\documentclass[12pt]{article}
\usepackage{harvard}
\usepackage{bibtexlogo}
\usepackage{html}
\usepackage{tcr}

\newcommand{\hlnk}[2]{\htmladdnormallink{#1}{#2}}

\newcommand{\hoturl}[1]{\hlnk{\texttt{#1}}{#1}}
\newcommand{\mailto}[1]{\hlnk{\texttt{#1}}{mailto:#1}}

\begin{document}

\tcrdate{30-Dec-1996}
\tcrvolume{01}
\tcrnumber{03}

\input{tcrfront}

\begin{latexonly}
\copyrightdate{1996}
% This must be manually echoed in the .latex2html
% also - l2h cannot (perversely) pick it up from here.
\end{latexonly}

\title{Einstein, Popper and the Theory of Relativity}
\author{\htmlref{Christoph von Mettenheim}{mettenheim}\\
\htmlref{\copyright \thecopyrightdate}{copyright}}

\maketitle

%\begin{quote}
%\textbf{Keywords:}
%\end{quote}
%\newpage
%\vspace*{\fill}

%\begin{abstract}
%\end{abstract}

%\vspace*{\fill}

\tableofcontents

\section*{Introduction}

\begin{latexonly}
\begin{verse}
Der Aberglaub\'{}, in dem wir aufgewachsen,\\
Verliert, auch wenn wir ihn erkennen, darum\\
Doch seine Macht nicht \"{u}ber uns. - Es sind\\
Nicht alle frei, die ihrer Ketten spotten.\\[3mm]
                                                     
\emph{G.E.~Lessing}
\end{verse}
\end{latexonly}

\begin{htmlonly}
\begin{rawhtml}
<PRE>
          Der Aberglaub', in dem wir aufgewachsen,
          Verliert, auch wenn wir ihn erkennen, darum
          Doch seine Macht nicht &uuml;ber uns. - Es sind
          Nicht alle frei, die ihrer Ketten spotten.
                                                     
                                G.E. Lessing
</PRE>
\end{rawhtml}
\end{htmlonly}


\tcrpar Karl Popper has found beautiful words for his
admiration of the period of enlightenment, which he
considered one of the most inspired in European
history\footnote{I have in mind his speech on
  \tcrquotes{Kant's Critique and Cosmology} at the
  150th.\fsp anniversary of Kant's death \cite[pp.~175
  ff., now also contained in the introduction to the
  German translation]{Popper:Conjectures}.}. In my
country this period is connected not only with the name
of the great philosopher and scientist Immanuel Kant,
but also with that of the poet Gotthold Ephraim
Lessing, one of the early outstanding figures in German
literature. Having acquired from Karl Popper a taste
for using mottoes I would like to put this paper under
a motto taken from Lessing's play \tcrquotes{Nathan der
  Weise}---\tcrquotes{Nathan the Wise}. This is a
beautiful play, staging a dramatic background from the
time of the crusades for a strong and very touching
plea for tolerance between the three great religions,
the Jewish religion, Christianity, and Islam, and
Lessing puts into the mouth of one of the main figures
of the play, the young templar, words which might, in a
clumsy attempt to preserve the metre, perhaps be
translated as follows:

\begin{latexonly}
\begin{verse}
The superstition in which we were brought up\\ 
will, though we have seen through it, thereby\\ 
not lose its power over us. - Not all\\
are free who mock at their own fetters.\\
\end{verse}
\end{latexonly}

\begin{htmlonly}
\begin{verbatim}
          The superstition in which we were brought up 
          will, though we have seen through it, thereby 
          not lose its power over us. - Not all
          are free who mock at their own fetters.
\end{verbatim}
\end{htmlonly}

\tcrpar
Of this last passage, \tcrquotes{Not all are free who mock at
their own fetters}---\tcrquotes{Es sind nicht alle frei, die
ihrer Ketten spotten},  I feel reminded time and again 
when reading Karl Popper's books, 
because I keep discovering new implications 
of ideas of his which I have accepted as true many 
years ago without, apparently, realising their full 
importance. This used to worry me a lot, but I have 
ceased to do so after reading, in \emph{Unended Quest},
that it took Popper himself years to discover the 
bearing which his demarcation criterion of 
falsifiability had on the problem of 
induction \cite[p.~52]{Popper:UnendedQuest}.
 
\tcrpar
I now believe this deficiency must be a normal
result of the fact that we are human beings, and not
computers. But it does mean, I think, that we must 
retain a very critical attitude even towards those of 
our own views which seem to be the clearest because 
they may be remnants of old superstitions which we 
believe to have left behind.

\section*{Part I}

\tcrpar
Einstein's theory of relativity is a striking example for this
observation. Most of you will know of Popper's 
admiration for Einstein, and how he was inspired by 
the theory of relativity (and by its high refutability 
in contrast to the irrefutability of psychoanalysis) 
more than by anything else to develop the criterion of 
falsifiability as a demarcation between science and 
metaphysics
\cite[pp.~37--38]{Popper:UnendedQuest}.
And in connection with this criterion he often quoted---and
amplified---Einstein's words from \emph{Geometrie und Erfahrung}
\cite{Einstein:GeometrieUndErfahrung}:

\begin{quote}
Insofar as the expressions of mathematics refer to 
reality they are not certain, and insofar as they are 
certain they do not refer to reality.
\end{quote}

\tcrpar
Going by this sentence it would appear that there
could hardly have been a more perfect understanding on 
methodology than that between Einstein and Popper. And 
they even agreed in this, that they both believed the 
theory of relativity \emph{not} to be the final truth.
Einstein spent the last thirty years of his life in 
search of a general field theory. And from Popper's 
\emph{Realism and the Aim of Science}
\cite[p.~xxviii]{Popper:Realism}
it can be seen that he considered the issue between 
Einstein and Lorentz, which is none other than the issue
between relativistic physics and non-relativistic 
physics, to be still open.

\tcrpar
Most amazing about this is that they both held the
key to the problem in their hands. Let us take a look 
at the general theory of relativity and at Einstein's 
concept of \tcrquotes{curved space}. It is reputed to be a
difficult concept, but Einstein's 
deduction is simple enough to be understood by 
everybody. In one of his famous \emph{Gedankenexperimente} he
assumes a large box, or lift, being accelerated 
through space at a constant rate outside any other 
field of gravitation. He then first discusses the 
situation of a generation of physicists being born and 
living in that box, without being able to \tcrquotes{look out of
the window}, and finds that, due to the constant rate 
of acceleration, they will be under the impression of 
living in a gravitational field. Permanent 
acceleration will permanently press them to the floor 
of the box. He then lets a ray of light travelling 
horizontally (parallel to the bottom, or x-axis) fall 
into the box through a hole in its side, and finds 
that, due to the finite velocity of light, and, again, 
due to the constant rate of acceleration of the box, 
this ray, which had been travelling in a straight line 
outside the box, will assume the shape of a parabolic 
curve inside the box. From the combination of these 
two considerations he infers that the scientists 
living in that box will ascribe the deflection of 
light to the influence of the gravitational field. And 
his further inference is that, since we, the human 
race, are living in the gravitational field of the 
earth, and the sun, without being able to \tcrquotes{look out of
the window}, we must also assume that the lines of 
light will be curved in this field. Going by his own 
explanation, this, and nothing else, lies at the 
bottom of his famous concept of 
\tcrquotes{curved space}.
%\footnote{I am referring to
%\cite[p.~84f]{Einstein:SpezielleUndAllgemeine}.}.

\tcrpar
Now, I fear that some of you may not believe this.
The lift-example is taken from the book \tcrquotes{\"{U}ber die
spezielle und die allgemeine Relativit\"{a}tstheorie} in which Einstein
tried to explain the theory of relativity to non-physicists in a popular
way \cite[chapter~20]{Einstein:SpezielleUndAllgemeine}. It seems unfair
to hold this attempt at a simple visualisation of a difficult scientific
theory against him. But I assure you that I have looked through
Einstein's papers very carefully, especially through the early ones, and
I have found many complicated discussions of the implications
\emph{resulting from} curved space, and also many \emph{adaptations} of
the theory to empirical results which surprised me because the
experiments in question might well have been taken as refutations of the
theory. I have also found the arguments used in the lift-example
formulated in a more scientific shape, referring not to lifts, or boxes,
but to systems of coordinates, and planets, and ellipses
\cite{Einstein:Grundlage}. But the gist of the argument always remained
the same. And nowhere have I found anything like a \emph{critical}
discussion of the fundamental premise that light will be deflected in
the gravitational field. The discussion of this premise did not take
place in Einstein's papers; nor have I found it anywhere else.

\tcrpar You may still not feel reassured because I am
only a lawyer, and admittedly know very little about
physics, and even less about mathematics. But perhaps a
quotation from Einstein's own text will convince you.
In \tcrquotes{Die Grundlage der allgemeinen
  Relativit\"{a}tstheorie}
\cite{Einstein:Grundlage}---and this is not a
simplified book, but one of his original publications
in \emph{Annalen der Physik}---he explicitly says (the
translation is mine, but the words are his):

\begin{quote}
For it is possible \emph{`to create'}
[he puts this in quotation marks] a gravitational 
field by merely  changing the system of  coordinates 
\ldots{} it is easy to see that the way of  
light relative to $K'$
[$K'$ is the system of coordinates that is being
accelerated, i.e.\fsp the \tcrquotes{lift}] will generally be curved
if light moves in a straight line and with constant 
velocity relative to $K$.
\end{quote}

\tcrpar This, I believe, shows quite clearly that, at
that time, he considered the deflection of light as a
necessary \emph{implication} of acceleration, resulting
from a purely mathematical operation, viz.\fsp the
\tcrquotes{changing of the system of coordinates}. To
him it was the result of a mathematical equation. In
fact he \emph{inferred} this deflection from two very
simple premises: (1) the constant velocity of light,
and (2) an identification of acceleration with
gravitation\footnote{\ftlabel{Zahar}Dr.~Elie Zahar has
  kindly pointed out to me that my description does not
  take into account the Riemann-Christoffel curvature
  of space which has nothing to do with the choice of
  the coordinates. I admit that I am making use of a
  simplification which is, however, necessary because
  otherwise my arguments, too, would become
  unintelligible. The following text will show that, as
  long as the concept of curved space---in whatever
  version---is based not on \emph{empirical
    assumptions}, but on \emph{mathematical inference},
  my arguments against it remain the same.}.

\tcrpar
Let us now look at this argument with Karl Popper's eyes.

\tcrpar According to Popper's \tcrquotes{Logic of
  Scientific Discovery} Einstein's reasoning is not
only \emph{very} strong, but in fact \emph{too strong}.
For it is, of course, true that the ray of light
falling into the accelerating box will form a parabolic
curve. But this is true not only for light, but also
for \emph{any} other ray travelling at constant speed,
and entering the box at an angle of $90^\circ$.  The
shape of the parabolic curve will, of course, vary
depending on the velocity of the ray and the
acceleration of the box, but if one of two intersecting
systems of coordinates is moving at constant velocity
while the other is being accelerated the function of
the intersection must necessarily be a parabolic curve.
This is not only true for rays of light; it is even
true for rays, or lines, that exist only in our
imagination, or for a box without top or bottom, and
therefore \emph{without gravitational field}. It is
independent of any physical properties of those rays,
and has nothing to do with gravitation, but simply
consists of a geometrical description of two bodies, or
systems of coordinates, moving at speeds relative to
each other, when one of them is being accelerated and
the other is not. It is the result of a \emph{valid}
mathematical inference which can therefore \emph{never}
be refuted. It is a simple truism of analytical
geometry and therefore \emph{belongs to mathematics,
  but not to physics}. So it falls victim to Einstein's
own criterion: because it is certain it does not refer
to reality. And if it does not refer to reality, then
it does not describe a physical property of light, or
of space. And in Popper's terminology the theory of
curved space is non-empirical because it cannot be
refuted by any conceivable experiment, or physical
property of
light\footnote{\ftlabel{SameArgument}The same
  argument will apply to \emph{any amendment} of this
  theory as, for example, by the Riemann-Christoffel
  curvature of space, as long as it is \emph{purely}
  mathematical and does not introduce new empirical
  (falsifiable) information.}.  Therefore we may
\emph{not} infer from this theory that space is curved
\emph{in reality}, and that light will be deflected in
the gravitational field.

\tcrpar
I wonder if any of  you will accept this. If you do not, I would be
eager to hear a counter argument because I know of none. I have
discussed this argument with physicists of my acquaintance without being
contradicted. I have tried to get a public discussion by publication in
serious periodicals of science, but was rejected. I then tried to get
support, or contradiction, from experts and sent a paper containing this
argument---and other arguments which I consider even more important, but
which are not directly related to Karl Popper's theories---to well known
physicists in England, France, the U.S.A.\fsp and Germany without being
deemed worthy of a reply\footnote{I am happy to be able to correct this
  statement now because it is no longer true: after writing first drafts
  of this article, but before reading it at the conference,
  I received a very kind reply from Dr.~Thomas Angelidis, London, who
  spared a tremendous amount of time to go into the details of my
  criticism and caused me to correct some serious mistakes in earlier
  versions of this article. Before the conference time was too short to
  reach a result in our discussion, but some of the footnotes refer to it.
  And Dr.~Elie Zahar, to whom the paper was sent by a common friend, also
  made some important---critical---comments. As far as possible I
  discuss these too in the
  footnotes (see already footnotes~\ref{Zahar}, \ref{SameArgument}).}.
So I have never been able to put
on trial my case \emph{Popper v.~Einstein}. Nobody seems to be
willing to believe that a practising lawyer can have anything sensible
to say on the theory of relativity; nor is anybody willing to discuss
the argument on its own merits. All I can do, therefore, is take
advantage of a defenseless audience and present my view at the
Annual Popper Conference, and via \emph{The Critical Rationalist},
and I am grateful to be able to do so. To me
it appears clear that some aspects of Einstein's \emph{general} theory
of relativity and Popper's \tcrquotes{Logic of Scientific Discovery}
contradict each other flatly and cannot, therefore, both be right.

\tcrpar
I could end my article at this point because this was
what I wanted to say. But you may ask: What about the 
empirical confirmations? Was not the general theory of 
relativity almost miraculously confirmed in Eddington's
famous experiments, made during the eclipses of 1919
and 1922?  Did he not observe that the light coming
from distant stars was in fact deflected in the 
gravitational field of the sun?---And did not
Einstein predict that the spectrum of light coming 
from other stars would be shifted towards the lower 
frequencies, which has since been confirmed by 
observation many times, in particular by the American 
astronomer Edwin Hubble?---Was it not even observed
that there is a time shift if we circumnavigate the 
earth in fast aeroplanes, which was predicted, and can 
only be explained, by the special theory of relativity?

\tcrpar
I have tried to look into all these empirical
confirmations. Some of them require more knowledge of 
mathematics than I have. This does not worry me too 
much because, as an admirer of Karl Popper, I am more 
interested in refutations than in confirmations, and 
to refute an empirical theory you do not necessarily 
need mathematics. It may be sufficient to discuss the 
premises on which it relies. According to the \emph{Logic
of Scientific Discovery} \cite{Popper:LScD}
one refutation will bring
down the theory of relativity, and so many 
confirmations will not set it up again if this 
refutation is valid and can be repeated.  But as far 
as my understanding goes I can say that \emph{all} the
\tcrquotes{confirmations} of the theory of relativity are
extremely dubious. I will explain this for the three 
most important ones, trying to be as brief as possible.

\tcrpar
Eddington's experiment is justly famous, not only for its
results but also for its magnanimity because, while 
the First World War was still going on, a team of 
British scientists meticulously designed an experiment 
to put to the test the theory of a German scientist. 
Anti-Semitism was not so important in 1919, and 
Einstein was still considered to be a German citizen. 
Eddington \emph{believed} in the theory of relativity and
tried to \emph{confirm} it. His experiment was carried out
during the eclipse of 1919, and repeated in 1922, and 
Eddington was able to show that, as Einstein had 
predicted, the light coming from distant galaxies was 
in fact deflected, as by a convex lens, in the 
gravitational field of the sun. It seems that this was 
the point when all resistance against the theory of 
relativity which had previously existed ultimately 
broke down. My impression is that the theory of 
relativity was never again seriously challenged after 
that.

\tcrpar
But looking into the publications of that time will produce some
surprising results. As mentioned, Eddington's experiment was carried out
in 1919/1922. But Einstein had already read, in 1915, a paper before the
Royal Prussian Academy of Science in which he stated that the deflection
of light passing the sun was 1.7 sec.\fsp of arc relative to the passing
distance from the sun \cite{Einstein:Erklarung}. And he \emph{adapted}
his general theory of relativity to this value. Therefore the value
cannot have been \emph{predicted} by the theory, but must have been
known \emph{before}, and independent of it. The theory was designed, and
adapted, so as to \emph{explain} this particular value.

\tcrpar
\emph{After} the publication of Eddington's
experiment Einstein \emph{still}
spoke of a deflection of 
1.7 sec.\fsp of arc. So the outcome of the experiment seems 
to have been exactly what he expected. But he now 
explained this value as having been caused \tcrquotes{one half
by the (Newtonian) attractive field of the sun, the 
other half by the geometric modification 
(\tcrquotes{curving}) of space caused by the sun}
\cite[p.~84f, my translation of his
words]{Einstein:SpezielleUndAllgemeine}.

\tcrpar At least this last explanation can certainly
not be valid. For, as I have tried to show before, the
concept of \tcrquotes{curved space} had been based, by
Einstein, \emph{exclusively} on the deflection of light
in the gravitational field. If (only) one half of the
deflection actually observed can be explained by the
curving of space, which had, in turn, been explained by
the gravitational field, then the other half cannot be
explained by the same gravitational field, this time in
the \tcrquotes{Newtonian} sense. You cannot arbitrarily
\tcrquotes{split} the effect of gravitation into one
half causing curved space, and the other half causing
attraction\footnote{This argument remains valid even
  after taking into account the Riemann-Christoffel
  curvature of space (cp.~footnotes~\ref{Zahar},
  \ref{SameArgument}) as long as this relies on purely
  mathematical inferences, i.e.\fsp without introducing
  new \emph{empirical} hypotheses.}.  This seems to
indicate that the deflection actually observed by
Eddington must have been much stronger than Einstein's
equations would permit\footnote{The same appears to
  have happened with his explanation of the rotation of
  the long axis of the Mercury-orbit. In
  \cite{Einstein:Erklarung} he reports that this
  implied the assumption of a curving of light in the
  gravitational field \emph{twice} as strong as
  previously calculated by him.  This was \emph{before}
  Eddington's experiment!}.

\tcrpar
Nevertheless there remains the indisputable fact that
the theory of relativity predicts a deflection of 
light by gravitation and that a deflection was 
actually observed. The inconsistency in Einstein's 
explanation does not affect this coincidence; it 
only puts a question mark behind the alleged 
correspondence of the exact values of theory and 
observation. But it gives additional weight to the 
question whether the theory of relativity can be a 
\emph{true} explanation of this observation.

\tcrpar
Almost equally famous is Einstein's
explanation of the redshift of light coming from 
distant galaxies which can be observed by the shift of 
the characteristic spectral lines. In special 
relativity Einstein explains that a clock which is 
being moved in a straight line will go more slowly 
than one that is stationary
\cite[p.~25]{Einstein:SpezielleUndAllgemeine}.
In general relativity he infers from this that a 
clock fastened on the periphery of a rotating disc 
will permanently work more slowly than one at 
rest
\cite[p.~53]{Einstein:SpezielleUndAllgemeine}.
And finally he reaches the general conclusion that
any oscillating process, as which he regards the atom 
emitting light, will be decelerated in the vicinity of 
inert mass\footnote{Also in \cite[p.~91]{Einstein:Grundzuge}}.
This implies that the spectrum of light coming from 
stars with great inert mass will be shifted towards 
the lower frequencies. That was Einstein's 
explanation of the redshift which had also already 
been observed when he wrote.

\tcrpar
But the American astronomer Edwin Hubble
discovered in or before 1937 that the redshift of 
light coming from distant galaxies is \emph{proportional to
their distance} from earth \cite[p.~35]{Hubble:Nebulae}.
According to Einstein's theory 
the extent of the redshift, being caused by 
gravitation, would be a function of inert mass. It 
should therefore vary \emph{in proportion to the inert mass}
of the object from which light is coming. So we should 
expect to observe different redshifts in heavier or 
less heavy bodies.  But the Hubble effect---in its
common interpretation by the Doppler principle\footnote{I
doubt the correctness of this interpretation, but
this does not affect the validity of the argument 
against the general theory of relativity since, at any 
rate, the Hubble effect must be explained
\emph{somehow}.}---shows that the extent of the redshift is in fact
graduated not in proportion to mass, but to distance. 
This would only conform with general relativity if we 
assume the stars, or galaxies, to be arranged 
precisely \emph{in the order of their size} (mass), which
conflicts with observation.

\tcrpar
\label{Clash}
So here we have a serious clash. We can either
accept the general theory of relativity or Hubble's 
explanation of the redshift in the light coming from 
distant galaxies. And in view of the strong, and 
repeated, empirical confirmation of Hubble's 
observation I think we must decide in favour of 
Hubble.

\tcrpar Another famous experiment is usually discussed
in terms of the special theory of
relativity\footnote{Strictly speaking the experiment
  discussed here must be considered in terms of the
  general theory of relativity because it deals with
  circular movements. But since the authors Hafele \&
  Keating (see footnote~\ref{Misunderstanding}) apply
  the special theory \tcrquotes{as a first
    approximation} I am doing the same in this article.
  The distinction between special and general
  relativity has no bearing on the arguments used in
  the following text as long as it is agreed that
  general relativity will also produce a time
  difference for the clock experiment.  Einstein
  explicitly said so
  \cite[p.~53]{Einstein:SpezielleUndAllgemeine}.}.  The
special theory of relativity rests on two fundamental
premises. One is the constant vacuum spreading
velocity of light, which Einstein accepted as the
result of various experiments, and the other is the
equivalence of all inertial systems, which he assumed
because we have no reason to believe the laws of
physics to be different on other stars from those here
on earth. From these two premises Einstein inferred,
in faultless mathematical deduction, that, if the
velocity of light is constant, time itself must be
variable. And he predicted that, therefore, a body in
motion will age more slowly than one that is
stationary.

\tcrpar
To test this theory a most interesting experiment
was carried out in October 1971 by two American physicists, 
Hafele \& Keating
\citeyear{Hafele:Predicted,Hafele:Observed}.
Four caesium beam atomic clocks were flown on
regularly scheduled commercial jet flights around the 
world twice, once eastward and once westward. It was 
observed that the flying clocks lost time (aged more 
slowly) during the eastward trip, and gained time (aged 
faster) during the westward trip relative to the 
corresponding time recorded by the reference atomic 
time scale at the U.S. Naval Observatory, MEAN (USNO). 
Hafele \& Keating explained this as a prediction of the
theory of relativity in the following words:

\begin{quote}
\ldots{} consider a view of the (rotating) earth as 
it would be perceived by 
\emph{an inertial observer looking down on the North Pole from a great
distance.} A clock that is stationary on the surface at 
the equator has a speed $R\Omega$ \emph{relative to nonrotating
space}, and hence runs slow relative to \emph{hypothetical
co-ordinate clocks of this space} in the ratio\ldots
\end{quote}

\tcrpar I will not continue this quotation because I
think the pivot of criticism is obvious: the clock of
comparison actually used in the experiment was
\emph{not} in the hands of \tcrquotes{an inertial
  observer looking down on the North Pole from a great
  distance}; it was at U.S. Naval Observatory, which
is, to my knowledge, situated at Richmond, Florida, and
therefore close to the equator. So it was itself
rotating with the equator. If there is no preferred
inertial system, as Einstein claims, then relative to
\emph{this clock} the velocity of both flying clocks,
eastward and westward, was \emph{equal}, and therefore,
if the theory of relativity had been right, the time
dilatation should have been equal
too\footnote{\ftlabel{Misunderstanding}At this point I
  have been accused of misunderstanding the special
  theory of relativity. Both Dr.~Angelidis and Dr.~Elie
  Zahar argued that the speed should be taken
  \tcrquotes{with respect to the same inertial frame}
  (Zahar's words), and that the east-west asymmetry
  would then be explained by \tcrquotes{the
    \emph{different} relative speeds involved}
  (Dr.~Angelidis' words). Their argument, I understand,
  is that the speed of the planes should not be taken
  relative to USNO where it would be equal, but
  relative to the axis of the earth, where it must be
  added to, or subtracted from, $R\Omega$ (the rotation
  velocity of the equator). I think they did not get my
  point, and I will therefore try to clarify my view
  with the following \emph{two arguments}:

  \begin{enumerate}
  \item According to Hafele \& Keating \tcrquotes{special relativity
  predicts that a moving standard clock will record less time compared
  with (real or hypothetical) coordinate clocks distributed at rest in
  an inertial reference space} (their words). Let us now assume three
  satellites, $S_g$,  $S_h$,  $S_l$, each with a clock on board. $S_g$
  is in a geostationary position, rotating with the equator and keeping
  its place relative to USNO and any other location on earth. $S_h$  is
  in a \tcrquotes{heliostationary} position, keeping its position
  between the sun and the earth (and therefore not rotating with the
  equator). And $S_l$ is in a \tcrquotes{lunarstationary} position,
  keeping its position between the earth and the moon, and relative to
  any location on the moon, and therefore rotating around the earth, but
  not with the equator. Why do Hafele \& Keating measure the time
  difference (only) relative to $S_h$? Taking $S_l$ would have produced
  something quite different, and taking $S_g$  would have produced
  equal relative velocities for both trips. Is there any reason to
  consider only $S_h$ to be \tcrquotes{at rest}, and the others not to
  be at rest? Since no observer was out in space, or on the North Pole,
  they could just as well have chosen $S_g$ or $S_l$ as their reference
  clock. Unless, for some reason, they \tcrquotes{prefer} $S_h$. But
  \tcrquotes{preferring} is not permitted by relativity.\par

  \item Another way of demonstrating the incoherence of the
  Hafele \& Keating experiment and the results obtained by relativity can
  be found in the following \emph{Gedankenexperiment}: let a satellite,
  having on board a caesium beam clock sending time signals to earth,
  fly on a course shaping a \emph{very long ellipse}, comparable to the
  course of Halley's comet relative to the sun, and circumnavigate the
  earth as its focal point \emph{from east to west}, i.e.\fsp against the
  direction of its rotation. According to special relativity the time
  dilatation on this satellite should be strongest (the ageing of the
  clock should be at its minimum) when its velocity is highest. This
  would be the point when its course is closest to the earth as its
  focal point. But the observation made by Hafele \& Keating shows that, at
  this point, when circumnavigating the earth in an east-west direction,
  the clock on the satellite will in fact be \emph{gaining time} (ageing
  faster).
  \end{enumerate}}.
But in fact the flying clocks lost time during the
eastward trip and gained time during the westward 
trip. So we have another clash, this time between 
observation and special relativity. We can either 
believe in Einstein's 
theory of special relativity or in the observations 
made by Hafele \& Keating. But we cannot believe in 
both.

\tcrpar
And the worst is yet to come: As Hafele \& Keating
explain, special relativity predicts a time difference 
for the clock coming back to its starting point after 
having circumnavigated the earth. But what is a \tcrquotes{time
difference}? This clearly depends on what meaning we 
give to the concept of \tcrquotes{time}.

\tcrpar
There have been endless discussions of this concept. Being a firm
believer in Popper's methodological nominalism, and in his view that any
discussion of \tcrquotes{concepts} will inevitably end in
\tcrquotes{empty verbiage and barren scholasticism}\footnote{Popper's
  words, \cite[volume~2, p.~9]{Popper:OpenSociety}.}, I have no intention
of joining them. I will therefore try to cut short any discussion on the
\tcrquotes{true}, or \tcrquotes{real} meaning of the concept of
\tcrquotes{time} by stating explicitly that, in this article, I am
\emph{not} discussing the \tcrquotes{existence} of time, or of the
\tcrquotes{reversibility} of natural events connected with the concept
of the \tcrquotes{arrow of time}, or anything of that sort, but will
simply use the term \tcrquotes{time} in the everyday sense which we
employ when we ask somebody: \tcrquotes{What time is it?} We then expect
an answer expressed in \emph{units} of time, i.e.\fsp in terms of
\tcrquotes{days}, \tcrquotes{hours}, \tcrquotes{minutes},
\tcrquotes{seconds} etc.

\tcrpar
The advantage of using the concept of \tcrquotes{time} in
this everyday sense is that we ourselves can \emph{define}
its meaning, for example by \emph{setting clocks}. This is an
implication of Popper's methodological
nominalism\footnote{I am referring to chapters 3 \& 11 of \tcrquotes{The Open
Society and its Enemies}, \cite{Popper:OpenSociety}.}.
And  if, as a matter of expedience, we define \tcrquotes{time}
as a \emph{unique} coordinate of singular events, meaning
that no set of space coordinates may have more than 
one time function, then there can never be two 
different \tcrquotes{times} for any singular physical location.
Let me explain this by an \emph{example}. We could, quite
arbitrarily, define noon time as the point when the 
sun reaches its zenith at, say, Greenwich. If our 
plane now starts from Greenwich and returns to 
Greenwich when the sun is in its zenith there, and the 
clock on board does not show 12.00 \emph{h} sharp at its
arrival then we do not have two different times at 
Greenwich but, according to our definition, \emph{the clock
is wrong}. And we should not blame the sun for this 
either, nor would it be wise to alter our definition 
of \tcrquotes{time} so that it will now permit two \tcrquotes{times} at
Greenwich because this would create an awful muddle. 
Our problem is, rather, to find out \emph{what happened to
the clock}, which is an empirical problem.

\tcrpar But relativity will, even starting from this
premise of time defined by the course of the sun,
predict a time difference for the plane surrounding the
equator\footnote{\ftlabel{Lorentz}This is a necessary
  implication of any theory involving the Lorentz
  transformation. For the expression:

\[ t' = \frac{t - \frac{v}{c^2}x}{\sqrt{1 - \frac{v^2}{c^2}}}  \]
                                                       
will yield $t' \neq t$ for any $c < \infty, v > 0$.
The arguments related to special relativity will
therefore remain valid for general relativity as long 
as general relativity relies on a transfer of the 
results obtained from an application of the
Lorentz transformation in special
relativity. This transfer Einstein undertook 
explicitly in \cite[p.~53]{Einstein:SpezielleUndAllgemeine},
when he assumed a time dilatation for a
clock mounted on the periphery of a rotating disc.}.
It is, as I understand it, a necessary theoretical 
implication of relativity that the clock starting and 
arriving exactly when the sun has reached its zenith 
at Greenwich will not show 12.00 \emph{h} sharp at its
arrival, but some fractions of a second less, and this 
although we have \emph{defined} 12.00 \emph{h} as being the moment
when the sun reaches its zenith at Greenwich. So, in 
spite of our nominalistic definition of time 
permitting no more than one time function for one 
location, and assuming only that we can circumnavigate 
the earth, we can infer from relativity that,

\begin{enumerate}
\item time will be 12.00 \emph{h} when the sun reaches its
zenith at Greenwich,
 and 
\item time will \emph{not} be 12.00 \emph{h} when the sun reaches its
zenith at Greenwich.
\end{enumerate}

\tcrpar
Assuming that a circumnavigation of the earth is
possible, the theory is therefore \emph{contradictory.}

\tcrpar
\emph{If this inference is correct, then special
relativity must be wrong. And if the inference is not 
correct it must still be wrong because, if it does not 
predict a time difference, it cannot account for the 
time difference observed by Hafele \& Keating in their
experiment.}


\tcrpar So we are faced with a truly enigmatic
situation. And since the mathematical formulae of
special relativity are faultless this implies that one
of its \emph{empirical} premises must be given up. But
special relativity rests on \emph{only two} empirical
premises. One of them is the assumption that the laws
of nature (physics) are universal; giving up this would
be the end of all explanation. And the other is the
hypothesis of the constant spreading velocity of light.
Therefore it must be this second premise which has to
be given up.  In other words: Special relativity does
not explain, or establish, the relativity of space and
time; but \emph{it does refute the empirical hypothesis
  of the constant spreading velocity of
  light}\footnote{\ftlabel{Doppler}At this point I
  should mention that the hypothesis of the constant
  spreading velocity of light is also incompatible with
  the common interpretation of the Hubble effect as an
  expression of the Doppler principle. Hubble's
  interpretation relies on the assumption that the
  redshift of light from distant galaxies is caused by
  their (assumed) flight movement which will, according
  to the Doppler principle, reduce frequency, as we can
  all observe when we hear the horn of a passing car.
  But this presupposes that the velocity of the
  (assumed) flight movement can be \emph{deducted} from
  the velocity of light coming from distant galaxies.
  The Doppler principle will never yield a difference
  of frequency without difference of velocity. But this
  clashes with Einstein's principle of the constant
  spreading velocity of light which explicitly
  \emph{excludes} an addition, or subtraction, to or
  from the velocity of light.  Therefore the common
  interpretation of the Hubble effect as an expression
  of the Doppler principle is also incompatible with
  the theory of relativity.}.  If we can travel around
the world and come back to our starting point, then the
velocity of light, if it is finite, cannot be constant.
And since the constancy of the spreading velocity of
light is the \emph{fundamental} premise both of
special and of general relativity this all boils down
to the thesis:

\begin{quote}
\emph{If the earth is round then the theory of relativity
must be wrong.}
\end{quote}

\tcrpar It also follows from this that the time
dilatation observed by Hafele \& Keating was not an
effect of special or of general relativity, but must
have been an effect of something else, at any rate
something empirical. My conjecture is that it was an
effect of \emph{ether}\footnote{This is a somewhat rash
  conclusion which, for time reasons, it was impossible
  to elaborate when actually reading the paper at the
  conference, and of which only a sketchy outline can
  be given here. It is clear that, even if the time
  difference observed by Hafele \& Keating cannot be
  explained by special relativity, as I have tried to
  show in the text (par.~\ref{Clash}), the refutation
  of special relativity will not explain it either. We
  need a new \emph{empirical} hypothesis, and time
  dilatation can be explained if we assume the earth
  and its surrounding atmosphere to be a vortex-like
  rotating field of energy (ether) in which matter is a
  form of appearance (comparable to a disturbance) of
  that field. The clock flying eastward is then flying
  \emph{with} the direction of rotation, the other one
  \emph{against} it, and all the time the field is
  \tcrquotes{flowing} \emph{through} both clocks, and
  their time difference is accounted for by the Doppler
  principle which would otherwise be incompatible with
  the assumption of a constant spreading velocity of
  light (see footnote~\ref{Doppler}). The
  \tcrquotes{time difference} is then, in fact, a
  difference of \emph{performance}: one of the clocks
  has gone through more oscillations of the field than
  the other.}, though of a somewhat different kind than
that expected by Michelson/Morley in their famous
experiment (1887).

\newpage
\section*{Part II}

\tcrpar I am almost certain that no readers of this
article will accept this. How can anyone presume to
know more about physics than Einstein? But remember
that I am not competing with Einstein in questions of
physics, or mathematics, but in questions of
methodology, and thanks to Karl Popper methodology has
made wonderful progress since Einstein's time. You may
have noticed that, so far, I did not use one single
mathematical formula in this text\footnote{The formula
  in footnote~\ref{Lorentz} is needed only to make the
  contradiction clearer for adherents of relativity; my
  criticism of the theory of relativity could do
  without it.}, and yet I have questioned all the
premises from which Einstein started. And I am not
only relying on my own arguments, but on those supplied
by Karl Popper, most of all on his methodological
nominalism\footnote{Here again I am referring to
  chapters 3 \& 11 of \emph{The Open Society and
    its Enemies}, \cite{Popper:OpenSociety}.}.  So I
may be wrong, but then, I think, Karl Popper would also
have been wrong. At any rate---since it is impossible
to reach a final result on this in one single
article---I challenge every reader of this article to
disprove my arguments \emph{in writing}. As a
practising lawyer I know only too well that oral
discussion always has an element of chance in it, and
that is why I am longing to be able to publish my
views---I have a lot more to say---and get a public
discussion, if only the physicists would let me. But
this is a \tcrquotes{Popperian Journal} and I have been
speaking on physics already too long. Let me therefore
return to the theory of scientific knowledge.

\tcrpar
I consider it as definitely established that
Einstein's theory of relativity and Popper's 
theory of scientific knowledge are incompatible. 
Einstein's theory of \tcrquotes{curved space} is non-empirical, and the
empirical parts of his theory concerning time and 
gravitation have been empirically refuted. If I am 
right, then physical science now has the task of 
re-interpreting a tremendous number of  experiments, 
and of finding new solutions for several important 
problems, most of all the problem of gravitation which 
has never been solved. The theory of the \tcrquotes{Big Bang}
and the theory of the \tcrquotes{Black Holes} will both break
down, and we may find that we know even less about our 
universe than we thought we did. What physics 
desperately needs, and apparently totally lacks at the 
moment, is a critical discussion of the premises on 
which the various theories rely which are at present 
happily co-existing despite all their contradictions. 
And to permit this discussion it seems that physics 
will first have to open its mind (and its 
institutions) to the possibility of error, and, 
instead of getting lost in mathematical formulae, find 
the courage to put forward new and daring ideas. And 
for the pupils and admirers of Karl Popper there 
remains the question why he himself did not see these 
implications---if they are as I see them.

\tcrpar
The answer, I believe, has been given by Lessing:
\tcrquotes{Not all are free who mock at their own fetters.} I
think Karl Popper was so impressed by Einstein's 
boldly stating the exact conditions under which his 
theory would break down, and by the contrast in which 
this stood to the dogmatism of psychoanalysis, and so 
overwhelmed by the outcome of Eddington's 
experiment, that he never extended his own criticism 
to Einstein himself. He was a boy of thirteen when 
Einstein published the general theory of relativity, 
and barely seventeen years old when he heard Einstein's 
lecture in Vienna just after Eddington's 
first experiment. At that time he had not yet 
developed the criterion of falsifiability as a 
demarcation between science and metaphysics, but 
Eddington's experiment gave the impulse from which it was 
eventually to emerge. It would have been almost 
superhuman for Popper to apply his criticism to the 
very theory which had inspired his own discovery. In 
later years he was critical of the theory of 
relativity in the sense that he did not consider it as 
the final truth, as was Einstein himself, who spent 
the last thirty years of his life in search of a 
better theory, and once wrote that:

\begin{quote}
\ldots{} the most beautiful lot of a scientific theory is if
it shows the way to a more comprehensive theory in 
which it continues to live as a borderline case.
\end{quote}

\tcrpar
But I think Popper simply never felt the necessity to
call the theory of relativity itself into question. He 
said that it took him years to discover the bearing 
which his demarcation criterion of falsifiability had 
on the problem of induction. In the case of the theory 
of relativity, I think, he continued to live with an 
inconsistency without being aware of it.

\tcrpar
But Einstein, I believe,  was less of a \emph{critical}
rationalist than Popper thought. He did say:

\begin{quote}
Insofar as the expressions of mathematics refer to 
reality they are not certain, and insofar as they are 
certain they do not refer to reality.
\end{quote}
 
So he held the instrument of criticism in his
hands.  But the superstition in which he had been
brought up had not lost its power over him. The most
important \tcrquotes{confirmations} of his theory were,
in fact, observations already known before, or at least
expected at, the time he wrote. The redshift of light
from distant galaxies, the deflection of light in the
gravitational field, and the perihelion-rotation of the
course of Mercury, were all known, or at least
surmised, before 1914, as can easily be seen from the
fact that Einstein himself discussed them in his early
papers \cite{Einstein:Grundlage,Einstein:Erklarung}.
They were the facts from which he \emph{started}, and
to which he \emph{adapted} his theory by logical or
mathematical inference. I am almost sure that if he had
been asked he would have said that he was working by
\emph{induction} because he said so almost
explicitly\footnote{There are many examples in his
  texts. In
  \cite[p.~13]{Einstein:SpezielleUndAllgemeine} he
  says: \tcrquotes{This is where the theory of
    relativity sets in. By an analysis of the physical
    concepts of \tcrquotes{time} and \tcrquotes{space}
    [he did not put quotation marks] it appeared that,
    \emph{in reality, there is no incompatibility of
      the relativity principle with the law of
      spreading velocity of light}, but that systematic
    application of both these laws will yield a
    logically faultless theory}---(his italics; my
  translation). I think this shows quite clearly that
  he \emph{thought} he was working exclusively by
  logical inference.}.  The power of his theory lay in
the fact that it was the first to offer one solution
for a whole set of different problems hitherto
unconnected. But Einstein did not know what we have
since learned from Karl Popper, that falsifiability of
a theory is no vice, but a virtue. So even after he had
published the general theory of relativity he kept
adapting it to whatever new observations came his way,
including Hubble's discovery
\cite[pp.~116ff.]{Einstein:Grundzuge}.  Thus he could
\emph{never} be refuted because he would always change
his theory. For a pupil of Karl Popper, Einstein's
early publications on the theory of relativity are
indeed quite astonishing to read because he was
permanently mending his theory in view of criticism and
new discoveries. But he never stooped to discussing the
compatibility of his premises with Hubble's new
observations. And the very fact that, in the above
quotation, he spoke of the \tcrquotes{expressions of
  mathematics} with respect to reality shows that he
did not distinguish between mathematics and physics as
sharply as Bertrand Russell, Alfred Tarski and Karl
Popper have since taught us to do. Today we know, in
Russell's famous words, that:

\begin{quote}
\ldots{} pure mathematics is the subject matter in which we do
not know what we are talking about, or whether what we 
are saying is true.
\end{quote}

\tcrpar
But we should not forget that it was a difficult
process to reach this point of knowledge, and that 
some of the most eminent scientists were involved in 
reaching it. Even today, and in spite of their 
teaching,  the view is widely spread that the rules of 
logic, and the calculi of arithmetic, are \tcrquotes{laws of
thought}. So we should learn to be still more critical 
of our own views and theories because even the best 
knowledge we have, or believe to have, will not 
automatically put right all those mistakes on which it 
has a bearing. Our brain may be superior to any 
computer because it can create new and revolutionary 
ideas. But it is also inferior to any computer because 
it may overlook inconsistencies which no computer 
would tolerate. If men like Einstein and Popper can 
fail then anyone can fail. That is why we should 
extend our criticism to everyone, to Einstein, and 
even to Popper, \emph{and most of all to ourselves}. And the
best service we can do to the memory of Karl Popper, 
as \emph{I} understood his intentions, is by keeping in mind
his words that:

\begin{quote}
\ldots{} there are experts, but no authorities,
\end{quote}

not even Einstein or Popper. Let us begin a new period
of enlightenment by daring to use our \emph{own}
intelligence: \emph{Sapere aude!} (in this context I
should probably be saying \tcrquotes{\emph{Sapere
    audiemus}}, but I am quoting Kant). Let us use
logic and mathematics where they can be helpful, but
let us also beware of being carried away by their
beauty. We must keep criticizing the premises of our
theories, and for this we need logic and mathematics.
But beyond that we should leave them to the computers
wherever possible, and use our wonderfully creative
human intelligence for developing new and daring
ideas---\emph{and take the risk of being wrong!}

\newpage
\bibliography{v01n03}
\bibliographystyle{tcr}

\newpage
\section*{Author Contact Information\label{authors}}

\label{mettenheim}
\begin{tabular}{ll}
    \multicolumn{2}{l}{\hlnk{\textbf{Christoph von Mettenheim}}
{http://www.eeng.dcu.ie/\~{}tkpw/people/vonmett.html}}\\[3mm]
    \multicolumn{2}{l}{Altes Badhaus,}\\
    \multicolumn{2}{l}{D-75203 Stein,}\\
    \multicolumn{2}{l}{}\\
    \multicolumn{2}{l}{Germany.}\\[3mm]
    \textbf{Telephone:} & \texttt{+49-7232-4576}\\
    \textbf{FAX:}       & \texttt{+49-721-495723} \\
%    \textbf{E-mail:}    & \mailto{}\\
    \textbf{Web:}       & 
      \hoturl{http://www.eeng.dcu.ie/\~{}tkpw/people/vonmett.html}
\end{tabular}
    
\section*{Copyright\label{copyright}}

This article is copyright \copyright~\thecopyrightdate{} by
\htmlref{Dr.~Christoph von Mettenheim}{mettenheim}.

Permission is hereby granted to private individuals to
access, copy and distribute this work, for purposes of
private study only, provided that the distribution is
complete and unmodified, is accompanied by this
copyright notice, and that no charges are levied.

The work may \emph{not\/} be accessed or copied, in
whole or in part, for commercial purposes, except with
the prior written permission of the author.

All other rights reserved.

\newpage
\section*{Retrieval\label{retrieval}}

The resources comprising this article are electronically
retrievable, in various formats, via the \hlnk{World Wide Web}
{http://www.w3.org/hypertext/WWW/TheProject}. The base
(directory) URL is:

\begin{quote}
\hoturl{http://www.eeng.dcu.ie/\~{}tkpw/tcr/volume-01/number-03}
\end{quote}

This URL directly accesses the root node of the HTML format
article.  Alternatively, an additional filename may be appended
to this URL, to retrieve one of the other formats, or related
resources, as follows:

\begin{itemize}
\item The complete set of files making up the HTML format of the
article is available in the single compressed archive file
\hoturl{v01n03.html.tar.gz} (30~KByte),
which can be conveniently downloaded for local and/or offline
browsing.
\item  A printable version of the article is available in 
postscript format
as file \hoturl{v01n03.ps} (194~KByte,
23~pages), or in compressed form as 
file \hoturl{v01n03.ps.gz} (69~KByte).
\item The original \LaTeX{} source of the article is available as file
\hoturl{v01n03.tex} (52~KByte), or in compressed form as
file \hoturl{v01n03.tex.gz} (19~KByte).
\item The \BibTeX{} source for the reference
list is available as file \hoturl{v01n03.bib}
(3~KByte).
\item Finally, a complete set of the files just listed
is available as a single compressed 
archive file \hoturl{v01n03.tar.gz} (141~KByte).
\end{itemize}

\input{tcrback}

\end{document}
