Conundrums of Classical Physics; http://arstechnica.com/science/2014/08/04/the-never-endingconundrums-of-classical-physics/
The never-ending conundrums of classical
physics
Even today, scientists still try to solve sprinkler brain teasers from the 19th century.
by Lee Phillips Aug 4, 2014 8:00pm CDT


Share
Tweet
180
Aurich Lawson
During its teenage and young adult years—what is now referred to as its “classical” period—
physics made a lot of mistakes.
In the old physics, mass and energy were separately conserved; particles’ positions and momenta
could be arbitrarily specified; gravity acted instantaneously at a distance; the equality of
gravitational and inertial mass was just a coincidence; and there was no speed limit. All these
ideas and assumptions are now known to be in some way untenable. They're either inaccurate or
theoretical dead-ends.
1
To put it plainly, classical physics is wrong. As such, there's really only one thing to do—
physicists have since abandoned the old, mistaken ideas, right? It's reminiscent of how doctors
discarded the system of humors, chemists chucked out phlogiston, and astronomers turned away
from astrology and geocentrism.
Not quite. Classical physics in fact survives as the foundation for most of engineering. At human
scales, when velocities are not too close to the speed of light, distances are less than
astronomical, and particles are not too small, modern physics reduces to classical physics, which
provides an excellent approximation under such conditions. (Experts may grumble here that I’m
not quite technically correct. Where I refer to particle size I should really speak of quantum
number, remembering the phenomena of superfluidity and superconductivity. Also, precise
measurements sometimes make the nonclassical nature of reality apparent even at human scales.
For example, the effect of gravity on the speed of light means that we have to use general
relativity in the design of the GPS navigation system.)
However, the practical utility of classical physics is not what this story is about. You see,
the incorrect hanger-on we call classical physics remains a vibrant arena of active research. Its
foundations and the fundamental problems posed by several of its subfields still engage the
imaginations of thousands of physicists throughout the world. And like all areas in active
development, it attracts contention and controversy to this very day.
What was old is new again
Physics is vast. In the first decades of the last century, a physicist could keep up with all the
relevant research by browsing a couple of journals. Now, due to the unabated exponential
growth in publications since WWII, most physicists can barely hope to keep up with progress in
their sub-subfield. Additionally, these subfields have become so specialized that it's nearly
impossible to glean more than a superficial understanding of a paper outside of one’s specialty.
The enormous scope and range and the proliferation of sub-specialties may lead to the
impression that physics is not a unified, nor even coherent discipline. At first glance, the study of
groundwater flows seems to exist in a different universe from the investigation of stellar
evolution. But what defines physics is not the subject matter so much as the set of tools and
general outlook one brings to bear upon the problems of the physical world.
Physicists are all brought up with a similar body of knowledge—techniques of applied
mathematics and computation and a similar approach to the root subjects within the field
(mechanics, electrodynamics, thermodynamics, quantum mechanics, etc.). Physics students tend
to learn this canon from a fairly small and stable set of classic texts. So if you’re currently near
the beginning of your graduate school career and taking a course in electrodynamics, there is an
excellent chance that you’re using the same thick, red book that was intimidating students 30
years ago. More than a catalog of subjects and problems, therefore, physics is a culture.
2
Public domain
Sir Godfrey Kneller’s portrait of Isaac Newton.
1.
3
2.
3.
This culture has adopted a consensus view of its own history and development according to
which there is a major division of physics into classical and modern. Classical physics had
precursors as far back as Archimedes. Its first powerful, compendious theories were consolidated
in the 17th century work of Newton, whom Neil deGrasse Tyson calls “The smartest person ever
to walk the face of this earth." (The quote continues: "The man was connected to the Universe in
spooky ways. He discovered the laws of motion, the laws of gravity, the laws of optics. Then he
turned 26.")
Classical physics advanced rapidly over the next couple of hundred years, acquiring conservation
laws, thermodynamics, and the various laws of electromagnetism, eventually bound together as a
coherent theory by Maxwell in the 1860s. Of course it wasn’t called “classical physics” at the
time—"Natural Philosophy" at first, then plain physics.
What eventually became modern physics had its first stirrings when Maxwell’s equations got a
careful examination. The period had its real beginning in 1905, with Einstein’s miraculous
papers on special relativity and quantum mechanics. Special and general relativity (the modern
theory of gravity) and quantum theory, accompanied by the surviving notions from classical
physics, remain the foundations of modern physical theory and all of its specialties, from
astrophysics to superconductivity.
As mentioned above, classical physics retains considerable utility as an excellent approximation
in most situations of practical interest. Neither relativity nor quantum theory is required to build
bridges or design cellphone antennas. While this provides a good reason to teach the subject to
engineers, it doesn’t explain why the classical versions of physical science continue to excite the
imaginations of physicists, or why they are still capable of generating controversy and debate.
A survey of the current issues of physics research journals leaves no doubt about the vibrant
status of classical physics. There are entire top flight journals devoted almost solely to classical
4
subjects:Fluid Mechanics, Physics of Fluids, Archive for Rational Mechanics and
Analysis, Physics of Plasmas, etc. Not to mention, there's a proliferation of more specialized
journals that also concentrate on classical subjects.
Physical Review Letters is the most prestigious venue for publication of cutting-edge physics
research, at least in the US. Articles are required to possess timely importance that advance the
state of the art, and the topic must be important enough to be of general interest to physicists.
Narrowly focused papers that would only be read by specialists are not accepted.
We took a census of the four most recent issues of PRL, which publishes weekly. The 207
articles in that sample have been sorted into the categories listed below, based on the titles and
abstracts of the articles:
Out of the 207 total papers, 71 percent treat subjects that exist within the frameworks of either
quantum mechanics or relativity, meaning they are part of modern physics. The other 29 percent
make no use of these modern developments; they are part of classical physics.
The categories used in the table were a natural fit for the particular journal issues surveyed, and a
different set of papers from another month of PRL would probably lead to a somewhat different
categorization. For example, although it is possible to have a paper in relativistic fluid dynamics,
all of the fluid dynamics papers in this set happened to be classical; therefore the fluids category
could be placed under the classical heading. Likewise, a biophysics paper that treated the
quantum physics of photosynthesis would have to go under modern physics, but all the
biophysics papers in this set happened to be classical. Someone else, it must be acknowledged,
would be likely to categorize a few of these papers slightly differently. But the general
conclusion, that roughly a third of current, state-of-the-art physics research is in fact some form
of classical physics, would survive.
Ultimately, physicists pursue problems that are interesting, beautiful, puzzling, and seem, in
some way hard to define, important. They seek out questions that get at the crux of how the
universe works. So aside from its immense practical utility, classical physics survives as a major
part of real physics because it still presents problems that tug at all the urges of a pure scientist.
There are some of us who are irresistibly drawn to the hypnotic intricacies of swirling liquids,
5
who are inspired by the challenge of taming a turbulent plasma with a succinct equation, or who
burn to understand how dragonflies fly so well.
We plow these fields with the best tools available, with the full knowledge that they are
incomplete, and in some senses faulty models of nature, because they are nevertheless the
appropriate levels of description. Well crafted tools are a pleasure to wield. We do not, however,
tolerate the occasional misguided complaint that classical physics is not “fundamental,”
especially if the complainer seeks to reserve that elusive quality to whatever happens to be at the
vanguard of physics at the moment.
The perennial puzzles
Enough of the background. Let's walk through some examples of classical physics as part of living
science. After 400 years this sect may be well established, but that doesn’t mean it’s well
understood. It is surprisingly easy to pose what appears to be a simple problem that manages to
stump not only students and amateurs, but the most brilliant physics minds as well. The following
classic puzzles are not only challenging and entertaining in their own right, but they periodically
reappear to ignite fresh rounds of controversy.
6
Ernst Mach’s drawing of a sprinkler.
Public domain.
Sprinkler suspense
If you force air out of the sprinkler depicted here by squeezing the bulb or connect it to a hose to
water your lawn, there is little doubt that, if the top part is allowed to rotate freely, it will spin in the
direction indicated by the large arrow. That's opposite to the direction of the expelled fluid.
But what happens when you release the bulb and suck air back in? Or, say, immerse the contraption
in a container and force water into the arms through pressure?
7
Since the sprinkler was first discussed by the great
physicist and philosopher of science Ernst Mach in the late nineteenth century, it has periodically
reappeared to challenge the intuition of physics students and Nobel prize winners alike. It is now
usually referred to as “Feynman’s sprinkler,” partly as a result of the popularity of Feynman’s
memoirs, where the young physicist exploded a tank full of water in Princeton’s cyclotron facility
while attempting to settle the question experimentally.
When encountering this puzzle for the first time, physicists tend to think the answer is clear. About
half think it’s obvious that the sprinkler will rotate counterclockwise, while the other half think it’s
obvious that it won’t move at all. This is a recurring theme in the classical physics of fluid motions:
intuition is often as unreliable a guide as it is when trying to predict the behavior of subatomic
particles. A systematic analysis shows that the sprinkler, when beginning to take in fluid,
experiences a transient torque, but it will not be observed to rotate as the intake continues. The
inflow situation is not the time-reversal of the outflow situation.
Fundamentals of flight?
Another puzzle that deals with the interaction between fluid flows and solid bodies is perhaps the
most widely contentious issue involving classical physics. How do planes fly?
Ask a group of physicists, if you dare. It’s a sure-fire way to ignite hours, or at least long minutes, of
hand-waving arguments and faulty explanations that defy logic, observation, and a basic knowledge
of fluid physics. You’re more likely to get a sensible explanation from a thoughtful grade schooler
than a grown-up scientist, whose mind will be clouded by half-remembered bogus explanations from
undergraduate textbooks, invariably invoking faulty applications of Bernoulli’s equation.
These unfortunate scientists have probably been confused by a popular demonstration that you can
do right now with a strip of paper: hold the paper so that it extends forth from your lips but is free to
bend downward due to gravity. Blow along the top surface and observe the paper lifting up and
assuming a nearly horizontal position. Clearly, there is a lifting force, caused by the flow of air along
the upper, curved surface of the paper. So far, so good. Invariably, the demonstrator will now offer
the “explanation” that this is due to something called Bernoulli’s law, which he thinks says pressure
is lower in faster air.
There are several things wrong with this explanation, which is intended to prepare the mind for an
even worse explanation of the lifting force on an airfoil. Perhaps the most obvious error is the fact
that the two flows—one expelled from the lungs, the other the motionless mass of air in the room,
8
separated by the discontinuity of the paper—are not related to each other in any way; not by
Bernoulli’s nor any other law.
The incorrect “equal transit time” explanation for lift.
Educators who think the Bernoulli explanation of the paper demonstration makes sense are also
fond of the faulty “equal transit time” explanation of the lifting force on a wing. This model, repeated
in innumerable textbooks, depends on the bizarre idea that two neighboring fluid particles at the
leading edge of the wing must, for some reason, meet up again at the trailing edge. Since the path
over the wing (see the picture) is longer than the path underneath it, the velocity of the flow above
the wing must be faster to allow the fluid particles to rejoin. Note that the diagram, as it is usually
drawn and as it appears here, shows the flow past the trailing edge horizontal, which means that
there can be no lifting force on the wing due to Newton’s third law. The final irony is that, while in the
lifting regime the upper flow is in fact faster than the lower flow, it is much faster than it would be if
the equal transit time picture were correct. The latter would give rise to an upper stream velocity too
slow to generate significant lift!
When faced with someone who thinks he has explained how airplanes fly, one satisfying gambit is to
ask innocently how it is that planes can fly upside down. Don’t judge the explainers harshly; it took
even experts in the field a long time to come to some kind of a consensus. The extended debate is
chronicled in The Enigma of the Aerofoil, a book that covers the decades of international bickering
over the correct explanation of lift.
A simple way to understand lift is to consider, again, Newton’s third law: for every action there is an
equal and opposite reaction. If the wings can be coerced to direct a stream of air downward, there
must be a reaction force upward. This explains how a plane can fly upside down, as long as an
angle of attack is maintained that forces air toward the ground. It also explains how model planes
with flat wings are able to fly. The reason real planes are not designed with flat airfoils is that the
resultant highly turbulent flow creates a large amount of drag, a problem partially solved by the
classic aerodynamic shape, borrowed from fish and dolphins. (Of course it’s not quite this simple.
9
The finite length of wings in the real world requires an analysis of the three-dimensional system of
vortices shed from their surfaces, for example.)
Hydrolics hijinx
Hydrostatics is the study of fluids at rest, and under this topic you will find simple but diabolical
questions about pressure and buoyancy. The fact that even innocent questions about floating bodies
have the counterintuitive power to stump experienced physicists and engineers makes Archimedes’
eureka moment all the more impressive.
Flickr user: Cinderellasg
Take a simple bottle of milk like the ones pictured here, with a cylindrical section below a tapering
segment. Gravity acting on the milk will cause a pressure to exist everywhere in the fluid, which will
be, according to Pascal’s law, the same in all directions. The milk will exert a force on the bottom of
the bottle; this force divided by its area is the pressure on the bottom.
I forgot to mention that this is old-fashioned, non-homogenized milk. Over time, the cream, which
initially exists as tiny suspended droplets uniformly distributed throughout the fluid, will separate and
float to the top, creating a layer of less-dense fat floating on the skim milk underneath. Let’s suppose
that the cream, after separating out, occupies part of the neck region near the top of the bottle.
Now the question: what happens to the pressure on the bottom of the bottle?
Most physicists think it’s obvious that the pressure can not change, because, if the bottle were on a
scale, the weight would obviously be observed to remain the same, as no matter has been lost or
10
added. But there is an implicit assumption here, that the weight of the fluid equals its pressure on the
bottle’s bottom multiplied by the latter’s area. This would indeed be true if the bottle were a cylinder,
with vertical sides. A correct analysis, which takes into account the forces on the bottle’s sloping
walls, leads to the conclusion that the pressure on the bottom actually decreases.
Deeper Issues
The examples so far are tricky problems that eventually gained accepted, correct solutions. But even
today, not every classical physics puzzle has a clear resolution. The following remain genuine
controversies years after these questions were first posed. They still have no universally accepted
solutions, often because some of these problems lie at the boundaries between physics and
philosophy.
The rotation problem
Think about an object rotating in space—say, two heavy balls connected by a rope. There must be a
tension in the rope holding the balls together and forcing them to follow circular paths,
otherwise they would fly apart. We have a special name for this tension: centripetal force.
But what do we mean by “rotating?” Rotating with respect to what? The two-sphere arrangement
was first considered systematically by Isaac Newton, who used the thought experiment as evidence
for the existence of an absolute space. The existence of centripetal forces was supposed to be
explained by rotation with respect to this absolute space, which establishes a fixed reference frame
that defines all motion.
11
Enlarge
The previously mentioned Ernst Mach would have none of this. He rejected all notions of an
absolute space and time. Instead, Mach suggested the apparently special status of the condition of
motion that can persist without centripetal forces exists because it is at rest with respect to the fixed
stars. For Mach, all motion had to be motion relative to something.
Mach’s rejections of Newton’s ideas of absolute space and time were a major influence on Einstein
when he was developing his theories of relativity. Today, most physicists would explain the situation
with the concept of “inertial frames.” All the reference frames that require the introduction of extra
forces not resulting from the observed rotation are not inertial. There is only one inertial frame, and
that happens to be at rest with respect to the fixed stars. We also know that the universe as a whole
is not rotating because of the observed properties of the microwave background radiation. The
explaining-away of the rotation problem by invoking inertial frames strikes some physicists as
circular reasoning, however, and so the problem of rotation cannot be considered entirely solved.
(Although, it might fairly be relegated to what is now called philosophy or metaphysics.)
Time tinkering
A movie of ordinary life played backward immediately looks absurd. Yet, the equations that describe
what we see have the same form if time is reversed inside them. For a concrete example, the
collisions of billiard balls are fully encompassed by mathematical laws that look the same if we
replace t by -t, reversing the direction of time. But if we see a recording that shows the scattered
balls flying about the table, colliding with each other and the bumpers and eventually settling into a
neat triangle, we know we’ve just seen the break backward.
How do laws of physics with no preferred direction for time give rise to our familiar reality with its
obvious future and past? (Note time reversal symmetry only works for pure mechanics; the related,
full symmetry known to modern physics is CPT invariance.) This is another issue with origins in
classical physics that still ignites debate and spawns research papers. A resolution is typically
sought with references to entropy, probability theory, the second law of thermodynamics, and even
the conditions prevailing during the Big Bang. Certainly these are the places one should look to
explain the arrow of time, but no single interpretation has satisfied everyone yet.
The definition of entropy and the proper mathematical foundations of thermodynamics and
continuum mechanics are a quagmire that has given rise to opposing, and not always friendly,
camps of mathematicians and physicists. This enterprise is far from over, and it has a guiding light in
Clifford Truesdell. From the end of WWII to his death in 2000, Truesdell led the effort to bring
mathematical order to these thorny subjects.
It may be surprising for a layman to learn that a small but persistent proportion of the theoretical
papers published in physics journals consists of little more than nonsense, and one of Truesdell’s
enduring contributions was to help to expose an entire strata of this misguided activity, mostly taking
the form of papers (and entire books) in the field of “irreversible thermodynamics.” As Truesdell’s
students and colleagues continue this work, it is impossible to say whether scientists will ever settle
12
on a notion of entropy and a formulation of thermodynamics that makes everyone happy. As such,
the problem of the arrow of time may never be considered resolved.
The transition to turbulence.
Dr. Gary Settles. CC BY-SA 3.0
Turbulence turmoil
One of the major unsolved problems in physics is the nature of turbulence. From Leonardo DaVinci’s
insightful observations and beautiful sketches to calculations on today’s most powerful
supercomputers, scientists have tried, with very limited success, to get a handle on understanding
the behavior of the chaotic fluid flows that are a part of everyday life. This is, as far as we know, a
purely classical problem. It's one that Feynman called the biggest unsolved problem of classical
physics. The Nobel prize-winning scientist—who helped create quantum electrodynamics, the model
for the interaction of matter and light that is the most accurate physical theory ever devised—spent
several years working on turbulence and got nowhere.
13
There are several related, unsolved problems that fall under the turbulence umbrella. One of the
most interesting is the prediction of the turbulent transition, depicted here. This is the question of
when, and why, an orderly, simple fluid flow suddenly becomes chaotic and unpredictable.
Today we still lack even a fundamental understanding of the kinds of solutions that might exist for
the most commonly used equations that describe fluid flow—the Navier-Stokes equations. Under
what circumstances do solutions exist and describe physically reasonable (finite) flows? One of the
million-dollar millennium Clay prizes awaits the mathematician who can figure this out.
Determinism
Classical physics is usually considered, by laymen and physicists alike, to be a deterministic theory.
Chance does not appear in its fundamental equations, which, assuming we know everything about
the present, predict the future with certainty. Many people also have the impression that quantum
mechanics is a non-deterministic theory that doesn’t allow us to know the future. Neither of these
views is completely immune to challenge.
In quantum mechanics, the evolution of the wavefunction that describes the state of a system is
governed by equations that are as deterministic as Newton’s laws. It is only in the measurement
process that probabilities become fundamental. Even here, there are several formulations of QM that
allow it to be interpreted as an essentially classical, deterministic theory, at least according to their
proponents.
Classical systems, although governed by deterministic laws, often exhibit a profound unpredictability.
Any uncertainty in the details of the motion at any time is hugely magnified by the system, leading to
a large degree of uncertainty a short time later. This characteristic of chaotic systems is the usual
explanation for the impossibility of predicting the weather more than a few hours into the future. Fluid
turbulence, the open problem that we visited above, is the canonical example of such a chaotic
system.
Stranger than that, though, is the fact that the familiar classical physics of Newton is not
even entirely deterministic. This can be demonstrated by considering certain problems involving a
small number of particles. In some cases, Newton’s laws permit an infinite velocity to develop, which
carries one of the particles “out of the world." Since the laws are time-reversible, when the system is
run backward the escaped particle can enter the world in a way that it is inherently impossible to
predict. This has come to be known as the problem of “space invaders,” and it was first studied in
the late nineteenth century. Another example involves three point particles approaching each other
along trajectories that are 120° apart. After they recoil, they may separate along any three paths that
are also 120° apart, but these need not be the original paths; the set of paths can be rotated by an
arbitrary angle, and all of classical physics is satisfied.
Perhaps the most interesting example of Newtonian non-determinism was discovered by John D.
Norton in 2003, when he constructed a dome-like surface with a particular shape and examined the
behavior of a mass subject to gravity and initially placed at rest at its apex. Norton showed that
14
classical physics allows the mass to begin sliding at an arbitrary time in an arbitrary direction; neither
the time nor the direction can be predicted. This is not the familiar instability of a ball at the top of a
hill; no infinitesimal force is required to get Norton’s mass moving. This system was critically
examined a few years later by David Malament, who pointed out that Norton’s dome has some
peculiar mathematical properties. Malament suggested that we might consider excluding such
constraint surfaces from what we would like to consider to be Newtonian systems. In this way, we
can “save” the determinism of classical physics by only admitting forces and boundaries that allow
Newton’s laws to be expressed as differential equations whose solutions are known to exist and be
unique.
It might be similarly argued that all the problems that seem to lead to indeterminism in classical
physics are either pathological or singular. While that might be true, these examples demonstrate
that Newtonian mechanics is not the straightforwardly deterministic theory that it’s usually assumed
to be. This leads to another area of the literature that shows, once again, that classical physics
remains a lively arena for debate and research, not only into its practical applications, but into its
fundamentals.
The "fundamental" question
Physicists interested in classical problems sometimes encounter condescension from colleagues.
They're informed these areas are “not fundamental,” or even, because of this, that they’re not “real
physics.”
The implication, of course, is that some areas of physics can be awarded the status of fundamental
theories. What that actually means is a question perennially engaging philosophers of science. No
single article can provide more than a light overview of the subject.
As we mentioned at the outset, we know that classical physics is wrong. It is even inconsistent with
part of itself: the theory of the electromagnetic field, embodied in Maxwell’s equations, has a different
kind of symmetry from Newton’s mechanics. When stepping from the platform onto the train,
Einstein showed that space and time change according to special relativity, using the Lorentz
transformation obeyed by Maxwell’s equations. Newton’s laws obey the simpler Galilean
transformation rules. In classical mechanics there is no speed limit, but Maxwell’s equations respect
the speed limit of the velocity of light and are consistent with special relativity.
Since we know that classical physics is not correct, we might agree that it’s not a fundamental
theory. However, neither is special relativity, which explains neither gravity nor reality at small
scales, for which we need, respectively, general relativity and quantum mechanics. However, since
these two magnificent accomplishments of modern physics have not yet been reconciled into a
unified theory of quantum gravity, neither of them is really fundamental either. Today we have an
assortment of models, each of which does remarkably well within its scale of length or energy but
can not be applied outside that realm.
15
We search physics in vain for a truly fundamental theory. High energy particle physics, which
sometimes describes itself as dealing with the fundamental constituents of matter, is constrained (as
is all physics) by the experiments that economics and politics make possible. Its predictions can only
be tested to the extent that we can afford to build more powerful accelerators, which makes any
claims for “fundamental” status at least problematic.
Hopefully, this provides some context for the claim that classical physics is “not fundamental.”
Whatever as yet undiscovered properties of nature can explain the transition to turbulence or the
other mysteries of chaotic motion, the appropriate level of description for their discovery is covered
by classical physics. Our pursuit of these problems arises from our desire to understand something
about how the universe works, and that motivation is what makes our science—the never-ending
conundrum of classical physics—fundamental.
16