THE KUYPERS EFFECT: ANGULARMOMENTUM CONSERVATION IMPLIES GLOBAL C IN GRAVITY

International Journal of Technical Research and Applications e-ISSN: 2320-8163,
www.ijtra.com Volume 3, Issue 1 (Jan-Feb 2015), PP. 120-122
120 | P a g e
THE KUYPERS EFFECT: ANGULAR-
MOMENTUM CONSERVATION IMPLIES
GLOBAL C IN GRAVITY
Otto E. Rossler
Institute for Physical and Theoretical Chemistry,
University of Tubingen, Auf der Morgenstelle 14, 72076 Tubingen, Germany
oeross00@yahoo.com
Abstract- A young astronomer’s by now ten years old
results are re-told and put in perspective. The implications are
far-reaching. Angular-momentum shows its clout not only in
quantum mechanics where this is well known, but is also a
major player in the space-time theory of the equivalence
principle and its ramifications. In general relativity, its
fundamental role was largely neglected for the better part of a
century. A children’s device – a friction-free rotating bicycle
wheel suspended from its hub that can be lowered and pulled
up reversibly – serves as an eye-opener. The consequences are
embarrassingly far-reaching in reviving Einstein’s original
dream.
Key words: Bicycle wheel, angular momentum
conservation, equivalence principle, gravitation, quantum
mechanics, global validity of c, black-hole theory, experimental
safety. (January 27, 2015)
I. INTRODUCTION
Angular momentum is a fundamental conserved
quantity in nature like energy. It is in the simplest case
defined by “rotation frequency times mass times squared
radius.” The most intuitive application is a bicycle wheel
suspended from its hub which can be vertically lowered and
pulled up again in a frictionless manner. The young
astronomer Heinrich Kuypers showed in an abstract
published in 2003 [1] and subsequently in a doctoral thesis
(unpublished) that an unfamiliar consequence is implicit in
angular-momentum conservation: a gravitational-redshift
proportional size change [2].
II. A REVIEW OF THE RESULT
The new effect is straightforward to derive. If the mass of
the frictionless bicycle wheel is for simplicity assumed to be
condensed into the ideally infinitely thin outer rim, then the
following simple textbook formula [3] suffices for a
description of the conserved angular momentum L when the
wheel is allowed to rotate frictionlessly at a constant – say
horizontal – orientation in space:
L = ω m r2
= const. (1)
In this easy-to-remember formula (“Lomrr” which almost
sounds like “l’hombre”), ω is the rotation rate and m the
mass and r the radius.
Next, look at Einstein’s epoch-making discovery of
gravitational redshift, or equivalently gravitational clock
slowdown downstairs, which is a monotonic function of the
height difference. It can be written as
ω’ = k ω, (2)
where k < 1 is the redshift factor relative to above. The
constant k depends on the height difference as originally
described by Einstein in the form
k=(1+Ф/c2
), (3)
with Ф the (by definition negative) gravitational potential as
a function of height assumed in the equivalence principle
and c the speed of light [4].
Eq.(2) means that the lowered wheel, while keeping its
original angular momentum L, cannot but rotate more slowly
downstairs in gravity in dependence on the height
difference. Its locally constant rotation rate makes the wheel
qualify as a “clock.” For example, the Schwarzschild metric
[5], described nine years later as an implication of the full
Einstein equation of 1916 [6], implies that k approaches zero
at the surface (“horizon” in Rindler’s terminology) of a
black hole. But we can stick here to the simpler context of
the equivalence principle proper without loss of generality.
What does the validity of Eq.(2) mean for Eq.(1)?
If ω is changed into ω’ downstairs by Eq.(2) in the
conservation law of Eq.(1), it is clear that either m or r or
both must undergo a compensatory change down there since
L is conserved. At first sight, infinitely many possibilities
open themselves up for an m’ or an r’ or both, in order to in
combination with ω’ keep the angular-momentum L of
Eq.(1) constant.
However, serendipity allows that another physical fact of
nature comes to the aid. It was discovered two decades later
and makes the consequences that Eq.(2) entails for Eq.(1)
well defined. This is the famous “creation and annihilation
operators” of Dirac’s quantum mechanics [7]. We saw
already that photons that are emitted downstairs possess – in
spite of their appearing normal locally – a reduced
frequency relative to above by virtue of Eq.(2) [4]. The
Dirac mechanism which allows particles to be
interconverted into photons and vice versa, constrains the
masses of elementary particles created downstairs. For
example, an electron and a positron which both have the
same mass can jointly get annihilated into two mutually
opposite 511 keV gamma-ray photons, a mechanism that has
medical applications (“PET scan”) and works at all height
levels on earth – despite the fact that the lower-level 511
keV photons have less energy (are redshifted) relative to
those above. As a consequence of Dirac’s result, all masses
locally at rest downstairs are reduced compared to above by
the relative gravitational redshift factor k of Eq.(2). This for
some reason almost never mentioned fact, cf. [8], means for
our wheel that its mass m’ valid downstairs is
m’ = k m. (4)
Now that m’ is fixed, so is r’: Inserting both Eq.(2) and
Eq.(4) into Eq.(1) yields the following astounding result:
r’ = r/k. (5)
That is, all material objects at rest downstairs in gravitation
are linearly increased in size by the gravitational redshift
factor k valid relative to above [9]. And so is space itself.

121 | P a g e
III. IMPLICATIONS
Eq.(5) when taken at face value (which is allowed to do
as we shall see in a moment) has an astounding consequence
when combined with Eq.(2): the speed of light in the
vacuum, c, proves to be an invariant across height levels. To
see this, it suffices to start out with ω = 2π/T with T the
rotation period of the wheel valid upstairs, and to then
multiply it with the unit length r valid upstairs. In this way a
constant linear velocity for the rim of the rotating wheel (a
certain small fraction of c) is obtained. Second, one can do
the same thing downstairs with the primed variables on the
right hand side of Eq.(1). Here one starts out with ω’ =
2π/T’ with T’ the rotation period valid downstairs, and
multiplies with the unit length r’ valid downstairs. In this
way a constant linear velocity for the rim of the rotating
wheel (a certain small fraction of c) is obtained again. In
both cases it is the same fraction of c. Therefore, both time
and space are concomitantly transformed in gravity such
that
c = globally constant. (6)
This result, implicit in Eqs.(2) and (5) combined,
rehabilitates c as a global constant in the equivalence
principle. Einstein’s so reluctantly arrived-at opposite
conclusion [4] is therefore no longer necessary and possible.
The finding that validity of Eq.(6) is enforced by angular-
momentum conservation in the equivalence principle is
Kuypers’ main result [1,2].
IV. A PROBLEM
The result of Eq.(6) obtained by Kuypers is maximally
astounding. It not only upsets more than a century old
wisdom, it also formally contradicts an indubitable result
valid in special relativity and by implication in the
equivalence principle: the fact that transversal distances are
conserved in special relativity. The latter fact, sometimes
called the “parallel railroad-tracks principle,” reads:
r-transversal’= r - transversal. (7)
Eq.(7) is bound to hold true in between the lowered wheel’s
radius r-transversal’ and the radius r-transversal of the
same wheel valid at the original higher-up position. It goes
without saying that Eq.(7) formally contradicts Eq.(5) and
hence also Eq.(6). Therefore, a logical impasse appears to
have been reached at first sight.
V. THE SOLUTION
Fortunately, the contradiction arrived at is not a logical
“contradictio in adjectu” because Eq.(7) is not a genuine
identity but rather involves a projection effect: It is only
“under vertical projection” that Eq.(7) holds true. That is to
say: even though lateral sizes do map upon each other under
vertical light rays in the equivalence principle by virtue of
special relativity [4], this projective constraint means
something new in light of Eq.(5).
It turns out that there is no contradiction. Both the
conserved projection of Eq.(7) and the size increase of
Eq.(5) are valid: Locally the objects are bigger downstairs,
but they do not look so from above. How can one be sure
that this is the solution? Answer: by letting the rotating
wheel rotate not about a vertical axis as before, but rather
about a horizontal axis. In this case, Eq.(5) is bound to
remain manifestly valid for the vertical wheel diameter of
the upright wheel, while simultaneously the horizontal
diameter of the upright wheel is “observationally
compressed” by virtue of Eq.(7). Thus the wheel looks like a
vertical ellipse under the influence of Eq.(5). This effect
may some day become empirically observable on a neutron
star where the ratio is almost 2:1.
VI. A PARALLEL CASE
If the described way out appears like a “last resort” to the
reader, there exists a “direct analog” familiar from special
relativity: the famous “garage paradox.” The latter
describes, not an optically masked expansion as is at stake
here, but rather an optically masked contraction.
Specifically, a fast-moving quadratic two-dimensional
“automobile” is well known to momentarily fit into a garage
that is shorter in its length by the Lorentz factor but has the
width of the same car at rest. (We neglect the subsequent
braking process inside the garage.) The point is that the
Lorentz-contracted automobile remained isotropic –
quadratic – in its own frame even though this fact is
optically masked. Analogously here: the Kuypers-expanded
upright bicycle wheel remains isotropic – circular – in its
own frame. That is, Kuypers’ “gravitational size expansion”
and FitzGerald and Lorentz’s “kinematic size contraction”
represent twin results in special relativity. One may even
speal of a duality.
This finishes the present account of Kuypers’ finding.
VII. AN UNUSUAL PLEDGE
I herewith pledge for the acceptance-at-long-last of
Kuypers’ thesis [2] as a doctoral dissertation. Its impact
appears comparable to that of Louis de Broglie’s likewise at
first unwelcome doctoral thesis. The impact of Kuypers’
main result, Eq.(5), is equally seminal. It for example
implies via the Schwarzschild metric [10] that the well-
known infinite temporal distance from the outside world of
the horizon of a black hole (at which k approaches zero) [11]
is matched by an infinite spatial distance valid from the
outside world.
A. Consequences of Equation Six, Part I
Since Eq.(6) is implicit in the Kuypers equation, Eq.(5), it
transpires that many “formally allowed” transforms of the
Einstein field equations get strongly constrained. Therefore
Eq.(6) profoundly changes the properties of lack holes. It
implies that nothing can enter the horizon in finite outer time
– not even by quantum tunneling – because the spatial
distance has become as infinite from the outside world as the
temporal distance has always been known to be [11]. Thus
at least one theoretically accepted if empirically
unconfirmed combined general-relativistic and quantum
effect, the famous Hawking radiation [12], ceases to be
physical in the wake of c-global. This could be seen as a
rather arcane opposition between a young man of 1963 and a
young man of 2005.
B. Consequences of Equation Six, Part II
There is a second clashing point between Kuypers’ c-global
and the “establishment” if you so wish. It concerns the
validity of the Friedmann solution to the Einstein equation
of 1924 [12]. This is because a “speed of global expansion”
can no longer be added to a “global c.” In light of this
unfamiliar fact, a certain “gut reaction” to Kuypers’ result
(Eq.6) is clearly understandable.

122 | P a g e
This explains in retrospect why the third referee chosen by
the faculty could single-handedly “kill” the promotion by
refusing to give a grade to the dissertation. The two “very
good” grades given by the supervisor and the maximally
prestigious external co-referee did not warrant a further
opinion, the faculty told me. This is understandable in light
of the novelty of Eq.(6): A student cannot be allowed to
upset a ruling paradigm.
C. Consequences of Equation Six, Part III
The collision with Hawking radiation implies that a
prestigious terrestrial experiment – announced to be re-
started at twice its former world-record energy in March
2015 – will be well advised to renew its now 7 years old
safety report [13] before the re-start. For Kuypers’ thesis
implies that the experiment relies on false physics (neglect
of c-global). Note that this would not be the first time that
false physics causes a catastrophe as the Eniwetak example
shows. But this time around, the falsity is known
beforehand. The proverb ex falso quodlibet (“from the false,
anything [can follow]”) includes even existential risks.
VIII. A PERSONAL CONCLUSION
To conclude, I apologize to my former student that it took
me ten years to rehabilitate him with the present paper. An
old textbook formula – Eq.(1) – proves to possess this
healing power. We all have to forgive the community for
having overlooked Kuypers’ thesis for so long. I know for
sure that the young generation – astronomer Kuypers works
as an esteemed highschool teacher – will be particularly
grateful to him: Can there be a simpler physical sentinel than
Eq.(1)? It is a long time that “high-school mathematics” led
to a major new insight in physics. This situation is bound to
change in the wake of l’hombre. The ultimate reason for this
is that physics is based on the “intrinsically simple” – or
synonymously the “beautiful” as Dirac called it [14].
IX. ACKNOWLEDGMENTS
I thank Dieter Fröhlich and Wolfgang Müller-Schauenburg
for discussions. For J.O.R.
REFERENCES
[1] H. Kuypers and O.E. Rossler, Matter-wave Doppler
effect (in German). WechselWirkung 120, 26-27 (2003).
[2] H. Kuypers, Atoms in the Gravitational Field: Heuristic
Hints at a Change of Mass and Size (in German). PhD
Thesis submitted to the Faculty of Chemistry and
Pharmacy, University of Tubingen 2005.
[3] P.A. Tipler, Physics for Scientists and Engineers, 3rd
edn., extended version. Worth Publishers, New York
1999, Eq. (8.32).
[4] A. Einstein, On the relativity principle and the
conclusions drawn from it (in German). Jahrbuch der
Radioaktivität 4, 411–462 (1907), p. 458; English
translation:
http://www.pitt.edu/~jdnorton/teaching/GR&Grav_2007/
pdf/Einstein_1907.pdf , p. 306.
[5] K. Schwarzschild, On the gravitational field of a mass
point according to Einstein's theory (in German).
Sitzungsberichte der Königlich-Preussischen Akademie
der Wissenschaften, Reimer, Berlin 1916, pp. 189-196.
English translation: http://arxiv.org/abs/physics/9905030
[6] A. Einstein, The Foundation of the Generalised Theory of
Relativity (in German). Annalen der Physik 354, 769-822
(1916). English translation:
http://en.wikisource.org/wiki/The_Foundation_of_the_G
eneralised_Theory_of_Relativity
[7] Dirac, P A M The Quantum theory of emission and
absorbtion of radiation. Proc. Roy. Soc. A 114, 243-265
(1927).
[8] J. Schwinger, Einstein’s Legacy. Scientific American
Books, New York 1986, pp.141-142.
[9] Remark: By coincidence, Eq.(5) follows from Eq.(4) also
via quantum mechanics. This is because atoms of a lower
mass m’ are by the factor m/m’ linearly enlarged via the
Bohr-radius formula of quantum mechanic, a0 = ħ/(mcα)
with ħ = h/(2π) and α the fine structure constant (see
P.A. Tipler and R.A. Llewellyn, Modern Physics, 4th
edn.
Freeman, San Francisco 1999, Eq. 4.33).
[10] O.E. Rossler, Abraham-like return to constant c in
general relativity: “ℜ-theorem” demonstrated in
Schwarzschild metric. Fractal Spacetime and
Noncommutative Geometry in Quantum and High
Energy Physics 2, 1-12 (2012). Preprint:
http://www.wissensnavigator.com/documents/chaos.pdf
[11] J.R. Oppenheimer and H- Snyder, On continued
gravitational contraction. Phys. Rev. 56, 455-459 (1939).
[12] S.W. Hawking, Black hole explosions. Nature 248, 30-31
(1963).
[13] A. Friedman, On the curvature of space (in German).
Zeitschrift für Physik 10, 377–386 (1922).
[14] S.B. Giddings and M.L. Mangano, Astrophysical
implications of hypothetical stable TeV-scale black
holes. Phys. Rev. D 78, 035009 (2008). “LSAG Report”
[15] G. Farmelo, The Strangest Man: The Hidden Life of Paul
Dirac, Quantum Genius. Faber and Faber, London 2009.

THE KUYPERS EFFECT: ANGULARMOMENTUM CONSERVATION IMPLIES GLOBAL C IN GRAVITY

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