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On implications of Newton’s law of gravity

Copyright Michael Schmiechen 2018

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Physics of gravity ‘deduced’ Implications of Newton’s law of gravity in terms of elementary classical dynamics, developed in axiomatic fashion

Michael Schmiechen

Third draft of work in progress

Berlin, June 04 / July 17, 2018

Michael Schmiechen

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Physics of gravity ‘deduced’

Implications of Newton’s law of gravity in terms of elementary classical dynamics, developed in axiomatic fashion 5

Michael Schmiechen Retired 1997 from the former Berlin Model Basin, VWS: Versuchsanstalt für Wasserbau und Schiffbau 10

Abstract

“The Missing Link: Classical Mechanics!”

Title of my explanatory note proposed for publi-cation in the Scientific American, September 11, 2003 (…/missg_lk.pdf). 15

The purpose of the present exercise is to provide a concise tutorial of a classical theory of gravity, reduced to the bare essentials and purged of my earlier mistakes and remaining superstition. The aim is to take almost all the ‘romance’ (John S. Bell concerning David Bohm’s quantum mechanics) not only out of classical general relativity. At least since 2001 I have 20

brought preliminary versions to the attention of colleagues working in the field in various letters and in many explanatory papers.

In the spirit of Maxwell I am concerned with the physics of the phenome-non of gravity as experienced on the macroscopic level of the Sun’s plane-tary system and in the mesoscopic, ‘sub-lunar’ (Aristotle) world we live in, 25

carrying our weights. And, last but not least, on the microscopic and on the sub- microscopic, the quantum level I am explaining gravity to be due to the dynamics of nucleons, the building bricks of ponderable matter.

Nota bene: I am not concerned with cosmology and not with Einstein’s relativistic theory, subject of textbooks and literatures of their own. An early 30

example, ‘Space – Time – Matter’ by the mathematician Hermann Weyl, the eighth German edition of which, edited, complemented and still recom-mended by Jürgen Ehlers in 1993, is being scrutinised in explanatory Notes at the end of the paper proper.

Keywords: momentum balance, momentum storage, momentum diffusion: 35

alias surface force, momentum production: alias body force; mass distribu-tion in the ‘world’: source field of the mass potential, pervading the Uni-verse; driving causes of momentum production: gradients of the mass po-tential; ‘production or reaction constant’: alias gravity constant.

For convenient reading this exposition may be printed (and bound) as 40

DIN A5 (21 x 15 cm) brochure.

Michael Schmiechen

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Table of Contents

1 Preface 5 1.1 Gravity field: arbitrary 5 5

1.2 Meta-physics: basics 6 1.3 Abstract: detailed 6

2 Opening operations 8 2.1 Background 8 2.2 Problem identified 10 10

2.3 Model conceived 11 2.4 Goal defined 11 2.5 Plan designed 12 2.6 Change organised 14

3 Meta-Mechanics 15 15

3.1 Meta-theory 15 3.1.1 Formal languages 15 3.1.2 Rules to be understood 16 3.1.3 Coherence required 16

3.2 Epi-Language 17 20

3.2.1 Categories necessary 17 3.2.2 Tales, metaphors, etc 18 3.2.3 Anschauen, Anschauung 19

3.3 Theory of knowledge 19 3.3.1 Philosophy aversion 19 25

3.3.2 Principles: prejudices 20 3.3.3 Serious pitfalls 21

4 Proto-mechanics 22 4.1 Theories interpreted 22

4.1.1 Facta: being made 22 30

4.1.2 Theories: dual structure 22 4.1.3 Magnitudes, ‘quantities’ 23

4.2 Measurements 26 4.2.1 Time and space 26 4.2.2 Reference frames 26 35

4.2.3 Notation, Terminology 27 5 Elementary mechanics 29

5.1 Elementary dynamics 29 5.1.1 Balance of momentum 29 5.1.2 Balance of mass 31 40

5.1.3 Theorems deduced 31



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5.2 Elementary kinematics 33 5.2.1 State equations 33 5.2.2 Relative motion 34 5.2.3 Notes on history 35

6 Gravity law 37 5

6.1 Universe 37 6.1.1 Reference ‘mollusc’ 37 6.1.2 Coherence of masses 38 6.1.3 Einstein’s ‘aether’ 39

6.2 Mass potential 40 10

6.2.1 Ideal universes 40 6.2.2 Law of gravity 42 6.2.3 Simplest universe 43

6.3 Applications 44 6.3.1 Interpretations 44 15

6.3.2 Obscure energy 45 6.3.3 Things that are not 46

7 Gravity constant 49 7.1 Explanations 49

7.1.1 Naked speculations 49 20

7.1.2 Body model 49 7.1.3 Pure mechanics 51

7.2 Structure of matter 52 7.2.1 Dynamics of nucleons 52 7.2.2 Mesoscopic approach 53 25

7.2.3 Grand unification 54 8 Concluding operations 56

8.1 Result evaluated 56 8.2 Values assessed 56 8.3 Decisions derived 57 30

9 Notes 58 9.1 World 58

9.1.1 Search of roots 58 9.1.2 World: filled, limited 58 9.1.3 Gravity field 60 35

9.2 Potentials 60 9.2.1 Potential theory 60 9.2.2 Gravity potential 61 9.2.3 Ritual references 62

9.3 Pulsars 63 40

9.3.1 True time 63 9.3.2 Gravity waves 63 9.3.3 Pulsar PSR 1913 64

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9.4 Global mechanics 65

10 References 66 10.1 My publications 66

10.1.1 Website since 1998 66 10.1.2 Opus magnum 2009 66 5

10.1.3 Papers and letters 66 10.2 Other publications 67

10.2.1 References in Opus 67 10.2.2 References in Notes 67

11 Indices 69 10

11.1 Name index 69 11.2 Subject index 70

12 Author 73 12.1 Biography 73 12.2 Moral rights 73 15

12.3 Contacts 73

Tables of axioms and theorems

Axiomatic theories: structure 15 20

One ‘world’: many aspects 17 Short hand notation adopted 28 Momentum balance: axioms 30 Mass balance: axioms 31 Momentum balance: theorems 32 25

State space: dynamics 32 State space: kinematics, dynamics 34 Mass potential deduced 41 Newton’s law of gravity 42 Spherical mass potential 46 30

Spherical potential integrated 46 Potential energy integrated 48 Solid body model: axioms 50 Solid body model: theorems 50

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1 Preface

1.1 Gravity field: arbitrary

Concerning the concept of field Einstein noted in his popular monograph ‘Über die spezielle und die allgemeine Relativitätstheorie’ (1997/42):

“Die Berechtigung dieses an sich willkürlichen Zwischenbegriffs wollen 5

wir hier nicht erörtern. Es sei nur bemerkt, daß man mit seiner Hilfe die elektromagnetischen Erscheinungen, insbesondere die der elektromagneti-schen Wellen, viel befriedigender theoretisch darstellen kann als ohne den-selben. Analog fasst man auch die Wirkung der Gravitation auf.” Italics: MS. 10

This ‘decision’ and the ‘argument’ for the assumption of a gravity force field is of course not sufficient to establish its physical existence. Note-worthy though is the phrase ‘an sich willkürlichen Zwischenbegriff’, from which I derive the ‘Berechtigung’, the ‘justification’ of my much more con-vincing alternative classical approach concerning the phenomenon of 15

‘weight’, Schwere, of gravity of bodies, getting along without the many ‘problems’ subject of current research.

In this context I also refer to my observation at the end of section ‘7.7.3 World geometry’ of my opus magnum (OM/514 ff) with an extended quota-tion from Georg Singer’s introduction to Alexander Friedmann’s mono-20

graph ‘The World as Space and Time’ (OM/516): “Most important is the conclusion: 'Wegen der prinzipiellen Willkürlich-

keit, die materielle Wirklichkeit geometrisch zu interpretieren, ist die geo-metrische, allgemeiner die logisch-mathematische Struktur der Welt empi-risch nur bedingt erforschbar.' “ 25

Einstein’s stringent analogy of the mechanical ‘metrical field’ with the electrical field, ‘so we must assume here’, and the clear distinction between the mechanical ‘metrical field’ of the ‘material content’ and the ‘world’, whatever the latter may be, ‘the material content filling the world’, will be questioned in the body of the paper and in the appended Notes, both not 30

concerned with Einstein’s theory of general relativity.

As I have demonstrated since 2001 classical dynamics does not support the assumption of a gravity field to exist outside bodies of matter.

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1.2 Meta-physics: basics

“In modern natural philosophy, the physical concepts themselves are made mathematical at the outset, and mathematics is used to formulate theories.”

Clifford A. Truesdell (1966/88). Italics in the ref-5

erence.

Based on the instinctive belief, that the foundations of classical mechanics cannot be found and reconstructed within mechanics itself, but only 'out-side', classical mechanics is 'understood' by embedding it into an adequate theory of knowledge and adequate meta- and proto-theories in terms of the 10

'language of dynamics' as demanded by Kant, Goethe and specifically by Maxwell in 1877. Evidence is produced, that available ‘philosophical’ expo-sitions, provided they are sufficient for the purpose at hand, are hopelessly inadequate.

My theory of gravity has ‘finally’ been published 2009 in my opus mag-15

num with the title ‘Newton’s Principia and related ‘principles’ revisited; Classical dynamics reconstructed in the spirits of Goethe, [Aristotle], Euler and Einstein`, taking twelve years of intense thinking, phrasing and re-phrasing its results.

The present thoroughly revised and rearranged exposition of the implica-20

tions of elementary classical dynamics and of Newton’s law of gravity is reduced to the bare essentials and is developed axiomatically, ‘more ge-ometrico’, as ‘requested’ by Clifford Truesdell.

Abundant pertinent remarks, some of them admittedly no longer up-held!, and related extensive quotations for ready reference are provided in my opus 25

magnum, freely available on my website.

1.3 Abstract: detailed

My basic model is the elementary balance of momentum, i. e. of the ex-tensity of translational motion, valid in any classical reference frame, de-fined in due course. In terms of this model the weight of bodies is material 30

momentum production, in engineering jargon aptly called ‘body force’.

According to Newton’s law of gravity momentum production ‘takes place’ in bodies prevented from moving freely as parts of the mass distribu-tion, the source field of the mass potential, both inseparable fields constitut-ing the physical Universe. 35

And the driving causes, not forces, of momentum productions are gradi-ents of the mass potential, the gravity constant being the ‘production or re-action’ constant, with which constrained bodies react with material momen-



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tum production on gradients of the mass potential.

And as the model of a body shows, this conception is in accordance with the structure of matter, with the standard model of nucleons, quarks sus-pended in gluons.

The belief of physicists and astronomers in force fields and thus potential 5

energy outside bodies of matter is solely based on the strictly formal multi-plication of the ‘immaterial’ mass potential with the gravity constant, a mesoscopic, alias macroscopic, property of matter. This procedure is a con-venient ‘standard’ at the surface of the Earth, where bodies are immersed in matter, in the atmosphere and ‘the seas’. 10

As long as scientists continue to talk about the gravity force field pervad-ing ‘empty’ space, they nolens volens imply, that the formal multiplication results in physical potential energy, not to say ‘dark’ energy. It remains an ‘incredible’ fact of the history of physics, that generation after generation fell into that pit, a trap concealed only for ‘unsuspecting’ persons. 15

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2 Opening operations

2.1 Background

“§43. So far concerning the principles: now we must speak of the nature of motion, and indeed this, since it is clearly perceived by the senses, is not rendered obscure 5

so much by its own nature as by the learned comments of philosophers.”

George Berkeley, Bishop of Cloney: De Motu. 1721 (1992/92).

My motivation as marine engineer, with a solid background in hydro-10

dynamics and thermo-dynamics, to reconstruct classical mechanics, dynam-ics in particular, after forty years of daily pertinent practice ab ovo and re-flection has not been the so far unsolved problem of gravity, but the same problem already d’Alembert referred to in his famous 'Traité de Dynamique' of 1743 (1997/6): 15

“Man ist aber nicht so sehr bedacht gewesen, die Principien dieser Wis-senschaften [Mathematik, Geometrie, Mechanik] auf die kleinstmögliche Zahl zurückzuführen, noch ihnen die volle Klarheit zu geben, wie sie wün-schenswert wäre. Die Mechanik vor allem scheint man in dieser Hinsicht am meisten vernachlässigt zu haben: So haben denn auch die meisten ihrer 20

Principien, theils an sich unklar, theils in unklarer Weise ausgesprochen und abgeleitet, Anlass zu mehreren heiklen Fragen Anlaß gegeben. Man ist allgemein bisher mehr bemüht gewesen, das Gebäude zu vergrößern, als den Eingang in dasselbe zu erhellen; und man hat vor allem daran gedacht, dasselbe aufzurichten, ohne seinen Grundlagen die nöthige Sicherheit zu 25

geben.”

And in my rational reconstruction of classical dynamics I have elaborated on the need for reconstruction and followed Mach’s advice to combine his-torical and systematical research (1883/237):

“Die historische Untersuchung des Entwicklungsganges einer Wissen-30

schaft ist sehr notwendig, wenn die aufgespeicherten Sätze nicht allmählich zu einem System von halb verstandenen Recepten oder gar zu einem Sys-tem von Vorurtheilen werden sollen. Die historische Untersuchung fördert nicht nur das Verständnis des Vorhandenen, sondern legt auch die Mög-lichkeit des Neuen nahe, indem sich das Vorhandene eben teilweise als 35

conventionell und zufällig erweist. Von einem höheren Standpunkt aus, zu dem man auf verschiedenen Wegen gelangt ist, kann man mit freierem Bli-cke ausschauen, und noch neue Wege erkennen.”



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On my way to a higher standpoint I carefully avoided the swamps of the historical literature, but resorted to the only way out, to efficient rational re-construction.

In his ‘Problems of Philosophy’ of 1912 Bertrand Russell has addressed the same problem and derived the goals and the limits of philosophy 5

(1981/25): “ …many instinctive beliefs have, by habit and association, become en-

tangled with other beliefs, not really instinctive” ,

or as Johann Wolfgang Goethe felt (Sepper, 1988/65): “Goethe had confidence that science could rectify its mistakes, overcome 10

its limitations, and contribute to the general welfare as long as it remained active and open to debate. The greatest threat to science was the sterile pas-sivity induced by the degenerate scientific scholasticism, the mere handing down of a written tradition without substantial criticism.”

Mach clearly stated the task (1883/228): 15

“ ... of the following two hundred years Newton could expect the founda-tions of his achievement to be further investigated and made sound.”

In fact Newton himself expected this explicitly at the end of his Preface to the first edition of his Principia (PM/XVIII).

Most impressive, even touching is the statement of Heinrich Hertz in his 20

‘Prinzipien der Mechanik’ of 1894 concerning the dignity of the subject (1984/76), in the authorised English translation of 1899 (1956/9):

“In this sense we admit, as every one does, the permissibility of the con-tents [and the predictive powers] of [classical] mechanics [and of the the-ory of general relativity]. But the dignity and the importance of the subject 25

demand, not simply that we should readily take for granted its logical clear-ness, but that we should endeavour to show it by a representation so perfect that there should be no longer any possibility for doubting it. [Not only does the dignity of the subject demand this, but the dignity of the students and their professors!]”. [New additions]: MS. 30

In response to Hertz’ demand I have asked in my reconstruction of classi-cal dynamics (OM/106):

“The present state of affairs is not different from that at the times of Mach and Hertz more than one hundred years ago. And the question arises: In view of the [dignity and] importance of the subject why has the task be-35

ing defined by [Newton,] Mach and Hertz not been taken up and solved ei-ther by Hertz himself or any of the followers?” [Additions]: MS.

In view of the present unsatisfactory state of research I explicitly drafted my present paper as another modest contribution towards the goal d’Alem-bert, Mach and Hertz have stated. 40

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2.2 Problem identified

“If relativistic electrodynamics is correct, then we are still far from having a dynamics for the translation of rigid bodies.”

Albert Einstein, 1907 (Pais, 1983/154). 5

Even more than hundred years after Einstein’s remark of 1907 and after his own fundamental work on the theory of general relativity of 1915 the situation is still unchanged.

My own work on gravity has been an unintended by-product of my recon-struction of classical dynamics, results having been published at least since 10

2001, starting with an extended correspondence of twenty mails with Chris-toph Appel.

My naïve belief at that time was, that physicist had all the data at hand to compute the mesoscopic gravity constant according to my microscopic the-ory. Firstly, it took me only short reflection to realise, that they cannot pos-15

sibly have obtained the data of interest by crudely destroying protons in col-liders of ever increasing power.

Secondly, I knew from the theory of continua, that only in the simplest cases the mesoscopic phenomenological parameters of the constitutive laws can be derived by ‘statistical’ mechanics from molecular and quantum data. 20

They have to be identified based on data obtained in mesoscopic experi-ments. And the same holds of course for the mesoscopic gravity constant. Hence my credo as marine engineer: If you want to know the properties of a hydro-mechanical system, you have to ‘talk to’ that system and not to ‘somebody’ else, e. g. your computer. 25

The plethora of the exotic, not to say esoteric ‘theories’ produced by ‘pro-fessional’ gravity researchers, remind me of the old lady, who ‘knew’ our Earth to be based not only on one turtle, but ‘on turtles all the way down’. But whom are cosmologists, physicists, engineers and naval architects laughing at (Mara Beller)? 30

It is ‘standard’ scientific practice to ‘support’ established theories, which are no longer adequate and/or acceptable, as long as possible by additional ‘turtles’. The most famous example is the Almagest dating back to Claudius Ptolemy (ca. 100 to 170 AD).

Thomas Kuhn, in his paradigm of disruptive changes of paradigms, pub-35

lished in 1962, has vividly described, what happens when these fragile ‘foundations’ can ‘no longer’ be stabilised, but new, stronger ‘turtles’ need to be introduced to ‘support’ the ‘scientific worlds’. More than hundred years earlier Ernst Mach has already expressed similar ideas concerning the evolution of science. 40



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Currently many cosmologists believe, the Universe to be based on the ‘turtles’ dark energy and dark matter. But it has long been felt and dis-cussed, that the conceptions of gravity underlying the current research at the pertinent large institutes worldwide, may be utterly misleading and thus causing an irresponsible waste of intellectual and financial resources. 5

2.3 Model conceived

“ ‘Nothing can be heavy, you know, except by trying to fall and being prevented from doing so. You all grant that?’

We all granted that.” 10

Lewis Carroll: Sylvie and Bruno. 1889 (1988/312). Complete quotation in my opus mag-num.

The logician Lewis Carroll, alias Charles Lutwidge Dodgson, has been Mathematical Lecturer at Christ Church College in Oxford for nearly half a 15

century. But he became famous not for his insights into Newton’s dynamics and law of gravity, but as the entertainer of the three Liddell daughters, as the author of ‘Alice and Wonderland’ and of ‘Sylvie and [her very little brother] Bruno’.

In accordance with Newton’s law of gravity and the motto taken from 20

Carroll’s ‘report’ of a party conversation all freely moving bodies of matter constitute the source field of the mass potential constituting physical space. Bodies prevented from moving freely as parts of the source field are ‘react-ing’ on gradients of the mass potential, the driving causes, not ‘driving forces’(!), with the production of momentum, with ‘gravity’, ‘weight’ or 25

‘inertia’.

‘Accordingly’ the ‘rate of reaction’, in analogy to chemical kinetics and process engineering, alias gravity constant, is a mesoscopic property of ponderable matter in accordance with the standard model of nucleons, the building blocks of ponderable matter, three quarks ‘suspended’ in gluons. 30

2.4 Goal defined

The present approach is different from Hertz’ 'analytical' and other formal expositions, more in line with James Clerk Maxwell's intention (1991/124):

“Our aim … is to cultivate our dynamical ideas. We therefore avail our-selves of the labours of the mathematicians, and retranslate their results 35

from the language of the calculus into the language of dynamics, so that our words may call up the mental image, not of some algebraical process, but of some property of moving bodies.” Italics: MS.

The present exposition is dedicated to that same goal, 'to cultivate our dy-

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namical ideas', but not by 'stripping' mechanics from mathematical con-structs and artifices or 're-translating' their results from the language of the axiomatic system into the language of dynamics. My reconstruction starts from 'first', from meta-mechanical principles in the process of understand-ing, in Goethe's way of treating science as aptly characterised by Novalis 5

(1981.a/302): “Er abstrahiert mit einer seltnen Genauigkeit, aber nie ohne das Objekt

zugleich zu konstruieren, dem die Abstraktion entspricht. Dies ist nichts als angewandte Philosophie ... .”

Specifically the goal of the present exposition is rigorously to discuss the 10

implications of the elementary balance of translational momentum, i. e. the extensity of translational motion. In my opus magnum published in 2009 and in numerous publications and communications, starting as early as 2001, I have already elaborated the implications of the viewpoint I am pro-moting. 15

And in the past ten years I have continued to ponder my model and can now more clearly and distinctly state the implications in question, omitting all the detailed, in meantime many no longer up-to date discussions and most of the references to be found in my opus magnum. According to Des-cartes clare et distincte are criteria of ‘true’ perceptions. 20

The change of view point promoted implies specifically, that some unre-solved ‘fundamental’ problems of cosmology are simply due to the fashion-able ignorance of basic ‘meta-physics’, i. e. the theory of theories, and of classical dynamics, claimed to be ‘wrong’ anyways since Einstein’s work, and, last but not least, of the classical literature, not only concerning phys-25

ics.

In a way my approach is strictly in the tradition of the Greek natural phi-losophers, who already ‘derived’ the existence of atoms, and the later ac-count of their insights by the Roman Lucretius Titus Carus. And my ap-proach is strictly in the tradition of Aristotle and Newton, the Principia of 30

1686 published under the appropriate title ‘Philosophiae naturalis principia mathematica’.

2.5 Plan designed

“Presence of synonymy, intuitive appeal, agreement with customary modes of speech, far from being the phi-35

losophical virtue, indicates that not much progress has been made and that the business of investigating what is commonly accepted has not even started.”

Paul Feyerabend: How to be a good empericist. (1999/101 f). 40



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Maxwell has further qualified his aim (1991/136): “In forming ideas and words relating to any science, which ... deals with

forces and their effects, we must constantly keep in mind the ideas appro-priate to the fundamental science of dynamics, so that we may, during the first development of the science, avoid inconsistencies with what is already 5

established, and also that when our ideas become clearer, the language we have adopted may be a help to us and not a hindrance.” Italics: MS.

According to the following exposition the ‘traditional’ language of dy-namics is definitely a major hindrance in understanding the essentials. Liter-ally repeating Newton’s dynamics and Lagrange’s analytical mechanics as 10

originally published is no longer acceptable.

Thus the first task in solving my problem has been to generate a suffi-ciently rich language, adequate for the purpose at hand and for the commu-nication among the parties taking seriously part in solving the problem.

In order to escape from the incoherent jargon of theoretical physics and 15

mechanics most of the present exposition is phrased in a coherent, rule driven terminology and the corresponding symbols, reflecting the underly-ing models.

This procedure is not only efficient, but reveals the nature of the concepts known under traditional names, many of them grossly misleading as docu-20

mented in the present literature, e. g., in Hermann Weyl’s exposition of Ein-stein’s theories as will be shown.

Newton already stated (Gleick, 2004/181): “I have discovered a new property of matter, one of the secrets of the

Creator. I have calculated and demonstrated its effects; should people quib-25

ble with me over the name I give it?” Italics: MS.

And Goethe in a discussion has referred to languages corresponding to un-derlying conceptions, ‘Anschauungen’ (1999/148):

“Derjenige dessen Lebensgeschäft es ist den geheimnisvollsten Kräften nachzuspüren, ..., muß ja wohl das Recht haben, diesen Kräften solche Na-30

men zu geben, die ihm am schicklichsten däuchten, und sich dieselben vor-zustellen, wie es seiner Denkart am gemäßesten ist.”

In order to avoid the traditional misconceptions the exposition will obey the grammar of formal languages, i. e. the structure of axiomatic theories, i. e. accepted inter-subjective, alias ‘objective’, conventions, considered as 35

‘final’ standard of expositions since Euclid’s ‘Elements’, hence ‘more ge-ometrico’, and since Newton’s Principia, “… the only science[s] it hath pleased God hitherto to bestow on mankind … " (Thomas Hobbes, 1588-1679 (WilliamsJT, 1996/58).

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Although both the extremely powerful and influential axiomatic systems mentioned do not meet current standards, they are often still quoted and used as originally published. In all textbooks of theoretical physics, I had the chance to inspect, Newton’s axioms are ritually repeated as Newton stated them, as if rational classical dynamics, the contributions of Euler, the 5

Bernoullis, d’Alembert, Lagrange and their followers are not existent.

2.6 Change organised

“There is then opportunity for reason to effect, with comparative speed, what otherwise must be left to the slow operation of the centuries amid ruin and reconstruc-10

tion.”

Alfred North Whitehead: Symbolism, Chapter III (1959/69).

A ‘disadvantage’ of traditions is their tendency to live longer than 'rea-sonable', even to live forever, perpetuated among others by ‘peer’ reviews. 15

In a recent paper on ‘Why we don't believe, what does not suit us’ Retzbach (2017) has quoted the results of research projects concerning this inherent feature of human psyche to deal with ‘cognitive dissonance’.

Thus it often takes two or three generations, even centuries, before a change takes place. But for competitive reasons that pace is often no longer 20

acceptable. Needs for change identified ‘every where’ resulted in a vast lit-erature concerning the management of change.

Two pertinent, very instructive animal fables, published in slim volumes, have found wide distribution, the first one by Spencer Johnson, author of the ‘One Minute Manager’: ‘Who moved my cheese?’, describing and promot-25

ing the strategy of mice for managers, and the later one by John Kotter 'Our iceberg is melting', describes the ‘Penguin principle’.



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3 Meta-Mechanics

3.1 Meta-theory

3.1.1 Formal languages

“ … Denn die Bücher ohne Formeln haben meistens keinen Sinn …” 5

From a paraphrase of Bertold Brecht's ‘Drei Groschen Oper’ (Beggars' Opera) for Max Born's fiftieth birthday (TellerE, 1991/1).

Underlying the present exposition of the axiomatic, in Newton’s terms ‘mathematical’ theory of elementary classical dynamics is the grammar of 10

formal languages, the structure of axiomatic theories.

As in natural languages two ‘calculi’ have clearly to be distinguished: the calculus of concepts and the calculus of propositions. In German grammar books these ‘calculi’ are called ‘Wort-Lehre’ and ‘Satz-Lehre’, respectively.

Axiomatic theories: structure 15

Calculus of … concepts propositions

Rules of … formation introduction

Basic … concepts

propositions: axioms

Rules of … definition deduction

Derived … concepts

propositions: theorems

In the calculus of concepts, the ‘constitutive’ calculus, the basic concepts are sometimes called ‘atoms’ and the formally defined concepts ‘molecules’ (Bocheński, 1975/81) of the limited universe of discourse. But after all I refrain from using these and other confusing terms proposed. 20

Concerning the calculus of propositions I shall use the ‘universally’ estab-lish names, ‘axioms’ for the basic propositions introduced and ‘theorems’ for the formally deduced propositions. In order not to shy my colleagues in naval architecture and marine engineering away I am also using the term ‘conventions’ for ‘axioms’. 25

Axiomatic systems are in fact perfect systems of conventions accepted by a community. Inter-subjective concepts, axioms and theorems are not ‘ob-jective’, but hopefully more than less suitable for the purposes at hand.

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The rules of definition and deduction are essentially the rules of syllo-gism, the logical calculus first developed and used by Aristotle to provide ‘valid’ explanations, not only in philosophy, but also extensively in biology.

In view of applications formal languages are most efficient jointly (!) to 5

deal with model-universes of discourse, as the present one.

3.1.2 Rules to be understood

In detailed, rigorous expositions the rules of formation and introduction and of definition and deduction, as well as the basic concepts and proposi-tions, must be introduced explicitly. While the latter may be completely 10

abstract and formalised, the rules cannot be completely abstract. It needs to be understood, what has be done and how it has to be done.

The rules used in the present exposition limited to elementary classical dynamics are only the simplest rules of vector algebra, assumed to be known. As a result of this convention the appearance of definitions and de-15

ductions is the same and thus these fundamentally different ‘equations’ can easily be and often are confused, if as usual the same equal sign is being used in definitions and deductions. In formal definitions I use the appropri-ate identity sign.

3.1.3 Coherence required 20

In view of applications it has to be noted and never to be forgotten: the abstract, ‘absolute’ in Newton’s terminology, concepts introduced derive their coherent meanings solely in the context of the formal language adopted. Any other usage leads to infinite regress and ‘research’.

And similarly the concepts introduced obtain their coherent operational, 25

‘relative’ in Newton’s terminology, interpretations in the world we live in, in the language currently agreed upon by the members of community con-cerned.

Words ‘outside’ languages are meaningless! George Berkeley, Bishop of Cloyne, has referred to this fact in his ‘De Motu’ of 1721 (1992/76): 30

“As long as they indulge so far in abstractions, it is necessary that even the greatest men persue terms endowed with no signification and which are mere shadows of scholastic things. Other examples, and indeed not a few, could be produced from the writings of more recent authors, by which it would be abundantly established that metaphysical abstractions have not 35

everywhere given way to mechanics and experiments, but still make trouble for philosophers.” Italics: MS.

Concerning the explicit, coherent reconstruction of languages Peter Janich



17

has pinpointed the current situation (1981/114): “Ein Beispiel für Gegenstände eines solchen reflektierten Selbstver-

ständnisses ist etwa die Rolle der Sprache für die Physik, ein Thema, zu dem es durchdachte theoretische Ansätze gibt, während Physiker in aller Regel wie eh und je ihre komplizierte[n] Fachsprache[n] (definitionstheore-5

tisch) so unreflektiert erlernen wie Kinder das Gehen.”

3.2 Epi-Language

3.2.1 Categories necessary

“ …, dass die syntaktischen Kategorien – entsprechend der allgemeinen Funktion der Sprache, die ein Abbild der 10

Realität sein will – die so genannten ontologischen Kate-gorien abbilden.”

I. M. Bocheński (1975/52).

The terminology used, to deal with the subject under consideration has not yet been standardised. The resulting confusion of ‘languages’ reminds of 15

the confusion at the building of the tower of Babel (Genesis, the first book of Moses, 11: 1 − 9).

To highlight the problem I have crudely drafted a table of classical cate-gories, in the spirit of Plato and Aristotle. ‘Categorisation is fundamental in languages, inference and decision making’ (Wikipedia). 20

In particular I have listed various ‘opposite’ and/or complementing as-pects or components of the objects in the limited universe of discourse, many of them important in the present exercise, e. g., in case of the dual structures of theories and of potentials. The choice of terms used strongly depends on the context and the point of view. 25

One ‘world’: many aspects

absolute relative Newton

abstract interpreted Newton

active passive

actual potential Aristotle

dynamical kinematical

explicit implicit

extensive intensive

factual conventional

genuine pseudo

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ideal real

material immaterial

mathematical common Newton

objective subjective

physical virtual

quantitative qualitative

real apparent

stringent conjectural

substantial formal

true apparent Newton

As has been explained, the only way to escape confusion is to use formal languages in ndealing with the model-universes under consideration.

Their contexts provide unambiguous ‘meanings’ for terms, different for the same terms in different languages, while in natural languages the mean-5

ing of a term may change drastically in only few decades, even to the oppo-site. ‘Consequently’ historical texts cannot be read without expert guidance, as explained in detail in my opus.

3.2.2 Tales, metaphors, etc

“The humour of the 'Disc-World' is based on meta-10

phors taken literally and the consequences pondered.”

Terry Pratchett in an interview (2002).

Humans are not communicating by stating propositions phrased as main clauses, often with intricate subordinate clauses, not only to be found in German ‘philosophical’ papers and books. But they use whole coherent sto-15

ries, tales, parables, fables, metaphors, persiflages and satires, which evolved as the most efficient ways to communicate and highlight complete intricate situations and strategies of ‘survival’. Usually the name of a tale et alii is sufficient to invoke the whole context for those initiated, i. e. edu-cated and/or trained. 20

Most famous are emperor Sun Tsu's, Sunzi's collection of stratagems, about 500 BC, the animal fables of Aesop, of the 6th century BC, and the Arab collection ‘Kalila and Dimna’, collected about 750 AD for the same educational purpose as Sunzi's stratagems.

Repeated claims, that stories, symbolical and/or allegorical narratives do 25

not ‘apply’ in modern science, ignore the wisdom of our forefathers and the



19

‘power’ of vivid conceptions, of Goethe's ‘Anschauung’.

In the present exposition I shall introduce and exploit the implications of only two very simple ‘stories’: of the balance of the translational momentum of bodies of matter, and of the mass potential and its source field, constitut-ing the physical space, the Universe. 5

3.2.3 Anschauen, Anschauung

“Gedanken ohne Inhalt sind leer, Anschauungen ohne Begriffe sind blind.”

Immanuel Kant: Kritik der reinen Vernunft, B 75 (LudwigR, 1996/58). 10

In the early stages of axiomatic systems their basic propositions were re-quired to be ‘self-evident’. The requirement of self-evidence has been dropped with the advent of non-Euclidean geometries. Without going into any detail the following quotation concerning the aspect of self-evidence is of interest here (Weyl, 1950/77): 15

“Proclus (A.D. 5) utters an emphatic warning against the abuse that may be practised by calling a proposition self-evident. This warning cannot be repeated too often; on the other hand we must not fail to emphasise the fact that, in spite of the frequency with which this property is wrongfully used, the 'self-evident' property is the final root of all knowledge, including em-20

pirical knowledge.” Italics: MS.

This remark repeats Bertrand Russell’s discussion of the initial problem and of the role of our instinctive beliefs in his ‘Problems of Philosophy’ of 1912 (1981). Odo Marquardt, who ‘discovered’ the ‘Inkompetenzkompen-sationskompetenz’, has stated this in a drastic metaphor (1981/118 f): 25

“Jede Philosophie ist metaphernpflichtig; so wie beim Grog gilt: Wasser darf, Zucker soll, Rum muss sein; sonst lohnt es sich nicht: dort nicht das Trinken und hier nicht das Philosophieren.” Italisc: MS.

3.3 Theory of knowledge

3.3.1 Philosophy aversion 30

“Problems can not be solved by the methods, which have caused them.”

Albert Einstein.

My mentioning and taking advantage of formal systems of conventions since 1980 has shied my naval colleagues away instead of inspiring them, 35

immediately to try the power tool themselves and solve problems impossi-ble to be solved before.

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Naïve 'philosophy' aversion, expressed by physicists and engineers, even by professors, misses the fact, that the aversion is based on 'philosophy' it-self, but on an implicit, particularly poor one, not meeting the current re-quirements and standards, established not in all fields of research.

Thus, when I started to reconstruct propulsion theory for the difficult 5

problems of ship powering trials and monitoring, I did not ask naval archi-tects, but rather 'architects' of theories.

There is no chance to pull yourself out of the 'morass of ignorance' (Pop-per), as the Anglo-Saxons try by their bootstraps or as the Germans try by their braids, following Baron von Münchhausen’s story, incidentally not of 10

German, but of English origin.

3.3.2 Principles: prejudices

“You cannot have a theory without principles.

'Principles' is another name for 'prejudices'.”

Mark Twain: ‘The Disappearance of Literature’ 15

Speech, 20 November 1900.

In my reconstruction of classical mechanics, kinematics and dynamics have consistently been based on invariance principles, i. e. postulates.

Part 2 starts with Chapter 9 on ‘Meta-mechanics: ad hoc’ (OM/553 ff), including among others a Sections on the important ‘Meta-principles’, in 20

particular dealing with ‘Relativity, objectivity’, ‘Material invariance, equivalence’ and ‘Material, immaterial production’ and ‘Similarity’.

Concerning the latter I just mention Edgar Buckingham’s Π-theorem of 1914, discussed e. g., by Garrett Birkhoff in his ‘Hydrodynamics. A Study in Logic, Fact, and Similitude’ of 1950. 25

Of particular interest is the inspectional analysis (1955/90): “In order to become convinced of the truth of the principle italicized

above − which always underlies the validity of Assumption IV in fluid me-chanics − one must resort to ‘inspectional analysis’. By this I mean, in gen-eral, testing for invariance under the transformation (1), every fundamental 30

mathematical equation on which a given theory is based. This test was in fact used by Fourier, Stokes, etc.; it was clearly in the back of Rayleigh’s mind when he referred to ‘similitude’; and it has been used very often since. It occurs most commonly in connection with ‘dynamic similitude’, as now defined for fluids.” 35

The method of normalising explicit models results among others in the parameters of dynamic similitude, the normalised phenomenological, i. e. physical parameters. In the present context of free motions the close rela-tionship between the Kepler number, to be defined further down, and the Froude number, governing gravity dominated phenomena and hence ship 40



21

model testing, is worth mentioning.

In Chapter 10 (OM/581 ff) on ‘Elementary mechanics: abstract’ the first Sections are dealing with ‘Elementary axioms’, ‘Principles of relativity, objectivity’ and ‘Principles of materiality’.

3.3.3 Serious pitfalls 5

“MEPHISTOPHELES, zum Schüler: Mein treuer Freund, ich rat' Euch drum, Zuerst Collegium Logicum.”

Johann Wolfgang Goethe: Faust I (BA 08/207).

Advising a student-to-be to start with the study of logics is strictly in the 10

Aristotelian tradition of teaching for two thousand years. That Goethe lets the devil offer the advice does not imply, that it is purposely misleading, quite to the contrary. Frequently the advice is ignored, ‘consequently’ re-sulting in more or less serious mistakes, not to say in plain nonsense.

The present exposition is essentially the story of some such fundamental 15

mistakes. The summary is to be found in a concluding sub-section on ‘things that are not’, a notion due to Jonathan Swift.

His fourth journey took Gulliver to the country of the Houyhnhnms, ra-tional, equine beings with an orderly and peaceful society, with a philoso-phy and a language, that are entirely free of political and ethical nonsense. 20

They have no word for a lie, but must substitute a circumlocution: ‘to say a thing which is not’. (Wikipedia).

The sarcasm of the following story, which nicely fits into the present con-text, is pin-pointing the situation of anybody exercising lateral thinking (2003/217; abstract: MS). 25

“On his fourth and last journey, Gulliver has been asked by the Houyhnhnms, from which part of the country he came and how he was taught to simulate a rational creature. His claim, that he came with other sailors over the sea in a ship built of wooden logs, was found not at all plausible by the Houyhnhnms, not only in view of the obscure building of 30

ships, but more so in view of the fact, that there were no countries beyond the sea.”

The case of interest here concerns the coherent interpretation of abstract concepts based on the abstract model adopted. Evidently this procedure may result in vicious circles, circuli vitiosi. Here I only mention the openly stated 35

suspicion, that ‘dark energy’ might only be ‘needed’, to account for the defi-ciencies of the currently widely accepted mis-conception of gravity.

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4 Proto-mechanics

4.1 Theories interpreted

4.1.1 Facta: being made

“Das Höchste wäre: zu begreifen, daß alles Faktische schon Theorie ist.” 5

Johann Wolfgang Goethe: Maximen und Reflexi-onen. Posthumously published 1833. (GA 17/723).

We describe the world in terms of currently, widely shared, thus inter-subjective, alias ‘objective’, conventions. This fundamental observation applies of course to all human ‘research’. 10

In the motto Goethe already explicitly referred to the fact, that all ‘facta’, as their name says, are ‘being made’, are theory-laden. Now it is a platitude in the philosophical literature to refer to this fact (Faye, 2000/171):

“It is part of the folklore of today’s philosophy of knowledge that per-ception is theory-laden.” 15

But the naïve belief in ‘solid’, ‘hard’ facts, not to forget the currently fash-ionable ‘alternative’ facts, is still widely entertained, not least concerning the subject of this exposition!

Goethe’s dictum about the falling stone belongs into this context (Leo-poldina I, 3/158): 20

“That the stone falls is factum, that it happens through attraction is the-ory, of which one may be most inwardly convinced, but which one can never experience, never see, never know.”

‘That the stone falls is factum’ is the title of an explanatory exposition of my conception dated 09.02.2010, i. e. soon after publication of my opus mag-25

num (09.09.2009).

4.1.2 Theories: dual structure

So far abstract, ‘pure’ theories have been considered. Any of them may serve as model of a limited universe of discourse, hopefully useful for the purposes at hand. 30

To demonstrate, that a theory is useful as model of natural phenomena, the concepts introduced and derived have to be ‘interpreted’ operationally, based on measurements directly or indirectly. If interpreted in terms of the language adopted the magnitudes measured will be coherent as the concepts.

At this stage it is important to note, that in order to measure mechanical 35



23

phenomena at a given system not only the abstract theory is necessary, but other mechanical systems, mechanical ‘instruments’ are necessary and oth-ers are used as well. Hence the pertinent details can and shall be dealt with only in due course, as the theory is being developed.

Even if these instruments are not perfect, they permit reliably to identify 5

the magnitudes of interest, if their deficiencies are accounted for, if appro-priate ‘corrections’ are applied. The typical example is the finite speed of ‘messengers’ transmitting signals, usually photons.

In accordance with the dual structure of ‘natural philosophy’ Newton clearly distinguished between the abstract theory and its operational inter-10

pretation.

In the Scholium Newton states explicitly (PM/11): “Wherefore relative quantities are not the quantities themselves, whose

names they bear, but those sensible measures of them (either .accurate or inaccurate), which are commonly used instead of the measured quantities 15

themselves. And if the meaning of words is to be determined by their use, then by the names time, space, place, and motion, their [sensible] measures are properly to be understood; and the expression will be unusual, and purely mathematical, if the measured quantities themselves are meant. On this account, those violate the accuracy of language, which ought to be kept 20

prec ise, who interpret these words for the measured quantities. Nor do those less defile the purity of mathematical and philosophical truths, who confound real quantities with their relations and sensible measures.”

Newton clearly states ‘absolute’, mathematical Euclidean or rather Carte-sian co-ordinates to be measured by corresponding ‘relative’ clocks and 25

frames, respectively, the simplest possible metrics meeting our ‘instinctive beliefs’ (Russell) in ‘objective’ configurations of bodies in time and space and our various purposes.

Though ‘absolutely’ clear for anybody capable to understand, what he reads, Newton’s views on space, time and motion have become the most 30

fertile branch of the ‘Newton industry’ (Truesdell). But the endless, ritual repetitions of foolish prejudices are not limited to Newton’s works.

Aristotle, the founder of many observation based sciences, suffers the same fate (James Gleick, 2004/26):

“Both men [Descartes and Galileo] defied Aristotle explicitly – Galileo 35

by claiming that all bodies are made of the same stuff, which is heavy, and therefore fall at the same rate.”

In the ‘sub-lunar’ sphere neither Aristotle lived, nor do we live in vacuo.

Magnitudes, ‘quantities’

For the coverage concerning the fundamental notion of 'Quantities', in 40

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German and French not less ambiguous 'Grössen' and 'Magnitudes', respec-tively, I refer to my website and to my Section ‘6.3.5 Measurable Magni-tudes’, from which I quote the following paragraphs (OM/356 ff).

“In his ‘Introduction to symbolic logic and its applications’, translated from the German original of 1954, in a very short chapter on ‘Quantitative 5

Concepts’ Rudolf Carnap has clarified some ‘absolutely’ fundamental issues concerning terminology, symbols and units (1958/168-170).

“Such concepts are most conveniently designated by functors; their value expressions are considered of greatest general usefulness when they are real number expressions. ... 10

In the terminology customarily employed by physicists (a terminology, by the way, which is not entirely clear) measurable magnitudes ... are sometimes termed 'variables'. According to the terminology of modern logic, however, it is signs and not their designata that are divided into vari-ables and constants. Each concept, therefore, is to be designated by a con-15

stant, not a variable; and in particular, measurable magnitudes are to be designated by functors constants.”

The unsatisfactory usage of the term ‘variable’ for ‘magnitude’ does not only occur ‘sometimes’, but is to be found ‘nearly universally’, in most textbooks inspected. The reason is, that the values of magnitudes, i. e., func-20

tions, may change with the value of the argument in the range of definition. This fundamental aspect has been made explicit further down:

“Nevertheless, physical laws are intended to refer to arbitrary space-time points or regions, hence (at least when completely formulated) must exhibit variables [for values] as well as functor constants. It is usual in physics, 25

however, to give not the complete formulation of a physical law, but only an abbreviated formulation in which the variables [for values] have been omitted. ... Of course the abbreviated formulation has important advan-tages; ... The functors character of the symbols that survive in the abbrevia-tion should, however, not be overlooked.” Italics: MS. 30

In the following paragraph, in small print, Carnap admits that the signs in the abbreviated formulation may also be taken as variables for the values of the functors, and that their interpretation may be incorporated into the ante-cedent of the complete law. This is standard practice in all computational procedures, where one is interested in values, in ‘numbers’ only. Weyl 35

makes a remark to the same effect (1950/98 f).

Carnap continues with a pertinent remark on units: “Another question deserves attention here. Values of a measurable mag-

nitudes are expressed in terms of some unit of measure; where and how should this unit be specified? Ordinary practice here is to add to the number 40

expression for the value of the magnitude a sign indicating the unit of measure, ... Strictly speaking, however, the specification of the unit is part of the definition of the functor; the value of the functor is always a pure



25

number.” Italics: MS.

This is again standard practice in computational procedures, although programming environments, like Mathcad, in simple cases offer the capabil-ity to handle units and thus to check the consistency of 'physical' equations, in German Grössen-Gleichungen, as is done efficiently in paper and pencil 5

derivations and computations.”

End of quotation.

In its name the standard ISO 8000-1 explicitly mentions ‘units’ of meas-urement. In classical mechanics they are coherently based on etalons of the fundamental magnitudes duration, distance and matter. As in case of the 10

interpretations the etalons will be introduced in due course.

In view of the importance of this subject I quote from my detailed pro-posal (…/din_raw_draft.pdf) of a revision of the standard DIN 1313 in the spirit of Carnap (2012/17 f):

“In der folgenden Teil-Norm wird, wie u. a. in der Norm ISO 80000-1, 15

die Betrachtung auf Grössen mit Werten aus den Werte-Bereichen der reel-len Zahlen und der komplexen Zahlen, mit reellen Real- und Imaginär-Teilen, sowie Matrizen solcher Zahlen und Matrizen solcher Matrizen (nested matrices), beschränkt. Grössen mit Werten aus anderen, z. B einge-schränkten Werte-Bereichen, wie denen der rationalen oder der ganzen 20

Zahlen, sind [in Zukunft evtl.] Gegenstände anderer Teile dieser Norm.

Die Vorteile dieses Ansatzes, der auf Arbeiten von Frege und Carnap fußt, und der gängigen Praxis entspricht, rein numerische Werte als Werte von Grössen-Funktionen zu benutzen, beruhen auf der Tatsache, dass die Modelle damit tatsächlich rein 'mathematische' Modelle sind und die ma-25

thematischen Strukturen und Kalküle ohne Weiteres so verwendet werden können, wie sie genormt und in Programmier-Sprachen und auf Rechen-Anlagen implementiert sind.

NOTIZ 5.2.1: Die von Carnaps Ansatz implizierte Rolle der Mathematik in den Wissenschaften ist im Einklang mit den Vorstellungen und Erkennt-30

nissen anderer Autoren, wie z. B. mit denen Wittgensteins im tractatus lo-gico-philosophicus von 1918, veröffentlicht 1921:

"6.211. Im Leben ist es ja nie der mathematische Satz, den wir brauchen, sondern wir benützen den mathematischen Satz nur, um aus Sätzen, welche nicht der Mathematik angehören, auf andere zu schließen, welche gleich-35

falls nicht der Mathematik angehören.”

NOTIZ 5.2.2: Diese Auffassung steht Im Gegensatz zu dem Konzept von Werten 'physikalischer Grössen', bestehend aus ihrer Einheit und ihrem Wert. Die Einheiten der Grössen sind nach Carnap logisch aber Bestandtei-le der Funktoren. Die Methoden zur Identifikation der Werte von Grössen 40

repräsentieren nämlich in der Praxis aufwändig kalibrierte Mess- und Re-chen-Systeme, die nur Zahlen-Werte liefern; s. 6.4 Kalibrierungen.”

Quantities in the original, narrow sense, in German ‘Mengen’, are exten-

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sities, extensive magnitudes, as distinguished from intensities, intensive magnitudes. While in English 'quantities' or ‘amounts’ are distinguished from ‘sets’ of elements, in German quantities of continuous ‘matters’, as well as quantities of ‘numbers’ of discrete ‘elements’ are called ‘Mengen’.

4.2 Measurements 5

4.2.1 Time and space

In dealing with Newtonian and classical mechanics the concepts of ‘time’ and ‘space’ are usually considered as given. Newton explicitly states in his Scholium appended to the Definitions at the beginning of his Principia (PM/6 ff): 10

“I do not define time, space, place, and motion, as being well known to all. ... Only I must observe, that the common people conceive those quanti-ties under no other notions but from the relation they bear to sensible ob-jects. And thence arise certain prejudices, for the removing of which it will be convenient to distinguish them into absolute and relative, true and ap-15

parent, mathematical and common.” Italics: MS.

And Newton further continues, explaining these notions very clearly for time and space in that sequence, quoted next for ready reference.

In this context it is interesting to note, that Aristotle in Book 4 of his ‘Physics’, devoted to proto-theory, discusses, between Place and Time, the 20

Void, the subject of current gravity research worldwide and of this modest exposition.

4.2.2 Reference frames

Elementary classical dynamics is conceived as the study of translational motions of compact ‘solid’, not (mathematically) ‘rigid’ bodies of matter, in 25

one-dimensional time and three-dimensional space.

In order to avoid the discussion of formal details, irrelevant for the pre-sent purpose, the bodies of matter subject of the formal deductions will of-ten be considered as ‘small’ and ‘homogneous’. The qualification ‘small’ will be specified in due course. 30

The advantage of small bodies, of ‘probes’, is that not only the field of motion intensity is uniform, but all other fields in the bodies to be intro-duced in due course. And further, that the mass potential due to the body itself is negligible.

To be specific, the translational motions of bodies are described in refer-35

ence frames, for the present purpose considered as right-handed orthogonal Cartesian frames in arbitrary translational motions relative to the body un-



27

der consideration and to each other. Special purposes may require more suitable frames; again details are irrelevant in the present context.

Contrary to Einstein’s remark, in classical dynamics there are no ‘good’ and no ‘bad’ reference frames, alias observation spaces. In the chapter on 'Mechanical reference systems' of 'The Evolution of Physics' (1950/180 ff) 5

Einstein states (1950/190, original 1938/165 f): “ ... im Falle des Schiffes, setzt schwerer Seegang ein, so ereignen sich

merkwürdige Dinge. ... an Bord des Schiffes kollern Tische und Stühle durcheinander, und die Passagiere werden seekrank. Physikalisch gesehen, liegt das einfach daran, daß die Gesetze der Mechanik für diese Systeme 10

nicht gelten, daß es sich also um 'schlechte' Systeme handelt.

Diese Erkenntnis kommt schon in dem Galileischen Relativitätsprinzip zum Ausdruck: Wenn die Gesetze der Mechanik in einem bestimmten Sys-tem gelten, so gelten sie auch für alle anderen Systeme, die sich relativ zu jenem gleichförmig bewegen. 15

Wenn wir es jedoch mit zwei Systemen zu tun haben, die sich ungleich-förmig gegeneinander bewegen, dann können die Gesetze der Mechanik keinesfalls in beiden herrschen. 'Gute' Koordinatensysteme, solche also, in denen die Gesetze der Mechanik gelten, nennen wir Inertialsysteme. Die Frage, ob es in Wirklichkeit gibt, bleibt noch offen. Wenn ja, dann gibt es 20

unendlich viele; denn jedes System, das sich relativ zu einem Solchen gleichförmig bewegt ist dann ebenfalls in Inertialsystem.” Italics: wide print in the reference!

The terms ‘System’ and ‘Koordinatensystem’ have to be interpreted as ‘ref-erence frame’. 25

This very strong statement is definitely different from later statements of Einstein. But this incredible opinion about ‘bad systems’ is still the ‘stan-dard’ in ‘theoretical’ mechanics and is turned against classical mechanics. But what type of mechanics is that, holding in 'good’ frames only?

This is not classical mechanics taught to and practised by scientists and 30

engineers. Historians of science will have to find out, how it could possibly happen, that this ridiculous caricature of classical mechanics is still en vogue. But maybe they will never find out, ritually repeating and thus per-petuating this incredible story themselves.

4.2.3 Notation, Terminology 35

Instead of starting with a list of symbols and terminology, which will be introduced step by step in due course, I start to explain the usage of inverted commas for paragraphs, which are not quotations proper, and further in line with the usage recommended in the 'Current English Usage' (Wood, 1970):

“(iv) Inverted commas may also denote, that a word is used in irony or 40

sarcasm, [or] in a sense which is not its generally accepted one; … how-

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ever, they should be employed only, if it is felt necessary to apologise for the use of such words. …”

and with italics used in the literature, e. g., by John T. Williams in his ‘Pooh and the philosophers’ (1996/152):

“Our author's italics warn us to look for special importance.” 5

or by Jonathan Swift in ‘Gulliver's Travels’ (Notes, 2003/279): “This and several other words in the passage were italicised … in order,

apparently, to help the reader appreciate Swift's irony.”

In the present exposition foreign words are italicised as well, mostly Latin words, and the negation not, often in the combination not only. 10

Further only the shorthand notation used in mathematical expressions is listed.

Short hand notation adopted

symbol definition epi-language

i, j, k i, j, k {1, 2, 3} operational indices denoting orthogonal vector components

d t d / d t substantial rate of change

i / x i partial derivatives, nabla operator

i i Σ i 2 / x

i 2 Laplace operator, using Ein-

stein’s summing convention, delta operator

For simplicity the operational indices will be omitted not only in the fol-15

lowing tables, but often where possible without causing ambiguity.

Further ‘bodies’ and ‘probes of ponderable matter’ will simply be referred to as ‘bodies’ and ‘probes’, respectively, and ‘reference frames’ will simply be referred to as ‘frames’.



29

5 Elementary mechanics

“Durch den ganzen logischen Apparat hindurch spre-chen die physikalischen Gesetze doch von den Gegen-ständen der Welt.”

Ludwig Wittgenstein: Tractatus logico-5

philosophicus, 6.3431.

5.1 Elementary dynamics

5.1.1 Balance of momentum

In my opus magnum I have developed elementary classical mechanics as an instance of the axiomatic meta-theory of quantities proper, the rules of 10

instantiation being straightforward.

The aspect under consideration is translational ‘motion’ of solid bodies of matter

A = mot

and thus the basic concepts can be introduced as follows. 15

The momentum in any given frame corresponds to the quantity or exten-sity of motion

A ext = mot ext = M i ,

the mass or quantity of matter corresponds to the capacity of motion A cap = mot cap m 20

and the translational speed in any given frame corresponds to the quality or intensity of motion

A int = mot int v i .

All magnitudes introduced are state magnitudes of the body’s motion un-der consideration. 25

Further the diffusive momentum flow into the solid body and the momen-tum production inside the body are introduced as basic concepts, in classical mechanics the only two mechanisms changing the state, the momentum stored inside the body of matter.

Newton used the term motus quantitas, ‘quantity of motion’ and ac-30

knowledged the conceptual leadership of Galileo. But he was not fully aware of the ‘quantitative nature’ of momentum, later exploited by Euler.

In the definitions VI, VII and VIII of the Principia Newton used the term ‘quantity’ for ‘magnitudes’ of any type as is still common practice in Eng-lish, though obscuring essential aspects. 35

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Momentum balance: axioms

basic concepts

basic propositions

epi-language

M i mot ext extensity of motion, quantity of motion, momentum

v i mot int intensity of motion, velocity, speed

m mot cap capacity of motion, quantity of matter, mass

M i = m v i momentum

d t M i rate of change of momen-tum, momentum storage

M D i material momentum diffu-

sion into a body, surface force

M P i material momentum pro-

duction in a body, body force

d t M i = M D i + M P

i momentum balance

This first, fundamental section of my exposition on basic concepts and propositions is the shortest compared to those following. But already this simple, fundamental model ‘upsets’ scientists, who are not familiar with 5

rational continuum mechanics and with global dynamics and who are un-aware of the unity of all quantitative sciences.

Further, it is noted, that even the simplest systems of axioms are the ‘roots’ of ‘trees’ of extremely useful theorems, though valid only in their contexts. The most prominent examples are Euclid’s and Newton’s simple 10

Definitions and Axioms, not to mention the more recent calculi of logics, the non-Euclidean geometries and, last but not least, the theory of probabil-ity, with their extremely simple axioms and their incredibly diverse applica-tions.

All these formal languages are different from natural languages and lo-15

gics based on inherited instinctive beliefs, daily experience and indoctrina-tion at school. Thus in order to avoid confusion each has to be treated for-mally. In the present exercise elementary vector algebra and simplest syllo-gisms are sufficient, but necessary!



31

Syllogisms are the core of the logic conceived by Aristotle in the fourth century BC and solely used until the 19th century.

5.1.2 Balance of mass

The mass, the quantity of matter, is another state magnitude, in fact it is the prototype of quantities proper. Hence the classical mass balance can be 5

set up as follows.

Mass balance: axioms

basic concepts

basic propositions

epi-language

m mot cap capacity of motion, quantity of matter, mass

m D material mass diffusion into a body

m P material mass production in-side a body

d t m = m D + m P mass balance

if m D = 0 and m P = 0

then d t m = 0

mass of the body remains unchanged

That even the mass etalon at Sèvres looses molecules is not relevant for the present purpose, but is of considerable concern at the large metrological 10

institutes.

5.1.3 Theorems deduced

“ ‘Nothing can be heavy, you know, except by trying to fall and being prevented from doing so. You all grant that?’ 15

We all granted that.”

Lewis Carroll, alias Charles Lutwidge Dodgson: Sylvie and Bruno. 1889 (1988/312).

Extensive state magnitudes represent storages. Thus, if in given frames the momentum diffusion balances the momentum production, the extensity 20

of motion does not change, the momentum remains constant, the state mag-nitude is stationary. Talking about conserved or of conservative magnitudes is non-sensical.

Stationary states imply, that the intensities of motion, that the velocities of

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bodies in the given frames are constant. Of particular interest are body fixed frames. In this case not only the extensity of motion is stationary, but the intensity of motion vanishes.

Momentum balance: theorems

defined concepts

deduced propositions

epi-language

if 0 = M D + M P then d t M = m d t v = 0

balance momentum diffu-sion and production in body frame, stationary states

if v = 0 then 0 = M D + M P

momentum balance in body frame

if v = 0 and M D = 0 then M P = 0

momentum production in body frame, if body is freely moving

5

As this exposition is devoted to the physics of motion the appropriate frames are body frames, in which no motion takes place!

The last two theorems are evidently in perfect agreement with the insight of Lewis Carroll of 1889.

The equation for the changes of the velocity, the dynamical state equation 10

has already been derived, the equation for the changes of the location, the kinematical state equation will be stated further down.

State space: dynamics

basic concepts


epi-language

d M D / m

momemtum diffusion per mass mass, surface force ‘per’ mass, not an intensity, not mass specific!, strictly formal, shorthand notation!

f M P / m

mass specific momentum production, intensity of body force, body force field

d t v = d + f momentum balance ‘per’ mass, the diffusive term is not a mass specific magni-tude



33

if d t v = 0 then 0 = d + f

at stationary motions the intensity of momentum production balances the momentum diffusion ‘per’ mass

if d = 0 then d t v = f

in freely moving bodies the rate of change of the inten-sity of motion equals the intensity of the momentum production Newton’s second axiom

hence if d = 0 and f = 0

then d t v = 0 e. g. v = 0

in the absence of surface and body forces the veloc-ity is constant, Newton’s first axiom

Newton’s first axiom, which has here been deduced as a theorem, implies an empty universe with a lonely body, a most unrealistic, not particularly useful model.

In this case the body frame is the reference for other frames. Free motions 5

of the body observed in any of the other frames and the corresponding body forces are strictly apparent, no material forces being involved, except for any forces inside lonely bodies, which are not ‘small’ and thus ‘producing’ a mass potential.

In this universe only distances can be measured using any extension of 10

the solid body as etalon. The mass of the body may serve as mass etalon, but there are no scales in an empty universe to measure masses. Further there are no events taking place in an empty universe, which might serve as clocks, as time etalons, thus durations and speeds cannot be ‘measured’, i. e. compared. 15

5.2 Elementary kinematics

5.2.1 State equations

Not only the momentum, the extensity of motion is a state magnitude, but the velocity, the intensity of motion in the reference frame chosen as well. Further the location of a body in the reference frame chosen is another state 20

magnitude, which in this paper is only of interest in the context of difference and displacements.

While the field of translational velocity in the body is uniform, is the same for all parts of the solid body, the translational location of the body has

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to be specified. Conveniently the location of the centre of mass is chosen.

State space: kinematics, dynamics

basic concepts


epi-language

s i location of the centre of mass in a frame chosen

d t s i = v i kinematical state equation, holonomic!

d t v i = d i + f i dynamical state equation as deduced before

if f i = 0 i

then d t v i = d i dynamical state equation for bodies subject to sur-face forces only as deduced before

if d i = 0 i

then d t v i = f i

dynamical state equation for freely moving body as deduced before

In general the dynamical and kinematical states of translational motion of 5

a solid body are representing a six-dimensional state space. By an appropri-ate choice of frame it can often be reduced to two dimensions. If a solid body is also performing rotational motions the state space is twelve dimen-sional and the kinematical state is no longer holonomical.

If the laws for the momentum diffusion and production ‘per’ mass are 10

functions of the location the state equations describe systems, which may oscillate. Both special cases are sufficient to ‘deduce’ the physics of gravity.

5.2.2 Relative motion

The only way adequately to address the phenomena observed on board of a ship in a seaway is to state, that the frame is moving relatively to a freely 15

moving object, that the ship is moving relative to that object, which serves as reference, as ‘gravity probe’.

And instead of waiting for painful collisions with bulkheads or deck houses, for a painful ‘impulsive’ force, sailors use one of their muscular arms and hands to restrain any uncontrolled, dangerous motions by a more 20

or less modest surface force. The chairs, mentioned by Einstein and other bodies on board, valuable containers in particular, are solidly lashed to decks to prevent dangerous situations, damages and/or losses.



35

The result of the deductions so far is only apparently ‘negative’. But with-out reference to any theory of inertia and/or gravity elementary classical dynamics arrives at fundamental insights concerning forces in bodies. In particular the body force called gravity has been identified as the mesoscopic production of momentum in bodies of matter. 5

5.2.3 Notes on history

As Aristotle, based on his observations, Newton two thousand years later clearly distinguished between the motions of celestial bodies in the celestial spheres of the ‘fixed’ stars and the planets, and those of bodies in the ‘sub-lunar sphere’ (Aristotle), dominated by convective and diffusive momentum 10

flows. The claim that Aristotle was ‘wrong’, is one of the many still ritually repeated foolish prejudices concerning his ‘theory’ of falling stones and his ‘Physics’ in general, the latter devoted to ‘Change’ in general.

And in fact nobody has ever been ‘most inwardly convinced’ (Goethe) of the existence neither of forces at a distance, nor of ‘body’ force fields out-15

side bodies, in ‘empty’ space surrounding them. In response to the questions of his contemporaries concerning the nature of the gravity Newton cau-tiously stated his famous ‘hypotheses non fingo’ (Principia, Book III, Gen-eral Scholium, PM/547), which for ready reference is quoted here again:

“But hitherto I have not been able to discover the cause of those proper-20

ties of gravity from phenomena, and I frame no hypotheses; for whatever is not deduced from the phenomena is to be called a hypothesis; and hypothe-ses, whether metaphysical or physical, whether of occult qualities or me-chanical, have no place in experimental philosophy. In this philosophy par-ticular propositions are inferred from the phenomena, and afterwards ren-25

dered general by induction. Thus it was that the impenetrability, the mobil-ity, and the impulsive force of bodies, and the laws of motion and of gravi-tation, were discovered. And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of celestial bodies, and of our sea.” 30

Italics: MS.

And Galileo was at least as cautious as Newton in writing about gravity. In his ‘Dialogue Concerning Two New Sciences’ Salviati declares (Jammer, 1999.f/94):

“The present does not seem the proper time to investigate the cause of 35

the accelerations of natural motion concerning which various opinions have been expressed by various philosophers, some explaining it by ..., while still others attribute it to a certain stress in the surrounding medium, which closes in behind the falling body, and drives it from one of its positions to another. Now all these fantasies, and others too, aught to be examined; but 40

it is not really worth while. At present it is the purpose of our Author merely to investigate and to demonstrate some of the properties of acceler-

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ated motion (whatever the cause of this may be).” Italics: MS.

In this context it is worth noting, that different from Newton in Book I of his Principia and Einstein Aristotle did not deal with the simple case of freely moving bodies, but with bodies moving in the atmosphere and thus subject to surface forces. Newton’s Book 2 on ‘Motions of bodies in resist-5

ing media’ is in large parts obsolete since the works of Leonhard Euler and of Jakob I, Johann I and Daniel Bernouli and their followers.

Today everybody watching TV is familiar with the final theorem. But the theorem applies not only to bodies in space stations and in gravity probes, but to any satellite, e. g., the Moon revolving around the Earth, and the 10

Earth revolving around the Sun.



37

6 Gravity law

6.1 Universe

6.1.1 Reference ‘mollusc’

“For it may be that there is no body really at rest, to which the places and motions of others may be referred.” 5

Isaac Newton: Principia (PM/8)

So far a single body has been considered. But in the Universe there are very many freely moving bodies providing references. Einstein has called this moving system ‘reference mollusc’. The planets and the ‘fixed’ stars have been used as references by mankind since time immemorial. 10

A caveat is in place here: The term ‘mollusc’ invokes a rather ‘tender’ image. Real molluscs and astronauts in their training tanks are not floating freely in ‘empty’ space, but in water, a medium of density comparable to their own. Thus their capacity of motion is no longer a scalar magnitude.

The reference mollusc, the present exposition and mankind are limited to, 15

is the planetary system of the Sun. The trajectories of the freely moving planets are not at all straight lines, but fancily curved depending on the choice of the frame.

Copernicus found the helio-centric ‘observation’ space, more aptly ‘de-scription’ space, which had already been proposed in antiquity, most appro-20

priate efficiently to deal with the motions in our planetary system. But he still considered the orbits of the planet as ‘perfect’ circles.

In his introduction to Copernicus’ ‘revolutionary’ work ‘De revolutioni-bus …’ Andreas Osiander was as cautious as Newton, claiming only the descriptive power of Copernicus’ work. But his ‘perfect’ pragmatism was 25

far too revolting, in fact not only at his time, but even today. So his intro-duction appeared only once, in the first edition. Extended quotations from the German translation are to be found in my opus magnum (OM/142 and 529).

Based on his more precise observations, Kepler found out that the plane-30

tary orbits are in fact (very nearly) elliptical. And Newton’s simple axio-matic theory of, elementary dynamics and his law of gravity permitted not only to explain Kepler’s findings, but even the much more precise observa-tions of the following astronomers.

An example for the precision of the findings and their explanations with-35

out present day computers is the precession of Mercury’s perihelion due to

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the influence of more than one ‘small’ planet. The total observed precession is 5600 arc seconds per century, of which 99.23 % are explained by New-ton’s theories, leaving only 43 arc sec per century for Einstein to explain.

These facts are mentioned here, to convince cosmologists, that the recon-struction of classical dynamics and the study of the implications of New-5

ton’s law of gravity are definitely worth their efforts, before trying to invent ‘new physics’.

6.1.2 Coherence of masses

The idea of the ‘coherence’ of all masses in the Universe is dating back to antiquity. Newton refers to verses of Lucretius’ ‘De rerum natura’, which 10

until fairly recently has been compulsory reading for any person, claiming to be educated (1989: II 1064-1066):

“Deswegen muß man stets aufs neue bekennen, daß sich auch anderswo weitere Massen des Urstoffs ballen, ähnlich unserer Welt, die der Äther weit ausholend festhält.” Italics: MS. 15

Thus the idea is much older than Treder mentions (1980/30 f): “ ..., since the time of Huygens, formulations of the relativity principle

have existed that relate the dynamical properties of particles to the motion of distant masses. Such principles are important for global formulations of the theory of gravitation. They underlie Mach’s principle of relativity of 20

dynamics, Poincaré’s postulate of relativity, and the Mach-Einstein doc-trine.”

Einstein is quoted by Borzeszkowski (1994.e/4): “Es legt die Vermutung nahe, daß die ganze Trägheit eines Massenpunk-

tes eine Wirkung des Vorhandenseins aller übrigen Massen sei, auf einer 25

Art Wechselwirkung mit den anderen beruhend.” Italics: MS.

And further it is mentioned that “ … he [Einstein, 1918] used even the notion ‘Mach’s principle’ and that

‘Mach’s principle said more about Einstein’s and other authors’ reading Mach than on Mach’s intention.” 30

In his ‘Speculative Remarks on Physics in General and Relativity in Par-ticular’ Mercier notes (1977/299):

“The fundamental field is gravitation and has no specific property other than that of being the fundamental field. As a matter of fact, the only fea-tures of that field are first its admitting of null-lines and thus null-cones lo-35

cally, and second its being the consensus of all matter.” Italics: MS.

What a beautiful, poetic phrase: ‘the consensus of all matter’!



39

6.1.3 Einstein’s ‘aether’

The term ‘Mach's principle’ has been introduced into the literature by Einstein in 1920 (Pais, 1983/287), five years after publication of his theory of general relativity! In that year Einstein published a number of papers re-lated to the topic, among them his inaugural lecture at the Rijksuniversiteit 5

Leiden under the title ‘Aether und Relativitätstheorie’, which is not really conclusive, but ends with the following most remarkable statement (Ein-stein, 1920):

“Zusammenfassend können wir sagen: Nach der allgemeinen Relativi-tätstheorie ist der Raum mit physikalischen Qualitäten ausgestattet; es exis-10

tiert also in diesem Sinne ein Äther. Gemäß der allgemeinen Relativitäts-theorie ist ein Raum ohne Äther undenkbar; denn in einem solchen gäbe es nicht nur keine Lichtfortpflanzung, sondern auch keine Existenzmöglich-keit von Maßstäben und Uhren, also auch keine räumlich-zeitlichen Entfer-nungen im Sinne der Physik. Dieser Äther darf aber nicht mit der für pon-15

derable Medien charakteristischen Eigenschaft ausgestattet gedacht werden, aus durch die Zeit verfolgbaren Teilen zu bestehen; der Bewegungsbegriff darf auf ihn nicht angewendet werden.”

In a footnote Pais states (1983/313): “By aether Einstein meant gravitational field (one may wonder if this 20

name was felicitously chosen). ‘The aether of the general theory of relativ-ity is a medium without mechanical and kinematical properties, but which codetermines mechanical and electromagnetic events.’ ”

And at another place (1983/287): “In later years, Einstein’s enthusiasm for Mach's principle waned and fi-25

nally vanished.”

And Einstein himself wrote in 1954 (Pais, 1983/288): “Von dem Machschen Prinzip sollte man eigentlich überhaupt nicht

mehr sprechen.”

My personal reaction concerning this remark is rather pragmatic: no longer 30

talking about a problem does not solve it.

The whole discussion up to this point is of course very closely related to the basic arguments of the theory of general relativity. (Einstein, 1979/61, 63):

“Now it came to me: the fact of the equality of inert and heavy mass, i. e. 35

the fact of the independence of the gravitational acceleration of the nature of the falling substance, may be expressed as follows: In a gravitational field (of small spatial extension) things behave as they do in a space free of gravitation, if one introduces in it, in place of an ‘inertial system’, a refer-ence system which is accelerated relative to an inertial system. 40

If then one conceives of the behaviour of a body, in reference to the lat-ter reference system, as caused by a real (not merely apparent) field, it is

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possible to regard this reference system as an ‘inertial system’ with as much justification as the original reference system.

So, if one regards as possible, gravitational fields of arbitrary extension which are not initially restricted by spatial limitations, the concept of the ‘inertial system’ becomes completely empty. The concept, ‘acceleration 5

relative to a given space’, then loses every meaning and with it the princi-ple of inertia together with the entire paradox of Mach.”

It has been shown before that all this is straightforward classical mechanics.

A final quotation is of interest here (Pais, 1983/288): “1954. Einstein writes to a colleague, ‘Von dem Mach’schen Prinzip 10

sollte man eigentlich überhaupt nicht mehr sprechen’; As a matter of fact, one should no longer speak of Mach’s principle at all.

It was to be otherwise.

After Einstein, the Mach principle faded but never died. In the post-Einsteinian era of revitalized interest in general relativity, it has become an 15

important topic of research. At GR9, a discussion group debated the issue, in particular what one has to understand by this principle. This question can arouse passion. I am told that the Zeitschrift für Physik no longer ac-cepts papers on general relativity on the grounds that articles on Mach’s principle provoke too many polemical replies. At stake is, for example, 20

whether a theory is then acceptable only if it incorporates this principle as a fundamental requirement (as Einstein had in mind in 1918) or whether this principle should be a criterion for the selection of solutions within a theory that also has non-Machian solutions. It must be said that, as far as I can see, to this day Mach’s principle has not brought physics decisively farther. It 25

must also be said that the origin of inertia is and remains the most obscure subject in the theory of particles and fields. Mach’s principle may therefore have a future - but not without quantum theory.” Italics: MS.

6.2 Mass potential

6.2.1 Ideal universes 30

Many bodies are ‘around’ in the Universe and ‘somehow’ all of them mu-tually influence their motions, even if not in bodily contact with each other, if no momentum diffusion takes place. The few informal, inconsistent, and thus not conclusive quotations so far do not provide any explanations, for-getting about the more recent esoteric proposals. 35

Thus, in order to escape the hopeless confusion, often only ritually re-peated ‘tribal lore’ (Truedell), the following provides an explicit, coherent model of Newton’s ideal universe and Einstein’s physical space, not to say ‘aether’.

In an ideal, ‘empty’ universe all bodies are moving freely and all the im-40



41

plications deduced for single bodies apply to any of them.

Further, bodies may be considered not only as small, but even as ‘singu-larities’, ‘mass points’. Newton already proved, that not only spherical bod-ies with spherical mass distributions may be ‘treated’ that way. The typical application is Newton’s inverse square law of gravity. 5

But this is not of interest in the following exposition. I agree with Clifford Truesdell’s strong verdict in an Editor’s Introduction (1966/35):

“Mass points are a children’s disease like the measles. One has to have been infected at some age, hopefully only for a short time. But they appear to have become chronical, ‘cancerous’ as some thermodynamic traditions.” 10

‘Instead’ I use the concept of the mass potential as model of the Universe.

The mass distribution in the Universe, conveniently described by the field of the mesoscopic mass density in space, Weyl’s ‘world’, is the mesoscopic source field of the mesoscopic mass potential, described by the Poisson equation, a non-homogeneous partial differential equation of the elliptic 15

type.

But this does not imply that the mass potential is merely a mathematical construct, quite to the contrary. The material source field and its ‘immate-rial’ potential field, the latter pervading the whole space, even where occu-pied by bodies or perfectly evacuated, are inseparable physical entities con-20

stituting the physical space, the Universe.

Mass potential deduced

defined concepts


epi-language

m / V matter density in the bodies of matter

u m V d m / r

= V dV / r

mass potential: defined

i i u m = 4 mass potential: equivalent dif-

ferential equation

The ‘history’ of the potential equation has been outlined by Max von Laue in his ‘Geschichte der Physik’ of 1947 (1959/36 f) 25

“…the in-homogeneous potential equation framed in 1812 by Siméon Denis Poisson after Joseph Louis Lagrange in 1777 had defined the mass potential, for which Pierre Simon Marquis de Laplace in 1782 had derived the homogeneous differential equation.”

The singularity in the integral has to be treated with care and, further, care 30

has to be taken at discontinuities, if any, e. g., at the surface of solid bodies,

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details to be found in the literature on potential theory.

Contrary to the classical equation of the mass potential Einstein deriva-tion of his field equation of general relativity is based on the formal equa-tion of the gravity potential. The ‘disadvantage’ of that equation is, that nei-ther its source field, nor the gravity potential correspond to physical fields. 5

Hence the ritually repeated claim, that Newton’s dynamics and law of gravity are special cases of Einstein’s much more ‘general’ theory is simply not true. Contrary to Einstein and his followers neither Newton in his hy-potheses non fingo claimed, nor classical dynamics implies the existence of a body force field outside bodies of matter. 10

6.2.2 Law of gravity

“Daß der Stein fällt ist ein Factum, daß es durch Attraction geschehe ist Theorie, von der man sich innigst überzeugen kann, die man aber nie erfahren, nie sehen, nie wissen kann.” 15

Johann Wolfgang Goethe: Über Newton. 1703.

In terms of the concepts introduced so far the simplest possible law of momentum production is Newton’s linear law of gravity.

In the following sections the symbols for the momentum diffusion into solid bodies and for the momentum production in solid bodies will refer to 20

the molecular and to the nuclear nature of the two processes, respectively.

Newton’s law of gravity

basic concepts

basic propositions

epi-language

M M ≡ M D referring to the molecular na-ture of momentum diffusion

M N ≡ M P referring to the nuclear nature of momentum production

G gravity constant

M N i = m G i u

m Newton’s law of momentum production, law of gravity

g i ≡ G i u m intensity of momentum produc-

tion in a body body force field

Newton’s law of gravity is the constitutive law of material momentum production and the phenomenological parameter, the gravity constant, is the 25



43

physical parameter of interest.

‘In principle’ the law of gravity does not need to be linear and in fact a modified law has been proposed by Mordehai Milgrom, explaining the ‘un-expected’ distribution of angular velocities in disk galaxies without ‘dark matter’. The theory is referred to as MOND, ‘Modified NewtOn Dynamics’. 5

In the light of the present exposition not Newton’s classical dynamics has been modified, but ‘only’ the law of the body force, of gravity.

The study of that law in the laboratory is extremely delicate, so that even the so called gravity ‘constant’ has not yet reliably been identified. And its study on cosmic scale is not less delicate. Orbital speed data have been pub-10

lished under title ‘SPARC: Mass Models for 175 Disk Galaxies with Spitzer Photometry and Accurate Rotation Curves’ by Federico Lelli, Stacy S. McGaugh, and James M. Schombert.

6.2.3 Simplest universe

“With gravity time comes into the Universe.” 15

August Föppl (1910/20.) Paraphrase: MS.

In the simplest case two bodies of equal, mass m with equal speeds v = d t r are approaching the joint centre of mass, the ‘origin’ of the frame. Instead of talking about two bodies it is much more enlightening to consider both as parts of one ‘system’, the simplest universe. 20

Consequently the momentum balances for the two parts of the system ap-proaching each other in the frame chosen are

+ [m d t v = M N = G m / (2 r) 2]

and − [m d t v = M N = G m / (2 r) 2] , 25

resulting in the vanishing sum. i. e. both, the storage and the production of momentum in the system vanish. In view of the interaction of any two bod-ies the same holds for any ideal universe.

In case of ideal elastic collisions the two bodies would provide a clock. More realistic are ‘clocks’ of two bodies revolving around their common 30

centre of mass. This model has the advantage, that it does not suffer from mistakes due to approximations adopted in case of ‘small’ bodies, e. g., planets, satellites or ‘gravity probes’, and resulting in endless inconclusive discussions.

TIME 35

The simplest of such ‘clocks’ are again two bodies of equal mass. In this case the radial acceleration is balanced by the gravity of the bodies

r ω 2 = G m / (4 r 2)

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with the circular frequencies of their revolutions ω = 2 π / T

and thus the Kepler number Ke ≡ G T 2 m / r

3 = 16 π 2.

The squares of circular frequencies are well known as ratios of the stiff-5

nesses and the masses of the simplest oscillating systems. Thus the result of this simple inspectional analysis does not come as a surprise, but is in ac-cordance with the property of the body model described further down.

Any such system may thus be used as clock. Typical systems of that type are the Sun with its planets, among them the Earth, and the Earth with its 10

Moon.

Newton in the Scholium refers to time measurements using such ‘relative’ clocks (PM/6):

"Absolute, true, and mathematical time of itself and from its own nature, flows equably without relation to anything external, and by another name is 15

called duration; relative, apparent, and common time, is some sensible and external (whether accurate or unequable) measure of duration by means of motion, which is commonly used instead of true time; such as an hour, a day, a month, a year." Italics: MS.

The pulsar PSR 1913 mentioned in the Notes provides a nearly perfect 20

clock.

6.3 Applications

6.3.1 Interpretations

After all the last two theorems deduced from the law of gravity finally permit coherently to identify the values of the quantity of matter and of the 25

constant of momentum production, alias gravity constant.

MASS Since times unknown masses have been determined by balancing their

weights in a given gradient of the mass potential with the weights of refer-ence masses. 30

In the simplest case this can be done by beam scales, balancing the sur-face forces caused by the weights, the momentum productions. In the equa-tion of the beam scales, the ‘equation’ of the surface forces

− m G i u m = M M

i = M M ref i = − m ref G i u

m ,

the local intensity of momentum production cancels out, thus resulting in 35

the ‘equation’ of masses. Evidently beam scales do not ‘work’, if the gradi-ent of the mass potential vanishes, as e. g., in space stations.



45

Modern scales, i. e. surface force meters, need to be re-calibrated with the reference masses, in order to account for changes in the gradients of the mass potential due to whatever causes. And the reference weights in (daily) use have to be regularly re-calibrated according to international and national laws and/or conventions, i. e. agreements. 5

While for most purposes the basic reference mass, the kilogram etalon at Sèvres serves the purpose, ‘a tous les temps, a tous le peuple’, two groups of physicist are presently replacing the simple platinum-iridium etalon by even more stable, though extremely elaborate ‘equivalents’ to serve the purposes of utmost precise measurements. 10

After all the term ‘ponderable’ matter is now ‘justified’ and even applied to celestial bodies, although these cannot be put onto calibrated surface force meters. Thus the values of the mass of celestial bodies are typical, examples of intricately theory laden facta.

GRAVITY CONSTANT 15

The only way to determine the value of the mesoscopic gravity constant are local measurements on laboratory scale, which first have been per-formed by Cavendish in 1798 and since developed to perfection.

But considerable ongoing research activities at various prominent insti-tutes resulted in differences between the values ‘measured’, much larger 20

than expected. The so-called ‘Controversy over Newton’s Gravitation con-stant’ with a literature of its own is covered in the Open Directory Project (http://dmoz.org) under ‘Classical Mechanics’.

An elaboration on the subject is to be found in my opus magnum. In the present exposition only the essentials are of interest. The last theorem de-25

rived permits to identify the value of the constant in question, provided the gradient of the mass potential is known. Wichard Pohl has demonstrated this approach successfully, even in one of his famous lecture hall experiments (1947/44).

As long as the gravity constant is not identified within a very narrow con-30

fidence range, it may be speculated to be slightly different for different ‘types’ of matter, depending on the ratio of the slightly different masses of protons and neutrons in the nuclei; see the subsection ‘4.3 Gravitation con-stant’ in my opus.

6.3.2 Obscure energy 35

Referring back to the various quotations, the mass potential may be con-sidered explicitly to represent the coherence of all bodies in the Universe and/or Einstein’s ‘aether’. The immaterial mass potential is not just a formal construct, but a physical reality.

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The term ‘aether’ has the advantage to refer to the 'immaterial' nature of the mass potential, but I shall not use that term in order to avoid any mis-conceptions and confusions. In my opus magnum various other concepts of ‘aether’, proposed since antiquity, some of them non-sensical as the aether at rest, constituting the Universe, have been discussed, but here only the 5

mass potential is left and of interest.

The present simple model-universe is restricted to the mechanical aspects of translational motions of bodies. But current research in astronomy is con-cerned with large bodies and with physics ‘in general’ at conditions far from those prevailing in our planetary neighbourhood, but somewhere ‘out’ in our 10

and in distant galaxies. ‘Our’ galaxy was the only one known at Einstein’s time, while in the meantime billions of galaxies, each with billions of solar systems, have been discovered.

Even in view of this Universe the simplest possible abstract universe, a single spherical body with its mass potential is of interest. 15

Spherical mass potential

basic concepts

basic propositions

epi-language

R radius of the spherical body

r = [R, ∞] surrounding space coordinate

Spherical potential integrated

defined concepts


epi-language

u m = m / r

mass potential of the spheri-cal mass outside the body

U m ≡ r u m 4 π r 2 dr integral over a space region

U m = 2 π m r 2 indefinite integral

U = ∞ integral over the whole mass potential outside the body

The result of this exercise will be important in the following discussion. 20

6.3.3 Things that are not

“There cannot be a better service done to the truth then to purge it of things spurious.”



47

Isaac Newton: Correspondence III, 358, (Gleick, 2004/149).

Jonathan Swift in his persiflage of past and current research organisation and projects already referred to ‘talking about things that are not’ and classi-fied it as ‘lying’. This was long before the fashionable concept of ‘alterna-5

tive facts’ had been introduced. At this stage I mention three notions, that do not correspond to existing things.

FORCE FIELD OUTSIDE BODIES

A body force field outside a body of matter is a self-contradictory notion, a contradictio in adjecto, even plain literal nonsense, subject of the elabora-10

tions in my opus magnum of 2009.

Contrary to the opinion widely held, mass specific momentum produc-tion, body force fields outside bodies do not exist, they ‘cannot’ exist, nei-ther conceptually, momentum production in ‘empty’ space is non-sensical, nor physically, there is no way to prove their existence, any probe intro-15

duced is another body of matter.

Accordingly Einstein states very cautiously, though explicitly referring to the ‘force of attraction’, Anziehungskraft, acting at a probe in the vicinity of a sphere, maybe the Sun, (1950/152 f):

“Die Strahlen in unserer Skizze sind die Kraft l inien des Schwere-20

feldes. Die Kraftlinien können nämlich im leeren Raum konstruiert wer-den, ohne daß Materie vorhanden zu sein braucht, und vorläufig zeigen sie alle – oder, wie man auch sagen kann -, zeigt das Feld lediglich an, wie sich ein Prüfkörper verhalten würde, den man in die Nähe der Kugel [, des Körpers] brächte, deren Feld wir konstruieren.” Italics: MS. 25

Most surprisingly Einstein does not further elaborate on this fundamental observation and does not exploit its full potential, but abruptly changes his ‘limited’ universe of discourse (1950/154; Italics: MS):

“Auf das Gravitationsproblem wollen wir nun allerdings noch nicht ein-gehen. Es sollte uns nur als Einführung dienen und das Verständnis ande-30

rer, ähnlicher Überlegungen aus der Elektrizitätslehre erleichtern.”

POTENTIAL ENERGY

If a body force field outside bodies of matter is claimed to exist, this im-plies nolens volens claiming a field of potential energy to exist, of which the body force field would be the gradient field. And in perfectly correct ex-35

periments the force field and the potential energy field will be demonstrated ‘to exist’.

For convenience humans at the surface of the Earth in the atmosphere, lo-cally in practically uniform gradient fields of the mass potential, may adopt the notion of potential energy. But this notion breaks down as soon as it is 40

applied to the space outside bodies of matter.

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In the simplest case a universe consists of only a single spherical body with a spherical gravity field and thus with a spherical field of potential en-ergy. Consequently the total potential energy around it is infinite.

Potential energy integrated

defined concepts


epi-language

e p ≡ G u m spherical potential energy field outside the body

E p ≡ r e p 4 π r 2 dr integral of the potential en-ergy over a spherical space region

E p = G U m indefinite integral

E = G U = ∞ potential energy in the whole space outside the body

5

This result shows, that the notion of potential energy is as non-sensical as the notion of a body force field outside bodies. The search for the ‘obscure’, not to say ‘dark’, potential energy is a particularly drastic instance of the pitfalls mentioned at the beginning.

INERTIA ET ALII 10

The notion of inertia does no longer occur in my exposition. The capacity of motion in the storage term of the momentum balance is the quantity of matter, the mass. There is no different magnitude ‘inertial mass’.

The same applies in case of the production term. Momentum production takes place in the quantity of matter, the mass. There is no different magni-15

tude ‘gravitational mass’.

While in Einstein’s expositions the so-called equivalence of ‘inertial mass’ and ‘gravitational mass’ played a prominent role (1972/43-46), in the present reconstruction the notions of ‘inertial mass’ and ‘gravitational mass’, active and passive, are ‘empty’ and thus obsolete. 20

That the values of the magnitudes mentioned and identified by Eötvös with utmost care, to be exactly the same, proves, that his measurements have been performed correctly and nothing else and nothing more. Thus the title of one of my explanatory papers is ‘What did Eötvös do?’ The result obtained by Eötvös is ‘true’ by definition, due to the conceptual identity of 25

the notions. This is another serious case of the pitfalls mentioned at the be-ginning.



49

7 Gravity constant

“Raffiniert ist der Herr Gott, aber boshaft ist er nicht. Subtle is the Lord, [but malicious He is not].”

Albert Einstein (Pais, 1982/113). Italics: Title of Abraham Pais’ indispensable Einstein-Biography. 5

7.1 Explanations

7.1.1 Naked speculations

So far the motions of bodies have been described mesoscopically. But Newton’s mesoscopic law of gravity, which at least permits successfully to predict the motions in our planetary system with very great precision, im-10

plies that all matter consists of the same building blocks, of the same ‘stuff’, as Galileo already concluded.

Thus a theory of gravity, which deserves this name, must be a theory of the gravity constant in terms of the structure of matter. As Copernicus’, Ke-pler’s and Newton’s theories Einstein’s theory of general relativity is a de-15

scriptive theory as far as gravity is concerned. And thus all related attempts to provide a ‘theory’ of the gravity constant, which I had the chance to in-spect up to now, proved to be ‘wild’, incoherent speculations, lacking any sound conception of dynamics and of the physics of gravity.

7.1.2 Body model 20

Analoga of a given system are any other systems behaving in the same way as the system under consideration. For the present purpose only a sim-ple abstract mechanical model is of interest and readily conceived. A solid body is modelled as a stiff, mass-less container with its highly ‘condensed’ mass suspended at its centre by a very stiff spring-system. 25

For simplicity a cubic container and springs from the centres of the sides of the container may be envisaged. Further this model is assumed ‘to be put’ onto a force meter measuring the surface force, in the direction of the gradi-ent of the mass potential.

The springs directly transmit the momentum inflow measured to the con-30

densed mass. Thus only the component of the momentum balance for the latter in the direction of the gradient of the mass potential needs to be con-sidered.

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Solid body model: axioms

basic concepts

basic propositions

epi-language

m agg

aggregate mass

m agg = m

ρ agg density of aggregate mass

M D aggi momentum inflow to the ag-

gregate mass

M Di agg = M D

i

c agg stiffness of the spring system

δ aggi deflection of the springs, dis-

placement of the aggregate mass relative to the reference container

M D aggi = c agg δ agg

i costitutive law of the surface force

Solid body model: theorems

defined concepts


epi-language

0 i = c agg δ agg

i

m agg G u m momentum balance of the aggregate mass

0 = c agg m agg G i i u

m partial derivative of the mo-mentum balance

i i u m = 4 agg mass potential in the aggre-

gate mass

hence 0 = c agg + m agg G 4 agg

agg 2 ≡ c agg / m

agg

natural circular frequency of the aggregate system

G = agg 2 / (4 agg) production or reaction con-stant of momentum produc-tion, gravity constant

This global ‘model of matter’, a perfect analogon of solid bodies, does of 5

course not permit to identify the production or reaction constant of momen-



51

tum production in bodies, alias gravity constant.

But already at this stage everybody with only the slightest bit of curiosity and imagination will guess how my ‘story’ will end. At this stage it is only noted, that the aggregate natural circular frequency of the model of solid bodies is extremely large. 5

Although the term ‘Ersatz-Modell’ is ‘standard’ in German engineering, I am not using it, ‘ersatz’ usually implying an inferior substitute. Further, I avoided the term ‘singularity’ in view of the finite mass density of the ag-gregate mass assumed, the aggregate mass is not a ‘mass point’, I am not playing with ‘glass beads’. 10

7.1.3 Pure mechanics

Peter Gummert has stated, that the law of gravity, among the laws of other ‘physical’ or ‘impressed’ forces, e. g., the constitutive laws of con-tinua, belongs to experimental physics, and thus not ‘primarily’ to classical mechanics (1994/468 f). More stringent and very strongly the idea of ‘pure’ 15

mechanics has repeatedly been expressed by Truesdell, e. g., in his ‘Essays in the History of Mechanics’ (1968/239). For my taste both authors underes-timated the dual structure of any ‘natural philosophy’.

Particularly enlightening concerning this aspect is the last of Truesdell’s ‘Six Lectures on Modern Natural Philosophy’, with the prestigious title 20

‘Method and taste in modern natural philosophy’ (1966/83-108). My modest claim is to have overcome the deficiencies, Truesdell has rightly pinpointed. But I do not consider ‘advanced’ mathematics as a (totally) ‘foreign’ lan-guage (1966/85), in the present case elementary mathematics serves the purposes at hand. 25

Elementary mathematics is sufficient for setting up the axiomatic system of classical mechanics and deducing the important theorems and, last but not least, to understand its implications. The situation is similar to that in probability theory, where measure theory is not necessary for understanding the essentials and for very many applications (Papoulis, 1965). 30

Concerning the action of forces on masses I have ‘not drawn figures and not repeated rituals’ (1966/88), but the forces I am talking about are ‘implic-itly’, axiomatically‘ defined, in the context of a formal language, and are ‘addressed’ by corresponding names, revealing their ‘natures’.

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7.2 Structure of matter

7.2.1 Dynamics of nucleons

“Dissecare naturam.”

Francis Bacon, 1561-1626.

As has been stated the simple model of a solid body is a mesoscopic, out-5

put equivalent model, an analogon. If the body is cut into pieces the model is ‘seen’ not to be ‘true’. But according to Plato’s parable of the prisoners in the cavern and the corresponding meta-mechanical theory of state space models need not to be 'true', they cannot be true ‘images’.

The fundamental observation at this stage is, that the model outlined 10

‘holds’ for any body and thus for any part of any body, down to the level of the microscopic description and subsequently further ‘down’ beyond the molecular structure, down to the nuclear structure, und further to the quan-tum structure of the building blocks of nuclei, the nucleons.

In order to ‘explain’ momentum production, puzzling mankind from its 15

childhood, the classical procedure has been invoked already in papers and letters by the author published on his website in 2001 and 2003 and brought to the attention of various experts with the negative response expected. A mail by Jürgen Ehlers is typical for the lack of any substantial communica-tion (2001): 20

“In Ihrem Entwurf zur klassischen Mechanik und Gravitationstheorie habe ich nichts entdeckt, was mein Verständnis der gegenwärtigen Proble-me der Grundlagenforschung fördern könnte. Bitte sehen Sie von weiteren Zuschriften ab; ich möchte mich anderen Fragen zuwenden.”

This is not an answer to the problems addressed: How does the current 25

basic research link-up with classical mechanics? And what may be wrong with the answer proposed? The same lack of communication, mostly just lack of any response, as documented in the Section on ‘Letters yet unan-swered’ on my website, has also been and is experienced with colleagues working in ship theory. 30

Internal sub-systems are being conceived and the momentum productions are expressed in terms of flows from these sub-systems into the system un-der consideration.

On the first level the sub-systems are molecules. No storage taking place in the space between the molecules for lack of matter, the 35

whole momentum diffusion into the matter ‘vanishes’ into the molecules. So the puzzle remains!



53

On the second level the sub-systems are nuclei. Due to the fact, that the mass of electrons is small compared to that of nuclei, ‘it is safe’ to assume that in the orbitals ‘hardly’ any momentum is stored. If storage happens momentarily it may be speculated to vanish on the average. Consequently the whole diffusive flow into the matter, at 5

least approximately, ‘vanishes’ in the nuclei. So again the puzzle remains! The question arises: Are there other levels of sub-systems and is there an end to ‘the way ‘down’?

On the third level the sub-systems are ‘singularities’ of mass in the nu-cleons, here assumed to be the final sub-systems in accordance with 10

the current standard model of nucleons, the stable building blocks of matter, already referred to by Lucretius.

In a popular account of the ‘Innenleben des Protons’ Robert Klanner notes 2001/66):

“Quarks und Elektronen sind offenbar tatsächlich Materiepunkte. Zu-15

mindest ist ihr Durchmesser nicht größer als ein tausendstel des Protonen-durchmessers, also etwa 10 18 Meter. Sind wir damit vielleicht am Ende der Kette aus immer weiter teilbaren Materieteilchen angelangt, ...? ”

Right after I had conceived the mechanical body model, I received my copy of ‘Spektrum der Wissenschaft’, the German edition of the ‘Scientific 20

American’, with Klanner’s paper, illustrated by an artist’s rendering of the internal structure of a proton, And immediately I ‘knew’, that my explanation of gravity was right.

But I do not agree with Klanner’s term ‘Materiepunkte’, whatever the term ‘Punkte’ may imply. In my body model I more cautiously use the term 25

‘aggregate’, finite mass density having been assumed, however that may be ‘understood’ on that level.

Consequently the whole diffusive flow into the matter finally 'vanishes' in the building blocks of the sources of the mass potential. Again on the aver-age there will be no storage between the 'container' of the nucleons and the 30

aggregate mass inside (Klanner, 2001/65): “Für die starke Kraft ist die Situation gänzlich anders. Deren Botenteil-

chen die Gluonen sind masselos; ... ”

7.2.2 Mesoscopic approach

All this can safely be stated without caring for the quantum mechanisms 35

involved, provided it is kept in mind, that the intuitive way of talking does not imply, the mesoscopic interpretations of the axiomatic theory to be valid any longer on the quantum level.

That the model of gravity is a mechanical model of solid bodies is not a defect, but a necessity! After all I am talking about mechanics, and any the-40

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ory of gravity, worth that name, has to 'reach up into' the mesoscopic me-chanical world we live in, ‘forced’ to carry our own weights.

Advanced studies on the structure of nucleons have already been men-tioned (DürrS, 2008):

“More than 99% of the mass of the visible universe is made up of pro-5

tons and neutrons. Both particles are much heavier than their quark and gluon constituents, and the Standard Model of particle physics should ex-plain this difference. We present a full ab initio calculation of the masses of protons, neutrons, and other light hadrons, using lattice quantum chromo-dynamics. Pion masses down to 190 mega-electron volts are used to ex-10

trapolate to the physical point, with lattice sizes of approximately four times the inverse pion mass. Three lattice spacings are used for a continuum ex-trapolation. Our results completely agree with experimental observations and represent a quantitative confirmation of this aspect of the Standard Model with fully controlled uncertainties.” 15

An example of microscopic reasoning on the quantum level are Schrödinger’s wave equation and Bohm’s mechanics, which have been shown to be instances of the local balance of ideal continua (OM/889-911).

7.2.3 Grand unification

“String theorists might be barking up the wrong tree. 20

'It could all be wrong’, Greene says. Could be a total waste of spacetime.”

Joel Achenbach: Interview with Brian Greene. Washington Post, March 11, 2004.

Everybody having followed the preceding exposition will immediately 25

start speculating about the ‘grand’ unification of forces, since Einstein’s unsuccessful attempts and still one of the unsolved problems of physics. The question is of course, what did Einstein try and why?

The model of matter developed in accordance with Newton’s axiomatic theory, implying the force free motion of the reference mollusc, suggests 30

that there is no force field outside bodies of matter, which ‘needs’ to be united with other forces of physics. ‘Are physicists barking up the wrong tree’ concerning gravity, as Brian Greene in his ‘Elegant Universe’ has sus-pected? In view of the preceding analysis this is no longer a vague, unquali-fied suspicion. 35

In order to ‘derive’ the gravity constant according to the rule stated for the global model, the dynamics of the nucleons have to be studied. Destroying protons in colliders of ever increasing power Goethe would have discarded as the wrong approach, as is killing living creatures in search of their souls.

Most of the particles, the debris of the collisions identified by physicists 40

that way, are unstable. Only free protons are stable. Even free neutrons are



55

unstable and each decays after a lifetime of about 15 minutes into a proton, an electron, and an anti-neutrino' (NIST Neutron Lifetime Experiment). Only if neutrons are part of nuclei of atoms, they are stable. More recent accounts of the research on the structure of nucleons have been published by Bass and Samulat (2008), by Dürr et alii (2008) and by Spillner (2009). 5

Concerning the standard model of matter, which is far beyond the horizon of my present exercise, a recent paper discussing ‘Die mathematische Zäh-mung des Standard-Modells’ is of interest (Bär, 2009). Its abstract reads as follows:

“Die moderne Theorie der Elementarteilchen, die Quanten-Yang-Mills-10

Theorie, vermag die experimentellen Befunde mit unerhörter Genauigkeit wiederzugeben, es fehlt ihr jedoch bisher ein solides mathematisches Fun-dament’. Über dessen Gestalt gibt es nur sehr nebelhafte Vorstellungen.” Italics: MS.

This abstract invokes the layman's conclusion: ‘Nebulous’ are the concep-15

tions concerning the physics!

Classical mechanics does not only link gravity to the whole Universe, but also to the standard model of matter in accordance with Newton's fourth ‘definition’ and d'Alembert’s principle, the fundamental observation con-cerning the material momentum diffusion and momentum production bal-20

ancing each other in bodies constrained in their free motions as part of the ‘reference mollusc’.

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8 Concluding operations

8.1 Result evaluated

Classical dynamics of solid bodies of matter, not ‘mass points’, and New-ton’s theory of gravity imply, that the mesoscopic phenomenon of gravity, i. e. of momentum production in bodies of matter, is due to the dynamics of 5

the nucleons.

After another ten years of pondering the ‘higher standpoint’ (Mach), which I had reached in my opus magnum, a rational reconstruction of classi-cal mechanics, the present rigorous scrutiny resulted in further considerable clarifications of the limited universe of discourse. 10

Although the present exposition cannot possibly be more than a draft, subject to further corrections based on the insights gained so far, it finally presents not only a certain state of maturity, but a formidable change of paradigm, a ‘Copernican turn’. Accordingly it may take generations before the necessary consequences will be effective: the change of minds and of 15

text books and, last but not least, the subsequent revision and re-evaluations of former and current research projects.

8.2 Values assessed

The present exercise is not only intellectually satisfactory, but it is in ac-cordance with current experience, purged of inherited superstition and de-20

void of non-sensical notions.

Popper, ‘ein dogmatischer Anti-Dogmatists’ (Treder), knew the reasons for the difficulty ‘to reforge the tradition of our forebears’ (1978/124):

“ ... ‘nothing seems less wanted than a simple solution of an age-old phi-losophical problem’. The view of many philosophers and, especially, it 25

seems, of Wittgensteinians, is that if a problem is soluble, it cannot have been philosophical. There are of course other ways of getting over the scandal of a solved problem. One can say that all this is old hat; or that it leaves the real problem untouched. And, after all, surely the solution must be all wrong, must it not? (And I am ready to admit that quite often an atti-30

tude like this is more valuable than one of excessive agreement.)”

The same ideas have been proposed much later by Thomas Kuhn in his all too popular paradigm of paradigms, the source of a literature of its own. Recently in ‘Re-thinking the Paradigm Paradigm’ Gary Taube (2001) noted that ‘the philosophers got it wrong: scientists love new ideas if they are 35

right’: “The reality is that vigorous scepticism aimed at a potential new para-



57

digm means one of two things and usually both: first, that the spectacular breakthrough or the wondrous paradigm is indeed too good to be true, and second, that the reasons to be sceptical are very good ones.

Progenitors of new paradigms extract the truth from the swamp of con-flicting data [and theories]. If the evidence supporting them reaches a high 5

enough pitch, scepticism will fade.”

My exposition clearly shows, that I have not drawn my truths from the ‘swamps’ of conflicting data and of incredible theories, but following Goethe and Russell, proceding carefully step by step, in methodical order (Peter Janich), based on coherent physical conceptions (Goethe, Maxwell) 10

and instinctive beliefs (Russell) and taking advantage of the elementary the-ory of knowledge. Hence the title of the present paper.

8.3 Decisions derived

At my age I shall ‘only’ continue to scrutinise my insights so far, to de-velop them further and to tell my ‘stories’ of the momentum balance and the 15

mass potential, as long as I shall feel fit.

Far ranging decisions concerning the departure from traditional preju-dices, not to say superstition, have to be taken by future generations evading the current ‘Trouble with Physics’ (Lee Smolin, 2006).

Wherever my wishes and my duties took me, I always gratefully enjoyed 20

the luxury of perfect freedom and sufficient leisure to identify and solve problems my way, based on the hierarchy of my instinctive believes, finally explicitly documented in my opus magnum. In 1912 Bertrand Russell had identified such hierarchies as sole foundations of our knowledge.

In May 1686 Newton concluded the Preface to the first edition of his 25

Principia with the very humble request (PM/XVIII): “I heartily beg that what I have here done may be read with forebear-

ance; and that my labors in a subject so difficult may be examined, not so much with the view to censure, as to remedy their defects.”

My claim is to have responded to Newton’s request and to Einstein’s re-30

mark of 1907, that ‘we are still far from having a dynamics for the transla-tion of rigid bodies.’ After hundred years I have provided more than a sketch of rational classical dynamics, in response to Truesdell’s request, and its implications concerning the translation and the gravity of bodies. And thus I modestly end this exposition as Newton ended his preface. 35

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9 Notes

9.1 World

9.1.1 Search of roots

Although the present paper is not concerned with cosmology and with Einstein’s theory of general relativity, that theory has of ‘cause’ repeatedly 5

more or less explicitly been referred to. And it has clearly and distinctly been stated, that Newton’s theories of dynamics in general and of gravity in particular are not special cases of the ‘general’ theory.

My following detailed notes have been triggered by a remark in bold print on ‘the world’ in my copy of Hermann Weyl’s ‘Space – Time – Matter’, a 10

translation of the fourth edition of the German edition of 1922.

In search of the ‘roots’, of the German original, I came across the eighth edition of ‘Raum – Zeit – Materie’ edited, complemented and after 100 years still recommended by Jürgen Ehlers (1993). As Newton, Einstein could have hoped, the following generations to have further developed his 15

theory in view of the evidently remaining problems.

CHAPTER IV, THE GENERAL THEORY OF RELATIVITY, The Relativity of Motion, Metrical Fields, Gravitation, in Weyl’s famous monograph provide a vivid impression of the incredible ‘style’ under scrutiny. My discussion in terms of classical dynamics refers only to the most ‘surprising’ statements. 20

In view of my axiomatic exposition of classical dynamics and of the im-plication of Newton’s law of gravity Hermann Weyl’s rather mathematical exposition of Einstein’s theory of general relativity looks like a draft, still confusing fundamental issues, as usual in early stages of projects and theo-ries according to my personal experience. 25

To be specific, in Weyl’s exposition I am missing the clear distinctions between fundamentally different notions, e. g. between the abstract, mathe-matical, ‘absolute’ (Newton) concepts and their operational, physical’, ‘rela-tive’ (Newton) interpretations, and, last but not least, between mechanical and electrical dynamics. 30

9.1.2 World: filled, limited

Concerning the situation of research on Mach’s principle described by Pais I found the following remarks in Hermann Weyl’s exposition of Ein-stein’s theory of general relativity, the fourth chapter in his ‘Space – Time – Matter’, first American Printing 1922 of the fourth edition of ‘Raum –35



59

Zeit – Materie’, first published in 1918 (1950/220): “ ... The behaviour of light-rays and measuring rods, besides being de-

termined by their own natures, as also conditioned by the ‘metrical field’, just as the behaviour of an electric charge depends not only on it, itself, but also on the electric field. Again, just as the electric field, for its part, de-5

pends on the charges and is instrumental in producing a mechanical interac-tion between the charges, so we must assume here that the metrical field (or, in mathematical language, the tensor with components g I k) is related to the material content filling the world. We again call attention to the principle of action set forth at the conclusion of the preceding paragraph; in 10

both of the parts which refer to substance, the metrical field takes up the same position towards mass as the electrical field does towards the electric charge. The assumption, which was made in the preceding chapter, con-cerning the metrical structure of the world (corresponding to that of Euclid-ean geometry in three-dimensional space), namely, that there are specially 15

favoured co-ordinate systems, ‘linear’ ones, in which the metrical ground-form has constant coefficients, can no longer be maintained in the face of this view.” Bold print: in the reference.

Without going here into the details I just want to refer to the statement:

“… just as the electric field, for its part, depends on the charges and is in-20

strumental in producing a mechanical interaction between the charges, so we must assume here that the metrical field is related to the material content filling the world.” Italics: MS.

In view of my following derivations two implications are remarkable here: the stringent analogy of the mechanical metrical field with the electri-25

cal field: “so we must assume here”, and the clear distinction between the mechanical ‘metrical field’ and the ‘material content’ of the ‘world’, what-ever the latter may be: “the material content filling the world”.

Before starting the discussion of the above quotation I mention Weyl’s remark concerning the ‘world’ (1950/173): 30

“The world is a four dimensional affine space whose metrical structure is determined by a non-definite quadratic form, which has one positive and three negative dimensions.” In the reference this sign convention is being changed as convenient for the discussion of different aspects.

This is of course not ‘the world’, but is providing a mathematical struc-35

ture, a ‘metrical field’, to describe the ‘world filled with material content’. A very surprising statement, a revival of the good old ‘stage’ of events?

Further, I provide Weyl’s remark concerning the classical limit of the general theory to which my discussion restricted (1950/243 f):

“… the planet of mass m moves according to the laws of classical me-40

chanics, if we assume that a force with the potential m Φ acts in it. In this way we have linked up the theory with that of Newton: Φ is the Newtonian potential that satisfies Poisson’s equation.

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The deviation of the metrical groundform from that of Euclid is thus considerable enough to make the geodetic world-lines differ from rectilin-ear uniform motion by the amount actually shown by planetary motion -although [!] the geometry which is valid in space … differs only very little from Euclidean geometry as far as the dimensions of the planetary system 5

are concerned. (The sum of the angles in a. geodetic triangle of these di-mensions differs very very slightly from 180°.) The chief cause of this is that the radius of the earth’s orbit amounts to about eight light-minutes whereas the time of revolution of the world in its orbit is a whole year.” Italics: Duplication of ‘very’ in the reference. 10

9.1.3 Gravity field

Concerning the gravity field Einstein notes in his ‘Über die spezielle und die allgemeine Relativitätstheorie’ (1997/42):

“Die Berechtigung dieses an sich willkürlichen Zwischenbegriffs wollen wir hier nicht erörtern, Es sei nur bemerkt, daß man mit seiner Hilfe die 15

elektromagnetischen Erscheinungen, insbesondere die der elektromagneti-schen Wellen, viel befriedigender theoretisch darstellen kann als ohne den-selben. Analog fasst man auch die Wirkung der Gravitation auf.” Italics: MS.

This ‘argument’ for the assumption of a gravity force field is of course 20

not sufficient to establish its physical existence. Noteworthy though is the phrase concerning the ‘an sich willkürlichen Zwischenbegriffs’, from which I derive the ‘Berechtigung’, the ‘justification’ of my clear cut classical ap-proach.

In this context I refer to my observation at the end of section ‘7.7.3 World 25

geometry’ (OM/514 ff) with an extended quotation from Georg Singer’s introduction to Alexander Friedmann’s monograph ‘The World as Space and Time’ (OM/516):

“Most important is the conclusion: ‘Wegen der prinzipiellen Willkür-lichkeit, die materielle Wirklichkeit geometrisch zu interpretieren, ist die 30

geometrische, allgemeiner die logisch-mathematische Struktur der Welt empirisch nur bedingt erforschbar.’ ”

Thus, in the quest for the physics of gravity, of the material reality, the materielle Wirklichkeit, subject of the present exercise, its various equiva-lent geometrical interpretations cannot provide the answer. 35

9.2 Potentials

9.2.1 Potential theory

The formal ‘fact’, that a source field and its potential are inseparable, is in



61

fact a basic proposition, an axiom of the abstract theory of potentials, not to say a trivial implication of the notion of potential.

The Poisson equation is an implicit definition of a potential. For a ‘given’ source field it permits to determine the potential field. But it does not permit to identify source fields and it does not imply the physical existence of an 5

assumed source field and ‘consequently’ it does not imply the physical exis-tence of the potential field, inseparably connected with the assumed source field.

9.2.2 Gravity potential

In case of the so-called ‘Newtonian’ or gravity potential 10

u g ≡ G u m

the Poisson equation becomes i i u

g = 4 G

with the source field g ≡ G 15

According to my understanding the gravity potential is Einstein’s guiding field described and discussed by Weyl (1950/219):

“But that which expresses itself as force must itself be real. We can, however, recognise the metrical structure as something real, if it is itself capable of undergoing changes and reacts in response to matter. Hence our 20

only way out of the dilemma − and this way, too, was opened up by Ein-stein’s …”

This is the most incredible, revealing statement. In terms of my classical theory it implies, that the ‘guiding’, metrical field acts like a ‘suspension’ preventing bodies to move freely as constitutive parts of the source field of 25

the mass potential. The conclusion to be drawn is, if the source field as-sumed by Einstein does not exist, the corresponding potential does not exist in ‘empty’ space of the Universe.

At the surface of the Earth, the space above its surface is not ‘empty’, but ‘filled’ with matter, with the atmosphere. And in this ‘sub-lunar’ sphere of 30

the Universe a body force field exists and its corresponding potential energy field as well. But Einstein was definitely not concerned with this regime of the Universe.

The adoption of the gravity source field everywhere implies, that gravity is treated like an extensive magnitude, like a quantity proper, in accordance 35

with the electrical analogy adopted by Einstein. But contrary to the electrical charge gravity is not an extensive magnitude, not a quantity proper, but one of the only two processes considered in classical dynamics causing changes of momentum.

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And as a ‘consequence’ of adopting this inadequate model no rational ex-planation of the gravity constant has been proposed up to date. In the con-text of the ‘universal’ balance the gravity constant, the only interesting physical parameter in classical dynamics, does not ‘show-up’ in the storage term, but in the production term as production or reaction constant. And it 5

can be explained as a nuclear property of ponderable matter, in terms of the dynamics of the nucleons.

9.2.3 Ritual references

Ritual references to ‘Newton’ are mostly unqualified, not based on New-ton’s own, very clear, explicit statements. The example of interest is the 10

‘gravity potential’. In the Wikipedia I found: “In mathematics, the gravita-tional potential is also known as the Newtonian potential.” Italics: MS. ‘Ac-cordingly’ the mathematician Weyl has used that terminology. The question arises: What is ‘known’? Or is the potential misleadingly called that way? And who introduced that name and the corresponding Poisson equation? 15

Newton himself to my knowledge did not refer to a potential! But Ein-stein has used the Poisson equation of the gravity potential as his starting point with Newton’s gravity constant, though to my knowledge without any explanation other than reference to the electrical analogy, forgetting about a theory of that fundamental mesoscopic physical property of ponderable mat-20

ter. In the indices of my copies of Einstein’s and of Weyl’s expositions of 1997 and 1950, respectively, the gravity constant is in both cases referred to only once.

In ‘An Historical and Explanatory Appendix’ to his Revision of Andrew Motte’s Translation of Newton’s ‘Principia’ Florian Cajori quotes various 25

pertinent remarks from Newton’s letters and ‘finally’ states: “The question of the nature of gravity has aroused new interest with the

advent of Einstein’s general theory of relativity, according to which gravity is looked upon not as innate to bodies, but rather as some modification of space. According to Einstein, the Earth produces in its surroundings a 30

gravitational field, which, acting on the apple brings about its motion of fall. In Einstein’s gravitational field, in general, a ray of light propagates curvilinearly. …” Italics: MS

Cajori refers to Lawson’s translation ‘Relativity, the Special and General Theory’ (New York, 1921/75 a. /88) of Einstein’s ‘Über die spezielle und 35

die allgemeine Relativitätstheorie’ of 1917, where I found the original statement (1997/42).



63

9.3 Pulsars

9.3.1 True time

In his lectures on ‘Technische Mechanik’ August Föppl stated in Volume 6, devoted to ‘Die wichtigsten Lehren der höheren Dynamik’ (1910/20):

“Die früheren Ausführungen über die Bedeutung des Trägheitsgesetzes 5

sind daher noch dahin zu ergänzen, daß es zugleich eine Anweisung dafür liefert, wie die Zeiten zu zählen sind, oder mit andern Worten, was wir in unserer Welt unter der wahren Zeit zu verstehen haben.” Italics: wide print in the reference.

In my opus I elaborated on this statement (OM/790): 10

According to Newton’s descriptive law of the interaction of bodies of matter the driving causes in case of the momentum production are gradients of the mass potential

f N i ≡ i e

N = a N i u m ,

at least this is the way thermo-dynamicists and chemists dealing with heat 15

and mass transfer and with chemical kinetics would describe the situation.

The phenomenological parameter a P = e P/ u m

has the dimension of frequency squared. With the interaction of masses ‘time comes into the Universe’. 20

According to Newton's law of gravity the intensity momentum production g i ≡ i e

N = a N i u m

is independent of the type of ponderable matter the body consists of. The gravitation constant

G = a N 25

is a universal mesoscopic phenomenological, a physical parameter of bod-ies, the quantity of the latter measured in units of mass, the invariant of translational inertia.

End of quotation.

9.3.2 Gravity waves 30

“And to us it is enough that gravity does really exist, and act according to the laws which we have explained, and abundantly serves to account for all the motions of celestial bodies, and of our sea.”

Isaac Newton (PM/547). 35

If the ‘gravity field’ does not exist, there are of course no ‘gravity’ waves,

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but there are waves of the mass potential due to the motions of its source field, the constitutive mass distribution, the bodies of matter freely moving, free of surface forces. Thus two bodies revolving around each other are causing waves in the mass potential.

Prominent examples are the waves caused by the Moon ‘revolving’ 5

around the Earth in any frame at rest at the surface of Earth. These are caus-ing in fact real gravity waves, resulting in tidal waves of ‘of our sea’, re-strained by the surface of Earth, and of Earth itself. Newton in explaining the tidal waves of the oceans did not yet refer to waves of the mass poten-tial. 10

9.3.3 Pulsar PSR 1913

‘Gravity’ waves predicted by Einstein have first indirectly been proved to ‘exist’, based on observations of the first binary pulsar PSR 1913 + 16 dis-covered by Russell A. Hulse and Joseph H. Taylor jr. in 1974 and jointly awarded the Nobel-Prize in 1993. 15

For ready reference I quote from the Press Release of the Royal Swedish Academy of Sciences, readily accessed via the link:

http://www.nobelprize.org/nobel_prizes/physics/laureates/1993/press.html.

In great detail the quotation describes the binary system far beyond the classical regime. 20

“… We have here two very small astronomical bodies, each with a of some ten kilometres but with a mass comparable with that of the sun, and at a short distance from each other, only several times the moon’s distance from the earth. Here the deviations from Newton’s gravitational physics are large. As an example may be mentioned that the periastron shift, the rota-25

tion of the elliptical orbit that the pulsar (according to Kepler’s first law from the beginning of the 17th century) follows in this system, is 4 degrees per year. The corresponding relativistic shift for the most favourable exam-ple in our solar system, the above-mentioned perihelion motion of Mercury, is 43 seconds of arc per century (this is less than a tenth of the very much 30

larger contributions to the perihelion motion caused by perturbations from other planets, chiefly Venus and Jupiter). The difference in size between the shifts is partly due to the orbital speed in the binary pulsar, which is al-most five times greater than Mercury’s, and partly due to the pulsar per-forming about 250 times more orbits a year than Mercury. The orbiting 35

time of the binary pulsar is less than eight hours, which can be compared with the one month our moon takes to orbit the earth.

A very important property of the new pulsar is that its pulse period, the time between two beacon sweeps (0.05903 see) has proved to be extremely stable, as opposed to what applies to many other pulsars. The pulsar’s 40

pulse period increases by less than 5% during 1 million years. This means



65

that the pulsar can be used as a clock which for precision can compete with the best atomic clocks, This is a very useful feature when studying the characteristics of the system.

The very stable pulse period is in fact a mean of the pulse period ob-served on earth over the time of one orbit of the pulsar system. The ob-5

served period actually varies by several tens of microseconds, i. e. by an amount that is much greater than the variation in the mean value. This is a Doppler effect, and led to the conclusion that the observed pulsar moves in a periodic orbit, meaning that it must have a companion. As the pulsar ap-proaches the earth, the pulses reach the earth more frequently; as it recedes 10

they arrive less frequently. From the variation in pulse period, conclusions can be drawn about the pulsar’s speed in its orbit and other important fea-tures of the system.” Italics: MS.

9.4 Global mechanics

Repeatedly I have stated, that the present paper is not concerned with 15

cosmology, with many bodies, but only with one or two bodies. Here I only mention, that I have treated global mechanics in extenso in my opus mag-num in Chapter ‘18 Elementary global balances’ OM/997 ff), and in Chap-ter ’19 Partial energy balances’, alias ‘Analytical mechanics’ (OM/1029 ff).

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10 References

10.1 My publications

10.1.1 Website since 1998

All related publications, including my opus magnum, explanatory papers, many letters and mails, have been published on my website. According to the Law of 5

2006 concerning the Deutsche Nationalbibliothek my website is a publication proper and is thus copyright protected and permanently archived by the DNB, collected every six months since October 2016.

The URL ‘http://d-nb.info/1078156735’ leads to the DNB Catalague entry and thus in future, after I passed away, to the website. If readers are not historians of 10

science, they should keep in mind, that only the content of the latest version is authorised and should be referred to. Further, in the meantime the archived web-site can now also be accessed worldwide via the DNB Webarchiv.

Annotated entries on the website ‘www.m-schmiechen.de’ including a ‘News flash’, ‘Preliminaries’, and eight Subject areas, may also be accessed via the 15

links provided or directly under …/HomepageClassic01/xxxx.yyy, e. g., …/news_flash.htm.

10.1.2 Opus magnum 2009

Schmiechen (2009, sigil: OM), Michael: Newton’s Principia and related ‘princi-ples’ revisited. Classical dynamics reconstructed in the spirits of Goethe, [Aris-20

totle,] Euler and Einstein. Elementary Mechanics from an advanced standpoint and vice versa. Second edition of work in progress. Berlin, Summer 2009.

In view of its volume published in three volumes Vol.1: Meta- and proto-mechanics Vol.2: Elementary and local mechanics 25

Vol.3: Global and propulsion mechanics

Still (individually) to be ordered at Books on Demand at Norderstedt/Hamburg, but now also freely available as a single pdf/a file of nearly 1500 pages on my website (…/opus magnum.pdf), though (still) without live Table of Contents.

The printed volumes are archived by Deutsche National-Bibliothek at Frankfurt/M 30

and Leipzig, by Staatsbibliothek zu Berlin (Stiftung Preussischer Kulturbesitz) and by University Libraries and pertinent Institutes at Berlin and Hannover.

10.1.3 Papers and letters

A complete coverage of published explanatory papers and letters up to mid 2014 with links is to be found in the Sections ‘News on relativity and gravitation’ and 35



67

‘Papers on relativity and gravitation’ on my website, so that there is no need for repetition (…/news_grav.htm and …/pap_grav.htm respectively).

More recent, even up-to date are the entries in the section ‘On the physics of grav-ity, cont’d!’ of the ‘News flash’.

For convenient reading many of the papers and letters are designed to be printed 5

and bound as DIN A5 (21 x 15 cm) brochures.

Further, during work on this exposition I found on my hard disc countless extended drafts, letters to editors of journals, to experts and friends, to be archived at the Archive of the Technical University Berlin, where my complete personal archive has already been archived and is accessible. 10

10.2 Other publications

10.2.1 References in Opus

During my work on this exposition it occurred to me, that I copied more and more arguments and references published in my opus magnum, now freely avail-able on my website, until I realised, that I was departing from my goal, to provide 15

just the bare essentials.

In the few copies taken from my opus I have freely up-dated the texts as neces-sary in view of my latest insights and scrutiny. All references are to be found in the opus magnum in the Section ‘29.2 Sources referred to’ (OM/1341-1390). Sources are identified by the year of publication of the reference, which I held in my 20

hands.

10.2.2 References in Notes

Einstein (1997), Albert: Über die spezielle und die allgemeine Relativitätstheorie. 1. Auflage 1917. Nachdruck der 23. Auflage 1988, nach der 22. Auflage 1972, die als Sonderausgabe der 21. Auflage 1969 erschien. Braunschweig: Vieweg, 25

1997.

Föppl (1910), August: Vorlesungen über Technische Mechanik. In sechs Bänden. Sechster Band: Die wichtigsten Lehren der höheren Dynamik. Leipzig: Teubner, 1910.

Friedmann (2000), Alexander: Die Welt als Raum und Zeit. 1923. Übersetzung aus 30

dem Russischen, Einführung und Anmerkungen von Georg SingerG. 1. Auflage. Frankfurt/M.: Deutsch, 2000. Ostwalds Klassiker der exakten Wissenschaften Band 287.

Hertz (1956), Heinrich: Die Prinzipien der Mechanik. In neuem Zusammenhang dargestellt. Leipzig: 1894. Gesammelte Werke Bd .3. Authorized English trans-35

lation by D. E. Jones and J. T. Walley: The Principles of Mechanics. Presented in a new form. 1899. New York: Dover, 1956. Paperback edition with a new in-troduction and a bibliography by Robert S. CohenRS.

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Hertz (1984), Heinrich: Die Prinzipien der Mechanik. In neuem Zusammenhang dargestellt. Mit drei Arbeiten von Heinrich Hertz einem Vorwort von Hermann von Helmholtz, einer Vorbemerkung von Philipp Lenard. Eingeleitet und mit Anmerkungen versehen von Josef Kuczera. Leipzig: Akademische Verlagsge-sellschaft, 1984. Ostwalds Klassiker der exakten Wissenschaften 263. 5

Maxwell (1991), James Clerk: Matter and Motions. 1887. Notes and appendices by Sir Joseph Larmor. New York: Dover, 1991. Reissue of the Dover edition of 1955, which was an unabridged, unaltered republication of the Lamor edition of 1920.

Newton (PM: 1966), Isaac: Philosophiae Naturalis Principia Mathematica. Mathe-10

matical Principles of Natural Philosophy And System of the World. Translated into English by Andrew Motte 1729, revised by Florian Cajori 1934. Sixth print-ing 1966 (First Paper-bound Edition in two volumes). Berkeley: University of California, 1966.

Volume One: Prefaces, Definitions /1-12, Axioms, or Laws of Motion /13-28, 15

Book I: The Motions of Bodies /29-233 and Book II: The Motions of Bodies (In resisting mediums) /235-396;

Volume Two: Book III: The System of the World (In mathematical treatment) /397-547, Rules of Reasoning in Philosophy /398-400, Phenomena /401-405, Propositions /406-542, General Scholium /543-547, followed by an anonymous 20

(Andrew Motte? /680) translation of The System of the World, 1728 /549-626, and An Historical and Explanatory Appendix by Florian Cajori, 1934 /627-680.

Weyl (1918), Hermann: Raum – Zeit – Materie. Mit Vorworten des Autors zur erste Auflage von 1918, zur dritten Auflage von 1919, zur vierten Auflage von 1920, zur fünfte Auflage von 1922. 25

Weyl (1993), Hermann: Raum – Zeit – Materie. Vorlesungen über allgemeine Re-lativitätstheorie. Herausgegeben und ergänzt von Jürgen Ehlers. Berlin: Sprin-ger, siebente Auflage 1988 mit einem Vorwort des Herausgebers, achte Auflage 1993 mit einem Geleitwort des Herausgebers.

Weyl (1950), Hermann: Space Time Matter. Translated from the German by 30

Henry L. Brose. New York: Dover, 1950. Dover Book S 267. First American Printing of the Fourth Edition (1922). With a Preface by the author to the first American Printing of 1950 and the Translator’s Note of 1921. My reference, which I bought in 1965.

35



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11 Indices

11.1 Name index

Aesop 18 Alice 11 5

Appel, C. 10 Aristotle 1, 6, 12, 15, 17, 23, 26, 30,

35, 66

Bacon, F. 52 10

Bell, J. F. 1 Beller, M. 10 Berkeley, G. 8, 16, 68 Bernoullis 13 Birkhoff, G. 20 15

Bocheński, I. M. 15, 17 Bohm, D. 1, 54 Born, M. 15 Borzeszkowski, H.-H. 38 Brecht, B. 15 20

Bruno 11, 31 Buckingham, E. 20

Cajori, F. 62, 68 Carnap, R. 24, 25 25

Cavendish, H. 45 Copernicus, N. 37, 49, 56

Descartes, R. 12, 23 Dodgson, C. L. 11, 31 30

Dürr, S. 54

Ehlers, J. 1, 52, 58, 68 Einstein, A. 1, 5, 6, 10, 12, 13, 19,

27, 28, 35, 37, 38, 39, 40, 41, 42, 35

45, 46, 47, 48, 49, 57, 58, 60, 61, 62, 64, 66, 67

Eötvös, R. v. 48 Euclid 13, 30, 60

Euler, L. 6, 13, 36, 66 40

Faye, J. 22 Feyerabend, P. K. 12 Föppl, A. 43, 63, 67 Fourier, J. B. J. 20 45

Frege, G. 25 Friedmann, A. 5, 60, 67 Froude, W. 20

Galilei, G. 23, 29, 35, 49 50

Gleick, J. 13, 23, 46 Goethe, J. W. 6, 9, 12, 13, 18, 21,

22, 35, 42, 57, 66 Gulliver. 21, 28 Gummert, P. 51 55

Hertz, H. 9, 11, 67, 68 Hulse, R. A. 64

Jammer, M. 35 60

Janich, P. 16, 57 Johnson, K. 14

Kant, E. 6, 19 Kepler, J. 20, 37, 44, 49, 64 65

Klanner, R. 53 Kotter, J. 14

Lagrange, J. L. de 13, 41 Laplace, P. S. de 28, 41 70

Laue, M. v. 41 Lawson, R. W. 62 Lelli, F. 43 Lucretius, T. C. 12, 38, 53

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Mach, E. 8, 9, 10, 38, 39, 40, 56, 58 Marquardt, O. 19 Mathcad 25 Maxwell, J. C. 1, 6, 11, 13, 57, 68 5

McGaugh, S. S. 43 Mercier, A. 38 Milgrom, M. 43 Motte, A. 62, 68

10

Newton, I. 1, 6, 9, 11, 12, 13, 15, 16, 17, 18, 23, 26, 29, 30, 33, 35, 37, 38, 40, 42, 43, 44, 45, 46, 49, 56, 57, 58, 59, 62, 63, 64, 66, 68

Novalis 12 15

Osiander, A. 37

Pais, A. 10, 39, 40, 49, 58 Plato 17, 52 20

Pohl, R. W. 45 Poisson, S. D. 41, 60, 61, 62 Popper, K. 20, 56 Pratchett, T. 18 Proclus 19 25

Ptolemaeus, C. 10

Rayleigh, J. W. S. 20 Retzbach, A. 14

30

Schombert, J. M. 43 Schrödinger, E. 54 Sepper, D. L. 9 Singer, G. 67 Smolin, L. 57 35

Stokes, G. G. 20 Sunzi 18 Swift, J. 21, 28, 47 Sylvie 11, 31

40

Taube, G. 56 Taylor, J. H. 64 Treder, H.-J. 38, 56 Truesdell, C. A. 6, 23, 41, 51, 57 Twain, M. 20 45

Weyl, H. 1, 13, 19, 24, 41, 58, 59,

61, 62, 68 Whitehead, A. N. 14

11.2 Subject index 50

Subject indices are notoriously difficult ‘subjects’, particularly in my opus mag-num and in the present exercise, purposely departing from the professional jargon and freely using journalistic jargon where felt enlightening and/or amusing.

Various concordance tables resulted in hopelessly clumsy subject indices. Fi-nally a section index was found to meet the purpose most adequately. 55

Abstract: detailed_1.3 6 Anschauen, Anschauung_3.2.3 19 Applications_6.3 44 60

Author_12 73 Background_2.1 8

Balance of mass_5.1.2 31 Balance of momentum_5.1.1 29 65

Biography_12.1 72 Body model_7.1.2 49 Categories necessary_3.2.1 17

On implications of Newton’s law of gravitation


71

Change organised_2.6 14 Coherence of masses_6.1.2 38 Coherence required_3.1.3 16 Concluding operations_8 56 Contacts_12.3 72 5

Decisions derived_8.3 57 Dynamics of nucleons_7.2.1 52 Einstein’s ‘aether’_6.1.3 39 10

Elementary dynamics_5.1 29 Elementary kinematics_5.2 33 Elementary mechanics_5 29 Epi-Language_3.2 17 Explanations_7.1 49 15

Facta: being made_4.1.1 22 Formal languages_3.1.1 15 Global mechanics_9.4 65 20

Goal defined_2.4 11 Grand unification_7.2.3 54 Gravity constant_7 49 Gravity field_9.1.3 60 Gravity field: arbitrary_1.1 5 25

Gravity law_6 37 Gravity potential_9.2.2 61 Gravity waves_9.3.2 63 Ideal universes_6.2.1 40 30

Indices_11 69 Interpretations_6.3.1 44 Law of gravity_6.2.2 42 35

Magnitudes, ‘quantities’_4.1.3 23 Mass potential_6.2 40 Measurements_4.2 26 Mesoscopic approach_7.2.2 53 Meta-Mechanics_3 15 40

Meta-physics: basics_1.2 6 Meta-theory_3.1 15 Model conceived_2.3 11 Moral rights_12.2 72 My publications_10.1 66 45

Naked speculations_7.1.1 49

Name index_11.1 69 Notation, Terminology_4.2.3 27 Notes_9 58 50

Notes on history_5.2.3 35 Obscure energy_6.3.2 45 Opening operations_2 8 Opus magnum 2009_10.1.2 66 55

Other publications_10.2 67 Papers and letters_10.1.3 66 Philosophy aversion_3.3.1 19 Plan designed_2.5 12 60

Potential theory_9.2.1 61 Potentials_9.2 61 Preface_1 5 Principles: prejudices_3.3.2 20 Problem identified_2.2 10 65

Proto-mechanics_4 22 Pulsar PSR 1913_9.3.3 64 Pulsars_9.3 63 Pure mechanics_7.1.3 51 70

Reference ‘mollusc’_6.1.1 37 Reference frames_4.2.2 26 References_10 66 References in Notes_10.2.2 67 References in Opus_10.2.1 67 75

Relative motion_5.2.2 34 Result evaluated_8.1 56 Ritual references_9.2.3 62 Rules to be understood_3.1.2 16 80

Search of roots_9.1.1 58 Serious pitfalls_3.3.3 21 Simplest universe_6.2.3 43 State equations_5.2.1 33 Structure of matter_7.2 52 85

Subject index_11.2 71 Tales, metaphors, etc_3.2.2 18 Theorems deduced_5.1.3 31 Theories interpreted_4.1 22 90

Theories: dual structure_4.1.2 22 Theory of knowledge_3.3 19 Things that are not_6.3.3 46 Time and space_4.2.1 26

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True time_9.3.1 63 Universe_6.1 37 Values assessed_8.2 56 5

Website since 1998_10.1.1 66 World_9.1 58 World: filled, limited_9.1.2 58

10

On implications of Newton’s law of gravitation


73

12 Author

12.1 Biography

See Biography on my website (…/biograph.htm)

12.2 Moral rights

The moral rights of Michael Schmiechen to be identified as the sole au-5

thor of this work are asserted by him in accordance with the Copyright, De-signs and Patents Act 1988.

12.3 Contacts

apl. Prof. Dr.-Ing. Michael Schmiechen 10

Bartningallee 16 DE 10557 Berlin Germany [email protected] www.m-schmiechen.de 15

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Notes

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