Time and Space Weight and Inertia. A Chronogeometrical Introduction to Einstein's Theory

198 Pages·1965·3.013 MB·English

by  A. D. Fokker and D. ter Haar (Auth.)

TIME and SPACE
WEIGHT and INERTIA
A chronogeometrical introduction to
Einstein s theory
BY
A. D. FOKKER
Formerly Teyler Professor,
University of Leyden
TRANSLATED BY
D. BIJL
Reader in Natural Philosophy,
University of St. Andrews
TRANSLATION EDITED BY
D. FIELD
Department of Mathematical Physics,
University of Birmingham
PERGAMON PRESS
OXFORD • LONDON • EDINBURGH • NEW YORK
PARIS • FRANKFURT
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Pergamon Press (Scotland) Ltd., 2 & 3 Teviot Place, Edinburgh 1
Pergamon Press Inc., 122 East 55th St., New York 22, N.Y.
Pergamon Press GmbH, Kaiserstrasse 75, Frankfurt-am-Main
Copyright © 1965
Uitgeversmaatschappij
W. de Haan, N.V.
This book is a translation of
TIJD EN RT7IMTE, TRAAGHEID EN ZWAARTE,
published by W. de Haan Ltd. (Publishers), Zeist
The Netherlands, with an appendix supplied by
the author in March 1965
Library of Congress Catalog Card No. 65-0000
Made in Great Britain at the Pitman Press, Bath
PREFACE TO THE ENGLISH EDITION
IT IS with great satisfaction that I may look forward to the English
edition of my chronogeometrical introduction to Einstein's theory.
I can no more attempt a full exposition of details and consequences.
Still, it may be worthwhile to present my general view to a wider
circle of physicists outside my own country.
I want to pay a tribute of admiration to the late Alfred A. Robb,
in Cambridge, who, as early as 1917, objected to the abuse of the
words relative and relativity. His early investigations led him to
study the absolute relations of time and space and he wrote a
stringent, consistent axiomatic geometry of time and space. That is
sound philosophy, contrasting with the all-too-popular, amateuristic
prate on relativity which has not yet died down even today.
The great truth underlying and emanating from Einstein's theory
is the recognition that our world is not a static existing entity, but
a kinetic and dynamic occurring history of events. Verba valent usu,
and therefore the meaning of common and usual words often lack
sharpness and definition. This much for words like to exist and to
be. Hamlet pondered: "to be or not to be, that is the question." I
may be allowed to venture the answer: "not to be, but to become";
perhaps better still: "to be coming." These subtleties in a language
not my own are beyond my power. I am all the more indebted to
Dr. D. Bijl for finding equivalents not too startling for English ears.
The French speak of Vespace and of le temps. The use of the
article le implies the conception of space as a real thing, something
like a hall or a vessel, and the conception of time as some mytho-
logical Chronos eating his own children or a road-roller crushing
the living present to dead past. It is an advantage that space and
time are used without an article. The English philosophic minds
are less liable to take them for existing entities and more ready to
see that events do not occur in pre-established time and space. On
the contrary, time and space are to be found in the occurring
universe as certain relations, perhaps as no more than the possibi-
lities of such relations between events. No events in time and space,
ix
X PREFACE TO THE ENGLISH EDITION
but time and space in the events. That is the lesson I learnt from
Robb. I praise the language which admits of such a formulation.
Sometimes in expert literature one meets curious atrocities. Such
is a combination of words like "time dilatation". Time is no entity
liable to be dilated. Neither can events called clocks be dilated.
Only events called measuring rods can be dilated. The offending
words are supposed to convey the idea of a slowing down of clocks.
I have tried to keep away from such mistakes.
Perhaps the deepest enigma brought to light by chronogeometry
is the occurrence of zero intervals, connecting events which are
located by observers with spatial distance and temporal duration
between them. Zero interval means no separation at all, an
immediate transmission of momentum and energy, as if there were
contiguity. Not only action at a distance, but action across a gap
in duration as well. We all know the experience of remembrance,
as the presence here and now (in our mind as we are inclined to
say, in the events constituting our mind) of an event, past and dis-
tant. That comes very close to zero intervals. The mathematical
formula is quite simple and plain, nevertheless it relates to one of
God's secrets and implies His sempiternal ubiquitous presence.
I have added to this English edition an appendix dealing briefly
with the latest terrestrial experiments on gravitational red-shift.
I am not going to write theology, nor do I want to anticipate what
is said in the translated preface to the home edition. May this
English edition speed well.
A. D. F.
PREFACE
IN 1905, nearly sixty years ago, during the most prolific period of
his scientific career, Einstein published his theory on the Elektro-
dynamik bewegter Korper. Almost thirty-five years ago the author
of this book wrote a textbook on the theory of relativity in the
Dutch language. This book followed the historic line of development,
which led to a new point of view. At present it is better to adopt
this new point of view right from the start.
It is often believed that space and time have a meaning indepen-
dent of events, in the sense that space and time as such are
recognizable entities. Space then is comparable with an empty
stage, which can be occupied by the actors, and time is something
like an empty pause, waiting for the beginning of the play.
This view however is not correct. Events do not take place in a,
pre-arranged space and time, but rather we find space and time
within events. Time and space are names for the possibilities of
certain relationships between occurring events, that is relationships
of the kind before-and-after, and of the kind neither-before-nor-after.
The reformation of our ideas begins with the recognition that if
the events do not occur in the same place, the relation neither-
before-nor-after between events does not imply the relation simulta-
neous-with. The relation before-and-after is a relation in terms of
time, whereas in the relation neither-before-nor-after we have a
relation in terms of space. These relations are fundamentally so
different, that measures in terms of time and measures in terms of
space cannot be treated alike.
The relationship between two events may be expressed in a
quantitative form; that is, a numerical value may be assigned to
their separateness. The general term for this numerical relation is
interval. The interval between two events of the kind before-and-
after is a time lapse, or duration, whereas the interval between two
events of the kind neither-before-nor-after is a distance.
We are quite used to expressing our ideas in geometric pictures.
Nevertheless, we have to watch our step. For instance, we may
xi
xii PREFACE
consider that three events determine a "triangle"; the "sides" of
the "triangle" are intervals. It is possible that the sum of any two
"sides" of the "triangle" is greater than the remaining one. This
would be in agreement with the Euclidean geometry of a triangle.
No objection then can be raised against the statement that the
events determine a plane. It is also possible, however, that the sum
of a pair of "sides" of the "triangle" is smaller than the third "side".
This is essentially different from the properties of Euclidean planes.
In this case the three intervals do not determine an ordinary two-
dimensional plane, but rather a one-plus-one-dimensional mani-
fold, which deserves to have another name. This was also felt by
Alfred A. Robb, who was the first to develop a strictly logical
discipline of time and space. From him I have taken over the
formulations before-and-after, and neither-before-nor-after. Robb
calls ordinary two-dimensional Euclidean planes: separation
planes. All intervals in such a plane are of the kind neither-before-
nor-after; they are distances. The other, unusual, manifolds he
calls "inertia planes", because they contain "inertia lines", which
are a sequence of events representing the existence of a (stationary
or moving) particle. In this respect I do not follow Robb, however
much I admire his discerning fundamental work. If we wish to
-express only the "duration" of a particle, it is premature to load the
term with a dynamical attribute, as is suggested by the term
"inertia plane". I prefer to call such a (1 + 1)-manifold: enduring
path.
New concepts require new terms. People with a visually inclined
intellect tend to think of visual pictures. Such pictures arouse
associations, which are no longer conscious; the majority are
useful, but some may be obstructive or even deceptive. If we use
the word "space", we imply something unique and permanent, in
fact we imply the totality of "enduring" points, even if we disregard
this "duration". Yet we need a term for a set of point-events,
vrhich is flat, three-dimensional, and without "duration". Such a
set I call a concurrence. There are many concurrences, they may
intersect and the intersection of two concurrences must be a plane.
I use several of such unusual words, which are nevertheless linguisti-
cally correct. For this I appeal to the reader's forbearance; perhaps
I even need his forgiveness. May the reader grant this to a teacher,
vrho believes in the vitality of the language of the community to
ivhich he belongs.
PREFACE xiii
The theory contains a certain conception which requires our
special attention. Durations are intervals, the square of which is
positive; distances are intervals whose square is negative. In
between these intervals there are other intervals the squares of
which are zero, and we must ask for the meaning of such intervals.
An example is a light ray which travels from the flame of a candle
towards us. Between the event of the radiating carbon particle in
the flame, and the event of the perception by our eyes, there is an
interval to which the theory assigns the magnitude zero. Taken
literally, this means that the numerical measure of the separateness
is zero, so that, strictly speaking, there is no separateness. The theory
teaches that if we see a nova flare up, somewhere inconceivably far
away in the galaxy (so that this happened innumerable years ago),
this flare-up is not separated from our eye. In fact, the action of the
flare is happening in our eye, by its action, here, and at this moment,
that is, now, and must be taken to be present in both senses of the
word; it belongs to our present.
From a point-instant here-now rays (with interval zero) may
strike out in all directions. Metaphorically speaking, this is called
the light cone in the point-instant. We picture spherical light waves
which diverge from that point-instant, and so we identify the
characteristic velocity associated with zero intervals, with the
velocity of light. Furthermore, this expanding spherical wave is
pictured as a cone in a one-plus-two dimensional manner. I propose
to call this cone the present of the point-instant.
The region of occurring events, before, around and after a point-
instant is divided as follows. There are events before the point-
instant ; these represent the absolute past of the point-instant and
not the present. There are the events after the point-instant; they
represent the absolute future of the point-instant, again not the
present. And there are the events of the kind neither-before-nor-
after, that is the absolute elsewhere of the point-instant as different
from here. On the boundary in between are the events of the absolute
present, of the kind here-now.
There is a difference between the ideas of a present, and of a
concurrence. The concurrences which contain a particular point-
instant spread out in different directions in the elsewhere of the
point-instant. It seems to me to be useful as well as necessary to
formulate such differences in a precise way by means of the above
terms.
xiv PREFACE
Not only the external relations, but also the content of events,
which has a bearing on their action, calls for a re-appraisal. We are
referring to the dynamic quantities, energy and momentum. Energy
and momentum are the temporal and the spatial component of one
entity, e.g. the mass associated with the event. A mass has certain
similarities with intervals in general, but in particular with duration
intervals. We may break up both the energy and momentum of a
given mass into two pieces; in general we obtain two masses, the
sum of which is less than the original mass. This corresponds to
the two "sides" of a "triangle" of durations which, together, are
shorter than the third side. We can recognize in this the loss of
mass during a nuclear fission.
Another similarity is, that for a certain ratio of energy and
momentum, the mass may be zero, just as there are zero intervals.
This applies to the energy and momentum which plays a role in the
interaction between events by means of photons. This mass gives
us a theoretical means of making a distinction between matter and
non-matter. Whenever the mass is finite, the events are material.
If the mass is zero, the events are non-material. Transitions from
matter to non-matter, and vice versa, are known.
There seem to be good reasons to be surprised at the fact that
the above-mentioned conclusions have been produced by a theory
which, according to its name, concerns itself with relativity. Indeed,
the considerations of relativity, which initially made a great impres-
sion, only touch on certain aspects of the theory. Several physicists
have remarked that the theory is rather more directed to the formu-
lation oiabsoluta (e.g. invariants) than to a basic notion of relativity.
The accepted name is truly obsolete; it has often been the cause of
grave misunderstandings, and only recently gave rise to an unpalat-
able controversy. The name "theory of relativity" serves as a
reminder of its historic development in the past. The name
chronogeometry offers a perspective for the future.
In order to help a clear understanding of chronogeometric
pronouncements, I have often used drawings in my academic
lectures, from which the case under consideration could be grasped
at a glance. Algebraic formulas are required for exactness, but only
drawings can illustrate the concrete content. This book is distinct
from others by the use of many figures. Another difference is the
attention paid to a number of special topics, such as the compound
axis of acceleration and rotation, the rigid body, the ideal clock, the
PREFACE XV
notions of a centre of mass and angular momentum, the current
velocity and the dynamical tensor of matter, the interpretation of
the curvature tensor and the events in the cabin of Jules Verne's
projectile on its interstellar journey.
I am fully conscious of many gaps and limitations, but I have to
resign myself to this. I hope to have made the main points clearly,
for physicists as well as for those who are somewhat removed from
the field, and for whom it would be a precarious undertaking to try
to follow the mathematics step by step. May some people benefit
from my work.
CHAPTER I
FOUNDATIONS
1. Reality, viewed as a stream of events. Space and
time within events
Our life, and in fact everything we experience, takes its course
or, more precisely, happens. Nothing remains unchanged, events
continually occur, they begin and come to an end. We get up from
our chair and we look round, we walk a step, followed by many
others, we listen to a warbling blackbird, we scent a flower, we pick
and taste a berry. These are all acts which alternate between
perception on the one hand, and intentional observations and acti-
vities on the other. Hamlet reflected: "to be or not to be, that is the
question." The answer is: actual existence is not "to be", but "to
happen". In this stream of reality we recognize distinct charac-
teristics, such as the fixed contours of houses and scenery. We
recognize the sun day after day and year after year. Recognitions
like these enable us to order our recollections in terms of actual
experiences. Furthermore, they enable us to assign locations and
instants to all events, that is to say, locations which indicate solely
here-and-there, and instants which indicate only before-and-after.
Here-and-there are locations like those of my two hands, or those
of two nails in a piece of wood. In the act of lifting a hammer,
swinging it down and hitting a nail, the lift occurs before the down-
swing and the hitting, the hit occurs after the lift and the down-
swing, and the down-swing after the lift, but before the hit. We
can remark immediately that the order of here-and-there in the
occurrence of events is essentially different from the order in terms
of before-and-after. The relation here-and-there has nothing to do
with the relation before-and-after, and, following Alfred A. Robb,
may be called neither-before-nor-after. The relation here-and-there
is the essential content of the concept of space. The relation before-
and-after is the essential content of the concept of time. We must
realize once and for all that it is exclusively in events that we discern
2 1