B), according as the Galilei transformation or the Lorentz transformation corresponds to the facts. Experiment 1 decides in favour of equation (B) derived from the theory of relativity, and the agreement is, indeed, very exact. According to recent and most excellent measurements by Zeeman, the influence of the velocity of flow v on the propagation of light is represented by formula (B) to within one per cent.

Nevertheless we must now draw attention to the fact that a theory of this phenomenon was given by H. A. Lorentz long before the statement of the theory of relativity. This theory was of a purely electrodynamical nature, and was obtained by the use of particular hypotheses as to the electromagnetic structure of matter. This circumstance, however, does not in the least diminish the conclusiveness of the experiment as a crucial test in favour of the theory of relativity, for the electrodynamics of Maxwell-Lorentz, on which the original theory was based, in no way opposes the theory of relativity. Rather has the latter been developed from electrodynamics as an astoundingly simple combination and generalisation of the hypotheses, formerly independent of each other, on which electrodynamics was built.

According to the principle of relativity the general laws are consistent with changes in uniform motion and inertia. This applies to the laws of electrodynamics too. Addition of velocities is valid when the inertia of the systems is of the same order of magnitude (both in the material domain). Lorentz transformation is valid when the inertia of the systems is many orders of magnitude apart (as in material and field domains).

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FINAL COMMENTS

According to the principle of relativity the general laws are consistent with changes in uniform motion and inertia. This applies to the laws of electrodynamics too.

Addition of velocities is valid when the inertia of the two systems is of the same order of magnitude (both in the material domain). Lorentz transformation is valid when the inertia of the systems is many orders of magnitude apart (as in material and field domains). Neither of the equations may apply when the inertia of both systems is in the field domain. This is because the inertia associated with the velocity of light is not exactly zero.

For equations to work flawlessly a relationship between velocity and inertia needs to be derived for material domain as well as for field domain.

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Einstein’s 1905 Paper on Relativity

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Reference: Disturbance Theory

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Einstein’s theory of relativity works for cosmological dimensions, but not when it comes to atomic dimensions. Einstein was critical of the quantum mechanics having no coherent theory, while he could not come up with a physical theory to explain quantum effects. This bothered him for the rest of his life.

An examination of Einstein’s postulates follows that led to his original paper on relativity. This 1905 paper of Einstein is available at the following link.

On the Electrodynamics of Moving Bodies

Parts of this paper are quoted below that show Einstein’s non-mathematical reasoning. Einstein’s statements are in black italics. My understanding follows in bold color italics.

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Basic Postulates

It is known that Maxwell’s electrodynamics—as usually understood at the present time—when applied to moving bodies, leads to asymmetries which do not appear to be inherent in the phenomena. Take, for example, the reciprocal electrodynamic action of a magnet and a conductor. The observable phenomenon here depends only on the relative motion of the conductor and the magnet, whereas the customary view draws a sharp distinction between the two cases in which either the one or the other of these bodies is in motion. For if the magnet is in motion and the conductor at rest, there arises in the neighbourhood of the magnet an electric field with a certain definite energy, producing a current at the places where parts of the conductor are situated. But if the magnet is stationary and the conductor in motion, no electric field arises in the neighbourhood of the magnet. In the conductor, however, we find an electromotive force, to which in itself there is no corresponding energy, but which gives rise—assuming equality of relative motion in the two cases discussed—to electric currents of the same path and intensity as those produced by the electric forces in the former case.

This introductory paragraph from the paper mentions asymmetry observed in the relative motion between a magnet and a conductor. This asymmetry occurs in the customary view, which uses the lab as its frame of reference. This results in different interpretation of the same phenomenon.

This “asymmetry” disappears when we use the magnetic lines of force, which are attached to the magnet, as the frame of reference. The conductor moves relative to these lines of force the same way in either case producing the same result. So, the problem has to do with how the frame of reference is selected.

Examples of this sort, together with the unsuccessful attempts to discover any motion of the earth relatively to the “light medium,” suggest that the phenomena of electrodynamics as well as of mechanics possess no properties corresponding to the idea of absolute rest. They suggest rather that, as has already been shown to the first order of small quantities, the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good. We will raise this conjecture (the purport of which will hereafter be called the “Principle of Relativity”) to the status of a postulate, and also introduce another postulate, which is only apparently irreconcilable with the former, namely, that light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body. These two postulates suffice for the attainment of a simple and consistent theory of the electrodynamics of moving bodies based on Maxwell’s theory for stationary bodies. The introduction of a “luminiferous ether” will prove to be superfluous inasmuch as the view here to be developed will not require an “absolutely stationary space” provided with special properties, nor assign a velocity-vector to a point of the empty space in which electromagnetic processes take place.

The Michelson-Morley’s experiment was very precise but it failed to discover any motion of earth relative to the light medium. That was because the inertia of light is imperceptibly small compared to the inertia of earth. But light does have inertia that causes its velocity to be finite (see the paper on The Problem of Inertia.)

Einstein suggests that there is no such thing as absolute rest. The fact is that motion reduces with increase in inertia. Only a body with infinite inertia shall come close to absolute rest.

Einstein postulates, “… the same laws of electrodynamics and optics will be valid for all frames of reference for which the equations of mechanics hold good.” Equations of mechanics hold good for frames of reference in which velocities correspond to the inertia of matter. They are many such frames for a range of inertia. The only value of inertia that would relate to all of them would be the reference value of zero inertia.

Einstein postulates light to provide such a reference point. This works for material frames of reference because inertia of light is imperceptibly small in comparison. However, it is questionable if Einstein’s postulate would work for particles of inertia in the atomic range, because inertia of light cannot be ignored in that range.

Einstein also postulates, “… light is always propagated in empty space with a definite velocity c which is independent of the state of motion of the emitting body.” The velocity of light is constant because it is determined by an inertia that is constant in the relatively small range of the frequencies of visible light. This inertia is not influenced by the inertia of emitting material body. Therefore, the velocity of light is independent of the state of motion of the emitting body.

The “luminiferous ether” was assumed to be a material-like medium of light waves. The inertial frame with the above two postulates then replaces the idea of “luminiferous ether”.

The theory to be developed is based—like all electrodynamics—on the kinematics of the rigid body, since the assertions of any such theory have to do with the relationships between rigid bodies (systems of co-ordinates), clocks, and electromagnetic processes. Insufficient consideration of this circumstance lies at the root of the difficulties which the electrodynamics of moving bodies at present encounters.

Rigidity of body corresponds to the material level of inertia. The systems of co-ordinates for space-time are designed with that rigidity in mind. So they apply to material bodies. It is questionable that these rigid space-time coordinates would apply to electromagnetic processes that have a level of inertia many orders of magnitude less than the inertia of matter.

Einstein’s theory of relativity is based on the dichotomy of “inertia – no inertia”.

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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.” 

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.

Einstein defines a “stationary system” in which the equations of Newtonian mechanics hold good. The space-time coordinates of this system have the rigid characteristics of the inertia applied to matter.

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”.

To describe the motion of a material point we give the values of its coordinates as functions of “time”. To represent this motion mathematically, we must define “time” with the understanding of simultaneity of events.

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 judgment of simultaneous events is possible only at the location of the event. Additional considerations are required to define simultaneity of events at different locations.

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 “time-value” comes from the position of the hand of the watch that is moving at a constant rate. The position of hands of watches at two different locations would have to be coordinated to achieve simultaneity. The communication between the two locations can be made through light signals.

If at the point A of space there is a clock, an observer at A can determine the time values of events in the immediate proximity of A by finding the positions of the hands which are simultaneous with these events. If there is at the point B of space another clock in all respects resembling the one at A, it is possible for an observer at B to determine the time values of events in the immediate neighbourhood of B. But it is not possible without further assumption to compare, in respect of time, an event at A with an event at B. We have so far defined only an “A time” and a “B time.” We have not defined a common “time” for A and B, for the latter cannot be defined at all unless we establish by definition that the “time” required by light to travel from A to B equals the “time” it requires to travel from B to A. Let a ray of light start at the “A time” tA from A towards B, let it at the “B time” tB. be reflected at B in the direction of A, and arrive again at A at the “A time” t’A.

In accordance with definition the two clocks synchronize if tB – tA = t’A – tB.

Simultaneity of clocks between two locations requires that light takes the same “time” of travel between the two locations in either direction.

We assume that this definition of synchronism is free from contradictions, and possible for any number of points; and that the following relations are universally valid:—

  1. If the clock at B synchronizes with the clock at A, the clock at A synchronizes with the clock at B.
  2. If the clock at A synchronizes with the clock at B and also with the clock at C, the clocks at B and C also synchronize with each other.

Thus with the help of certain imaginary physical experiments we have settled what is to be understood by synchronous stationary clocks located at different places, and have evidently obtained a definition of “simultaneous,” or “synchronous,” and of “time.” The “time” of an event is that which is given simultaneously with the event by a stationary clock located at the place of the event, this clock being synchronous, and indeed synchronous for all time determinations, with a specified stationary clock.

In agreement with experience we further assume the quantity 2AB/( t’A – tA) = c to be a universal constant—the velocity of light in empty space.

Einstein is assuming that light provides the fastest means of coordination to ascertain simultaneity of mechanical events. This is probably the case when mechanical systems are used for detection.

But for reasonable synchronization of clocks only a synchronization of tempo is needed. The rest is taken care of by the knowledge of distance between the two locations and the speed of light.

It is essential to have time defined by means of stationary clocks in the stationary system, and the time now defined being appropriate to the stationary system we call it “the time of the stationary system.”

The above concept of “time” may be understood in the following two ways:

  1. We take the velocity of light as our reference point. This velocity is so large that compared to it the differences in velocities of material objects are negligible. This allows us a constant rate of change (tempo) with which to measure the motion of material bodies.

  2. We take the inertia of light as our reference point. It is so small that we can treat it as the “zero” for the range of inertia for material bodies. This allows us a basis from which to measure the inertia, and therefore, the motion of all material bodies.

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§ 2. On the Relativity of Lengths and Times

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

The theory of relativity stipulates that the laws by which the states of physical systems undergo change are not affected by translatory motion of frames of reference. This stipulation applies to material systems only, and not to rest of the physical systems covered in The Spectrum of Substance.

The theory of relativity stipulates the velocity of light ‘c’ to be a universal constant. This is true only for the range of frequencies that describe visible light. It is not certain that ‘c’ would apply to the whole range of frequencies on The Spectrum of Substance because ‘c’ represents the “drift velocity” that varies with inertia of the substance.

In this section Einstein develops his mathematical model to determine the relationship between two systems of coordinates that are moving at a uniform velocity relative to each other. In both coordinate systems the velocity of light ‘c’ is assumed to be the same. Einstein did not know that this relationship was already calculated by Lorentz earlier.

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§ 3. Theory of the Transformation of Co-ordinates and Times from a Stationary System to another System in Uniform Motion of Translation Relatively to the Former

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

This section is purely mathematical. It derives the relationship between two co-ordinate systems that are moving uniformly relative to each other when the principle of relativity is applied. According to this principle, the velocity of light is constant in both stationary and moving frames of reference.

The mathematical stipulations are as follows:

  • Both “stationary” and “moving” frames of references are rigid like matter. They are homogenous throughout. In other words, the units of space and time maintain the same characteristics throughout.

  • The moving frame moves at a uniform velocity in the same direction.

  • Simultaneity of clocks at the two ends of a distance requires that light takes the same “time” of travel between the two locations in either direction.

  • The velocity of the moving frame is negligibly small compared to the speed of light.

Einstein then comes up with the same relationship that Lorentz had come up earlier.

Lorentz Boost

Lorentz used the following assumptions:

  • The speed of light is the same in all inertial systems.

  • The gamma “fudge” factor is the same for all inertial systems.

  • The above assumptions are good for a “v/c ratio” of 1/10,000 or less. This is the ratio of the velocity of earth to the velocity of light.

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§ 4. Physical Meaning of the Equations Obtained in Respect to Moving Rigid Bodies and Moving Clocks

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

The equations obtained above predict that length and time shrink with motion. But it is not stated how a velocity is introduced to the moving frame. In reality, velocity can be introduced only through acceleration, which then increases the inertia of the system. This is similar to the observation that wavelength and period of an electromagnetic wave shrink with frequency with resulting increase in inertia.

Force must be applied to generate acceleration or a frequency gradient. The application of force raises the inertia of the system to a new level. Thus, Einstein’s exercise with the “principle of relativity” indirectly supports a continuum of inertia. This continuum has been presented as The Spectrum of Substance. Here substance is primarily represented as an electromagnetic field. With decreasing inertia, substance regresses back to emptiness. With increasing inertia, substance advances towards matter.

Einstein’s own interpretations of the relativity of time have raised many interesting speculations, such as, “time travel”. But such interpretations assume that the principle of relativity works without limitation. This is not so. The workability of the principle of relativity is limited to the upper band of matter in The Spectrum of Substance, where the drift velocities are very low compared to the velocity of light, and any influence on length and time is infinitesimal.

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§ 5. The Composition of Velocities

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

Einstein’s theory assumes that inertia of light is zero, because only then can light be treated the same in all inertial frames of reference. In other word, Einstein’s theory implicitly assumes the velocity of light to be infinite.

In reality, light has a very small amount of inertia as evidenced by a very large, but finite, velocity (see The Problem of Inertia). This inertia may be ignored because Einstein’s frames of reference are limited to matter, but we cannot ignore the implicit assumption of “infinite velocity” for light when dealing with composition of velocities.

Therefore, the following conclusions of Einstein are correct only when ‘c’ is infinite.

  1. “It follows from this equation that from a composition of two velocities which are less than c, there always results a velocity less than c.”

  2. “It follows, further, that the velocity of light c cannot be altered by composition with a velocity less than that of light.”

These conclusions are incorrect when ‘c’ is given a finite value. Thus, we see that math can be fallible when the assumptions are ignored.

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II. ELECTRODYNAMICAL PART

§ 6. Transformation of the Maxwell-Hertz Equations for Empty Space. On the Nature of the Electromotive Forces Occurring in a Magnetic Field During Motion

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

Einstein uses his theory to modify the explanation of forces that are acting on an electric charge, which is moving in a magnetic field. This helps explain the asymmetry observed in the relative motion between a magnet and a conductor mentioned at the beginning of this paper.

We need to reexamine this explanation in the light of the understanding that “empty space” is essentially an electromagnetic field (see The Problem of “Empty Space”).

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§ 7. Theory of Doppler’s Principle and of Aberration

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

From the mathematics in this section Einstein concludes that to an observer approaching a source of light with the velocity c, this source of light must appear of infinite intensity.

In this case the conclusion might be correct because the applicable assumption that inertia of light is negligible compared to the inertia of the source of light is valid.

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§ 8. Transformation of the Energy of Light Rays. Theory of the Pressure of Radiation Exerted on Perfect Reflectors

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

Einstein concludes, “It is remarkable that the energy and the frequency of a light complex vary with the state of motion of the observer in accordance with the same law.”

That means we have a relationship between frequencies of a substance, which represent inertia in some way, and its ‘drift velocity’. See The Problem of Inertia. It may be possible to work out these relationships mathematically.

The weakness of Einstein’s theory is that it assumes the inertia of light to be zero. Once this is corrected, we may be able to achieve some groundbreaking result.

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§ 9. Transformation of the Maxwell-Hertz Equations when Convection-Currents are Taken into Account

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

We don’t really know the exact nature of charge. It could result from the misalignment of frequency gradients in the electromagnetic field, but this needs to be researched further.

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§ 10. Dynamics of the Slowly Accelerated Electron

[Please refer this section in the original paper. I am only writing my comments on the contents of this section here.]

It is not certain if Newton’s laws of force can apply to an electron, which is a particle just forming out of electromagnetic field. The nature of electron appears to be more like a whirlpool in an electromagnetic field. It’s inertia is not comparable to the inertia of a material point.

Einstein’s analysis of the motion of electron is, therefore, inconclusive.

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The Problem of Relativity

Lorentz Boost

Reference: Disturbance Theory

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Einstein regarded space as a physical reality for the following reason. From Einstein’s essay, Relativity & Problem of Space [1]:

But in this [Newton’s] theory, acceleration can only denote “acceleration with respect to space”. Newton’s space must thus be thought of as “at rest”, or at least as “unaccelerated”, in order that one can consider the acceleration, which appears in the law of motion, as being a magnitude with any meaning.

In Newton’s theory, acceleration is a motion relative to the object itself and not to other objects in space. On a smoothly flying plane, we do not feel the velocity, but the moment there is acceleration we feel it instantly in our bones. The idea of acceleration is tied closely with the concept of inertia, which is the property of all substance.

Newton defined inertia in his book “Philosophiæ Naturalis Principia Mathematica”as follows:

The vis insita, or innate force of matter, is a power of resisting by which every body, as much as in it lies, endeavors to preserve its present state, whether it be of rest or of moving uniformly forward in a straight line.

In the cosmic background all bodies have some “uniform motion” or a drift velocity. This velocity shall be small for stars of very large inertial mass because the larger is the inertial mass the more force it takes to move it. On the same account, the drift velocity  for bodies of small inertial mass shall be large. Theoretically, a body with infinite inertia may have zero velocity; and a body with zero inertia may have infinite velocity. A finite drift velocity of a body shall mean that it has finite inertia. This shall apply to all substances whether matter or field.

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Light

Though light has a very large speed in the cosmic background, it is not infinite. Michelson-Morley’s experiment determined this speed quite accurately, but it was unable to detect any inertia for light. However, the large but finite velocity of light means that it must have finite inertia. This inertia maybe infinitesimally small but it is not zero.

Light is made up of electromagnetic cycles. Each cycle consists of dynamic interchange between electrical and magnetic fields. But there is innate resistance to the formation of these fields. The inertia of light comes from such resistance to its cycles. Permittivity (ε0) is the measure of resistance that is encountered when forming an electric field in emptiness. Permeability (μ0) is a measure of how easily a magnetic field can pass through emptiness. Therefore, the resistance to the formation of an electromagnetic cycle is (μ0ε0). This may provide a measure of inertia for light.

The relationship between this inertia and the speed of light is,

c = 1/√(μ0ε0)

We may say that the speed of light is inversely proportional to the square root of its inertia.

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The Lorentz Transformation

In physics, the Lorentz transformations are coordinate transformations between two coordinate frames that move at constant velocity relative to each other. Historically, the transformations were the result of attempts by the Dutch physicist Hendrik Lorentz and others to explain how the speed of light was observed to be independent of the reference frame.

The derivation of Lorentz transformation from purely mathematical considerations may be found at Reference from Khan Academy and Reference from Yale University.

The derivation of Lorentz transformation assumes the following.

Assumption #1: The speed of light is the same in all inertial systems.

Based on Michelson-Morley’s experiment, the speed of light of 3 x 108 meters/second was not affected by the velocity of the earth, which is 3 x 104 meters/second relative to the sun. The “v/c ratios” in this case is 1/10,000, which is of the same order of magnitude as most material bodies in the universe. Therefore, this assumption is good for a “v/c ratio” of 1/10,000 or less.

Assumption #2: The gamma “fudge” factor is the same for observers in different inertial systems.

In this cosmos, each body is drifting in space under a balance of inertial forces. These drift speeds are as different as their respective inertia. This may influence the gamma factor. But this difference may not be significant for a “v/c ratio” of 1/10,000 or less.

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Special Theory of Relativity

The special theory visualizes inertial systems to be boundless inertial “spaces” that move rigidly relative to each other. These inertial systems are equivalent for the formulation of natural laws. In other words, the natural laws are invariant with respect to the transition from one inertial system to another.

The special theory further assumes that the speed of light is a natural law. It is, therefore, invariant with respect to the transition from one inertial system to another. This allows the Lorentz transformations to be used in the special theory. According to Einstein,

The whole content of the special theory of relativity is included in the postulate: The laws of Nature are invariant with respect to the Lorentz transformations. The important thing of this requirement lies in the fact that it limits the possible natural laws in a definite manner.

The Lorentz transformations have been successful in explaining the “aberration” of the fixed stars in consequence of the annual motion of the earth; and the “Doppler effect”, i.e. the influence of the relative motion of stars on the frequency of the light. This success depends on the “v/c” ratio being within the limits of assumption of the Lorentz transformation in these cases. In other words, the validity of Lorentz transformations depends on the inertia of light being negligible compared to the inertia of matter.

Thus, the special theory produces valid results as long as the inertia of light is negligible compared to the inertia of the system under consideration.

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The Atomic Systems

The atomic systems have inertia comparable to the inertia of light. Therefore, when it comes to the application of special theory to systems of atomic dimensions, the inertia of light can no longer be considered negligible. So, the special theory of relativity does not produce valid results in such cases.

The success of the theory of relativity can be assured across the board only by reformulating it with a reference point of zero inertia instead of the velocity of light.

Such a reference point is provided by the concept of emptiness.

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