LECTURE NO. IIIc

EINSTEIN'S ACCOUNT OF HIS THEORY

Copyright © Harold Aspden, 2001

INTRODUCTION

As indicated in the Physics Note introduction to Part III of 'The Physics of Creation', our object here is to work through, chapter by chapter, Einstein's own book 'Relativity', this being his authoritative explanation of the full scope of his theory as updated to 1952. We will, however, exclude the five appendices from this step by step commentary, as, apart from a comment already made in the Physics Note introduction they do not affect the case which Einstein presents in the main text.

I will, however, stress from the outset why I have decided to proceed in this way. If the reader has understood the basis of the aether theory on which I rely and which I have presented in Part II of 'The Physics of Creation', it will be evident that I require the basic electric charges, which define the structure of the aether by sitting each at a lattice site in that structure, to share a concerted motion by moving all in synchronism with no relative retardation.

This is tantamount to saying that the electric field of each of those charges moves as if it asserts an instantaneous action at a distance on charges located elsewhere. I justify this simply by noting that the relative spacing between any two charges sharing a common component of motion is not a changing quantity and so the mutual energy potential attributable to their electrostatic field interaction is invariant regardless of the fact that both charges experience a cyclic jitter motion that provides the performance platform for quantum theory.

As I see it, if energy does not have to deploy to be redistributed as between those moving aether charges, there is no reason to expect any retardation owing to a finite speed of propagation of the field action. Accordingly, I accept the possibility that an electric field action at a distance can be instantaneous, and, indeed, I may note further that this 'instantaneous "Coulomb Gauge" action' is an assumption that is sometimes voiced by theoretical physicists specializing in quantum theory.

So I begin by declaring quite openly that my aether theory stands on ground that is totally incompatible with any doctrine advocated by Einstein that purports to limit electric field action to a speed of light propagation restriction.

That said, I will now list 'look-out-for' topics that I shall keep in mind as I progress, chapter by chapter, through Einstein's book. These are:

1. Does Einstein discuss the property he calls the 'field' for the separate cases of (a) the electric field, (b) the magnetic field and (c) the gravity field or are we to assume that his 'field' concept is all-embracing?

2. Whereas Einstein in his Special Theory will be very concerned with the speed of light theme, which, as we know from Clerk Maxwell's theory, concerns propagating waves involving lateral oscillations of magnetic and electric fields, where in his book will he discuss the possibility of different speeds at which the electric, magnetic and gravitational fields propagate in a longitudinal sense? Here I have in mind the field strengths which diminish in an inverse square of distance relationship, whereas the field strengths in the Maxwell wave diminish as a direct inverse relationship with distance travelled.

3. How does Einstein's theory address the specific case of the field interaction of two spaced-apart bodies of similar mass and similar electric charge? I have in mind here the gravitational interaction of two stars in a binary pair as a case which contrasts with the action between the sun and a planet, where the mass of the sun far outweighs that of the planet.

With this introduction I now proceed the chapter by chapter commentary on the Einstein book. There are 32 short chapters, all of which Einstein thought were needed to present his 'clear explanation' of his theory.

PART I: THE SPECIAL THEORY OF RELATIVITY

I
Here in this first chapter, on the subject of GEOMETRICAL PROPOSITIONS, Einstein refers to Euclidean geometry and debates 'truth' in the context of distance measurements on a 'rigid' body. His concluding words are: 'Of course, the conviction of the "truth" of geometrical propositions in this sense is founded exclusively on experience or rather incomplete experience. For the present we shall assume the "truth" of the geometrical proposition, then at a later stage (in the general theory of relativity) we shall see that this "truth" is limited, and we shall consider the extent of its limitation.' Accordingly, here the reader is warned that what is true according to our experience may not remain so if we accept his theory as it develops! So one reads on with an open mind to see how Einstein's story develops.

II
Here Einstein writes under the heading THE SYSTEM OF COORDINATES and explains the use of the Cartesian system of coordinates in the physics of measurement by particular reference to the specification of position and measurement by indirect means where one seeks clarity in observation in the field of astronomy. He concludes that, in describing events in space, 'the resulting relationship takes for granted that the laws of Euclidean geometry hold for "distances", the "distances" being represented physically by means of the convention of two marks on a rigid body'. Here one can but begin to wonder how this argument will now advance.

III
The Subject now is SPACE AND TIME IN CLASSICAL MECHANICS. Einstein explains how the purpose of mechanics is to describe how bodies change their position in space with "time". He then finds he needs to clarify what is meant by "position" and by "time". He then argues his way forward, coming to a point which he says is self-evident, which is that 'we entirely shun the vague word "space", of which, we must honestly acknowledge, we cannot form the slightest conception and we replace it by "motion relative to a practically rigid body of reference".' He then moves on to replace these words "body of reference" by "system of co-ordinates", which he says: 'is a useful idea for mathematical description'. After this he then reverts to the notion of time, as measured by clocks, concluding the chapter with the words: 'In this connection we have not taken account of the inaccuracy involved by the finiteness of the velocity of propagation of light. With this and with a second difficulty prevailing here we shall have to deal in detail later.' By this stage we see that we are being prepared for the inclusion of speed-of-light limitations in measurements pertaining to the realm of classical mechanics.

IV
Under the title THE GALILEAN SYSTEM OF CO-ORDINATES Einstein here opens his argument by referring to 'the fundamental law of the mechanics of Galileo-Newton, which is known as the law of inertia. He explains that if we use a system of co-ordinates rigidly fixed to body Earth then every fixed star in the universe must every day describe a circle of immense radius, which he accepts is contrary to the law of inertia. Accordingly, he concludes that the law of inertia can only apply in a system of co-ordinates relative to which the fixed stars do not move in a circle. Here then he has introduced what is called a "GALILEAN system of co-ordinates". Physics has now come into the act, and we are no longer concerned with mere geometry.

V
This chapter has the title THE PRINCIPLE OF RELATIVITY. After explaining how the mechanical laws of Galileo-Newton must be the same in two different GALILEAN co-ordinate systems, these being devoid of rotation, he then declares that 'this statement is called the principle of relativity'. Then follows a discussion of relative motion by reference to moving carriages, following which he converges on the theme of the Earth's motion in orbit around the sun. Einstein then concludes that 'if the principle of relativity were not valid we should therefore expect that the direction of motion of the Earth at any moment would enter into the laws of nature'. Then he points out that the most careful observations have never revealed such a property in terrestrial space and declares this as being a 'very powerful argument in favour of the principle of relativity'.

After five chapters we have reached a position where the classical understanding of mechanics and the inertial properties of matter has been adopted and renamed as 'The Principle of Relativity', but we have yet to see how this has any bearing upon the broader spectrum of physics pertaining to the propagation of light, the force of gravity and the electric and magnetic forces that span distances across space or, to be more specific in our use of Einstein's terminology, across the distance separating different co-ordinate positions in a GALILEAN frame of reference.

VI
Here, in his chapter VI entitled THE THEOREM OF THE ADDITION OF VELOCITIES EMPLOYED IN CLASSICAL MECHANICS, Einstein merely explains how a man walking along a railway carriage at a certain speed would be seen to move faster by a stationary observer standing at the side of the railway. The speed of the train has to be added. Then, however, he declares that this result cannot be maintained in reality as, he states, 'we shall see later'.

VII
Chapter VII: THE APPARENT INCOMPATIBILITY OF THE LAW OF PROPAGATION OF LIGHT WITH THE PRINCIPLE OF RELATIVITY. Einstein opens this chapter with the words:

'There is hardly a simpler law in physics than that according to which light is propagated in empty space. Every child at school knows, or believes he knows, that this propagation takes place in straight lines with a velocity c=300,000 km/s.'

At the end of his first paragraph he writes:

'Who would imagine that this simple law has plunged the conscientiously thoughtful physicist into the greatest intellectual difficulties?'

Einstein then discusses a ray of light transmitted from the stationary observer by the side of the railway track and considers the velocity of light relative to the moving carriage, concluding that it must be smaller than the velocity c, a result which he says: 'comes into conflict with the principle of relativity'. It would appear, he declares: 'that another law of propagation of light must necessarily hold with respect to the carriage'.

Now here I confess that I had never in my school years been told that the constancy of the speed of light ranked as a 'law', nor would I have expected the laws of classical mechanics to be applicable to the reference frame governing the propagation of light as an electromagnetic wave. Indeed, on this point raised by Einstein, I would, as a schoolboy have merely presumed that a similar problem would arise if the speed of sound was considered instead of the speed of light, which would have brought the presence and movement of air into the picture and posed no problem for classical mechanics or the principle of relativity as so far defined. But one needs to recognize that 'air' exists as a real medium of reference.

Einstein had spent five chapters to reach a point where he avoided the notion that space might contain a real medium and reducing it to a single system of coordinates defining one frame of reference, but in no way dealing with the possibility that the medium that fills space could be of dual or multiple composition and so defined by more that one overlapping frame of reference. The speed of light, not just sound, is affected by the presence of air and so depends on the motion of the co-ordinate system of that air and, for all Einstein knew, the medium that occupies what we call the vacuum could well itself have two co-ordinate systems in relative motion which, with air, puts three co-ordinate systems into the picture, not to mention the co-ordinate systems of the light transmitting source and the light receiving sensor. After all, the properties of the vacuum were not something one could found on mere assumption. Are we to expect the 'principle of relativity' to tell us that the physics of the problem are the same as viewed separately relative to all five of these overlapping co-ordinate reference frames?

To say the speed of light is constant one has no option but to define the frame of reference, which means probing the properties of the aether by experiment in search of that reference frame. Surely one cannot just dismiss the aether, adopt a so-called 'law', fix an arbitrary reference frame, and then find fault with what one has just assumed.

So how did Einstein then proceed in his chapter VII? He declared:

'The law of the propagation of light in vacuo would then have to be replaced by a more complicated law conformable with the principle of relativity.'

In the same paragraph he goes on to refer to the 'epoch-making theoretical investigations of H. A. Lorentz on the electrodynamical and optical phenomena', without explaining what these are, only then to add, still in the same paragraph:

'Prominent theoretical physicists were therefore more inclined to reject the principle of relativity, in spite of the fact that no empirical data had been found which was contradictory to this principle.'

He then rapidly in his next paragraph concludes by giving his own verdict on the subject by, without further ado, renaming the principle under discussion, which is now extended to include speed of light considerations, as 'the special theory of relativity'.

So we arrive at a point where the principles of classical mechanics have embraced speed of light considerations and a 'principle' is adopted for the sole reason that no contradictory empirical data had been found.

Now, here, I want to point out that the student reading Einstein's book has not, to this stage of the work, been given any background concerning the experimental physics that triggered the interest in what is under discussion. Guided by the book's index reference to the 'Michelson Morley experiment' one sees that it will not be until chapter XVI that there is reference to any experimental data bearing on the subject. Evidently, Einstein is more intent on pursuing a kind of 'brainwashing' exercise to nurse the student into believing his 'principle' before saying: 'Eureka, here is some evidence to prove it.'

So here I will digress a little to refer to events of record in the pre-1905 period, 1905 being the year of publication of Einstein's paper introducing his 'Special Theory of Relativity'.

In 1895 a paper by H. A. Lorentz entitled 'Michelson's Interference Experiment' was published in German. It began by explaining that Clerk Maxwell has pointed out that if two points A and B 'undergo displacement without carrying the ether with them' the time required for light to travel to and fro between A and B must vary slightly with the speed of that displacement. Lorentz then discusses the experiment on this as carried out by Michelson in 1881 and the more conclusive experiment performed later (1887) by Michelson in collaboration with Morley. Mirrors, with light reflected to and fro many times between the mirrors were involved in the test, which failed to detect any motion of body Earth through the aether. Lorentz realized that this could mean that the aether within the apparatus could be sharing the Earth's motion and so deny the possibility of measuring the speed of that motion.

Lorentz did not mention an alternative possibility, which is that light waves in their passage through light waves of similar intensity moving in the opposite direction would entrap wave energy between those mirrors and that presence of energy could mean that the waves were not travelling freely as was assumed. If the wave nodes and antinodes set up by such standing wave conditions are carried along with the mirror system, as is in fact the case, then one might see why there is little or no evidence of wave interference fringes in such an experiment.

In the event, however, Lorentz, duly acknowledging a similar interpretation of earlier date by Fitzgerald, went on to discuss the possibility that motion of the mirror apparatus through the aether might involve some structural distortion that could be direction-sensitive in a way which would account for the null finding in the Michelson-Morley experiment.

In short, Lorentz contended that there is a general contraction, what we now call the 'Fitzgerald contraction', of any apparatus by a fractional amount (v/c)²/2 in the direction of motion at speed v and this, it works out, will exactly compensate for the effect which otherwise would have given a positive outcome from the Michelson-Morley experiment.

Now, although Lorentz did imply that his argument here was based indirectly on the effect of electrodynamic interaction forces as between the molecules comprising the apparatus, I find this poses its own problems. Here I have in mind that two electrons, for example, carried along at the same speed side by side will experience a force of mutual attraction superimposed on their normal mutual electrostatic repulsion, but if they move along a common line, one behind the other, standard electrodynamic theory as it applied in the Lorentz era says there is no such mutual electrodynamic force. In other words one might expect a body composed of numerous particles which are electrically charged at the sub-atomic level to suffer a measure of contraction lateral to the direction of motion. This is exactly the opposite to contraction in the direction of motion. However, taking into account that the forces in the body are in equilibrium and that its molecules are somehow held together to form a solid body, the dominant force attributable to electrostatic action has to be one of mutual attraction. Accordingly, the electrodynamic action, as an offset action must be act primarily between moving charge of opposite polarity, and so will be repulsive overall lateral to the direction of motion. This is in accord with the Lorentz interpretation. My own misgivings here arise from my research into the true law of electrodynamics that was described in Part I of 'The Physics of Creation', where I took the Trouton-Noble experimental result as the definitive last item of evidence needed to derive the true emprical law of electrodynamics. This rules out any scope for anisotropy of electrodynamic action such as would support the notion of a Fitzgerald contraction.

However, as to Einstein, he obviously was influenced by the contraction argument of the 1895 publication by Lorentz as giving a mathematical insight into the interpretation of the Michelson-Morley null result. One can only wonder if he was influenced further by the events of 1903 and 1904. In 1903 the experiment reported by Trouton and Noble, the one just referred to in the preceeding paragraph, demonstrated that the Earth's motion through the aether could not be detected on the basis of the then-accepted law of electrodynamics. Two electrical charges as part of a suspended capacitor system when conveyed through space with body Earth should, given an electrodynamic frame of reference set by the aether, have caused a deflection as a measure of speed through the aether. This experiment gave a null result, again raising the issue of aether drag, but all it really proved was that parallel motion of two charges involves electrodynamic interaction forces that act directly along the line joining the two charges. It was not seen in this context at the time and was duly regarded as providing evidence supplementing the result of the Michelson-Morley experiment.

Accordingly, Lorentz in 1904 presented a paper in which he formalized his theory bringing the contraction hypothesis into a theory and, along with that, the notion that time itself is also modified by a change of motion of the reference frame. Quoting the words of Lorentz:

'I shall show that, if we start from any given state of motion in a system without translation, we may deduce from it a corresponding state that can exist in the same system after a translation has been imparted to it.'

His object was clear. Guided mainly by theory related to electrodynamics he aimed to develop an algorithmic formulation by which he could explain electromagnetic phenomena in a way which embraced the outcome of the Michelson-Morley and Trouton-Noble experiments as well as the experimental findings of Kaufmann (1903) on the increase of mass of the electron at speeds approaching the speed of light. The formulation which pertained to the fundamental equations of classical electron theory came to be known as the 'Lorentz transformation', and one must wonder what Einstein had to offer, given this 1904 account by Lorentz.

VIII
Moving on from his preoccupation with velocity addition involving the speed of light, Einstein in this chapter VIII address THE IDEA OF TIME IN PHYSICS. Here he talks about lightning strikes and how two flashes seen as occurring simultaneously at A will be seen as simultaneous at B. Time, it seems, depends upon the readings given by clocks and a definition of 'simultaneous' applies if what is observed at A and B occur when the pointers on clocks at A and B read the same. This is hardly impressive reading for our notional student who is encouraged to study Einstein's book.

IX
On much the same theme Einstein's chapter IX entitled THE RELATIVITY OF SIMULTANEITY takes us back onto the railway carriage as we watch lightning strikes and wonder if they occur simultaneously. Einstein here explains how 'before the advent of the theory of relativity it had always tacitly been assumed in physics that the statement of time had an absolute significance, i.e that it is independent of the motion of the body of reference.' He then goes on to say that the reader has been shown 'that this assumption is incompatible with the most natural definition of simultaneity' and 'if we discard this assumption, then the conflict between the law of the propagation of light in vacuo and the principle of relativity as developed in chapter VII disappears.' The reader evidently must now believe that time itself has to depend upon the state of motion of a frame of reference and is being eased into a mould which looks remarkably close to that indicated by the Lorentz 1904 paper, as just discussed.

X
Here under the title ON THE RELATIVITY OF THE CONCEPT OF DISTANCE Einstein goes further by suggesting that if an observer on a train measures the length of the train and an observer on an embankment measures the length of the train as it passes, then the two lengths as measured may well be different. Einstein does not enlarge on this but resumes the discussion in the next chapter.

XI
This chapter is entitled; THE LORENTZ TRANSFORMATION, where we find that Einstein has simply adopted the teachings of Lorentz that emerged from his 1904 paper. The expression 'principle of relativity' is not mentioned in this chapter, but time and the measure of distance are seen to be subject to modification as a function of velocity v, all in order to assure conformity with the argument that v cannot ever be measured from within a frame of reference which moves at the velocity v.

XII
Having introduced the Lorentz transformation and its mathematical formulation in chapter XI, Einstein here applies this to the BEHAVIOUR OF MEASURING RODS AND CLOCKS IN MOTION. He merely declares that had he used the Galilean transformation, rods would not contract as a function of their motion and goes on to argue that the Lorentz transformation implies that as a consequence of its motion a clock will slow down.

XIII
Here we find that Einstein explores how the speed of light is affected in its passage through a liquid moving in a tube and he derives two equations, firstly that of the classical theory and secondly one according to the Lorentz transformation, which he finds 'corresponds to the theorem of addition for velocities in one direction according to the theory of relativity'. He is writing under the heading: THEOREM OF THE ADDITION OF VELOCITIES: THE FIZEAU EXPERIMENT and at this stage states: 'The question now arises as to which of these two theorems is the better in accordance with experience. He refers to a mid-19th century experiment by Fizeau and declares that it decides in favour of the equation derived from the principle of relativity. His concluding paragraph begins with the sentence: '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.'

It seems that we have here fourteen chapters written by Einstein only to be told something that was evident from the earlier writings of Lorentz apart from the use of the expression 'the theory of relativity'. Now, as to this Fizeau experiment, Fresnel and later (in 1846) Stokes derived on the basis of aether theory the formula that is an experimental match for the one Einstein claims as supporting the principle of relativity. This was before Fizeau performed the experiment (1859) and so Fizeau's experiment really supports aether theory. Einstein had simply been misleading his student readers into thinking that one can expect, on classical theory, that the speed of light in a column of moving water will be c plus the velocity of that water. In fact, the nineteenth century physicist knew full well that light moves through water at rest at a fraction of its speed in vacuo and allowing for this, by bringing its refractive index into the equation, one gets the Fresnel expression that was duly confirmed by the experiment of Fizeau.

So all we have is that, guided by Lorentz, Einstein is adopting what he calls the principle of relativity to explain an experimental result which supports aether theory, and when I search for any distinction I can draw between the Lorentz and Einstein arguments I can find this is summed up in a book on 'Einstein's Theory of Relativity' by Max Born (1962 Edition of earlier 1924 publication). It is that Einstein eliminated the aether and replaced it by a 'mathematical device for conveniently describing processes in matter and their relationships', whereas 'Lorentz was so attached to his assumption of an aether at absolute rest that he did not acknowledge the physical significance of the equivalence of infinite numbers of systems of reference which he had proved'. [Quotation from page 224 of the Dover 1962 Edition]

To me that statement by Max Born is somewhat emotive and not justified. Just because one can infer that physical phenomena in general can manifest themselves equally in many different frames of reference that does not mean that central to all such frames there cannot be one having special character that somehow keeps the act together. The aether and its real existence is a nucleus from which to work in interpreting what we experience in physics. It cannot be eliminated if we are to see ourselves as belonging to a universe governed by order rather than one immersed in chaos. Yes, one can try to justify explanations that avoid the use of the word 'aether' but, if the price is to distort space and time to satisfy our mental needs, then I urge that we rely a little more on the intuition of Lorentz rather than that of Einstein and, in an rescue effort, explore where Lorentz went off course, leaving Einstein to flounder and sink in a sea of his own making.

XV
In this chapter, GENERAL RESULTS OF THE THEORY, Einstein moves on to discuss how 'the special theory of relativity has rendered the Maxwell-Lorentz theory so plausible, that the latter would have been generally accepted by physicists even if experiment had decided less unequivocally in its favour.' That is quite a statement and one which I trust no serious physicist could endorse. All the chapter deals with is the increase in energy by mass in motion and how that energy augments the mass more and more as the speed converges on the limiting value c.

Anyone who has read about the discovery of the electron and the measurement of its charge/mass ratio at speeds approaching the limiting speed of light, coupled with the early theory on these matters prior to 1904, will know that the formula E=Mc² and the related formula for increase of mass with speed were inevitable consequences stemming from experiments using the cathode ray tube and not a theory that was more concerned with the experiments of Michelson-Morley and Trouton-Noble.

Both of the latter experiments conform with the assumption that the relevant electromagnetic frame moves with the test apparatus. The theory needed was one which explored the make-up of that electromagnetic reference frame and not one which reduced it to the mathematics of a geometrical set of co-ordinates.

XVI
EXPERIENCE AND THE SPECIAL THEORY OF RELATIVITY. This is a penultimate summary chapter for the first part of Einstein's book, before he moves on to General Relativity. It surveys what has already been said but ends with an interesting note on the subject of contraction: '.. the prime factor in this contraction we find, not the motion itself with respect to the body of reference chosen in the particular case in point. Thus for a co-ordinate mirror system moving with the Earth the mirror system of Michelson and Morley is not shortened, but it is shortened for a co-ordinate system which is at rest relative to the sun.'

One is tempted here to say that this is a way of telling that High School student for whom the book is intended, that to do any worthwhile experiments that can actually verify what Einstein is saying one has to be in a laboratory elsewhere in the universe and perform from there tests based on remote observation of the effects of motion in causing distortion of something in an laboratory here on Earth.

XVII
Here with the title: MINKOWSKI'S FOUR-DIMENSIONAL SPACE the student reader is introduced to something that depends upon time being a function of an imaginary number, the square root of minus 1. The topic really does not warrant further comment. In Einstein's own concluding words (Page 57):

'These inadequate remarks can give the reader only a vague notion of the important idea contributed by Minkowski. Without it the general theory of relativity, of which the fundamental ideas are developed in the following pages, would perhaps have got no farther than its long clothes.'

So here, at the end of PART I of Einstein's book, the reader is prepared for the task ahead. Shall we see Einstein adopting notions proposed by Minkowski, much as he has adopted those of Lorentz? And what are we to make of that reference to 'long clothes'? Dare I suggest here that I am reminded of the 19th century Hans Christian Andersen fairytale of the emperor's invisible clothes? Will the student then come to see the truth of what is implied and openly declare that Einstein has clothed his theory in an invisible web of mystery which, in spite of its alleged elegance, is really a theory having no substance.

PART II: THE GENERAL THEORY OF RELATIVITY

XVIII Einstein in this chapter, SPECIAL AND GENERAL PRINCIPLE OF RELATIVITY, again considers railway carriages moving relative to an embankment but now he extends the ambit of his enquiry beyond frames of reference that are in 'uniform rectilinear motion and non-rotary motion'. He concludes by stating that 'the Galileian law does not hold with respect to the non-uniformly moving carriage', a result which he says compells us 'to grant a kind of absolute reality to non-uniform motion, in opposition to the general principle of relativity'. This he then qualifies with the final comment that in 'what follows we shall soon see that this conclusion cannot be maintained.' Again, here, one must wonder if that High School student finds this easy to understand.

XIX
THE GRAVITATIONAL FIELD is a chapter in which Einstein introduces a law according to which the gravitational mass of a body is equal to its inertial mass. He concludes the chapter by saying 'It is true that this important lawhad hitherto been recorded in mechanics, but it had not been interpreted'. A satisfactory explanation is then promised in the next chapter.

XX
THE EQUALITY OF INERTIAL AND GRAVITATIONAL MASSAS AN ARGUMENT FOR THE GENERAL POSULATE OF RELATIVITY tells us that how an observer standing inside a large enclosing box senses forces if the box is accelerated by application of an external force. The point he makes is that, given a uniform rate of acceleration produced by that force, the observer in the box will sense a 'kind of gravitational field' which 'we might easily suppose is only an apparent one'.

XXI
Next, under the heading IN WHAT RESPECTS ARE THE FOUNDATIONS OF CLASSICAL MECHANICS AND OF THE SPECIAL THEORY OF RELATIVITY UNSATISFACTORY', Einstein talks about two pans on a gas range, one emitting steam and the other not, but the one producing steam having a bluish glow beneath it. The assumption is that the glow and the steam are connected in some way, one being a consequence of the other. However, he says, if no bluish glow exists in either case and steam comes from one pan and not the other, then one would be dissatisfied pending discovery of some circumstance to which one can attribute the different behaviour of the two pans. By analogy he then declares how, looking at two bodies of reference, and given an observable difference, whatever their states of motion, he cannot find an explanation within the scope of classical mechanics, or in the special theory of relativity. Accordingly, he states one needs 'a physics which is conformable to the general principle of relativity, since the equations hold for every body of reference, whatever may be his state of motion.'

Our student must surely now be wondering what kind of magic governs this so-called 'general principle of relativity' and be curious as to where we are headed.

XXII Here Einstein presents A FEW INFERENCES FROM THE GENERAL PRINCIPLE OF RELATIVITY one being that one can produce what is effectively a gravitational field merely by accelerating a frame of reference. He also refers to a body in rectilinear motion with respect to frame K that is executing an accelerated and in general curvilnear motion with respect to an accelerated reference frame K'. In the context of gravitation there is, he says, 'nothing essentially new' to consider here, but there is a 'new result of fundamental importance when we carry out the analogous consideration for a ray of light'. His conclusion is that 'in general,rays of light are propagated curvilearly in gravitational fields' and adds 'This result is of great importance'. Elaborating on this he says that the general theory of relativity tells us that light rays grazing past the sun should be deflected through 1.7 seconds of arc.

Here, at last, is something testable by observation that his theory has predicted! However, to this stage, Einstein has not shown how he arrives at this result. So far it is a mere statement intended to encourage onward study by the reader. Indeed, as he reaches the end of the chapter he states 'we may entertain the hope that the general law of gravitation will be derivable from such gravitational fields of a special kind.' Then he goes on to say 'This hope has been realised in a most beautiful manner' but 'it was necessary to surmount a serious difficulty' and, to progress, 'we require to extend our ideas about the space-time continuum still farther.'

XXIII This chapter opens by Einstein stating that he has so far 'purposely refrained from speaking about the physical inyterpretation of space-time-data in the case of the general theory of relativity', as a consequence of which he says he is 'guilty of certain slovenliness'. To rectify this he goes back to the subject of BEHAVIOUR OF CLOCKS AND MEASURING RODS ON A ROTATING BODY OF REFERENCE.

It seems that clocks carried by rotating discs near their perimeter upset one's 'definition of time with respect to the body of reference'. By the end of the chapter Einstein declares: 'Thus all our previous conclusions based on general relativity would appear to be called into question. In reality we must make a subtle detour in order to be able to apply the postulate of general relativity exactly. I shall prepare the reader for this in the following paragraphs'. To read these paragraphs we have to turn to the next chapter.

By now, even I am getting a little bewildered by what is billed on the book cover of this 15th edition of Einstein's work as being 'A clear explanation of the famous theory that brought about the atomic age'!

XXIV
Einstein now speaks about EUCLIDEAN AND NON-EUCLIDEAN CONTINUUM and is back on the theme of 'measuring rods'. He concludes that '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.'

XXV
Einstein here introduces GAUSSIAN CO-ORDINATES and sums up this chapter by saying that 'Gauss invented a method for mathematical treatment of continua in general, in which distance between neighbouring points are defined.' A crucial statement then follows: 'To every point of a continuum are assigned as many numbers (Gaussian co-ordinates) as the continuum has dimensions.' The four-space dimensional continuum has been introduced and Einstein is now ready in his next chapter to bring Minkowski's four-dimensional space with its square root of minus 1 time component into the Gaussian system.

XXVI The stage just mentioned is implemented in this chapter under the heading THE SPACE-TIME CONTINUUM OF THE SPECIAL THEORY OF RELATIVITY CONSIDERED AS A EUCLIDEAN CONTINUUM.

XXVII
Here the chapter heading states the conclusion: THE SPACE-TIME CONTINUUM OF THE GENERAL THEORY OF RELATIVITY IS NOT A EUCLIDEAN CONTINUUM. Somehow from what is said in this chapter Einstein finds he can say: 'Every physical description resolves itself into a number of statements, each of which refers to two events A and B. In terms of Gaussian co-ordinates, every such statement is expressed by the agreement of their four co-ordinates x₁, x₂, x₃, x₄. Thus, in reality, the description of the space-time contin by means of Gauss co-ordinates completely replaces the description with the aid of a body of reference.

So we now see that Einstein, after building his scheme by reference to the geometry of rigid bodies has opted for a system of mathematics which adds the fourth dimension to our more familar world of three space dimensions and we have to contend with that imaginary square root of minus 1 term disguised as a space dimension denoted x₄.

XXVIII We are now ready to digest the EXACT FORMULATION OF THE GENERAL THEORY OF RELATIVITY. Einstein tells us that the idea of the general theory of relativity is that:

'All Gaussian co-ordinate systems are essentially equivalent for the formulation of the general laws of nature.'

In this chapter Einstein describes each of his four-dimensional worlds as a "mollusc" and says:

'The general principle of relativity requires that all these molluscs can be used as reference-bodies with equal right and equal success in the formulation of the general laws of nature; the laws themselves must be quite independent of the choice of mollusc.'

He then follows this with the concluding words:

'The great power possessed by the general principle of relativity lies in the comprehensive limitation which is imposed on the laws of nature in consequence of what we have seen above.'

XXIX
We have arrived at chapter 29 which bears the title: THE SOLUTION OF THE PROBLEM OF GRAVITATION ON THE BASIS OF THE GENERAL PRINCIPLE OF RELATIVITY and find that Einstein opens with the following assurance:

'If the reader has followed all our previous considerations, he will have no further difficulty in understanding the methods leading to the solution of the problem of gravitation.'

Clearly we are on the downhill slope headed for enlightenment on how Einstein explains the force of gravity!

Now here we find that Einstein lists three 'demands' governing his theory. They are

'(a) The required generalisation must likewise satisfy the general postulate of relativity.
(b) If there is any matter in the domain under consideration, only its inertial mass, and thus only its energy is of importance for its effect in exciting a field.
(c) Gravitational field and matter together must satisfy the law of conservation of energy (and of impulse).'

He then goes on, without giving any account of his theory of gravitation, to point to the weakness of Newton's theory in not giving an explanation for the progressive motion of the elliptical orbit of a planet around the sun, most evident for the planet Mercury and he declares that the anomalous 43 seconds of arc advance per century evident from astronomical observation is explained by his theory. Our hypothetical High School student reader is not destined to be shown how this 43 seconds of arc figure is derived within the framework of general relativity. He will have wait until in later years he can tackle the complexities of tensor analysis as applied to the four-dimensional space-time world of Einstein's imagination. However, I must here comment on Einstein's words, as quoted from page 103 of his book:

'This effect can be explained by means of classical mechanics only on the assumption of hypotheses which have little probability, and which were devised solely for this purpose.

Here I wonder if Einstein is making reference here to Paul Gerber, a German school teacher who, in 1898, published a paper showing how the 43 seconds of arc figure is explained exactly if the force of gravity governing the radial component of oscillation of the planet Mercury in its orbit around the sun was subject to changes in gravitational action propagating at the finite speed of light. All Gerber had done was to incorporate into classical mechanics the notion that transfer of energy between the gravitational potential of the sun and planet interaction is subject to the limitations set by the speed of light. That is hardly 'an assumption having little probability' and it certainly is one that a High School student would appreciate from his knowledge of classical mechanics. If the period of radial oscillation is slowed down slightly by retardation of gravity propagation then that period will slip relative to the normal orbital period and so there will be a progressive advance of the perihelion of the elliptical motion.

However, it seems that Einstein chose to ignore Gerber's explanation and it so remains for those interested in the subject to form their own opinions. One can formulate a law of gravitation based on the Gerber theme, albeit by taking note that energy transfer is not strictly along a pencil thin line joining sun and planet, and so one can show how the planetary motion is explained, along with the deflection of light by the sun and a third consideration, the sun's red shift. The law of gravitation found is the same as that derived by Einstein after struggling through the complicated maze which besets his theory. This has all been dealt with earlier in Part I of 'The Physics of Creation', but let us now see what Einstein's further chapters have to say.

PART III: CONSIDERATIONS ON THE UNIVERSE AS A WHOLE

XXX
This chapter entitled COSMOLOGICAL DIFICULTIES OF NEWTON'S THEORY is one of three chapters which conclude the main text of Einstein's book. It offers nothing constructive, it being a mere criticism of Newton's hypothesis concerned how stellar matter is distributed in the universe.

XXXI
THE POSSIBILITY OF A "FINITE" AND YET "UNBOUNDED" UNIVERSE is a chapter that talks again about geometry and measuring rods, going on from there to discuss curved space. It all leads Einstein to the statement 'closed spaces without limits are conceivable'. Our High School reader might see this as meaning that if you go around in circles for ever you will still be within a closed space!

XXXII
THE STRUCTURE OF SPACE ACCORDING TO THE GENERAL THEORY OF RELATIVITY talks about Euclidean geometry as implying an infinite universe and concludes that if there is to be matter in the universe then the universe is of necessity finite. I note that Einstein throughout his work does not question the possibility that gravity might have a limited range of action in no way connected with his notion that space is curved.

FINAL DISCUSSION

In opening this LECTURE I listed three what I called 'look-out-for' topics. These first of these was:

1. Does Einstein discuss the property he calls the 'field' for the separate cases of (a) the electric field, (b) the magnetic field and (c) the gravity field or are we to assume that his 'field' concept is all-embracing?

The answer to this I found was negative. His attention seemed to be focussed only on his obsession with his theme 'the principle of relativity' with speed of light being the dominant factor in Part I, the special theory of relativity section of his book, and the gravity field being the exclusive theme of Part II, the general theory of relativity section of his book. I am, therefore, left to ponder on the problem of how the general theory of relativity treats the Coulomb interaction, whereas and the magnetic field, as a feature of electrodynamic interaction, is presumably dismissed as being covered by the Lorentz transformation and the Lorentz interest in the Trouton-Noble experiment. Nowhere did I see a glimmer of interest in trying to account for gravity as an electrodynamic interaction force, this surely being essential in moving forward towards a Unified Field Theory.

The second topic was:

2. Whereas Einstein in his Special Theory will be very concerned with the speed of light theme, which, as we know from Clerk Maxwell's theory, concerns propagating waves involving lateral oscillations of magnetic and electric fields, where in his book will he discuss the possibility of different speeds at which the electric, magnetic and gravitational fields propagate in a longitudinal sense? Here I have in mind the field strengths which diminish in an inverse square of distance relationship, whereas the field strengths in the Maxwell wave diminish as a direct inverse relationship with distance travelled.

So far as I could see Einstein did not discuss the velocity at which perturbations of the magnetic field or the electric field propagate. Even concerning the gravity field one must assume that he regards that as being attributable to curvature of his space medium or 'molusc' to use his own term. How that curvature might alter transiently as the source mass accelerates is somewhat of a mystery, though it is probably inferred that speed of light constraints apply.

The third topic is:

3. How does Einstein's theory address the specific case of the field interaction of two spaced-apart bodies of similar mass and similar electric charge? I have in mind here the gravitational interaction of two stars in a binary pair as a case which contrasts with the action between the sun and a planet, where the mass of the sun far outweighs that of the planet.

This is certainly not discussed by Einstein but it is a vital consideration for any serious cosmologist interested in binary stars. Here one cannot expect Einstein himself to have had the foresight to discuss this topic, but one must point an accusing finger at the modern cosmologist who is unwilling to depart from the doctrines of Einstein's theory. This is a subject, however, that we have addressed by reference to the binary star DI Herculis in LECTURE NO. IIIb and, as the reader will have already seen or may there see, Einstein's theory stands defeated on the test of what is obseerved astronomically.

This ends the Part III section of this work 'The Physics of Creation'.

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