Posts by David D. Nolte

E. M. Purcell Distinguished Professor of Physics and Astronomy at Purdue University

Chandrasekhar’s Limit

Arthur Eddington was the complete package—an observationalist with the mathematical and theoretical skills to understand Einstein’s general theory, and the ability to construct the theory of the internal structure of stars.  He was Zeus in Olympus among astrophysicists.  He always had the last word, and he stood with Einstein firmly opposed to the Schwarzschild singularity.  In 1924 he published a theoretical paper in which he derived a new coordinate frame (now known as Eddington-Finkelstein coordinates) in which the singularity at the Schwarzschild radius is removed.  At the time, he took this to mean that the singularity did not exist and that gravitational cut off was not possible [1].  It would seem that the possibility of dark stars (black holes) had been put to rest.  Both Eddington and Einstein said so!  But just as they were writing the obituary of black holes, a strange new form of matter was emerging from astronomical observations that would challenge the views of these giants.

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.

White Dwarf

Binary star systems have always held a certain fascination for astronomers.  If your field of study is the (mostly) immutable stars, then the stars that do move provide some excitement.  The attraction of binaries is the same thing that makes them important astrophysically—they are dynamic.  While many double stars are observed in the night sky (a few had been noted by Galileo), some of these are just coincidental alignments of near and far stars.  However, William Herschel began cataloging binary stars in 1779 and became convinced in 1802 that at least some of them must be gravitationally bound to each other.  He carefully measured the positions of binary stars over many years and confirmed that these stars showed relative changes in position, proving that they were gravitational bound binary star systems [2].  The first orbit of a binary star was computed in 1827 by Félix Savary for the orbit of Xi Ursae Majoris.  Finding the orbit of a binary star system provides a treasure trove of useful information about the pair of stars.  Not only can the masses of the stars be determined, but their radii and densities also can be estimated.  Furthermore, by combining this information with the distance to the binaries, it was possible to develop a relationship between mass and luminosity for all stars, even single stars.  Therefore, binaries became a form of measuring stick for crucial stellar properties.

Comparison of Earth to a white dwarf star with a mass equal to the Sun. They have comparable radii but radically different densities.

One of the binary star systems that Hershel discovered was the pair known as 40 Eridani B/C, which he observed on January 31 in 1783.  Of this pair, 40 Eridani B was very dim compared to its companion.  More than a century later, in 1910 when spectrographs were first being used routinely on large telescopes, the spectrum of 40 Eridani B was found to be of an unusual white spectral class.  In the same year, the low luminosity companion of Sirius, known as Sirius B, which shared the same unusual white spectral class, was evaluated in terms of its size and mass and was found to be exceptionally small and dense [3].  In fact, it was too small and too dense to be believed at first, because the densities were beyond any known or even conceivable matter.  The mass of Sirius B is around the mass of the Sun, but its radius is comparable to the radius of the Earth, making the density of the white star about ten thousand times denser than the core of the Sun.  Eddington at first felt the same way about white dwarfs that he felt about black holes, but he was eventually swayed by the astrophysical evidence.  By 1922 many of these small white stars had been discovered, called white dwarfs, and their incredibly large densities had been firmly established.  In his famous book on stellar structure [4], he noted the strange paradox:  As a star cools, its pressure must decrease, as all gases must do as they cool, and the star would shrink, yet the pressure required to balance the force of gravity to stabilize the star against continued shrinkage must increase as the star gets smaller.  How can pressure decrease and yet increase at the same time?  In 1926, on the eve of the birth of quantum mechanics, Eddington could conceive of no mechanism that could resolve this paradox.  So he noted it as an open problem in his book and sent it to press.

Subrahmanyan Chandrasekhar

Three years after the publication of Eddington’s book, an eager and excited nineteen-year-old graduate of the University in Madras India boarded a steamer bound for England.  Subrahmanyan Chandrasekhar (1910—1995) had been accepted for graduate studies at Cambridge University.  The voyage in 1930 took eighteen days via the Suez Canal, and he needed something to do to pass the time.  He had with him Eddington’s book, which he carried like a bible, and he also had a copy of a breakthrough article written by R. H. Fowler that applied the new theory of quantum mechanics to the problem of dense matter composed of ions and electrons [5].  Fowler showed how the Pauli exclusion principle for electrons, that obeyed Fermi-Dirac statistics, created an energetic sea of electrons in their lowest energy state, called electron degeneracy.  This degeneracy was a fundamental quantum property of matter, and carried with it an intrinsic pressure unrelated to thermal properties.  Chandrasekhar realized that this was a pressure mechanism that could balance the force of gravity in a cooling star and might resolve Eddington’s paradox of the white dwarfs.  As the steamer moved ever closer to England, Chandrasekhar derived the new balance between gravitational pressure and electron degeneracy pressure and found the radius of the white dwarf as a function of its mass.  The critical step in Chandrasekhar’s theory, conceived alone on the steamer at sea with access to just a handful of books and papers, was the inclusion of special relativity with the quantum physics.  This was necessary, because the densities were so high and the electrons were so energetic, that they attained speeds approaching the speed of light. 

Something wonderful, but also a little scary, happened when Chandrasekhar included the relativistic effects in his calculation.  He discovered that electron degeneracy pressure could balance the force of gravity if the mass of the white dwarf were smaller than about 1.4 times the mass of the Sun.  But if the dwarf was more massive than this, then even the electron degeneracy pressure would be insufficient to fight gravity, and the star would continue to collapse.  To what?  Schwarzschild’s singularity was one possibility.  Chandrasekhar wrote up two papers on his calculations, and when he arrived in England, he showed them to Fowler, who was to be his advisor at Cambridge.  Fowler was genuinely enthusiastic about  the first paper, on the derivation of the relativistic electron degeneracy pressure, and it was submitted for publication.  The second paper, on the maximum sustainable mass for a white dwarf, which reared the ugly head of Schwarzschild’s singularity, made Fowler uncomfortable, and he sat on the paper, unwilling to give his approval for publication in the leading British astrophysical journal.  Chandrasekhar grew annoyed, and in frustration sent it, without Fowler’s approval, to an American journal, where “The Maximum Mass of Ideal White Dwarfs” was published in 1931 [6].  This paper, written in eighteen days on a steamer at sea, established what became known as the Chandrasekhar limit, for which Chandrasekhar would win the 1983 Nobel Prize in Physics, but not before he was forced to fight major battles for its acceptance.

The Chandrasekhar limit expressed in terms of the Planck Mass and the mass of a proton. The limit is approximately 1.4 times the mass of the Sun. White dwarfs with masses larger than the limit cannot balance gravitational collapse by relativistic electron degeneracy.

Chandrasekhar versus Eddington

Initially there was almost no response to Chandrasekhar’s paper.  Frankly, few astronomers had the theoretical training needed to understand the physics.  Eddington was one exception, which was why he held such stature in the community.  The big question therefore was:  Was Chandrasekhar’s theory correct?  During the three years to obtain his PhD, Chandrasekhar met frequently with Eddington, who was also at Cambridge, and with colleagues outside the university, and they all encouraged Chandrasekhar to tackle the more difficult problem to combine internal stellar structure with his theory.  This could not be done with pen and paper, but required numerical calculation.  Eddington was in possession of an early electromagnetic calculator, and he loaned it to Chandrasekhar to do the calculations.  After many months of tedious work, Chandrasekhar was finally ready to confirm his theory at the 1934 meeting of the British Astrophysical Society. 

The young Chandrasekhar stood up and gave his results in an impeccable presentation before an auditorium crowded with his peers.  But as he left the stage, he was shocked when Eddington himself rose to give the next presentation.  Eddington proceeded to criticize and reject Chandrasekhar’s careful work, proposing instead a garbled mash-up of quantum theory and relativity that would eliminate Chandrasekhar’s limit and hence prevent collapse to the Schwarzschild singularity.  Chandrasekhar sat mortified in the audience.  After the session, many of his friends and colleagues came up to him to give their condolences—if Eddington, the leader of the field and one of the few astronomers who understood Einstein’s theories, said that Chandrasekhar was wrong, then that was that.  Badly wounded, Chandrasekhar was faced with a dire choice.  Should he fight against the reputation of Eddington, fight for the truth of his theory?  But he was at the beginning of his career and could ill afford to pit himself against the giant.  So he turned his back on the problem of stellar death, and applied his talents to the problem of stellar evolution. 

Chandrasekhar went on to have an illustrious career, spent mostly at the University of Chicago (far from Cambridge), and he did eventually return to his limit as it became clear that Eddington was wrong.  In fact, many at the time already suspected Eddington was wrong and were seeking for the answer to the next question: If white dwarfs cannot support themselves under gravity and must collapse, what do they collapse to?  In Pasadena at the California Institute of Technology, an astrophysicist named Fritz Zwicky thought he knew the answer.

Fritz Zwicky’s Neutron Star

Fritz Zwicky (1898—1874) was an irritating and badly flawed genius.  What made him so irritating was that he knew he was a genius and never let anyone forget it.  What made him badly flawed was that he never cared much for weight of evidence.  It was the ideas that mattered—let lesser minds do the tedious work of filling in the cracks.  And what made him a genius was that he was often right!  Zwicky pushed the envelope—he loved extremes.  The more extreme a theory was, the more likely he was to favor it—like his proposal for dark matter.  Most of his colleagues considered him to be a buffoon and borderline crackpot.  He was tolerated by no one—no one except his steadfast collaborator of many years Ernst Baade (until they nearly came to blows on the eve of World War II).  Baade was a German physicist trained at Göttingen and recently arrived at Cal Tech.  He was exceptionally well informed on the latest advances in a broad range of fields.  Where Zwicky made intuitive leaps, often unsupported by evidence, Baade would provide the context.  Baade was a walking Wikipedia for Zwicky, and together they changed the face of astrophysics.

Zwicky and Baade submitted an abstract to the American Physical Society Meeting in 1933, which Kip Thorne has called “…one of the most prescient documents in the history of physics and astronomy” [7].  In the abstract, Zwicky and Baade introduced, for the first time, the existence of supernovae as a separate class of nova and estimated the total energy output of these cataclysmic events, including the possibility that they are the source of some cosmic rays.  They introduced the idea of a neutron star, a star composed purely of neutrons, only a year after Chadwick discovered the neutron’s existence, and they strongly suggested that a supernova is produced by the transformation of a star into a neutron star.  A neutron star would have a mass similar to that of the Sun, but would have a radius of only tens of kilometers.  If the mass density of white dwarfs was hard to swallow, the density of a neutron star was billion times greater!  It would take nearly thirty years before each of the assertions made in this short abstract were proven true, but Zwicky certainly had a clear view, tempered by Baade, of where the field of astrophysics was headed.  But no one listened to Zwicky.  He was too aggressive and backed up his wild assertions with too little substance.  Therefore, neutron stars simmered on the back burner until more substantial physicists could address their properties more seriously.

Two substantial physicists who had the talent and skills that Zwicky lacked were Lev Landau in Moscow and Robert Oppenheimer at Berkeley.  Landau derived the properties of a neutron star in 1937 and published the results to great fanfare.  He was not aware of Zwicky’s work, and he called them neutron cores, because he hypothesized that they might reside at the core of ordinary stars like the Sun.  Oppenheimer, working with a Canadian graduate student George Volkoff at Berkeley, showed that Landau’s idea about stellar cores was not correct, but that the general idea of a neutron core, or rather neutron star, was correct [8].  Once Oppenheimer was interested in neutron stars, he kept going and asked the same question about neutron stars that Chandrasekhar had asked about white dwarfs:  Is there a maximum size for neutron stars beyond which they must collapse?  The answer to this question used the same quantum mechanical degeneracy pressure (now provided by neutrons rather than electrons) and gravitational compaction as the problem of white dwarfs, but it required detailed understanding of nuclear forces, which in 1938 were only beginning to be understood.  However, Oppenheimer knew enough to make a good estimate of the nuclear binding contribution to the total internal pressure and came to a similar conclusion for neutron stars as Chandrasekhar had made for white dwarfs.  There was indeed a maximum mass of a neutron star, a Chandrasekhar-type limit of about three solar masses.  Beyond this mass, even the degeneracy pressure of neutrons could not support gravitational pressure, and the neutron star must collapse.  In Oppenheimer’s mind it was clear what it must collapse to—a black hole (known as gravitational cut-off at that time). This was to lead Oppenheimer and John Wheeler to their famous confrontation over the existence of black holes, which Oppenheimer won, but Wheeler took possession of the battle field [9].

Derivation of the Relativistic Chandrasekhar Limit

White dwarfs are created from the balance between gravitational compression and the degeneracy pressure of electrons caused by the Pauli exclusion principle. When a star collapses gravitationally, the matter becomes so dense that the electrons begin to fill up quantum states until all the lowest-energy states are filled and no more electrons can be added. This results in a balance that stabilizes the gravitational collapse, and the result is a white dwarf with a mass density a million times larger than the Sun.

If the electrons remained non-relativistic, then there would be no upper limit for the size of a star that would form a white dwarf. However, because electrons become relativistic at high enough compaction, if the initial star is too massive, the electron degeneracy pressure becomes limited relativistically and cannot keep the matter from compacting more, and even the white dwarf will collapse (to a neutron star or a black hole). The largest mass that can be supported by a white dwarf is known as the Chandrasekhar limit.

A simplified derivation of the Chandrasekhar limit begins by defining the total energy as the kinetic energy of the degenerate Fermi electron gas plus the gravitational potential energy

The kinetic energy of the degenerate Fermi gas has the relativistic expression


where the Fermi k-vector can be expressed as a function of the radius of the white dwarf and the total number of electrons in the star, as

If the star is composed of pure hydrogen, then the mass of the star is expressed in terms of the total number of electrons and the mass of the proton

The total energy of the white dwarf is minimized by taking its derivative with respect to the radius of the star

When the derivative is set to zero, the term in brackets becomes

This is solved for the radius for which the electron degeneracy pressure stabilizes the gravitational pressure

This is the relativistic radius-mass expression for the size of the stabilized white dwarf as a function of the mass (or total number of electrons). One of the astonishing results of this calculation is the merging of astronomically large numbers (the mass of stars) with both relativity and quantum physics. The radius of the white dwarf is actually expressed as a multiple of the Compton wavelength of the electron!

The expression in the square root becomes smaller as the size of the star increases, and there is an upper bound to the mass of the star beyond which the argument in the square root goes negative. This upper bound is the Chandrasekhar limit defined when the argument equals zero

This gives the final expression for the Chandrasekhar limit (expressed in terms of the Planck mass)

This expression is only approximate, but it does contain the essential physics and magnitude. This limit is on the order of a solar mass. A more realistic numerical calculation yields a limiting mass of about 1.4 times the mass of the Sun. For white dwarfs larger than this value, the electron degeneracy is insufficient to support the gravitational pressure, and the star will collapse to a neutron star or a black hole.


[1] The fact that Eddington coordinates removed the singularity at the Schwarzschild radius was first pointed out by Lemaitre in 1933.  A local observer passing through the Schwarzschild radius would experience no divergence in local properties, even though a distant observer would see that in-falling observer becoming length contracted and time dilated. This point of view of an in-falling observer was explained in 1958 by Finkelstein, who also pointed out that the Schwarzschild radius is an event horizon.

[2] William Herschel (1803), Account of the Changes That Have Happened, during the Last Twenty-Five Years, in the Relative Situation of Double-Stars; With an Investigation of the Cause to Which They Are Owing, Philosophical Transactions of the Royal Society of London 93, pp. 339–382 (Motion of binary stars)

[3] Boss, L. (1910). Preliminary General Catalogue of 6188 stars for the epoch 1900. Carnegie Institution of Washington. (Mass and radius of Sirius B)

[4] Eddington, A. S. (1927). Stars and Atoms. Clarendon Press. LCCN 27015694.

[5] Fowler, R. H. (1926). “On dense matter”. Monthly Notices of the Royal Astronomical Society 87: 114. Bibcode:1926MNRAS..87..114F. (Quantum mechanics of degenerate matter).

[6] Chandrasekhar, S. (1931). “The Maximum Mass of Ideal White Dwarfs”. The Astrophysical Journal 74: 81. Bibcode:1931ApJ….74…81C. doi:10.1086/143324. (Mass limit of white dwarfs).

[7] Kip Thorne (1994) Black Holes & Time Warps: Einstein’s Outrageous Legacy (Norton). pg. 174

[8] Oppenheimer was aware of Zwicky’s proposal because he had a joint appointment between Berkeley and Cal Tech.

[9] See Chapter 7, “The Lens of Gravity” in Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press, 2018).



George Green’s Theorem

For a thirty-year old miller’s son with only one year of formal education, George Green had a strange hobby—he read papers in mathematics journals, mostly from France.  This was his escape from a dreary life running a flour mill on the outskirts of Nottingham, England, in 1823.  The tall wind mill owned by his father required 24-hour attention, with farmers depositing their grain at all hours and the mechanisms and sails needing constant upkeep.  During his one year in school when he was eight years old he had become fascinated by maths, and he had nurtured this interest after leaving school one year later, stealing away to the top floor of the mill to pore over books he scavenged, devouring and exhausting all that English mathematics had to offer.  By the time he was thirty, his father’s business had become highly successful, providing George with enough wages to become a paying member of the private Nottingham Subscription Library with access to the Transactions of the Royal Society as well to foreign journals.  This simple event changed his life and changed the larger world of mathematics.

French Analysis in England

George Green was born in Nottinghamshire, England.  No record of his birth exists, but he was baptized in 1793, which may be assumed to be the year of his birth.  His father was a baker in Nottingham, but the food riots of 1800 forced him to move outside of the city to the town of Sneinton, where he bought a house and built an industrial-scale windmill to grind flour for his business.  He prospered enough to send his eight-year old son to Robert Goodacre’s Academy located on Upper Parliament Street in Nottingham.  Green was exceptionally bright, and after one year in school he had absorbed most of what the Academy could teach him, including a smattering of Latin and Greek as well as French along with what simple math that was offered.  Once he was nine, his schooling was over, and he took up the responsibility of helping his father run the mill, which he did faithfully, though unenthusiastically, for the next 20 years.  As the milling business expanded, his father hired a mill manager that took part of the burden off George.  The manager had a daughter Jane Smith, and in 1824 she had her first child with Green.  Six more children were born to the couple over the following fifteen years, though they never married.

Without adopting any microscopic picture of how electric or magnetic fields are produced or how they are transmitted through space, Green could still derive rigorous properties that are independent of any details of the microscopic model.

            During the 20 years after leaving Goodacre’s Academy, Green never gave up learning what he could, teaching himself to read French readily as well as mastering English mathematics.  The 1700’s and early 1800’s had been a relatively stagnant period for English mathematics.  After the priority dispute between Newton and Leibniz over the invention of the calculus, English mathematics had become isolated from continental advances.  This was part snobbery, but also part handicap as the English school struggled with Newton’s awkward fluxions while the continental mathematicians worked with Leibniz’ more fruitful differential notation.  The French mathematicians in the early 1800’s were especially productive, including works by Lagrange, Laplace and Poisson.

            One block away from where Green lived stood the Free Grammar School overseen by headmaster John Topolis.  Topolis was a Cambridge graduate on a minor mission to update the teaching of mathematics in England, well aware that the advances on the continent were passing England by.  For instance, Topolis translated Laplace’s mathematically advanced Méchaniqe Celéste from French into English.  Topolis was also well aware of the work by the other French mathematicians and maintained an active scholarly output that eventually brought him back to Cambridge as Dean of Queen’s College in 1819 when Green was 26 years old.  There is no record whether Topolis and Green knew each other, but their close proximity and common interests point to a natural acquaintance.  One can speculate that Green may even have sought Topolis out, given his insatiable desire to learn more mathematics, and it is likely that Topolis would have introduced Green to the vibrant French school of mathematics.             

By the time Green joined the Nottingham Subscription Library, he must already have been well trained in basic mathematics, and membership in the library allowed him to request loans of foreign journals (sort of like Interlibrary Loan today).  With his library membership beginning in 1823, Green absorbed the latest advances in differential equations and must have begun forming a new viewpoint of the uses of mathematics in the physical sciences.  This was around the same time that he was beginning his family with Jane as well as continuing to run his fathers mill, so his mathematical hobby was relegated to the dark hours of the night.  Nonetheless, he made steady progress over the next five years as his ideas took rough shape and were refined until finally he took pen to paper, and this uneducated miller’s son began a masterpiece that would change the history of mathematics.

Essay on Mathematical Analysis of Electricity and Magnetism

By 1827 Green’s free-time hobby was about to bear fruit, and he took out a modest advertisement to announce its forthcoming publication.  Because he was an unknown, and unknown to any of the local academics (Topolis had already gone back to Cambridge), he chose vanity publishing and published out of pocket.   An Essay on the Application of Mathematical Analysis to the Theories of Electricity and Magnetism was printed in March of 1828, and there were 51 subscribers, mostly from among the members of the Nottingham Subscription Library who bought it at 7 shillings and 6 pence per copy, probably out of curiosity or sympathy rather than interest.  Few, if any, could have recognized that Green’s little essay contained several revolutionary elements.

Fig. 1 Cover page of George Green’s Essay

            The topic of the essay was not remarkable, treating mathematical problems of electricity and magnetism, which was in vogue at that time.  As background, he had read works by Cavendish, Poisson, Arago, Laplace, Fourier, Cauchy and Thomas Young (probably Young’s Course of Lectures on Natural Philosopy and the Mechanical Arts (1807)).  He paid close attention to Laplace’s treatment of celestial mechanics and gravitation which had obvious strong analogs to electrostatics and the Coulomb force because of the common inverse square dependence. 

            One radical contribution in Green’s essay was his introduction of the potential function—one of the first uses of the concept of a potential function in mathematical physics—and he gave it its modern name.  Others had used similar constructions, such as Euler [1], D’Alembert [2], Laplace[3] and Poisson [4], but the use had been implicit rather than explicit.  Green shifted the potential function to the forefront, as a central concept from which one could derive other phenomena.  Another radical contribution from Green was his use of the divergence theorem.  This has tremendous utility, because it relates a volume integral to a surface integral.  It was one of the first examples of how measuring something over a closed surface could determine a property contained within the enclosed volume.  Gauss’ law is the most common example of this, where measuring the electric flux through a closed surface determines the amount of enclosed charge.  Lagrange in 1762 [5] and Gauss in 1813 [6] had used forms of the divergence theorem in the context of gravitation, but Green applied it to electrostatics where it has become known as Gauss’ law and is one of the four Maxwell equations.  Yet another contribution was Green’s use of linear superposition to determine the potential of a continuous charge distribution, integrating the potential of a point charge over a continuous charge distribution.  This was equivalent to defining what is today called a Green’s function, which is a common method to solve partial differential equations.

            A subtle contribution of Green’s Essay, but no less influential, was his adoption of a mathematical approach to a physics problem based on the fundamental properties of the mathematical structure rather than on any underlying physical model.  Without adopting any microscopic picture of how electric or magnetic fields are produced or how they are transmitted through space, he could still derive rigorous properties that are independent of any details of the microscopic model.  For instance, the inverse square law of both electrostatics and gravitation is a fundamental property of the divergence theorem (a mathematical theorem) in three-dimensional space.  There is no need to consider what space is composed of, such as the many differing models of the ether that were being proposed around that time.  He would apply this same fundamental mathematical approach in his later career as a Cambridge mathematician to explain the laws of reflection and refraction of light.

George Green: Cambridge Mathematician

A year after the publication of the Essay, Green’s father died a wealthy man, his milling business having become very successful.  Green inherited the family fortune, and he was finally able to leave the mill and begin devoting his energy to mathematics.  Around the same time he began working on mathematical problems with the support of Sir Edward Bromhead.  Bromhead was a Nottingham peer who had been one of the 51 subscribers to Green’s published Essay.  As a graduate of Cambridge he was friends with Herschel, Babbage and Peacock, and he recognized the mathematical genius in this self-educated miller’s son.  The two men spent two years working together on a pair of publications, after which Bromhead used his influence to open doors at Cambridge.

            In 1832, at the age of 40, George Green enrolled as an undergraduate student in Gonville and Caius College at Cambridge.  Despite his concerns over his lack of preparation, he won the first-year mathematics prize.  In 1838 he graduated as fourth wrangler only two positions behind the future famous mathematician James Joseph Sylvester (1814 – 1897).  Based on his work he was elected as a fellow of the Cambridge Philosophical Society in 1840.  Green had finally become what he had dreamed of being for his entire life—a professional mathematician.

            Green’s later papers continued the analytical dynamics trend he had established in his Essay by applying mathematical principles to the reflection and refraction of light. Cauchy had built microscopic models of the vibrating ether to explain and derive the Fresnel reflection and transmission coefficients, attempting to understand the structure of ether.  But Green developed a mathematical theory that was independent of microscopic models of the ether.  He believed that microscopic models could shift and change as newer models refined the details of older ones.  If a theory depended on the microscopic interactions among the model constituents, then it too would need to change with the times.  By developing a theory based on analytical dynamics, founded on fundamental principles such as minimization principles and geometry, then one could construct a theory that could stand the test of time, even as the microscopic understanding changed.  This approach to mathematical physics was prescient, foreshadowing the geometrization of physics in the late 1800’s that would lead ultimately to Einsteins theory of General Relativity.

Green’s Theorem and Greens Function

Green died in 1841 at the age of 49, and his Essay was mostly forgotten.  Ten years later a young William Thomson (later Lord Kelvin) was graduating from Cambridge and about to travel to Paris to meet with the leading mathematicians of the age.  As he was preparing for the trip, he stumbled across a mention of Green’s Essay but could find no copy in the Cambridge archives.  Fortunately, one of the professors had a copy that he lent Thomson.  When Thomson showed the work to Liouville and Sturm it caused a sensation, and Thomson later had the Essay republished in Crelle’s journal, finally bringing the work and Green’s name into the mainstream.

            In physics and mathematics it is common to name theorems or laws in honor of a leading figure, even if the they had little to do with the exact form of the theorem.  This sometimes has the effect of obscuring the historical origins of the theorem.  A classic example of this is the naming of Liouville’s theorem on the conservation of phase space volume after Liouville, who never knew of phase space, but who had published a small theorem in pure mathematics in 1838, unrelated to mechanics, that inspired Jacobi and later Boltzmann to derive the form of Liouville’s theorem that we use today.  The same is true of Green’s Theorem and Green’s Function.  The form of the theorem known as Green’s theorem was first presented by Cauchy [7] in 1846 and later proved by Riemann [8] in 1851.  The equation is named in honor of Green who was one of the early mathematicians to show how to relate an integral of a function over one manifold to an integral of the same function over a manifold whose dimension differed by one.  This property is a consequence of the Generalized Stokes Theorem, of which the Kelvin-Stokes Theorem, the Divergence Theorem and Green’s Theorem are special cases.

Fig. 2 Green’s theorem and its relationship with the Kelvin-Stokes theorem, the Divergence theorem and the Generalized Stokes theorem (expressed in differential forms)

            Similarly, the use of Green’s function for the solution of partial differential equations was inspired by Green’s use of the superposition of point potentials integrated over a continuous charge distribution.  The Green’s function came into more general use in the late 1800’s and entered the mainstream of physics in the mid 1900’s [9].

Fig. 3 The application of Green’s function so solve a linear operator problem, and an example applied to Poisson’s equation.

[1] L. Euler, Novi Commentarii Acad. Sci. Petropolitanae , 6 (1761)

[2] J. d’Alembert, “Opuscules mathématiques” , 1 , Paris (1761)

[3] P.S. Laplace, Hist. Acad. Sci. Paris (1782)

[4] S.D. Poisson, “Remarques sur une équation qui se présente dans la théorie des attractions des sphéroïdes” Nouveau Bull. Soc. Philomathique de Paris , 3 (1813) pp. 388–392

[5] Lagrange (1762) “Nouvelles recherches sur la nature et la propagation du son” (New researches on the nature and propagation of sound), Miscellanea Taurinensia (also known as: Mélanges de Turin ), 2: 11 – 172

[6] C. F. Gauss (1813) “Theoria attractionis corporum sphaeroidicorum ellipticorum homogeneorum methodo nova tractata,” Commentationes societatis regiae scientiarium Gottingensis recentiores, 2: 355–378

[7] Augustin Cauchy: A. Cauchy (1846) “Sur les intégrales qui s’étendent à tous les points d’une courbe fermée” (On integrals that extend over all of the points of a closed curve), Comptes rendus, 23: 251–255.

[8] Bernhard Riemann (1851) Grundlagen für eine allgemeine Theorie der Functionen einer veränderlichen complexen Grösse (Basis for a general theory of functions of a variable complex quantity), (Göttingen, (Germany): Adalbert Rente, 1867

[9] Schwinger, Julian (1993). “The Greening of quantum Field Theory: George and I”: 10283. arXiv:hep-ph/9310283

The Wonderful World of Hamiltonian Maps

Hamiltonian systems are freaks of nature.  Unlike the everyday world we experience that is full of dissipation and inefficiency, Hamiltonian systems live in a world free of loss.  Despite how rare this situation is for us, this unnatural state happens commonly in two extremes: orbital mechanics and quantum mechanics.  In the case of orbital mechanics, dissipation does exist, most commonly in tidal effects, but effects of dissipation in the orbits of moons and planets takes eons to accumulate, making these systems effectively free of dissipation on shorter time scales.  Quantum mechanics is strictly free of dissipation, but there is a strong caveat: ALL quantum states need to be included in the quantum description.  This includes the coupling of discrete quantum states to their environment.  Although it is possible to isolate quantum systems to a large degree, it is never possible to isolate them completely, and they do interact with the quantum states of their environment, if even just the black-body radiation from their container, and even if that container is cooled to milliKelvins.  Such interactions involve so many degrees of freedom, that it all behaves like dissipation.  The origin of quantum decoherence, which poses such a challenge for practical quantum computers, is the entanglement of quantum systems with their environment.

Liouville’s theorem plays a central role in the explanation of the entropy and ergodic properties of ideal gases, as well as in Hamiltonian chaos.

Liouville’s Theorem and Phase Space

A middle ground of practically ideal Hamiltonian mechanics can be found in the dynamics of ideal gases. This is the arena where Maxwell and Boltzmann first developed their theories of statistical mechanics using Hamiltonian physics to describe the large numbers of particles.  Boltzmann applied a result he learned from Jacobi’s Principle of the Last Multiplier to show that a volume of phase space is conserved despite the large number of degrees of freedom and the large number of collisions that take place.  This was the first derivation of what is today known as Liouville’s theorem.

Close-up of the Lozi Map with B = -1 and C = 0.5.

In 1838 Joseph Liouville, a pure mathematician, was interested in classes of solutions of differential equations.  In a short paper, he showed that for one class of differential equation one could define a property that remained invariant under the time evolution of the system.  This purely mathematical paper by Liouville was expanded upon by Jacobi, who was a major commentator on Hamilton’s new theory of dynamics, contributing much of the mathematical structure that we associate today with Hamiltonian mechanics.  Jacobi recognized that Hamilton’s equations were of the same class as the ones studied by Liouville, and the conserved property was a product of differentials.  In the mid-1800’s the language of multidimensional spaces had yet to be invented, so Jacobi did not recognize the conserved quantity as a volume element, nor the space within which the dynamics occurred as a space.  Boltzmann recognized both, and he was the first to establish the principle of conservation of phase space volume. He named this principle after Liouville, even though it was actually Boltzmann himself who found its natural place within the physics of Hamiltonian systems [1].

Liouville’s theorem plays a central role in the explanation of the entropy of ideal gases, as well as in Hamiltonian chaos.  In a system with numerous degrees of freedom, a small volume of initial conditions is stretched and folded by the dynamical equations as the system evolves.  The stretching and folding is like what happens to dough in a bakers hands.  The volume of the dough never changes, but after a long time, a small spot of food coloring will eventually be as close to any part of the dough as you wish.  This analogy is part of the motivation for ergodic systems, and this kind of mixing is characteristic of Hamiltonian systems, in which trajectories can diffuse throughout the phase space volume … usually.

Interestingly, when the number of degrees of freedom are not so large, there is a middle ground of Hamiltonian systems for which some initial conditions can lead to chaotic trajectories, while other initial conditions can produce completely regular behavior.  For the right kind of systems, the regular behavior can hem in the irregular behavior, restricting it to finite regions.  This was a major finding of the KAM theory [2], named after Kolmogorov, Arnold and Moser, which helped explain the regions of regular motion separating regions of chaotic motion as illustrated in Chirikov’s Standard Map.

Discrete Maps

Hamilton’s equations are ordinary continuous differential equations that define a Hamiltonian flow in phase space.  These equations can be solved using standard techniques, such as Runge-Kutta.  However, a much simpler approach for exploring Hamiltonian chaos uses discrete maps that represent the Poincaré first-return map, also known as the Poincaré section.  Testing that a discrete map satisfies Liouville’s theorem is as simple as checking that the determinant of the Floquet matrix is equal to unity.  When the dynamics are represented in a Poincaré plane, these maps are called area-preserving maps.

There are many famous examples of area-preserving maps in the plane.  The Chirikov Standard Map is one of the best known and is often used to illustrate KAM theory.  It is a discrete representation of a kicked rotater, while a kicked harmonic oscillator leads to the Web Map.  The Henon Map was developed to explain the orbits of stars in galaxies.  The Lozi Map is a version of the Henon map that is more accessible analytically.  And the Cat Map was devised by Vladimir Arnold to illustrate what is today called Arnold Diffusion.  All of these maps display classic signatures of (low-dimensional) Hamiltonian chaos with periodic orbits hemming in regions of chaotic orbits.

Chirikov Standard Map
Kicked rotater
Web Map
Kicked harmonic oscillator
Henon Map
Stellar trajectories in galaxies
Lozi Map
Simplified Henon map
Cat MapArnold Diffusion

Table:  Common examples of area-preserving maps.

Lozi Map

My favorite area-preserving discrete map is the Lozi Map.  I first stumbled on this map at the very back of Steven Strogatz’ wonderful book on nonlinear dynamics [3].  It’s one of the last exercises of the last chapter.  The map is particularly simple, but it leads to rich dynamics, both regular and chaotic.  The map equations are

which is area-preserving when |B| = 1.  The constant C can be varied, but the choice C = 0.5 works nicely, and B = -1 produces a beautiful nested structure, as shown in the figure.

Iterated Lozi map for B = -1 and C = 0.5.  Each color is a distinct trajectory.  Many regular trajectories exist that corral regions of chaotic trajectories.  Trajectories become more chaotic farther away from the center.

Python Code for the Lozi Map

"""
Created on Wed May  2 16:17:27 2018
@author: nolte
"""
import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt

B = -1
C = 0.5

np.random.seed(2)
plt.figure(1)

for eloop in range(0,100):

    xlast = np.random.normal(0,1,1)
    ylast = np.random.normal(0,1,1)

    xnew = np.zeros(shape=(500,))
    ynew = np.zeros(shape=(500,))
    for loop in range(0,500):
        xnew[loop] = 1 + ylast - C*abs(xlast)
        ynew[loop] = B*xlast
        xlast = xnew[loop]
        ylast = ynew[loop]
        
    plt.plot(np.real(xnew),np.real(ynew),'o',ms=1)
    plt.xlim(xmin=-1.25,xmax=2)
    plt.ylim(ymin=-2,ymax=1.25)
        
plt.savefig('Lozi')

References:

[1] D. D. Nolte, “The Tangled Tale of Phase Space”, Chapter 6 in Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford University Press, 2018)

[2] H. S. Dumas, The KAM Story: A Friendly Introduction to the Content, History, and Significance of Classical Kolmogorov-Arnold-Moser Theory (World Scientific, 2014)

[3] S. H. Strogatz, Nonlinear Dynamics and Chaos (WestView Press, 1994)

Top 10 Topics of Modern Dynamics

“Modern physics” in the undergraduate physics curriculum has been monopolized, on the one hand, by quantum mechanics, nuclear physics, particle physics and astrophysics. “Classical mechanics”, on the other hand, has been monopolized by Lagrangians and Hamiltonians.  While these are all admittedly interesting, the topics of modern dynamics that monopolize the time and actions of most physics-degree holders, as they work in high-tech start-ups, established technology companies, or on Wall Street, are not to be found.  These are the topics of nonlinear dynamics, chaos theory, complex networks, finance, evolutionary dynamics and neural networks, among others.

Cover

There is a growing awareness that the undergraduate physics curriculum needs to be reinvigorated to make a physics degree relevant to the modern workplace.  To that end, I am listing my top 10 topics of modern dynamics that can form the foundation of a revamped upper-division (junior level) mechanics course.  Virtually all of these topics were once reserved for graduate-student-level courses, but all can be introduced to undergraduates in simple and intuitive ways without the need for advanced math.

1) Phase Space

The key change in perspective for modern dynamics that differentiates it from classical dynamics is the emphasis on the set of all possible trajectories that fill a “space” rather than emphasizing single trajectories defined by given initial conditions.  Rather than study the motion of one rock thrown from a cliff top, modern dynamics studies an infinity of rocks thrown from every possible point and with every possible velocity.  The space that contains this infinity of trajectories is known as phase space (or more generally state space).  The equation of motion in state space becomes the dynamical flow, replacing Newton’s second law as the central mathematical structure of physics.  Modern dynamics studies the properties of phase space rather than the properties of single trajectories, and makes rigorous and unique conclusions about classes of possible motions.  This emphasis on classes of behavior is more general and more universal and more powerful, while also providing a fundamental “visual language” with which to describe the complex physics of complex systems.

2) Metric Space

The Cartesian coordinate plane that we were all taught in high school tends to dominate our thinking, biasing us towards linear flat geometries.  Yet most dynamics do not take place in such simple Cartesian spaces.  A case in point, virtually every real-world dynamics problem has constraints that confine the motion to a surface.  Furthermore, the number of degrees of freedom of a dynamical system usually exceed our common 3-space, expanding to hundreds or even to thousands of dimensions.  The surfaces of constraint are hypersurfaces of high dimensions (known as manifolds) and are almost certainly not flat hyperplanes. This daunting prospect of high-dimensional warped spaces can be surprisingly simplified through the concept of Bernhard Riemann’s “metric space”.  Understanding the geometry of a metric space can be as simple as applying Pythagoras’ Theorem to sets of coordinates.  For instance, the metric tensor can be taught and used without requiring students to know anything of tensor calculus.  At the same time, it provides a useful tool for understanding dynamical patterns in phase space as well as orbits around black holes.

3) Invariants

Introductory physics classes emphasize the conservation of energy, linear momentum and angular momentum as if they are special cases.  Yet there is a grand structure that yields a universal set of conservation laws: integrable Hamiltonian systems.  An integrable system is one for which there are as many invariants of motion as there are degrees of freedom.  Amazingly, these conservation laws can all be captured by a single procedure known as (canonical) transformation to action-angle coordinates.  When expressed in action-angle form, these Hamiltonians take on extremely simple expressions.  They are also the starting point for the study of perturbations when small nonintegrable terms are added to the Hamiltonian.  As the perturbations grow, this provides one doorway to the emergence of chaos.

4) Chaos theory

“Chaos theory” is the more popular title for what is generally called “nonlinear dynamics”.  Nonlinear dynamics takes place in state space when the dynamical flow equations have terms that algebraically are products of variables.  One important distinction between chaos theory and nonlinear dynamics is the occurrence of unpredictability that can emerge in the dynamics when the number of variables is equal to three or higher.  The equations, and the resulting dynamics, are still deterministic, but the trajectories are incredibly sensitive to initial conditions (SIC).  In addition, the dynamical trajectories can relax to a submanifold of the original state space known as a strange attractor that typically is a fractal structure.

5) Synchronization

One of the central paradigms of nonlinear dynamics is the autonomous oscillator.  Unlike the harmonic oscillator that eventually decays due to friction, autonomous oscillators are steady-state oscillators that convert steady energy input into oscillatory behavior.  A prime example is the pendulum clock that converts the steady weight of a hanging mass into a sustained oscillation.  When two autonomous oscillators (that normally oscillator at slightly different frequencies) are coupled weakly together, they can synchronize to the same frequency.   This effect was discovered by Christiaan Huygens when he observed two pendulum clocks hanging next to each other on a wall synchronize the swings of their pendula.  Synchronization is a central paradigm in modern dynamics for several reasons.  First, it demonstrates the emergence of order when a collective behavior emerges from a collection of individual systems (this phenomenon of emergence is one of the fundamental principles of complex system science).  Second, synchronized systems include such critical systems as the beating heart and the thinking brain.  Third, synchronization becomes a useful tool to explore coupled systems that have a large number of linked subsystems, as in networks of nodes.

6) Network Dynamics

Networks have become one of the driving forces of our modern interconnected society.  The structure of networks, the dynamics of nodes in networks, and the dynamic growth of networks are all coming into focus as we live our lives in multiple interconnected webs.  Dynamics on networks include problems like diffusion and the spread of infection and connect with topics of percolation theory and critical phenomenon.  Nonlinear dynamics on networks provide key opportunities and examples to study complex interacting systems.

7) Neural Networks

Perhaps the most enigmatic network is the network of neurons in the brain.  The emergence of intelligence and of sentience is one of the greatest scientific questions.  At a much simpler level, the nonlinear dynamics of small numbers of neurons display the properties of autonomous oscillators and synchronization, while larger sets of neurons become interconnected into dynamic networks.  The dynamics of neurons and of neural networks is a  key topic in modern dynamics.  Not only can the physics of the networks be studied, but neural networks become tools for studying other complex systems.

8) Evolutionary Dynamics

The emergence of life and the evolution of species stands as another of the greatest scientific questions of our day.  Although this topic traditionally is studied by the biological sciences (and mathematical biology), physics has a surprising lot to say on the topic.  The dynamics of evolution can be captured in the same types of nonlinear flows that live in state space.  For instance, population dynamics can be described as a large ensemble of interacting individuals that are born, flourish and die dependent on their environment and on their complicated interactions with other members in their ecosystem.  These types of problems have state spaces of extremely high dimension far beyond what we can visualize.  Yet the emergence of structure and of patterns from the complex dynamics helps to reduce the complexity, as do conceptual metaphors like evolutionary fitness landscapes.

9) Economic Dynamics

A non-negligible fraction of both undergraduate and graduate physics degree holders end up on Wall Street or in related industries.  This is partly because physicists are numerically fluent while also possessing sound intuition.  Therefore, economic dynamics is a potentially valuable addition to the modern dynamics curriculum and easily expressed using the concepts of dynamical flows and state space.  Both microeconomics (business competition, business cycles) and macroeconomics (investment and savings, liquidity and money, inflation, unemployment) can be described and analyzed using mathematical flows that are the central toolkit of modern dynamics.

10) Relativity

Special relativity is a common topic in the current upper-division physics curriculum, while general relativity is viewed as too difficult to expose undergraduates to.  This is mostly an artificial division, because Einstein’s “happiest thought” occurred when he realized that an observer in free fall is in a force-free (inertial) frame.  The equivalence principle, that states that a frame in uniform acceleration is indistinguishable from a stationary frame in a uniform gravitational field, opens a wide door that connects special relativity to general relativity.  In an undergraduate course on modern dynamics, the metric tensor (described above) is introduced in simple terms, providing the foundation to develop Minkowski spacetime, and the next natural extension is to warped spacetime—all at the simple level of linear algebra combined with partial differentiation.  General relativity ties in many of the principles of the modern dynamics curriculum (dynamical flows, state space, metric space, invariants, nonlinear dynamics), and the students can simulate orbits around black holes with ease.  I have been teaching General Relativity to undergraduates for over ten years now, and it is a highlight of the course.

Introduction to Modern Dynamics

For further reading and more details, these top 10 topics of modern dynamics are defined and explored in the undergraduate physics textbook “Introduction to Modern Dynamics: Chaos, Networks, Space and Time” published by Oxford University Press (2015).  This textbook is designed for use in a two-semester junior-level mechanics course.  It introduces the topics of modern dynamics, while still presenting traditional materials that the students need for their physics GREs.

 

Dark Matter Mysteries

There is more to the Universe than meets the eye—way more. Over the past quarter century, it has become clear that all the points of light in the night sky, the stars, the Milky Way, the nubulae, all the distant galaxies, when added up with the nonluminous dust, constitute only a small fraction of the total energy density of the Universe. In fact, “normal” matter, like the stuff of which we are made—star dust—contributes only 4% to everything that is. The rest is something else, something different, something that doesn’t show up in the most sophisticated laboratory experiments, not even the Large Hadron Collider [1]. It is unmeasurable on terrestrial scales, and even at the scale of our furthest probe—the Voyager I spacecraft that left our solar system several years ago—there have been no indications of deviations from Newton’s law of gravity. To the highest precision we can achieve, it is invisible and non-interacting on any scale smaller than our stellar neighborhood. Perhaps it can never be detected in any direct sense. If so, then how do we know it is there? The answer comes from galactic trajectories. The motions in and of galaxies have been, and continue to be, the principal laboratory for the investigation of  cosmic questions about the dark matter of the universe.

Today, the nature of Dark Matter is one of the greatest mysteries in physics, and the search for direct detection of Dark Matter is one of physics’ greatest pursuits.

 

Island Universes

The nature of the Milky Way was a mystery through most of human history. To the ancient Greeks it was the milky circle (γαλαξίας κύκλος , pronounced galaktikos kyklos) and to the Romans it was literally the milky way (via lactea). Aristotle, in his Meteorologica, briefly suggested that the Milky Way might be composed of a large number of distant stars, but then rejected that idea in favor of a wisp, exhaled like breath on a cold morning, from the stars. The Milky Way is unmistakable on a clear dark night to anyone who looks up, far away from city lights. It was a constant companion through most of human history, like the constant stars, until electric lights extinguished it from much of the world in the past hundred years. Geoffrey Chaucer, in his Hous of Fame (1380) proclaimed “See yonder, lo, the Galaxyë Which men clepeth the Milky Wey, For hit is whyt.” (See yonder, lo, the galaxy which men call the Milky Way, for it is white.).

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Hubble image of one of the galaxies in the Coma Cluster of galaxies that Fritz Zwicky used to announce that the universe contained a vast amount of dark matter.

Aristotle was fated, again, to be corrected by Galileo. Using his telescope in 1610, Galileo was the first to resolve a vast field of individual faint stars in the Milky Way. This led Emmanual Kant, in 1755, to propose that the Milky Way Galaxy was a rotating disk of stars held together by Newtonian gravity like the disk of the solar system, but much larger. He went on to suggest that the faint nebulae might be other far distant galaxies, which he called “island universes”. The first direct evidence that nebulae were distant galaxies came in 1917 with the observation of a supernova in the Andromeda Galaxy by Heber Curtis. Based on the brightness of the supernova, he estimated that the Andromeda Galaxy was over a million light years away, but uncertainty in the distance measurement kept the door open for the possibility that it was still part of the Milky Way, and hence the possibility that the Milky Way was the Universe.

The question of the nature of the nebulae hinged on the problem of measuring distances across vast amounts of space. By line of sight, there is no yard stick to tell how far away something is, so other methods must be used. Stellar parallax, for instance, can gauge the distance to nearby stars by measuring slight changes in the apparent positions of the stars as the Earth changes its position around the Sun through the year. This effect was used successfully for the first time in 1838 by Fredrich Bessel, and by the year 2000 more than a hundred thousand stars had their distances measured using stellar parallax. Recent advances in satellite observatories have extended the reach of stellar parallax to a distance of about 10,000 light years from the Sun, but this is still only a tenth of the diameter of the Milky Way. To measure distances to the far side of our own galaxy, or beyond, requires something else.

Because of Henrietta Leavitt

In 1908 Henrietta Leavitt, working at the Harvard Observatory as one of the famous female “computers”, discovered that stars whose luminosities oscillate with a steady periodicity, stars known as Cepheid variables, have a relationship between the period of oscillation and the average luminosity of the star [2]. By measuring the distance to nearby Cepheid variables using stellar parallax, the absolute brightness of the Cepheid could be calibrated, and the Cepheid could then be used as “standard candles”. This meant that by observing the period of oscillation and the brightness of a distant Cepheid, the distance to the star could be calculated. Edwin Hubble (1889 – 1953), working at the Mount Wilson observatory in Passedena CA, observed Cepheid variables in several of the brightest nebulae in the night sky. In 1925 he announced his observation of individual Cepheid variables in Andromeda and calculated that Andromeda was more than a million light years away, more than 10 Milky Way diameters (the actual number is about 25 Milky Way diameters). This meant that Andromeda was a separate galaxy and that the Universe was made of more than just our local cluster of stars. Once this door was opened, the known Universe expanded quickly up to a hundred Milky Way diameters as Hubble measured the distances to scores of our neighboring galaxies in the Virgo galaxy cluster. However, it was more than just our knowledge of the universe that was expanding.

Armed with measurements of galactic distances, Hubble was in a unique position to relate those distances to the speeds of the galaxies by combining his distance measurements with spectroscopic observations of the light spectra made by other astronomers. These galaxy emission spectra could be used to measure the Doppler effect on the light emitted by the stars of the galaxy. The Doppler effect, first proposed by Christian Doppler (1803 – 1853) in 1843, causes the wavelength of emitted light to be shifted to the red for objects receding from an observer, and shifted to the blue for objects approaching an observer. The amount of spectral shift is directly proportional the the object’s speed. Doppler’s original proposal was to use this effect to measure the speed of binary stars, which is indeed performed routinely today by astronomers for just this purpose, but in Doppler’s day spectroscopy was not precise enough to accomplish this. However, by the time Hubble was making his measurements, optical spectroscopy had become a precision science, and the Doppler shift of the galaxies could be measured with great accuracy. In 1929 Hubble announced the discovery of a proportional relationship between the distance to the galaxies and their Doppler shift. What he found was that the galaxies [3] are receding from us with speeds proportional to their distance [4]. Hubble himself made no claims at that time about what these data meant from a cosmological point of view, but others quickly noted that this Hubble effect could be explained if the universe were expanding.

Einstein’s Mistake

The state of the universe had been in doubt ever since Heber Curtis observed the supernova in the Andromeda galaxy in 1917. Einstein published a paper that same year in which he sought to resolve a problem that had appeared in the solution to his field equations. It appeared that the universe should either be expanding or contracting. Because the night sky literally was the firmament, it went against the mentality of the times to think of the universe as something intrinsically unstable, so Einstein fixed it with an extra term in his field equations, adding something called the cosmological constant, denoted by the Greek lambda (Λ). This extra term put the universe into a static equilibrium, and Einstein could rest easy with his firm trust in the firmament. However, a few years later, in 1922, the Russian physicist and mathematician Alexander Friedmann (1888 – 1925) published a paper that showed that Einstein’s static equilibrium was actually unstable, meaning that small perturbations away from the current energy density would either grow or shrink. This same result was found independently by the Belgian astronomer Georges Lemaître in 1927, who suggested that not only was the universe  expanding, but that it had originated in a singular event (now known as the Big Bang). Einstein was dismissive of Lemaître’s proposal and even quipped “Your calculations are correct, but your physics is atrocious.” [5] But after Hubble published his observation on the red shifts of galaxies in 1929, Lemaître pointed out that the redshifts would be explained by an expanding universe. Although Hubble himself never fully adopted this point of view, Einstein immediately saw it for what it was—a clear and simple explanation for a basic physical phenomenon that he had foolishly overlooked. Einstein retracted his cosmological constant in embarrassment and gave his support to Lemaître’s expanding universe. Nonetheless, Einstein’s physical intuition was never too far from the mark, and the cosmological constant has been resurrected in recent years in the form of Dark Energy. However, something else, both remarkable and disturbing, reared its head in the intervening years—Dark Matter.

Fritz Zwicky: Gadfly Genius

It is difficult to write about important advances in astronomy and astrophysics of the 20th century without tripping over Fritz Zwicky. As the gadfly genius that he was, he had a tendency to shoot close to the mark, or at least some of his many crazy ideas tended to be right. He was also in the right place at the right time, at the Mt. Wilson observatory nearby Cal Tech with regular access the World’s largest telescope. Shortly after Hubble proved that the nebulae were other galaxies and used Doppler shifts to measure their speeds, Zwicky (with his assistant Baade) began a study of as many galactic speeds and distances as they could. He was able to construct a three-dimensional map of the galaxies in the relatively nearby Coma galaxy cluster, together with their velocities. He then deduced that the galaxies in this isolated cluster were gravitational bound to each other, performing a whirling dance in each others thrall, like stars in globular star clusters in our Milky Way. But there was a serious problem.

Star clusters display average speeds and average gravitational potentials that are nicely balanced, a result predicted from a theorem of mechanics that was named the Virial Theorem by Rudolf Clausius in 1870. The Virial Theorem states that the average kinetic energy of a system of many bodies is directly related to the average potential energy of the system. By applying the Virial Theorem to the galaxies of the Coma cluster, Zwicky found that the dynamics of the galaxies were badly out of balance. The galaxy kinetic energies were far too fast relative to the gravitational potential—so fast, in fact, that the galaxies should have flown off away from each other and not been bound at all. To reconcile this discrepancy of the galactic speeds with the obvious fact that the galaxies were gravitationally bound, Zwicky postulated that there was unobserved matter present in the cluster that supplied the missing gravitational potential. The amount of missing potential was very large, and Zwicky’s calculations predicted that there was 400 times as much invisible matter, which he called “dark matter”, as visible. With his usual flare for the dramatic, Zwicky announced his findings to the World in 1933, but the World shrugged— after all, it was just Zwicky.

Nonetheless, Zwicky’s and Baade’s observations of the structure of the Coma cluster, and the calculations using the Virial Theorem, were verified by other astronomers. Something was clearly happening in the Coma cluster, but other scientists and astronomers did not have the courage or vision to make the bold assessment that Zwicky had. The problem of the Coma cluster, and a growing number of additional galaxy clusters that have been studied during the succeeding years, was to remain a thorn in the side of gravitational theory through half a century, and indeed remains a thorn to the present day. It is an important clue to a big question about the nature of gravity, which is arguably the least understood of the four forces of nature.

Vera Rubin: Galaxy Rotation Curves

Galactic clusters are among the largest coherent structures in the observable universe, and there are many questions about their origin and dynamics. Smaller gravitationally bound structures that can be handled more easily are individual galaxies themselves. If something important was missing in the dynamics of galactic clusters, perhaps the dynamics of the stars in individual galaxies could help shed light on the problem. In the late 1960’s and early 1970’s Vera Rubin at the Carnegie Institution of Washington used newly developed spectrographs to study the speeds of stars in individual galaxies. From simple Newtonian dynamics it is well understood that the speed of stars as a function of distance from the galactic center should increase with increasing distance up to the average radius of the galaxy, and then should decrease at larger distances. This trend in speed as a function of radius is called a rotation curve. As Rubin constructed the rotation curves for many galaxies, the increase of speed with increasing radius at small radii emerged as a clear trend, but the stars farther out in the galaxies were all moving far too fast. In fact, they are moving so fast that they exceeded escape velocity and should have flown off into space long ago. This disturbing pattern was repeated consistently in one rotation curve after another.

A simple fix to the problem of the rotation curves is to assume that there is significant mass present in every galaxy that is not observable either as luminous matter or as interstellar dust. In other words, there is unobserved matter, dark matter, in all galaxies that keeps all their stars gravitationally bound. Estimates of the amount of dark matter needed to fix the velocity curves is about five times as much dark matter as observable matter. This is not the same factor of 400 that Zwicky had estimated for the Coma cluster, but it is still a surprisingly large number. In short, 80% of the mass of a galaxy is not normal. It is neither a perturbation nor an artifact, but something fundamental and large. In fact, there is so much dark matter in the Universe that it must have a major effect on the overall curvature of space-time according to Einstein’s field equations. One of the best probes of the large-scale structure of the Universe is the afterglow of the Big Bang, known as the cosmic microwave background (CMB).

The Big Bang

The Big Bang was incredibly hot, but as the Universe expanded, its temperature cooled. About 379,000 years after the Big Bang, the Universe cooled sufficiently that the electron-nucleon plasma that filled space at that time condensed primarily into hydrogen. Plasma is charged and hence is opaque to photons.  Hydrogen, on the other hand, is neutral and transparent. Therefore, when the hydrogen condensed, the thermal photons suddenly flew free, unimpeded, and have continued unimpeded, continuing to cool, until today the thermal glow has reached about three degrees above absolute zero. Photons in thermal equilibrium with this low temperature have an average wavelength of a few millimeters corresponding to microwave frequencies, which is why the afterglow of the Big Bang got its CMB name.

The CMB is amazingly uniform when viewed from any direction in space, but it is not perfectly uniform. At the level of 0.005 percent, there are variations in the temperature depending on the location on the sky. These fluctuations in background temperature are called the CMB anisotropy, and they play an important role helping to interpret current models of the Universe. For instance, the average angular size of the fluctuations is related to the overall curvature of the Universe. This is because in the early Universe not all parts of it were in communication with each other because of the finite size and the finite speed of light. This set an original spatial size to thermal discrepancies. As the Universe continued to expand, the size of the regional variations expanded with it, and the sizes observed today would appear larger or smaller, depending on how the universe is curved. Therefore, to measure the energy density of the Universe, and hence to find its curvature, required measurements of the CMB temperature that were accurate to better than a part in 10,000.

 

Andrew Lange and Paul Richards: The Lambda and the Omega

In graduate school at Berkeley in 1982, my first graduate research assistantship was in the group of Paul Richards, one of the world leaders in observational cosmology. One of his senior graduate students at the time, Andrew Lange, was sharp and charismatic and leading an ambitious project to measure the cosmic background radiation on an experiment borne by a Japanese sounding rocket. My job was to create a set of far-infrared dichroic beamsplitters for the spectrometer.   A few days before launch, a technician noticed that the explosive bolts on the rocket nose-cone had expired. When fired, these would open the cone and expose the instrument at high altitude to the CMB. The old bolts were duly replaced with fresh ones. On launch day, the instrument and the sounding rocket worked perfectly, but the explosive bolts failed to fire, and the spectrometer made excellent measurements of the inside of the nose cone all the way up and all the way down until it sank into the Pacific Ocean. I left Paul’s comology group for a more promising career in solid state physics under the direction of Eugene Haller and Leo Falicov, but Paul and Andrew went on to great fame with high-altitude balloon-borne experiments that flew at 40,000 feet, above most of the atmosphere, to measure the CMB anisotropy.

By the late nineties, Andrew was established as a professor at Cal Tech. He was co-leading an experiment called BOOMerANG that flew a high-altitude balloon around Antarctica, while Paul was leading an experiment called MAXIMA that flew a balloon from Palastine, Texas. The two experiments had originally been coordinated together, but operational differences turned the former professor/student team into competitors to see who would be the first to measure the shape of the Universe through the CMB anisotropy.  BOOMerANG flew in 1997 and again in 1998, followed by MAXIMA that flew in 1998 and again in 1999. In early 2000, Andrew and the BOOMerANG team announced that the Universe was flat, confirmed quickly by an announcement by MAXIMA [BoomerMax]. This means that the energy density of the Universe is exactly critical, and there is precisely enough gravity to balance the expansion of the Universe. This parameter is known as Omega (Ω).  What was perhaps more important than this discovery was the announcement by Paul’s MAXIMA team that the amount of “normal” baryonic matter in the Universe made up only about 4% of the critical density. This is a shockingly small number, but agreed with predictions from Big Bang nucleosynthesis. When combined with independent measurements of Dark Energy known as Lambda (Λ), it also meant that about 25% of the energy density of the Universe is made up of Dark Matter—about five times more than ordinary matter. Zwicky’s Dark Matter announcement of 1933, virtually ignored by everyone, had been 75 years ahead of its time [6].

Dark Matter Pursuits

Today, the nature of Dark Matter is one of the greatest mysteries in physics, and the search for direct detection of Dark Matter is one of physics’ greatest pursuits. The indirect evidence for Dark Matter is incontestable—the CMB anisotropy, matter filaments in the early Universe, the speeds of galaxies in bound clusters, rotation curves of stars in Galaxies, gravitational lensing—all of these agree and confirm that most of the gravitational mass of the Universe is Dark. But what is it? The leading idea today is that it consists of weakly interacting particles, called cold dark matter (CDM). The dark matter particles pass right through you without ever disturbing a single electron. This is unlike unseen cosmic rays that are also passing through your body at the rate of several per second, leaving ionized trails like bullet holes through your flesh. Dark matter passes undisturbed through the entire Earth. This is not entirely unbelievable, because neutrinos, which are part of “normal” matter, also mostly pass through the Earth without interaction. Admittedly, the physics of neutrinos is not completely understood, but if ordinary matter can interact so weakly, then dark matter is just more extreme and perhaps not so strange. Of course, this makes detection of dark matter a big challenge. If a particle exists that won’t interact with anything, then how would you ever measure it? There are a lot of clever physicists with good ideas how to do it, but none of the ideas are easy, and none have worked yet.

[1] As of the writing of this chapter, Dark Matter has not been observed in particle form, but only through gravitational effects at large (galactic) scales.

[2] Leavitt, Henrietta S. “1777 Variables in the Magellanic Clouds”. Annals of Harvard College Observatory. LX(IV) (1908) 87-110

[3] Excluding the local group of galaxies that include Andromeda and Triangulum that are gravitationally influenced by the Milky Way.

[4] Hubble, Edwin (1929). “A relation between distance and radial velocity among extra-galactic nebulae”. PNAS 15 (3): 168–173.

[5] Deprit, A. (1984). “Monsignor Georges Lemaître”. In A. Barger (ed). The Big Bang and Georges Lemaître. Reidel. p. 370.

[6] I was amazed to read in Science magazine in 2004 or 2005, in a section called “Nobel Watch”, that Andrew Lange was a candidate for the Nobel Prize for his work on BoomerAng.  Around that same time I invited Paul Richards to Purdue to give our weekly physics colloquium.  There was definitely a buzz going around that the BoomerAng and MAXIMA collaborations were being talked about in Nobel circles.  The next year, the Nobel Prize of 2006 was indeed awarded for work on the Cosmic Microwave Background, but to Mather and Smoot for their earlier work on the COBE satellite.

How to Weave a Tapestry from Hamiltonian Chaos

While virtually everyone recognizes the famous Lorenz “Butterfly”, the strange attractor  that is one of the central icons of chaos theory, in my opinion Hamiltonian chaos generates far more interesting patterns. This is because Hamiltonians conserve phase-space volume, stretching and folding small volumes of initial conditions as they evolve in time, until they span large sections of phase space. Hamiltonian chaos is usually displayed as multi-color Poincaré sections (also known as first-return maps) that are created when a set of single trajectories, each represented by a single color, pierce the Poincaré plane over and over again.

The archetype of all Hamiltonian systems is the harmonic oscillator.

MATLAB Handle Graphics

A Hamiltonian tapestry generated from the Web Map for K = 0.616 and q = 4.

Periodically-Kicked Hamiltonian

The classic Hamiltonian system, perhaps the archetype of all Hamiltonian systems, is the harmonic oscillator. The physics of the harmonic oscillator are taught in the most elementary courses, because every stable system in the world is approximated, to lowest order, as a harmonic oscillator. As the simplest dynamical system, one would think that it held no surprises. But surprisingly, it can create the most beautiful tapestries of color when pulsed periodically and mapped onto the Poincaré plane.

The Hamiltonian of the periodically kicked harmonic oscillator is converted into the Web Map, represented as an iterative mapping as

WebMap

There can be resonance between the sequence of kicks and the natural oscillator frequency such that α = 2π/q. At these resonances, intricate web patterns emerge. The Web Map produces a web of stochastic layers when plotted on an extended phase plane. The symmetry of the web is controlled by the integer q, and the stochastic layer width is controlled by the perturbation strength K.

MATLAB Handle Graphics

A tapestry for q = 6.

Web Map Python Program

Iterated maps are easy to implement in code.  Here is a simple Python code to generate maps of different types.  You can play with the coupling constant K and the periodicity q.  For small K, the tapestries are mostly regular.  But as the coupling K increases, stochastic layers emerge.  When q is a small even number, tapestries of regular symmetric are generated.  However, when q is an odd small integer, the tapestries turn into quasi-crystals.

#!/usr/bin/env python3
# -*- coding: utf-8 -*-
“”
@author: nolte
“””

import numpy as np
from scipy import integrate
from matplotlib import pyplot as plt
plt.close(‘all’)
phi = (1+np.sqrt(5))/2
K = 1-phi     # (0.618, 4) (0.618,5) (0.618,7) (1.2, 4)
q = 4         # 4, 5, 6, 7
alpha = 2*np.pi/q

np.random.seed(2)
plt.figure(1)
for eloop in range(0,1000):

xlast = 50*np.random.random()
ylast = 50*np.random.random()

xnew = np.zeros(shape=(300,))
ynew = np.zeros(shape=(300,))

for loop in range(0,300):

xnew[loop] = (xlast + K*np.sin(ylast))*np.cos(alpha) + ylast*np.sin(alpha)
ynew[loop] = -(xlast + K*np.sin(ylast))*np.sin(alpha) + ylast*np.cos(alpha)

xlast = xnew[loop]
ylast = ynew[loop]

plt.plot(np.real(xnew),np.real(ynew),’o’,ms=1)
plt.xlim(xmin=-60,xmax=60)
plt.ylim(ymin=-60,ymax=60)

plt.title(‘WebMap’)
plt.savefig(‘WebMap’)

 

References and Further Reading

D. D. Nolte, Introduction to Modern Dynamics: Chaos, Networks, Space and Time (Oxford, 2015)

G. M. Zaslavsky,  Hamiltonian chaos and fractional dynamics. (Oxford, 2005)

 

 

 

Wave-Particle Duality and Hamilton’s Physics

Wave-particle duality was one of the greatest early challenges to quantum physics, partially clarified by Bohr’s Principle of Complementarity, but never easily grasped even today.  Yet long before Einstein proposed the indivisible quantum  of light (later to be called the photon by the chemist Gilbert Lewis), wave-particle duality was firmly embedded in the foundations of the classical physics of mechanics.

Light led the way to mechanics more than once in the history of physics.

 

Willebrord Snel van Royen

The Dutch physicist Willebrord Snel van Royen in 1621 derived an accurate mathematical description of the refraction of beams of light at a material interface in terms of sine functions, but he did not publish.  Fifteen years later, as Descartes was looking for an example to illustrate his new method of analytic geometry, he discovered the same law, unaware of Snel’s prior work.  In France the law is known as the Law of Descartes.  In the Netherlands (and much of the rest of the world) it is known as Snell’s Law.  Both Snell and Descartes based their work on Newton’s corpuscles of light.  The brilliant Fermat adopted corpuscles when he developed his principle of least time to explain the law of Descartes in 1662.  Yet Fermat was forced to assume that the corpuscles traveled slower in the denser material even though it was generally accepted that light should travel faster in denser media, just as sound did.  Seventy-five years later, Maupertuis continued the tradition when he developed his principle of least action and applied it to light corpuscles traveling faster through denser media, just as Descartes had prescribed.

HuygensParticle-02

The wave view of Snell’s Law (on the left). The source resides in the medium with higher speed. As the wave fronts impinge on the interface to a medium with lower speed, the wave fronts in the slower medium flatten out, causing the ray perpendicular to the wave fronts to tilt downwards. The particle view of Snell’s Law (on the right). The momentum of the particle in the second medium is larger than in the first, but the transverse components of the momentum (the x-components) are conserved, causing a tilt downwards of the particle’s direction as it crosses the interface. [i]

Maupertuis’ paper applying the principle of least action to the law of Descartes was a critical juncture in the development of dynamics.  His assumption of faster speeds in denser material was wrong, but he got the right answer because of the way he defined action for light.  Encouraged by the success of his (incorrect) theory, Maupertuis extended the principle of least action to mechanical systems, and this time used the right theory to get the right answers.  Despite Maupertuis’ misguided aspirations to become a physicist of equal stature to Newton, he was no mathematician, and he welcomed (and  somewhat appropriated) the contributions of Leonid Euler on the topic, who established the mathematical foundations for the principle of least action.  This work, in turn, attracted the attention of the Italian mathematician Lagrange, who developed a general new approach (Lagrangian mechanics) to mechanical systems that included the principle of least action as a direct consequence of his equations of motion.  This was the first time that light led the way to classical mechanics.  A hundred years after Maupertuis, it was time again for light to lead to the way to a deeper mechanics known as Hamiltonian mechanics.

Young Hamilton

William Rowland Hamilton (1805—1865) was a prodigy as a boy who knew parts of thirteen languages by the time he was thirteen years old. These were Greek, Latin, Hebrew, Syriac, Persian, Arabic, Sanskrit, Hindoostanee, Malay, French, Italian, Spanish, and German. In 1823 he entered Trinity College of Dublin University to study science. In his second and third years, he won the University’s top prizes for Greek and for mathematical physics, a run which may have extended to his fourth year—but he was offered the position of Andrew’s Professor of Astronomy at Dublin and Royal Astronomer of Ireland—not to be turned down at the early age of 21.

Hamilton1

Title of Hamilton’s first paper on his characteristic function as a new method that applied his theory from optics to the theory of mechanics, including Lagrangian mechanics as a special case.

His research into mathematical physics  concentrated on the theory of rays of light. Augustin-Jean Fresnel (1788—1827) had recently passed away, leaving behind a wave theory of light that provided a starting point for many effects in optical science, but which lacked broader generality. Hamilton developed a rigorous mathematical framework that could be applied to optical phenomena of the most general nature. This led to his theory of the Characteristic Function, based on principles of the variational calculus of Euler and Lagrange, that predicted the refraction of rays of light, like trajectories, as they passed through different media or across boundaries. In 1832 Hamilton predicted a phenomenon called conical refraction, which would cause a single ray of light entering a biaxial crystal to refract into a luminous cone.

Mathematical physics of that day typically followed experimental science. There were so many observed phenomena in so many fields that demanded explanation, that the general task of the mathematical physicist was to explain phenomena using basic principles followed by mathematical analysis. It was rare for the process to work the other way, for a theorist to predict a phenomenon never before observed. Today we take this as very normal. Einstein’s fame was primed by his prediction of the bending of light by gravity—but only after the observation of the effect by Eddington four years later was Einstein thrust onto the world stage. The same thing happened to Hamilton when his friend Humphrey Lloyd observed conical refraction, just as Hamilton had predicted. After that, Hamilton was revered as one of the most ingenious scientists of his day.

Following the success of conical refraction, Hamilton turned from optics to pursue a striking correspondence he had noted in his Characteristic Function that applied to mechanical trajectories as well as it did to rays of light. In 1834 and 1835 he published two papers On a General Method in Mechanics( I and II)[ii], in which he reworked the theory of Lagrange by beginning with the principle of varying action, which is now known as Hamilton’s Principle. Hamilton’s principle is related to Maupertuis’ principle of least action, but it was more rigorous and a more general approach to derive the Euler-Lagrange equations.  Hamilton’s Principal Function allowed the trajectories of particles to be calculated in complicated situations that were challenging for a direct solution by Lagrange’s equations.

The importance that these two papers had on the future development of physics would not be clear until 1842 when Carl Gustav Jacob Jacobi helped to interpret them and augment them, turning them into a methodology for solving dynamical problems. Today, the Hamiltonian approach to dynamics is central to all of physics, and thousands of physicists around the world mention his name every day, possibly more often than they mention Einstein’s.

[i] Reprinted from D. D. Nolte, Galileo Unbound: A Path Across Life, the Universe and Everything (Oxford, 2018)

[ii] W. R. Hamilton, “On a general method in dynamics I,” Phil. Trans. Roy. Soc., pp. 247-308, 1834; W. R. Hamilton, “On a general method in dynamics II,” Phil. Trans. Roy. Soc., pp. 95-144, 1835.