[1] I have in mind his speech on "Kant's Critique and Cosmology" at the 150th. anniversary of Kant's death (Popper 1965, pp. 175 ff., now also contained in the introduction to the German translation).

[2] 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 empirical
    assumptions, but on mathematical inference,
  my arguments against it remain the same.
[3] The same
  argument will apply to any amendment of this
  theory as, for example, by the Riemann-Christoffel
  curvature of space, as long as it is purely
  mathematical and does not introduce new empirical
  (falsifiable) information.
[4] 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 2, 3).
[5] This argument remains valid even
  after taking into account the Riemann-Christoffel
  curvature of space (cp. footnotes 2,
  3) as long as this relies on purely
  mathematical inferences, i.e. without introducing
  new empirical hypotheses.
[6] The same appears to
  have happened with his explanation of the rotation of
  the long axis of the Mercury-orbit. In
  (Einstein 1915) he reports that this
  implied the assumption of a curving of light in the
  gravitational field twice as strong as
  previously calculated by him.  This was before
  Eddington's experiment!
[7] Also in (Einstein 1990, p. 91)
[8] 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
[9] 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 10) apply
  the special theory "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
  (Einstein 1917, p. 53).
[10] 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
  "with respect to the same inertial frame"
  (Zahar's words), and that the east-west asymmetry
  would then be explained by "the
    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,  tex2html_wrap_inline622  (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 two arguments:

  1. According to Hafele & Keating "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, tex2html_wrap_inline626 , tex2html_wrap_inline628 , tex2html_wrap_inline630 , each with a clock on board. tex2html_wrap_inline626 is in a geostationary position, rotating with the equator and keeping its place relative to USNO and any other location on earth. tex2html_wrap_inline628 is in a "heliostationary" position, keeping its position between the sun and the earth (and therefore not rotating with the equator). And tex2html_wrap_inline630 is in a "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 tex2html_wrap_inline628 ? Taking tex2html_wrap_inline630 would have produced something quite different, and taking tex2html_wrap_inline626 would have produced equal relative velocities for both trips. Is there any reason to consider only tex2html_wrap_inline628 to be "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 tex2html_wrap_inline626 or tex2html_wrap_inline630 as their reference clock. Unless, for some reason, they "prefer" tex2html_wrap_inline628 . But "preferring" is not permitted by relativity.

  2. Another way of demonstrating the incoherence of the Hafele & Keating experiment and the results obtained by relativity can be found in the following Gedankenexperiment: let a satellite, having on board a caesium beam clock sending time signals to earth, fly on a course shaping a very long ellipse, comparable to the course of Halley's comet relative to the sun, and circumnavigate the earth as its focal point from east to west, i.e. 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 gaining time (ageing faster).
[11] Popper's
  words, (Popper 1966, volume 2, p. 9).
[12] I am referring to chapters 3 & 11 of "The Open
Society and its Enemies", (Popper 1966).
[13] This is a necessary
  implication of any theory involving the Lorentz
  transformation. For the expression:


will yield tex2html_wrap_inline652 for any tex2html_wrap_inline654 . 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 (Einstein 1917, p. 53), when he assumed a time dilatation for a clock mounted on the periphery of a rotating disc.

[14] 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 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
  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.
[15] 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. 22), the refutation
  of special relativity will not explain it either. We
  need a new 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
  with the direction of rotation, the other one
  against it, and all the time the field is
  "flowing" 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 14). The
  "time difference" is then, in fact, a
  difference of performance: one of the clocks
  has gone through more oscillations of the field than
  the other.
[16] The formula
  in footnote 13 is needed only to make the
  contradiction clearer for adherents of relativity; my
  criticism of the theory of relativity could do
  without it.
[17] Here again I am referring to
  chapters 3 & 11 of The Open Society and
    its Enemies, (Popper 1966).
[18] There are many examples in his
  texts. In
  (Einstein 1917, p. 13) he
  says: "This is where the theory of
    relativity sets in. By an analysis of the physical
    concepts of "time" and "space"
    [he did not put quotation marks] it appeared that,
    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 thought he was working exclusively by
  logical inference.

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TCR Issue Timestamp: Mon Dec 30 17:41:04 GMT 1996