fredag 31 juli 2009

The Importance of Pretending to Understand

The expertize of a concert pianist or tennis player is very clearly demonstrated to the public at each performance, and the public can on good grounds express appreciation by applause. 

But the general public cannot directly judge the expertize of a scientist or a priest, because the nature and depth of the expertize is beyond comprehension for non-specialists. The Nobel Prize ceremony illustrates the difficiulty: Scientific priests of lower rank convey a message to the Royal family and the general public in presentation speeches, that they truely understand the remarkable scientific achievements of the Laurates, while it is understood that nobody outside the inner circle can understand. It is a spectacle which works under the assumption that some few understand something which is hidden to the many. 
A theoretical a subject without obvious practical results is difficult to judge, since understanding of the theory is required, which however is restricted to the inner circle. 

But pretending to understand can be a substitute for real understanding. From outside, it can be difficult to judge if someone really understands a theory without practical conseuqences, or just pretends to understand. It is like judging if a priest claiming to believe, really believes or just pretends to believe.

The theoretical subjects of theoretical physics and pure mathematics are particularly prone to pretending. Theoretical physicists (have to) pretend to understand
pure mathematicians (have to) pretend to understand
and theoretical fluid dynamicists pretend to understand  
while understanding of these subjects is not possible for the general public. 

The highest priests of theoretical physics can admit that they don't understand, which is to be taken as a sign of an even deeper understanding. Compare with Science vs Religion and Presentation Speech Nobel Prize in Physics 2004: Asymptotic Freedom
  • Nature has chosen the only theory we have found to be possible.
Like Newton, modern physicists dictate Nature, not the other way around.

How do you show that you understand something? By nodding your head! If a group of scientists collectively nod their heads, a scientific truth has been established.

The education in mathematics and theoretical physics/mechanics aims at implanting a strict codex of basic beliefs, which is also the goal of religious schools, and there is little room for critical independent thinking. While practice can harbor multivalued truth, it seems that theoretical science and much of religion only can accept the one truth preached by the high priests. If you cannot make yourself pretend to understand, then you are in trouble, and maybe you choose to study something else than mathematics and physics...or religion...

torsdag 30 juli 2009

Science or PseudoScience?

According to Karl Popper, astrology definitely represents pseudoscience, faked science, while marxism and psychoanalysis belong to the risk-zone.  The basic principle of astrology is:
The relative position of celestial bodies influences your life.

What makes astrology into pseudoscience is that a cause-effect description of the influence of the positions of celestial bodies on your fortune or misfortune, is lacking. It could be that there is such an influence, but what is required is a mechanism for the action and as long as this mechanism is lacking, astrology will remain pseudoscience.

Let us compare with the Boundary Layer Theory BLT by Ludwig Prandtl, the father of modern fluid mechanics. BLT states that the drag and lift of a body moving through a fluid with very small viscosity, such as air and water, is influenced by a very thin boundary layer close to the surface of the body, where the flow velocity changes quickly from zero on the boundary to the free stream value just outside. In short BLT states:
  • A very thin boundary layer influences drag and lift in a fluid with very small viscosity.
Is BLT science or pseudoscience? What is the mechanism of how the boundary layer influences drag/lift?  BLT claims that the mechanism is generation of vorticity from strong shear in the boundary layer, which somehow is transported into the fluid and then causes drag/lift. The basic argument is that even if the boundary layer gets thinner with decreasing viscosity, the total vorticity from shear does not decrease to zero and thus possibly can influence drag/lift. 

But BLT does not describe a mechanism of how the boundary layer vorticity can get into the fluid and influence drag/lift. 

Similarly, astrology does not describe a mechanism of how a far away celestial object can influence your life.

BLT and astrology both involve a large effect from a small cause, but a corresponding mechanism is unstable and thus delicate to work with. It is like saying that anything can happen from nothing, which is not science; science requires stable mechanisms with substantial effects from substantial causes.

If you claim that you have a theory which allows you to predict the outcome of a roulette
or single throw of a dice, possibly based on the position of Pluto or state of a very thin boundary layer, then you have a problematic probably pseudoscientific theory. Compare with the following confession by K. Stewartson in his survey article on D'Alembert's paradox:
  • ...great efforts have been made during the last hundred or so years to explain how a vanishingly small frictional force can have a significant effect on the flow properties.
We conclude that BLT seems to be more pseudoscience than science, a conclusion supported by the survey article Laminar Boundary Layer Theory: A 20th Century Paradox?

Let's subject some other theories to a quick test as concerns the presence of a stable cause-effect mechanism:
  • planetary system: mechanism: gravitational force: science
  • acoustics: mechanism: wave propagation in material medium: science
  • light: mechanism: electromagnetic wave propagation in vacuum: science
  • evolution: mechanism: genetic variation + selection: science, possibly incorrect
  • gravitation: mechanism: unknown: pseudoscience
  • quantum mechanics: mechanism: electrostatic repulsion/attraction: science
  • string theory: mechanism: unknown: pseudoscience
  • special relativity: mechanism for spacetime dilation: unknown; pseudoscience
  • general relativity: mechanism for spacetime warp: unknown: pseudoscience
  • circulation theory of lift: mechanism for circulation: unknown: pseudoscience
where we consider a theory without stable cause-effect mechanism as pseudoscience.

The purpose of a scientific theory is to make predictions of effects of certain causes through certain mechanisms, while the mechanisms behind the causes can be left without explanation.
Thus the motion of a planetary system or a space rocket, can be predicted in a scientific theory based on Newton's law of gravitation, which itself is left without explanation.

The correctness of a scientific theory including a stable cause-effect mechanism, like Darwin's evolution theory, depends on the correctness of the mechanism. It may be that genetic variation + selection, is not the mechanism of evolution. But intelligent design is pseudoscience as long as no design mechanism is presented.

In the hierarchy of sciences, theoretical physics/mathematics is viewed to be the foundation carrying the different levels of science and engineering: physics-mechanics-chemistry-biology-....cosmology. Each level above the foundation contributes a specific cause-effect mechanism
starting from the previous level, while it seems that the foundation has to come out of the blue, so to speak.

onsdag 29 juli 2009

Simulation of the Great Ocean Conveyor by CFD?

The “pump” in question is in the western North Atlantic Ocean, where pools of cold, dense water form in winter and sink beneath less-dense warmer waters. The sinking water feeds into the lower limb of a global system of currents of the Great Ocean Conveyor. To replace the down-flowing water, warm surface waters from the tropics are pulled northward along the Conveyor’s upper limb.

What will be the future of the Conveyor under different scenarios of global warming? In principle we should be able to get the answer by Computational Fluid Dynamics CFD, but what is state-of-the-art of CFD? Can you simulate the Conveyor by CFD, today or in 50 years? Compare with Global Climate Modeling.

Would Europeans Freeze Without the Gulf Stream?

People in Britain and Northern Europe commonly believe that a relatively comfortable life is possible because of the Gulf Stream. which brings warm water from Sargasso causing the temperature to be 15-20 degrees warmer than in in eastern North America/Alaska.

Some fear the fresh water from melting ice in the Arctic can slow down the Gulf Stream or even reverse its direction, thus causing a form ice age in Northern Europe.

What does science tell? Richard Seager at Columbia University reports:
  • The notion that the Gulf Stream is responsible for keeping Europe  anomalously warm turns out to be a myth.
  • Fifty percent of the winter temperature difference across the North Atlantic is caused by the eastward atmospheric transport of heat released by the ocean that was absorbed and stored in the summer.
  • Fifty percent is caused by the stationary waves of the atmospheric flow.
  • The ocean heat transport contributes a small warming across the basin.
  • The seasonal ocean heat storage and pattern of atmospheric heat transport add up to make winters in western Europe 15 to 20 degrees C warmer than those in eastern North America. A very similar process occurs across the Pacific Ocean. The ocean heat transport warms the North Atlantic Ocean and the land on both sides by a modest few degrees C. The only place where the ocean heat transport fundamentally alters climate is along the coast of northern Norway which would be sea ice-covered were it not for the warm northward flowing Norwegian Current. 
  • Any cooling tendency would probably be overwhelmed by the direct radiatively-driven warming by rising greenhouse gases
So, except for northern Norway, we don't have to freeze if the Gulf Stream slows down and even changes direction by freshening of the waters caused by anthropogenic climate change via melting glaciers and increased water vapor transport into high latitudes.

But what is the truth? Can Computational Fluid Dynamics CFD help to give an answer? Now or in 50 years? What is state-of-the-art of CFD? According to Journal of Fluid Mechanics, the leading scientific journal? Compare with Gulf Stream Shut Down and Abrupt Climate Change.

Global Climate Modeling: Fate of Humanity?

Policies to control climate change must be based on global climate models which are mathematical models of the circulation of the atmosphere and the ocean of our planet in the form of the Navier-Stokes equations of fluid flow plus thermodynamics with various energy sources, referred to as General Circulation Models GCM.

Recently G8-leaders have decided to put a limit to global warming of 2C.

The key question is the reliability/accuracy GCM. Since GCM is based on computational solution of the Navier-Stokes equations, also referred to as Computational Fluid Dynamics CFD, it is of interest to investigate what is state-of-the-art of CFD.

The areodynamics of flight is governed by the Navier-Stokes equations, and one may ask if today CFD allows computational modeling of the take-off of a jumbo-jet, for example. The answer is a disappointing NO: We have to wait 50 years before computers are powerful enough! Says the state-of-the-art expert Parviz Moin.

Of course you ask: If we cannot computationally model the flight of an airplane, how can we expect to model the immensely much more complicated system of the the global climate?

Or maybe more is possible than what state-of-the-art says, see Fear of Flying and  Why It Is Possible to Fly.

Maybe it is possible to predict global climate with better Navier-Stokes solvers, which can be used today on existing computers. Waiting 50 years for the predictions may not be very meaningful, like waiting one week to get the prediction of tomorrows weather.

As concerns the state-of-the-art of CFD and fluid mechanics in general, see the Interview with Editors of Journal of Fluid Mechanics JFM.

Since the fate of humanity critically depends on the reliability and accuracy of computational global climate modeling, it would seem that no efforts should be spared to bring forward the best possible competence in this area. It does not seem to be represented by JFM, which poses a serious problem since JFM is supposed to be the leading journal of fluid mechanics. What can be done? But who cares? Do you?

tisdag 28 juli 2009

Theory vs Practice: Science as Religion

In science theory and practice/reality should ideally come together into a synthesis with theory representing understanding of reality. However, modern theoretical physicists work intensely with concepts such as: 

  • many-dimensional wave function (quantum mechanics
  • curved spacetime (relativity theory) 
  • 10-dimensional space (string theory) 
which defy physical reality. Similarly, modern mathematicians work intensely with concepts such as 
  • infinitely small 
  • infinitely large 
  • infinite set
  • countable/uncountable infinity
which again defy reality.  The role of the professional theoretical physicist/pure mathematicians is then to claim to understand and work with these concepts, which are beyond grasp for non-professionals. This is analogous to the the religious postulate stating that only those who properly believe, the high priests, can understand the reasoning. 

Once theory has cut the ties to reality, the field is open to a Theory of Everything or to transfinite number theory. The lack of contact with reality can then be compensated by inflating the scope of the theory, which now is without bound.  This is like boosting practical religion about how to live and what to eat, by concepts such as Eternal Life, Heaven, Hell, God and Creation of the World.

The difference between science and religion of the Enlightenment thus steadily diminished during the 20th century and today science based on theoretical mathematics/physics can serve as the religion of our time.

What is crucial is then the dimension of the theory, not its actual content, construction and credibility. In this paradigm, string theory of the infinitely small meets the cosmology of the infinitely large in Big Bang theory  illustrated in the above picture, which surpasses all religious scenarios in precision and fantastery, stating that 
  • Approximately 10−37seconds into the expansion, a phase transition caused a cosmic inflation, during which the universe grew exponentiallyAfter about 10−11 seconds, the picture becomes less speculative, since particle energies drop to values that can be attained in particle physics experiments. At about 10−6 seconds, quarks and gluons combined to form baryons such as protons and neutrons.

Read and think!

Why Are There So Few Science Blogs?

The blogosphere is quickly inflating, like the early Universe according to Big Bang theory, but few scientists write blogs, in particular few mathematicians and physicists, not to speak of applied mathematicians and fluid dynamicists, who are indeed very few. Nature reports that out of 46.7 million blogs indexed by the Technorati blog search engine, five scientists' sites make it into the top 3,500. 

Among the few in basic science, mostly represented by pure mathematicians and theoretical physicists, you find prestigious Nobel Laureates and Fields medalists insiders such as: 
and outsiders such as
but the large number of scientists between inside and outside do not write blogs. Why? Does it tell us something about how science is organized and performed? 

Yes, it seems to be a result of the strong hierarchy in science, where independent thinking either propels you to a top Prize position or to an outsider position. The large bulk of normal science in the terminology of Kuhn performed by normal scientists, cannot harbour too much of independent thinking, and thus normal scientists are not encouraged to write blogs, except possibly very normal blogs. 

On the other hand, authors and writers have an incentive to write blogs and also do, since for them expressing independent thinking is most essential.

The autocracy of science is illustrated in the Interview with Editors of Journal of Fluid Mechanics about the state-of-the-art concerning the basic problem of d'Alembert's paradox, where nobody from the large body of Associate Editors dares to express an opinion, only one of the two Editors who has the role of expressing the dogma of normal science, in this case that the paradox is not a paradox.

Pure mathematicians and theoretical physicists have found a common interest in string theory, which is both incomprehensible and mind-boggling and thus perhaps more attractive for mind-blogging than e.g. solid mechanics.

måndag 27 juli 2009

Fear of Flying?

Are you afraid of flying? If so, you should not read the following, except the very last paragraph. If you are not afraid of flying, and want to know the whole thruth, you should read the following and see if there is any need to change your mind.

Summing up my blog-poll with aerodynamics experts on why it is possible to fly and sail, or how a wing generates lift at small drag, I find that no convincing theory is presented. Either the expert flatly refuses to say anything whatsoever about the basic question of the expert's area of expertize, which is common but a bit disappointing as an expression of expertize, or the expert including NASA presents nonsense-theory such as
  • lift comes from the pressure distribution around the wing
  • lift comes from downwash
  • lift comes from Newton's law 
  • lift comes form Bernuoilli's law
or refers to a "mathematical" circulation theory stating that
  • lift comes from large scale circulation around the wing
  • lift comes from wing-tip vortices
  • lift comes from a Prandtl lifting line including a starting vortex.
The circulation theory for lift of a wing was suggested by Kutta-Zhukovsky in the beginning of the 20th century as an analog to an earlier circulation theory for the Magnus effect causing a top-spin tennis ball to curve down. 

However, according to state-of-the-art fluid dynamics, the Magnus effect results from unsymmetric separation and not from large scale circulation. Likewise, circulation is not the origin of lift of a wing. The failing belief in circulation theory is expressed by the fact that in state-of-the-art it is described as a "mathematical" theory, which means that it is an un-physical theory, which does not correctly describe physics; it is only a "mathematical trick" without physical meaning, see The Spell of Kutta-Zhukovsky's Circulation Theory.

The net result is that according to experts there is no physical theory of flying. Surprised? How do you feel now? No fear of flying even if experts don't know what keeps an airplane in the air?

Should I then cancel my next flight, you ask? No, you don't have to. Take a look at Why It Is Possible to Fly and you will discover an understandable new correct physical theory of lift and drag of a wing. You can take it as medication against fear of flying! 

Turtle Theory

The Turtles All the Way Down Theory states that the Earth rests on turtles upon turtles, which however are invisible and thus in no way can be detected. 

The Turtle Theory is experimentally verified by the observation that the Earth does not fall down, as illustrated by the picture.

Convinced? If not, then you may have some difficulty of buying modern physics in the form of of relativity theory, quantum mechanics, the standard model and string theory:

  • relativity theory: curved spacetime root of gravitation
  • experiment: perihelion of Mercury
  • quantum mechanics: smallest quantum of energy
  • experiment: black-body radiation
  • standard model: protons and neutrons made of quarks
  • experiment: atomic kernels made of protons and neutrons 
  • string theory: quarks made of tiny vibrating strings
  • experiment:  not available - no observation of quarks.
The experimental evidence is here indirect/missing: curved spacetime, quantum of energy, quarks and strings are not directly observable, just as invisible turtles are not directly observable. 

The epistemology of modern physics characterized by indirect experimental evidence was laid in a fierce debate between Mach and Boltzmann about matter as atoms and molecules in the late 19th century, before the existence of atoms and molecules was experimentally verified. 

Boltzmann developed a statistical mechanics gas dynamics theory based on an assumption of a gas as a large collection of little elastic spheres representing hypothetical atoms or molecules assumed to have statistically independent velocities before collision. Mach objected claiming that the theory was empty as long as the basic assumption of statistically independent 
elastic collision of atoms/molecules, was not experimentally verified. 

Bolzmann first vigorously defended his theory, despite its lack of direct experimental support, but eventually Mach's criticism was effective: Boltzmann came to doubt his own theory to the degree of taking his own life in 1906.  Shortly thereafter the existence of atoms was experimentally verified by Rutherford, which ended also Mach's career, even if Boltzmann's assumption of statistical independence before collision, was not experimentally verified. 

The curtain of the tragic opera of classical physics thus closed with everybody left dead on the scene, and out of the turmoil modern physics was born with a new modern epistemology: 
  • Direct experimental evidence of the basic axioms of a theory is not necessary.
  • Indirect evidence is enough
motivated by the following argument:
  • See how wrong Mach was! 
  • Atoms and molecules do exist! 
This opened the door to modern physics allowing you:
  • to assume any basic laws for quarks, strings, turtles...
  • escape experimental verification of basic laws,
  • consider indirect experimental evidence to be enough.
But there is a weak point in the argument: OK, atoms and molecules apparently do exist, in one form or the other, and thus Mach's criticism missed the point. But this does not imply that all forms of Machian criticism can be refuted: The key question is still if quarks, strings and invisible turtles do exist or not. 


fredag 24 juli 2009

The Beginning of the End of Physics

Modern physics and fluid mechanics was born out of a crisis of classical physics at the turn into the 20th century caused by the seemingly unsolvable problems: 

  1. Second Law of Thermodynamics.
  2. Cut-off of high-frequency spectrum of black-body radiation.
  3. Michelson-Morley experiment showing observer-independent speed of light.
  4. Nature of gravitation.
  5. D'Alembert's paradox of non-zero drag in fluids with very small viscosity.
The problems were "solved" by introducing new non-classical basic laws:
  1. Boltzmann: Molecular chaos.
  2. Planck:  Smallest quantum of energy.
  3. Einstein: Special relativity based on Lorentz transformation of space-time coordinates.
  4. Einstein: General relativity based on curved spacetime and equivalence of heavy and inertial mass.
  5. Prandtl: Substantial effects from vanishingly thin boundary layers.
The nature of these basic laws is "modern" in the sense that they cannot be verified experimentally. Classical basic laws such as 
  • Hooke's law of elasticity
  • Newton's law of gravitation
  • Newton's law of viscous stress
  • Fourier's law of heat conduction
  • Coulomb's, Faraday's and Ampere's law of electromagnetics,
can all be tested experimentally. But the new laws cannot, because they are either 
  • referring to experimentally untestable microscopics (molecular chaos, quantum of energy, vanishing viscosity) 
  • simply definitions (Lorentz transformation, equivalence of heavy and inertial mass).
But basic laws of physics based on microscopics or definitions which cannot be tested experimentally, are laws of pseudo-science. This is like playing poker without ever being called; it is not a fair game.

The price to take classical physics out of its crisis was high, for science and for the scientists, who like Faust sold their souls: Boltzman hanged himself, Einstein became depressed and lonely, and Planck and Prandtl had to carry the glory of German science through two World Wars, as illustrated in The Desperation of Planck.

The ultimate microscopics of physics is represented by string theory dominating contemporary physics, based on basic laws way way beyond any possibility of direct experimental verification.

Interview with Glenn Research Center

Interview with Glenn Research Center.

CJ: On your aeronautics educational site Beginners Guide to Aerodynamics you present three incorrect theories for lift of a wing, but no theory claimed to be correct. Is the correct theory classified, or is GRC not aware of any correct theory? 

GRCHi Claes: The correct theory of lift is fairly complex.  We mention on the page about Newton and Bernoulli that the real of theory of lift is included in the work of Bernoulli's student, Euler:

The generation of lift occurs because of the pressure variation around the airfoil. When you integrate the pressure times the area times the local normal base vector around the entire surface, you obtain a single aerodynamic force. The component of that force perpendicular to the flight direction is the lift.

Now why does the pressure vary? Because the velocity around the airfoil varies and pressure and velocity are related along a streamline by Bernoulli's equation.

Why does the velocity vary? Because the flow can't go through the surface of the airfoil. In an ideal flow situation, the flow is tangent to the surface of the airfoil. In reality, the flow at the surface sticks to the surface and the velocity is zero, but the external flow reacts to the edge of the boundary layer from the surface to free stream. The free stream flow must simultaneously conserve mass, momentum and energy (that's where the Euler equations come in)

Solutions of the Euler equations introduce the theory of bound vorticity within a lifting airfoil

Extensions of the bound vorticity theory to three dimensions leads to the Prandtl lifting line theory and the vortex lattice method.

None of this is classified; it is just complex. Undergraduate aero engineering students learn all of this and how to derive the necessary equations that describe flow around an airfoil.

Tom Benson, Editor.

CJ: Thanks for the information. Saying that the correct theory is complex, is another way of saying that GRC/you does/do not know how a wing generates lift, only that it does, right?

GRC/TB: No ... saying that the answer is complex means just that ... it is complex, it isn't simple. But we know what it is.

Many people look for simple answers to questions, when the answer may not be simple. When the answer is really complex, people make simplifying assumptions so that they can get a simple answer. Unfortunately, with fluid mechanics, when you make simplifying assumptions you can get the wrong answer.  That's what has happened with the incorrect theories and their inability to produce meaningful results.
Fluid mechanics comes down to satisfying the conservation laws for mass, momentum and energy. That's what the Euler equations (and the more complete Navier-Stokes equations) express. There are some limited solutions to the Euler equations and the bound vortex / Prandtl lifting line theory are some of them. These solutions correctly predict observed flow phenomenon.

CJ: I hear you say that you know, but I don't think you do. If you want to know, take a look at Why It Is Possible to Fly showing in particular that Kutta-Zhukovsky's circulation theory for lift, which you refer to, is un-physical: An airplane does not take off by shedding transversal vorticity as indicated in GRCs picture. This is a common misconception which GRC unfortunately contributes to transmit to the young generation. Often a 2d bathtub foam surface experiment is used to support the circulation theory, but this is misleading since an airplane wing acts in 3d. In fact, 2d flight is impossible. There are no pictures of shed transversal vorticity from airplane wings, because they don't exist. 3d streamwise vorticity: Yes! 2d transversal vorticity: No! Right?

Interview with Ulf Ringertz: Flightdynamicist KTH

Interview with Prof. Ulf Ringertz, Head of Division of Flight Dynamics KTH

CJ: How is lift generated by a wing?

UR: I refrain, from giving any answer.

CJ: Does this mean that you don't know?

UR: ???

Interview with Swedish Defense Research Agency

Interview with Swedish Defence Research Agency, one of Europe’s leading research institutes in the defence and security area.

CJ: Which theory for the generation of lift and drag of a wing is by the Swedish Defence Research Agency considered to be correct?

SDRA: Sorry, we have nothing to say.

torsdag 23 juli 2009

Interview with Bryon D. Anderson: Sailing Expert

CJ: In your book and talk you refer to classical wing theory including Bernouilli's principle, downwash and wing tip vortices, as the explanation of the generation of lift of a sail. Is this correct?

BDA: The physical origin of lift for an airplane wing or a sail or a keel has been discussed for about 100 years now.  What I presented in my short book on the physics of sailing is "classical" lift theory.  The basic physical understanding is hard to arrive at.  I refer people to the excellent book by Ross Garrett entitled, "The Symmetry of Sailing" for a detailed attempt to do this.  In Chapter 3 Garrett outlines three ways for understanding lift.  First is the "flow line method," which describes classical lift theory and arrives at Bernoulli's principle applied to a foil.  Garrett's second way, "momentum change," emphasizes that macroscopically a foil must have the net effect of deflecting the fluid flow in order to derive lift.  That is obvious, but it must be appreciated. His third way to understand lift is the "mathematical approach," which introduces circulation, using several fluid flow theorems leading to the Kutta-Joukowski theorem. This approach is what engineers use to calculate lift; but it does not provide a clear physical description of lift. A good website discussing lift for a wing (or sail or keel) is provided by A. Gentry, The Origins of Lift.

CJ: I get your point. But isn't it difficult to explain the Physics of Sailing, when "the basic physical understanding is hard to arrive at"?

BDA: What I tried to do in my book entitled "The Physics of Sailing Explained" was to talk about how sailboats work including the limitation of hull speed, the basic kinds of resistance, the basic principles of how sails and keels work, and how all this comes together to enable a sailboat to move through water, including moving up to within 30 to 45 degrees off the direction of the wind.  The detailed description of lift is both beyond what I was trying to do there and is something many existent books already discuss in more appropriate detail.  This is also a topic of research and discussion in great detail for airplane wings of course, and even more attention has been applied to this question in that context. See, for example, the text "Foundations of Aerodynamics," by Kuethe and Chow.  I believe my book works well at pointing out and explaining a number of important physical considerations for sailing and seems to have been well received in that way.

I have found that a number of persons have very strong opinions on how sails work, including those who feel that it is all "Newtonian Mechanics" and is just due to air molecules bouncing off the sails.  This effect is certainly involved, but saying that it is only this ignores the fact that air is a fluid with interactions between the molecules (called Van der Waals forces).  If one can reduce the pressure on one side of the sail, the higher pressure on the other side, without any wind needed, will cause there to be a force exerted on the sail toward the low pressure side.  Because a moving fluid produces a drop in pressure, as measured experimentally and explained by Bernoulli, causing the air to move faster over one side of the sail will produce this kind of difference in pressure.  

For a sailboat moving directly downwind, the Bernoulli effect is small and most of the driving force comes simply from air molecules hitting the back of the sail.  As the direction of sail moves more into the wind, the Bernoulli effect becomes larger and eventually dominates.  As Fig. 3.2 in my book shows, for moving at an angle forward of 90 degrees into the wind direction, the drop in pressure on the leeward (downwind) side of the sail is much greater than the rise in pressure on the windward side of the sail. 

Persons advocating the "Newtonian" approach will note that airplanes can sail upside down, even with asymmetrical wing shapes; however, they do this by flying at a bit of an angle, called the angle of attack, which has the effect of still producing a longer path over the "up" side of the wing and leading to the Bernoulli effect again. (See my description of how this works for symmetrical keels in water flow.)  Clearly, as speed increases, say to jet plane speeds, it does not need much of an angle of attack and the wings become more symmetrical in cross section (just as keels are).  Early descriptions of air flow around a wing assumed that the flow over the curved top, with the longer path had to arrive at the back of the wing at the same time as the flow under the bottom of the wing, with a shorter path.  Since it is a longer path over the top, the air flow had to be faster to arrive at the same time as the flow beneath the wing.  Wind tunnel demonstrations with "smoke" in the air clearly shows that the air flow over the top is faster, but that it still does not arrive at the back of the wing at the same time as the air flow under the bottom; in fact the difference in time leads to the idea of "circulation" of air flow around the wing.

Putting all of this together, in order for a wing (or sail or keel) to provide lift, there must be a net deflection of the air downward behind the wing; it is then the reaction force to this downward deflection of the air that is the lift provided to the wing.  This downwash is seen in the photo of Fig. 2.8 in my book. The reaction force is applied to the wing through the pressure increase on the bottom and the pressure decrease on the top.  The downward deflection of the air and the reaction force on the wing is a "macroscopic" description, while the pressure differentials on the wing is a more "microscopic" description of how the reaction force is applied.

CJ: Thanks for detailed account of your standpoint. You connect lift to reaction from downwash, which is obvious, but you don't explain why there is downwash, right? 

BDA: Downwash occurs from the redirection of the air flow past the wing (or sail or keel).  In usual wing theory (see again the text by (Kuethe and Chow, Chaps 4-6) this is explained in terms of the circulation of fluid flow around the wing.  Because the air flows faster over the top and slower past the bottom, there is a net circulation of air around the wing.  This is called the "bound vortex."  (Bound in the sense that it stays with the wing.  When the wing starts moving, a reverse vortex is formed called the Starting vortex.)  The circulation of the bound vortex provides the net downward flow of air from the wing which we call the downwash.  

Now again, those who advocate the Newtonian approach will say this is just due to the air molecules bouncing off the bottom of the wing; however, a downwash occurs even if the wing is moving through the air with the flat bottom exactly parallel to the oncoming air.  The air moving over the curved top will move faster and create the circulation to provide the downwash.  Now, at the same time, one can talk about the air following the shape of the wing because it is a viscous fluid, with interactions between the air and the wing and between the air molecules themselves.  These forces (the Van der Waals forces again) will cause the air to follow the shape of the wing, resulting in a downward flow from the back of the wing, all in a way consistent with the circulation theory.  If the angle of attack is made too steep, or there are too many obstructions on the wing surface, the flow will leave the surface of the wing (it is said to "separate") and the circulation pattern, and hence the downwash, will be reduced and the lift will decrease.  The overall downwash pattern is affected also by the tip vortices at the end of the wing (or sail or keel) reducing the lift and this is called "induced drag," and is discussed also in my book. 

CJ: OK, so you say that lift comes from downwash, which comes from circulation, right? From where does then circulation come? Compare with Why It Is Possible to Sail.

BDA: I indicated in my last response how circulation arises. Your website shows that you have thought a lot about how wings actually work.  It will take me awhile to digest all that you present there.

Interview with WB-Sails: Quality from Design

Interview with WB-Sails.

CJ: On your web site you state "In the last few years, advances in CFD (Computer Fluid Dynamics) and FSI (Fluid Structure Interaction) have changed the way we perceive sail aerodynamics. Old beliefs are proven wrong and new features found."
What old beliefs are wrong and what new features are found, in short?

WB-Sails/Mikko Brummer: These are listed on WB-Sails website. In short
  • The mast is not just a drag-device: it can contribute significantly to the  driving force of the sails.
  • On the windward side of the sails, flow separation occurs rather  through luff separation vortices than bubbles.
  • The turbulence always present in natural wind may lead to different performance on different days & weather.
  • The presence of the hull effects the flow around sails, so it needs to be included in the simulation.
  • On leeward side of the main, the leech vortex from the head of the jib "rips" the flow loose from the mainsail surface, causing a conical separation area on the main at the hounds level.
  • On the leeward side of the jib, separation propagates upwards from the tack, as the foot vortex bursts from the deck.
CJ: Thanks for interesting information. Would you say that you now can realistically simulate the action of sails + mast + hull computationally? If not, what computational simulations would be of interest to you?

MB: All computational simulations around a sailboat would be very much of interest to me ;-). Hard to say how realistic or accurate our simulations are... there are plenty of issues in collecting experimental data from sailing, too, both in the windtunnel & in the real world, so we hardly know exactly what we are comparing against. While the inviscid methods (vortex lattice/panel methods coupled with BL theory) could provide very similar forces to those in windtunnel tests (for upwind cases), the recent N-S solvers certainly have brought qualitative flow prediction to a new level of realism, when compared to windtunnel tests or real world observations (with tell tails etc). On the quantitative (forces) level, N-S results may differ from windtunnel, but there are issues with windtunnels too: turbulence level, inflow twist & shear, R-number, model geometry etc.

CJ: Interesting!

onsdag 22 juli 2009

Interview with Arvel Gentry: Sailing Expert

Interview with Arvel Gentry author of several technical papers on the theory of sailing.

CJ: In your technical papers you present classical circulation theory as the explanation of the generation of lift of a sail. Is this correct?


tisdag 21 juli 2009

Interview with John D. Anderson: Flight Expert

Interview with John D. Anderson, author of several books on aerodynamics and flight including Introduction to Flight, now Curator of Aerodynamics at the National Air and Space Museum.

CJ: In your book Introduction to Flight you state on page 252:  
  • As Curator of Aerodynamics at the Smithonian's National Air Space Museum the author is frequently asked by visitors how a wing produces lift - a natural and perfectly innocent question. Unfortunately there is no satisfactory one-liner for an answer. Even a single paragraph does not suffice. After a hundred years since the Wright Flyer, different people take different points of view about what is the most fundamental mechanism that produces lift, some pressing their views with almost religious fervor.
Since no scientific consensus is reached, does it mean that nobody including yourself really knows how lift is produced by a wing? A natural and perfectly innocent question, right?


Interview with Anderson-Eberhardt: Flight Experts

Interview with David F. Anderson and Scott Eberhardt serving as scientific experts on Aeoronautics Learning Laboratory funded by NASA and on The Aviation History Online Museum with the article

: In your book Understanding Flight and the related article The Newtonian Description of Lift-Revised from February 2009, you state that what is often taught as the physical explanation of lift of a wing, is incorrect; most explanations do not even make sense. You claim that a correct explanation is that lift comes from viscosity. Could you briefly explain in more detail how lift is generated by viscosity?

DFA: Lift doesn't come from viscosity. Viscosity is a characteristic of air that makes flight possible. But then, non -viscous air would be an oxymoron. I don't want to go there. It really is complicated, the concept of zero viscosity. To say lift comes from viscosity is like saying a race car accelerates because of friction with the road. It is part of the answer but not the most important part. Let me give you some quotes from the new book:

A jet engine and a propeller produce thrust by blowing air back. A helicopter’s rotor produces lift by blowing air down. In the same way, a wing produces lift by diverting air down. A jet engine, a propeller, a helicopter’s rotor, and a wing all work by the same physics: Air is accelerated in the direction opposite the desired force.So why does the water bend around the glass or air over a wing? 

First, consider low-speed flight (subsonic). In low-speed flight, the forces on the air and the associated pressures are so low that not only is the air considered a fluid but it is also considered an incompressible fluid. This means that the volume of a mass of air remains constant and that flows of air do not separate from each other to form voids (gaps). 

A second point to understand is that streamlines communicate with each other. A streamline, in steady-state flight, can be looked at as the path of a particle in the moving air. It is the path a small, light object would take in the airflow over the wing. The communication between streamlines is an expression of pressure and viscosity. Pressure is the force per area that the air exerts on the neighboring streamline. Viscosity in a gas or liquid corresponds to friction between solids.  Think of two adjacent streamlines with different speeds. Since these streamlines have different velocities, forces between them try to speed up the slower streamline and slow down the faster streamline. The speed of air at the surface of the wing is exactly zero with respect to the surface of the wing. This is an expression of viscosity. The speed of the air increases with distance from the wing. Now imagine that the first non-zero-velocity streamline just grazes the high point of the top of the wing. If it were to go straight back initially and not follow the wing, there would be a volume of zero-velocity air between it and the wing. Forces would strip this air away from the wing, and without a streamline to replace it, the pressure would lower. This lowering of the pressure would bend the streamline until it followed the surface of the wing. 

In brief, lift is created by the wing pulling air down from above producing an upward lift. It is not by the air striking the bottom of the wing. For a well designed wing, non of the lift is produced by air striking the bottom of the wing. In fact, the pressure is slightly lowered on the bottom of the wing to prevent air striking causing high drag. 

I find it better to understand lift by looking at it in the rest frame where the air is standing still and the wing flies by. A bit of air sees the slope of the top side of the wing as a receding surface. Not unlike a piston being pulled down. This lowers the pressure and causes the air to start moving down. It chases the wing until it gets out of the way and then continues straight down. This is what happens.  Hope this is of some help.

CJ: Thanks for response. It is interesting to follow your argument of the role of viscosity for lift. Would you say that the more viscous a fluid is, the better it is for flying? Is gliding flight possible for little insects?

DFA:  Lift is possible in the limit of zero viscosity. Air is just fine. The more viscous the medium has the greater the parasite drag. Can you imagine flying through honey? Viscosity is the communication of information in the medium. If there were zero viscosity the molecules would not know about the others and they would all end up on the ground. 

I've never thought about insects gliding. Induced power goes as load squared. That is why hummingbirds can hover but not eagles. I suspect that an insect takes so little energy to fly, and distances so short,  that it is not necessary for their wings to be efficient. On the other hand monarch butterflies fly thousands of miles and have large wings. And the space shuttle "glides" with a glide ratio of 4:1. 

CJ: Thanks for valuable information. If you are interested in my own thoughts, please take a look at Why It Is Possible to Fly and related material. Any comments are wellcome.

DFA: Claes,   I started to look at your web site. But it is 5 am and I am about to leave on a trip for a week. There is too much stuff for the moment. I can see that you've collected a lot of material. I'll give the site more time when I get back next week. 

A few key points on understanding flight are: The shape of the wing has nothing to do with lift. Any explanations that rely on the shape of the wing are just wrong. The shape only effects drag and stall characteristics.  All wings have the same lift as a function of effective angle of attack inverted.  Lift requires work. A jumbo jet is diverting about its own weight in air per second. Before the plane came by the air was standing still. Afterwards there is a great deal of air with energy heading down. Energy is not conserved in real flight and so air pressure and speed are not related by the Bernoulli principle. This is not so for the infinite wings that calculations are make on. Stick an Rudder got it right when it says to forget Bernoulli. In fact, until after the WWII only Newton was taught. Then the engineers got computers and forgot how to think. Any explanation the says Bernoulli is wrong. Likewise any explanation that start out with the acceleration of the air causing the lowering of the pressure is violating Newton's first law: A body at rest will remain at rest, and a body in motion will continue in straight-line motion unless acted on by an external applied force. Lift is not a surface effect. A great deal of air from above the wing is involved. Lift on a real wing is from lowering the pressure on the top of the wing with little or no (often negative) increase in pressure on the bottom. So forget the bottom of the wing. All the action is on the top. 

So much for your morning sermon from Reverend Anderson Ciao, David 

PS. Where in the world are you located? I forget where .se is.

CJ: Great that you will take a look. Sweden is the place, the country where the name of most people is Andersson.

SE: I'd be happy to give an interview.  We just completed the second edition of the book, which should be coming out in August.  In the first edition and the paper, we made the point that viscosity is a necessary evil for lift.  But, we didn't say what we meant correctly  and the interpretation many take home is that we say lift comes from viscosity.  Lift is not generated by viscosity, it is generated by pushing air down.  However, without intermolecular communication, which shows itself as pressure and viscosity, there would be no lift.

CJ: Can you comment on the fact that NASA, according to the Interview with NASA, presents three incorrect theories for lift but no theory claimed to be correct.

DFA: I looked at "How Airplanes Work" on this site. It is much better than I have ever seen before on a NASA web site. Their description is essentially correct.

CJ: Correct description? NASA presents three incorrect theories, but which is the correct
theory you indicate is presented? Circulation theory? 

DFA: When NASA says: These velocity variations are caused by the disruption and turning of the air flowing past the wing. They are right on. It is the first time I've seen this in print somewhere and I wish we'd used this line in the book. It neatly sumes up an important part of our book.

CJ: This is hand-waving intended to confuse the reader and the general public and give the impression that NASA understands.  If you know a correct answer to a certain question, there is no reason to present three incorrect answers; in fact there are infinitely many wrong answers, but only one correct. NASA does not claim to present a correct theory, unless handwaving is a theory. Do you really mean that to just say that the turning of the flow of air around a wing is what causes lift, is a theory?

DFA: The air curves because air has the characteristics given it by viscosity. The air is bent by the lowering of the pressure. If it didn't bend there would be a vacuum which would bend the air. This lowering of the pressure propagates out accelerating down much more air above the wing. In the rest frame where the air is moving and the wing is standing still, their explanation is a heck of a lot better than and Bernoulli explanation. It may not be complete but for a change it isn't wrong.

CJ: Maybe it is not wrong, but it does not explain anything either; it only says that what happens, happens. It is not a theory, according to any reasonable scientific standard. The real problem is not only to explain lift, but to explain why the lift/drag quotient is of the order 10-20.Right?

DFA: You've given me something to think about. The reason that this is a mystery to you is that you are thinking in drag not power. If you just look at the drag associated with producing lift, I'm sure that there are gliders with lift to parasite drag on the order of 100:1. When the airplane is sitting on the ground the lift due to the wheels is still the weight of the airplane but the drag is now zero. Lift to drag is just the glide ratio, nothing more profound. It is not like saying the work out divided by the work in is 20:1. You could lift the airplane with a 20:1 ratio with a block and tackle. 

CJ: I don't understand: If lift/drag was only = 1-2, then you would see no airplanes or birds in the sky. Why can lift/drag be as large as 10-20 or even up 70 for extreme glider wings? Have you acted as NASA's flight expert?

DFA: I have spent 35 years investigating flight and never saw anyone make a point that L/D is 20. L and D may have the same units but in reality they are apples and oranges. Flight is not fairly efficient. Yes, if L/D were 1 or 2 there would be no powered flight. The question is why should it be 1? As for NASA the answer is No. I can tell you that I'm sure that the statements that have about the bending of air is probable due to Scott and my work.

CJ:I am amazed: Have you not realized that L/D = 10-20 is what makes flying possible?

DFA: L/D (glide ratio) of the space shuttle is 4. That of a high performance glider is 60. It just relates to the size of the wing compared to the weight. You can make an aircraft with almost any glide ration you want. It's all in Newton's laws. Lift goes as mv and power goes as mv^2. Since lift is mv equals weight (a constant) if you could divert an infinite amount of air (VERY large wings) you would set v to zero and thus power to zero. That would be an infinite L/D. Saying l/D of 1 or 2 is what make flight impossible is just saying that an eagle can't fly with humming bird wings. L/D = 20 is not a miracle by any means. Just physics and engineering.

CJ: I am even more amazed: The space shuttle can land but not take off! A barn door at 45 degrees angle of attack has L/D = 1, and you can't fly on barn door unless you have a very powerful engine. Yes, you can design very slender large wing gliders with L/D = 70 flying very slowly at a small angle of attack, but birds and commerical airflight cannot afford that. To say that flying is just physics and engineering is an insult to the clever birds, right?

It is a common misconception that D scales like L*L, but that is not true for real birds and normal airplanes for which L/D = 10-20 for normal angles of attack say 4-12.

DFASo insult the birds. A bullet follows a correct trajectory without having taken a physics class. But it is still physics. I'm getting a little tired of this conversation. Induced power does scale as load squared for fixed speed. That is a fact. Since drag is just power divided by speed and speed is a constant in this conversation, induced drag varies as load squared. Parasite drag only changes with load because the angle of attack of the whole airplane changes with load.

CJ: The flight of a a bullet is trivial physics, but the flight of a bird is not trivial.

DFA: Either is string theory but it's still physics.

CJ: Maybe. Anyway, many thanks for giving an interview concerning state-of-the-art. Your input is useful in a continued discussion.

måndag 20 juli 2009

Interview with David Gross: Physicist

Interview with David Gross.

CJ: In your Opening Remarks to Strings09 you pose the questions:
  • 1. Can we imagine physics with more than one dimension of time?
  • 2. What are the rules of physics without spacetime?
To most people including myself these questions seem utterly strange. Can you give some indication why you are posing these questions and where answers possible can be found?

DG: ???

söndag 19 juli 2009

Interview with Frank Wilczek: Physicist

Interview with Frank Wilczek.

CJ: The incompatibility of general relativity and quantum mechanics has caused a crisis of modern physics. What makes you insist on a geometric theory for gravitation such as general relativity at a very high scientific cost? How sure are you that general relativity is correct, and cannot be abandoned?

CJ: What do you consider to be the quantum mechanical ground state of helium? Orthohelium, parahelium or some other state?

FW: ???

fredag 17 juli 2009

Interview with Editors of Journal of Fluid Mechanics

Interview with S. H. Davis and M. G. Worster Editors of Journal of Fluid Mechanics including also Associate Editors.

CJ: d'Alembert's paradox represents a basic problem of fluid mechanics: Zero drag of theoretical potential flow but substantial drag in real slightly viscous flow. As Editors of the most influential journal of fluid mechanics, I ask you about the current status of the paradox. Is it resolved and if so how and by whom?

SHD: This has not been a "paradox" for perhaps 100 years. See for example the fluid mechanics book by George Batchelor.

CJ: Thanks for response. The discussion is most essential and important for the science of fluid mechanics. As Editor of the main fluid dynamics journal you have an important position in the discussion. However, I don't see that Batchelor, founder of Journal of Fluid Mechanics, in his book An Introduction fo Fluid Mechanics presents a resolution of the paradox, nor claims that there is no paradox: On pages 332, 338 and 342 in his book he states:
  • The result that an inviscid fluid offers no resistance to steady translational motion of a rigid body when the flow is irrotational is sometimes referred to as d'Alembert's paradox, since rigid bodies do experience a resistance to motion through a real fluid. The result is in serious disagreement with observation in the case of bluff bodies, which is hardly surprising, since the flow at the rear of a bluff body is far from having the assumed irrotational form...
  • Analysis of the flow due to a bluff body moving steadily through fluid is effectively prohibited by the large scale unsteadiness of the flow behind the body...Knowledge of this type of flow is mostly empirical...the part of the boundary of this irrotational region formed by the separating streamlines is of complex, fluctuating and unknown shape and as a consequence the irrotational flow cannot be determined...
  • Prandtl suggested in 1914 that the explanation lies in the behaviour of the boundary layer...
Is this your evidence that d'Alembert's paradox "has not been a "paradox" for 100 years"? If not, what is it? It is not even clear that Batchelor was a great admirer of Prandtl and believed in his boundary layer resolution. Right? Too bad that Batchelor is not among us so that he can tell us. 

SHD: d'Alembert's paradox was resolved by the inclusion of viscosity in the fluid.

CJ: Oh, really? How? Please be explicit: d'Alembert's paradox is a central problem in fluid mechanics! It is interesting that Batchelor on page 338 states as his opinion that
  • Although the flow behind a bluff body is unsteady in practice, there is no reason to doubt that a steady (unstable) solution of the equations of motion does exist.
Unstable! This is precisely what is shown in the knol D'Alembert's Paradox based on the article Resolution of d'Alembert's Paradox in Journal of Mathematical Fluid Mechanics JMFM. Batchelor was on the right track = instability of potential flow, which is completely different from Prandtl's suggestion indicating mysterious effects from vanishing skin friction in a very thin boundary layer! What do you say as a successor of Batchelor? Prandtl or Batchelor??

SHD: ???

CJ: Summing up the interviews so far we have: The entire editorial board of the leading journal of fluid mechanics combined with the fluid mechanics expertize at KTH, altogether 22 experts of fluid mechanics, has produced two statements concerning the present status of d'Alembert's paradox, one of the most basic problems of fluid mechanics: 
  • There is no paradox.
  • If there is a paradox, it can somehow be resolved by viscosity.  
Conclusion? Is fluid mechanics not a science?  Is fluid mechanics still split into hydraulics observing phenomena which cannot be explained, and theoretical fluid mechanics explaining phenomena which cannot be observed? What will be the effect on future funding, when the truth gets revealed?  When readers of Journal of Fluid Mechanics read also Journal of Mathematical Fluid Mechanics? How is it possible that so many experts have so little to say? What prevents them?

Note that the official JFM standpoint presented in the Wikipedia article about d'Alembert's paradox, which ranks 1st on Google, does not mention my knol D'Alembert's Paradox, which ranks 2nd on Google. Anyone who finds this incorrect can correct the Wiki article.

Galileo's Dialogue Revisited

Simplicio: I am told that Einstein's special/general theory of relativity together with quantum mechanics are the two pillars of modern physics, which however are incompatible. Isn't this strange? But I am probably too stupid to understand. It seems that very few people can understand relativity theory?

Salvatio: Yes, this is the standard view propagated by the physics community, but as you say it is strange. No, you are not too stupid. It seems that nobody really understands relativity, because there is nothing to understand. It is empty! It is a non-physical theory, which says nothing about physics.

Simplicio: Empty? You must be kidding? How can I understand that it does not say anything about physics?

Salvatio: You see this from the fact that there is no physics input, and thus there can be no physics output. Einstein's special theory of relativity is based on the following two assumptions:
  • (i) There is no aether (aether = material medium for light propagation).
  • (ii) The speed of light is 1 (lightsecond/second).
To understand the nature of these axioms, let us compare with a theory of economy based on the following axioms:
  • (i) There is no free lunch.
  • (ii) There are 100 cents on each dollar.
We understand that (i) is a negative statement stating the non-existence of something, and that (ii) is a definition. There is no physics/economics input in these assumptions, just absence or non-existence of physics/economics.

Simplicio: I see what you are driving at: You ask if a theory about physics can come out from two basic axioms stating non-existence of something + a definition, which have no positive content of physics? It seems to me that you need something more substantial, something positive and not just non-existence? 

Salvatio: Very clever! You are right, you cannot get something out of nothing. No physics input means that there is no physics output. Einstein's special theory of relativity is non-physical. 

Simplicio: I am shocked, but I see what you are saying! But is there then some alternative theory of relativity based on something instead of nothing. A physical theory which can accomodate the Michelson-Morley experiment indicating that the same Maxwell equations should hold for all observers independent of motion.

Salvatio: Yes, you can develop various positive alternative theory of relativity: An example is many-minds relativity based on the following basic axiom:
  • All observers assume that light propagates according to the same Maxwell's equations in a vacuum fixed to the observer.
In a theory of economy, this could be like an agreement among all actors to follow the Bretton Woods system.

Simplicio: I see that the basic axiom has a substantial positive physical content, and I understand that in this theory the Michelson-Morley experiment will turn out the same for all observers, as was observed. In this theory there are many vacui, one attached to each observer. In the same way as each economic actor has an individual perspective, while following Bretton Woods.

Salvatio: Exactly! And this was suggested already in 1914 by the British mathematician Ebenezer Cunningham, which unfortunately was overshadowed by Einstein's non-physical theory.

Simplicio: Is many-minds relativity compatible with quantum mechanics?

Salvatio: Yes, it so seems. 

Simplicio: Well, I now have something to think about.. Replacing Einstein by Cunningham can help modern physics out of its 20th century trauma? I will return with some questions after studying the documents...I will start with Theory of Relativity...