I. The Fritz Zwicky Paradox: the fourth failure of Newton (MOND, MOG, etc)

The radical conclusion that Cold- or Warm-Darm-Matter cosmology ought to be discarded as a viable description of physical reality would imply, as a strict logical process, that this physical reality must be non-Newtonian in a certain physical regime which is found on galactic scales and beyond. This goes hand-in-hand with discarding Newtonian dynamics in these regimes.

Indeed, changing dynamics away from Newtonian dynamics is actually already a very well established tradition in physics and results from the desire to understand how objects with “mass” move about and influence each other in our four-dimensional “space-time” world.

Before proceeding, the distinction between mechanics and dynamics in physics ought to be clarified: Mechanics is the science “concerned with the set of physical laws governing and mathematically describing the motions of bodies and aggregates of bodies geometrically distributed within a certain boundary under the action of a system of forces.” Dynamics, on the other hand, is the study of the “causes of motion and changes in motion”. In what follows we do not distinguish these two, but interchangeably refer to Newtonian dynamics and Newtonian mechanics (or in short just Newton) as meaning the classical mathematical description of the motions and changes of motions. These are Newton’s three laws and his 1/r^2 law of gravitation.

Now, lets take a time machine and go back about 120 years. On arrival we find that after centuries of applying pure Newtonian mechanics to describing natural phenomena highly successfully indeed and after a brilliant history of applying this theoretical knowledge to practical utilisation (industrial revolution, planetary motions, long-range travel with trains pulled by steam engines, building ever bigger and more powerfull machines), problems had begun to emerge:


The first failure of Newton (on small spatial scales)

The first famous example was the break down of Newtonian mechanics on small scales.

The historical events leading to the full-fledged theory of (non-Newtonian) Quantum Mechanics we know today appear to share some parallels with our problem at hand (speculative dark matter versus non-Newtonian gravitational dynamics):

As noted here:

“In 1802, William Wollaston in England had discovered that in fact the solar spectrum itself had tiny gaps – there were many thin dark lines in the rainbow of colors. These were investigated much more systematically by Joseph von Fraunhofer, beginning in 1814. He increased the dispersion by using more than one prism. He found an “almost countless number” of lines.”

Johann Balmer, a math and Latin teacher at a girls’ school in Basel, Switzerland, had done no physics before, and made his great discovery (in about 1885) when he was almost sixty. He decided that the most likely atom to show simple spectral patterns was the lightest atom, hydrogen. Angstrom had measured the four visible spectral lines to have wavelengths 6562.10, 4860.74, 4340.1 and 4101.2 in Angstrom units (10-10 meters). Balmer derived the empirical formula to describe the wavelength of spectral lines in the Hydrogen atom.”

These were thus the first empirical hints for quantisation of energy.

In 1901 Max Planck solved the long standing unsolved problem of how the shape of the black body energy spectrum comes to be.

“At first Planck considered that quantisation was only ‘a purely formal assumption … actually I did not think much about it…’; nowadays this assumption, incompatible with classical physics, is regarded as the birth of quantum physics and the greatest intellectual accomplishment of Planck’s career.”

Balmer’s empirically derived formula (with a constant that had no explanation at that time) was later explained by the Nils Bohr model of the hydrogen atom of 1913 which showed that electrons were arranged in discrete shells about the nucleus.

Albert Einstein’s explanation of the photoelectic effect in 1905 further gave strong evidence for the quantum nature of matter and energy.

These discoveries show how the long-cherished concept of a Newtonian world broke down fundamentally on small scales. The (nowadays famous) scientists of the day had bits and pieces of partially computable theoretical understanding (e.g. Planck’s spectral energy distribution, Bohr’s atomic model, Balmer’s spectral line rules). But the Theory of Quantum Mechanics we know today did not exist yet. This theory was developed largely in the late 1920s by Werner Heisenberg and Erwin Schroedinger.

Note that this failure of Newton implies that particles can “magically” appear at a place where Newton would not have allowed them to. A physicist stating this may be referred to as being nuts, but tunneling is in fact an established physical process of quantum mechanics which also happens to be quite fundamental to important technologies we are using without even being aware of it.

Thus we had the following historical development away from Newtonian mechanics to Quantum Mechanics:

  • 1802 – spectral lines exist – William Wollaston.
  • 1885 – Balmer lines – Johann Balmer.
  • 1901 – Black body spectrum explained by quantisation of energy – Max Planck.
  • 1905 – Photoelectric effect as a quantum effect – Albert Einstein.
  • 1913 – mathematical model of Hydrogen atom based on quantised states – Nils Bohr.
  • late 1920s – development of the mathematical theory of quantum mechanics – Erwin Schroedinger and Werner Heisenberg.


The second failure of Newton (at large velocities)

A famous case of the break down of Newtonian mechanics occured at large velocities. This was first hinted at by the inexplicable discovery that the speed of light is constant independent of the velocity of the light source (the Michelson-Morley experiment). And directly related to this, after a major research effort to constrain its properties, the Ether had to be abolished as a physical reality. The outcome, in 1905, was the Non-Newtonian Theory of Special Relativity by Albert Einstein. As we all know, this theory was not the final word in physics.


The third failure of Newton (at large gravitational accelerations)

The precession of Mercury’s perihelion is a famous example of the break-down of Newtonian dynamics within the SolarSystem. We read:

 “The anomalous rate of precession of the perihelion of Mercury’s orbit was first recognized in 1859 as a problem in celestial mechanics, by Urbain Le Verrier.”


“Einstein showed that general relativity agrees closely with the observed amount of perihelion shift. This was a powerful factor motivating the adoption of general relativity.”

In strong space-time curvature, space and time are distorted such that locally clocks go faster and lengths are contracted. This leads to objects moving on different trajectories than an observer interpreting the world through Newtonian eyes would expect. The Theory of General Relativity was able to precisely account for Mercury’s perihelion shift, and also correctly predicted the deflection of light by the Sun.


The fourth failure of Newton (at small accelerations)

Today we have a similar situation as in the first case above, but now in the realm of very small accelerations (i.e. vanishing space-time curvature), but the job of finding the underlying mathematical theory is not yet done:

The Fritz Zwicky Paradox arises under the hypothesis that (1) Newtonian/Einsteinian dynamics is valid on galaxy-cluster scales (Zwicky 1937) and that (2) tidal dwarf galaxies are formed in galaxy encounters (Zwicky 1956). Both together lead to a logical inconsistency. This Fritz Zwicky Paradox can only be  solved by discarding the assumption that Newontian dynamics is valid (Kroupa et al. 2010).

Note that Bekenstein’s TeVeS theory includes Einstein’s Theory of General Relativity, Milgrom’s dynamics and Newtonian dynamics as limiting cases in the relevant physical conditions (e.g. TeVeS gives normal Newtonian dynamics on Earth, but leads to MOND dynamics in the outer regions of galaxies and to General Relativity near black holes).

Today it is known that MOND (Modified Newtonian Dynamics, e.g. Milgrom & Bekenstein 1984) or some other modification of gravity such as MOG (Modified Gravity, Moffat 2006) are the preferred descriptions of galactic dynamics in the weak field limit (small curvature case). Does this mean that gravity per se differs from the Newtonian law (i.e. space-time curvature as resulting from General Relativity), or does it mean that the principle of equivalnce between inertial and gravitating mass is violated in the regime of very small space-time curvature (see Bekenstein & Milgrom 1984)? We do not know.


Concluding Remarks

Thus, in the past it was realised that Newtonian mechanics and dynamics broke down on small spatial scales, at high velocities and at large gravitational accelerations. Today we are learning that it also breaks down when the accelerations are small. Note this progression of the time derivative of physical quantities describing our four-dimensional world (scale=zeroth derivative, velocity=first derivative and acceleration=second derivative). Does this mean something? No idea.

But, it is clear that Newton’s mechanics was found to not work when technological advance, based on Newtonian mechanics, allowed us to probe large velocities and small spatial scales. The new theories allowed us to build better machines (e.g. telecommunication being based on Einstein’s theories and computers and measuring devices being based on quantum mechanical concepts). The theoretical insights into the properties of matter via quantum mechanics and relativity gave us access to an unprecedented control over matter and to unimaginable amounts of energy.

Today we can probe regions of physics where space-time curvature is extremely small finding that, again, Newtonian mechanics fails. This may mean that the Theory of General Relativity needs an extension or that we do not yet understand what “space-time” and “mass” are nor how they are fundamentally related. Perhaps it just boils down to the problem of us not understanding the vacuum.

The four failures of Newtonian dynamics can be rephrased as follows: The real physical world, when interpreted strictly through Newtonian eyes, misbehaves badly on small spatial scales, at high velocities and at large and small accelerations. Clearly, Newtonian mechanics and dynamics are a useful, but limited, mathematical description of the physical world.

Similarly as above we can state that (some) scientists of the day have bits and pieces of partially computable theoretical understanding (e.g. galactic rotation curves, the Bullet Cluster, tidal dwarf galaxies, the Local Group of galaxies, modified dynamics).  But a full theory of gravitation and mass is still lacking.

We can thus rest in comfort that in the realm of low accelerations dynamics is MOdified rather than Newtonian and that we live in a world without Cold or Warm Dark Matter. But, the fundamental theory underlying MOND or MOG (the final word in physics explaining space-time and mass) is yet to be discovered. This is a great present-day chance for mathematical physicists to get to play.


Exciting Times – Our Times

The present epoch is an extraordinarily exciting time indeed: How was it to live, as a physicist, during the legendary time of the theoreticians Einstein, Bohr, de Broglie, Planck and Heisenberg and Schroedinger? Well, perhaps much like today which is, perhaps, the time of Milgrom, Bekenstein, Moffat, Blanchet, Zhao, Famaey and others perhaps yet to emerge.

by Anton Ippendorf, Pavel Kroupa and Marcel Pawlowski (01.09.2010, “The Fritz Zwicky Paradox: the fourth failure of Newton  (MOND, MOG, etc)in “The Dark Matter Crisis – the rise and fall of a cosmological hypothesis” on SciLog. See the overview of topics in  The Dark Matter Crisis.


Author: Prof. Dr. Pavel Kroupa

I am a Czech-Australian teaching and researching at the University of Bonn on dynamics and stellar populations. After studying physics at The University of Western Australia, Perth, I obtained my PhD from Cambridge University, UK, as an Isaac Newton Scholar at Trinity College. After spending eight years in Heidelberg I habilitated at the University of Kiel, Germany. I then took up a Heisenberg Fellowship and later accepted the position as a professor at Bonn University in 2004. I was awarded a Leverhulme Trust Visiting Professorship (2007, Sheffield, UK) and a Swinburne Visiting Professorship (2007, Melbourne, Australia). In 2013 I received the Silver Commemorative Medal of the Senate of the Czech Republic, and I took-up an affiliation with the Charles University in Prague in 2016. Pure innovative science can only truly thrive in non-hierarchical societies in which competition for resources is not extreme. Therefore I see the need for the German academic system to modernise (away from its hierarchies) and warn of academic systems that are based on an extreme competition for resources (USA), as these stifle the experimentation with new ideas.

5 thoughts on “I. The Fritz Zwicky Paradox: the fourth failure of Newton (MOND, MOG, etc)”

  1. Local Group Test of Dark matter Concordance CosmologyDark matter, giving no clear signature in any laboratory experiment is not very appealing for a  physicist work in atomic or condensed matter physics.
    Therefore, the every idea to solve the current problems in our understanding of the universe without dark matter seems to me very attractive.
    On the other hand, I got aware of three recent publications which question alternative laws of gravitation (1), provide an understanding why dark and normal matter are intimately coupled (2) and give some insight into the problem of missing satellite galaxis (3).
    (1) Confirmation of general relativity on large scales from weak lensing and galaxy velocities
    Author(s): Reyes, R; Mandelbaum, R; Seljak, U, et al.
    Source: NATURE   Volume: 464   Issue: 7286   Pages: 256-258   Published: 2010
    (2) Universal Properties of Dark Matter Halos
    Author(s): Boyarsky, A; Neronov, A; Ruchayskiy, O, et al.
    Source: PHYSICAL REVIEW LETTERS   Volume: 104   Issue: 19   Article Number: 191301   Published: 2010
    (3) Satellite galaxies in hydrodynamical simulations of Milky Way sized galaxies
    Authors M. Wadepuhl, V. Springel
    Could you please comment on these new findings which are obviously not mentioned in your recent article (they published later) on Local-Group tests of the dark matter Concordance Cosmology proposing a new paradigm for structure formation. What does it mean for your claims?
    With many thanks in advance!

  2. Local Group Test of Dark matter Concordance Cosmology (comment by Thomas Moller)Dear Thomas,

    Thank you for your comment and your questions. These are very relevant indeed.

    *** Concerning your point (1): _consistency_ of data with a theory does not prove this theory to be correct, as there may be another theory just as consistent with the same data. It is falsification which matters. Thus, the LCDM model is falsified on Local-Group-of-Galaxies scales, while modified gravitation theories are not excluded with large-scale data. The Reyes et al. (2010) paper is actually already discussed in the Introduction and the Conclusions of our paper.

    Reyes et al. (2010) write in their Conclusions:
    “Nevertheless, even with the current limits, we can tentatively rule out particular models. For example, a particular tensor–vector–scalar model1 predicts that EG = 0.22, which is lower than the observed value by more than 2.5σ. Whether this result rules out the entire class of tensor–vector–scalar models is an open issue,…”

    So they do not exclude all classes of TeVeS theories. In fact, in the Conclusions of our paper we note that the particular version of TeVeS tested by Reyes et al. was already known to be unstable and thus not to be a viable model.

    Interesting for you may be the following paper which claims to rule out LCDM with 99 per cent confidence based on the way galaxies are distributed in galaxy clusters:

    The Misalignments between Matter and Galaxy Distributions in Triaxial Clusters: A Signature of a Possible Fifth Force?
    Authors: Jounghun Lee (Seoul National University)
    submitted to ApJL

    We need to wait to see if it holds up to the refereeing process.

    *** Concerning your point (2), the Boyarsky, A; Neronov, A; Ruchayskiy, O, et al. paper is only concerned with universal properties of dark-matter halos. These are challenging for the LCDM theory, as is discussed in that paper, and may be obtained via secondary infall of dark matter. A final, convincing conclusion is not provided.

    In their conclusions they write
    “The universal S–M scaling in the DM halos,
    found in the observational data and in pure DM N-body simulations
    can be analytically understood in the secondary infall
    model. It seems to be insensitive to the presence of baryons
    and to the details of DMdensity distributions. This shows that
    this relation has pure DM origin (in contrast with its interpretation
    in favor ofModified Newtonian Dynamics, as discussed
    e.g. in [8]).”

    Now, this statement appears to be in total disagreement with the other observational finding that baryons and dark matter are perfectly coupled within the dark-matter core radius:

    In our paper we write in Section 6.4.1:
    “Notably, the realisation (Gentile et al. 2009; Milgrom 2009a) that the ratio of DMmass to baryonic mass within the DM core radius is constant despite the large variation in the DM–to–baryonic-matter ratio globally
    within galaxies cannot be understood within the DM hypothesis.”

    Thus, the failure of the LCDM model is not amended by the Boyarsky et al. paper you mention, and in fact, the Boyarsky et al. work appears to be in conflict with the baryon–dark-matter coupling observed in galaxies.

    *** Concerning your point (3) on the Wadepuhl & Springel (2010) contribution:

    Their luminosity–mass relation (their fig.17) also fails (see Section 2 in our paper where many other models are discussed). That is, the theoretical dark-matter-mass–luminosity relation of satellite galaxies has a slope, as expected from elementary energy arguments (more massive dark matter halos can hold more gas and form more stars), while the observations do not show such a slope. That is, the real satellite galaxies do not behave as they should if they had their own dark matter halos. The dark-matter approach therewith fails.

    Furthermore, their promise in the abstract that their models fit the spatial distribution of the satellites is not shown to be true in the text (I was not able to find any statements on the 3D distribution of the satellites).

    This point raises a worrying issue concerning how research results are being dissiminated by parts of the LCDM community. Another example of the way the abstract of a LCDM paper suggests that a major problem has been solved, whereby the text of the paper demonstrates this not to be the case, is the following:

    Libeskind et al. (2009, MNRAS)
    suggest in their abstract that 30 per cent of Milky-Way-type galaxies have a disk-of-satellite type alignment as is observed around the Milky Way. This is being quoted, by the LCDM community, as having solved the disk-of-satellites issue.

    Reading their paper it turns out however, that a given dark matter halo mass which is similar to that of the Milky Way has a chance of about 0.5 per cent of having such a disk-of-satellite system (see Section 5.4 in our paper). Together with the fact that our neighbour galaxy, Andromeda, is very similar to the Milky Way, this basically excludes these models as viable representations of the Local Group, in stark contrast to the statement in the abstract.

    Finally, the results of Libeskind et al. (2009) are not revoked by the Wadepuhl & Springel paper (our Section 5.5). Of course, the Libeskind et al. results on the spatial distribution of satellite galaxies cannot be affected by any additional study, since the excellent Libeskind et al. work is a consequence of LCDM.

    So the failure of LCDM is not mended.

    I hope this helps a little to clarify some of the issues. On a personal note: As I stressed in the first contribution in our series on SciLogs, I was all for dark matter when I entered the field of galactic dynamics. But, the excellent large body of computational and observational work that has accumulated since 15 years has now quite clearly shown that the hypothesis of a cold or warm dark matter dominated Einsteinian universe does not hold.

    Pavel Kroupa

  3. I. The Fritz Zwicky Paradox: the fourth failure of Newton (MOND, MOG, etc)Newtonian physics alone might be enough to explain dark matter, without the need for MOND or TeVeS.
    But the speed of light has to be taken into account for the speed of gravity. Dark matter can in that way be explained in an “ordinary” fassion.
    See for more details:
    If someone wants to comment, please respond.
    Fritz Zwicky and Ernst Mach were two outstanding German physicists, with a sharp eye for the unknown.

  4. Three big questions …There are some at first sight very different phenomen that are still not understood and there is a chance, that they are interconnected. The first one is Gravity: We still don’t understand its nature. This is especially true in Quantum Mechanics, but obviously also not really understood on large scales. The second one is the measurement process in QM. Many scientists suspect that gravity directly contributes to decoherence but how exactly is an open question. The third one is intelligent behaviour related to a concept of consciousness. There are reasonable papers by Penrose, Tegmark and others relating this to long standing superpositions within a brain thus looking at decisions as a result of a measurement process of some sort.
    I personally researched for a reasonable model of intelligent behavior. I had in mind Penrose analysis in “The emperors new mind” where he tried to prove that classical computer architecture never would be able to produce real intelligent behavior. I experimented with self referencing mathematical structures (basically matrices viewed as operators and as vectors). These investigations came out with swarms of stateless software agents providing some amazing features. The swarm simulation exactly resembles a measurement process for a spin 1/2 particle. The underlying stochastic process at the same time shows dynamics that basically are a random walk in a curved spacetime.
    So what does that mean for physics? Maybe nothing – but if I apply the findings to physics there are some implications to our understanding of nature overall:
    1) Gravity is not a QM effect. Stop looking for Higgs bosons and the graviton. It governs the internals of a measurement process. This means all the dynamics observed within our universe cannot be described by a wave function evolving according to a Schroedinger equation. It actually evolves according to a measurement process.
    2) GR can only be understood as a statistical model that works fine near (whatever “near” means) masses but not in an nearly empty space.
    There is a paper available describing the maths behind this model at http://bzus.de/dual_reality.html .

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