Archive for the ‘Great Discoveries’ Category

Reporter and Curator: Dr. Sudipta Saha, Ph.D.


MRI-guided focused ultrasound (MRgFUS) surgery is a noninvasive thermal ablation method that uses magnetic resonance imaging (MRI) for target definition, treatment planning, and closed-loop control of energy deposition. Ultrasound is a form of energy that can pass through skin, muscle, fat and other soft tissue so no incisions or inserted probes are needed. High intensity focused ultrasound (HIFU) pinpoints a small target and provides a therapeutic effect by raising the temperature high enough to destroy the target with no damage to surrounding tissue. Integrating FUS and MRI as a therapy delivery system allows physicians to localize, target, and monitor in real time, and thus to ablate targeted tissue without damaging normal structures. This precision makes MRgFUS an attractive alternative to surgical resection or radiation therapy of benign and malignant tumors.


Hypothalamic hamartoma is a rare, benign (non-cancerous) brain tumor that can cause different types of seizures, cognitive problems or other symptoms. While the exact number of people with hypothalamic hamartomas is not known, it is estimated to occur in 1 out of 200,000 children and teenagers worldwide. In one such case at Nicklaus Children’s Brain Institute, USA the patient was able to return home the following day after FUS, resume normal regular activities and remained seizure free. Patients undergoing standard brain surgery to remove similar tumors are typically hospitalized for several days, require sutures, and are at risk of bleeding and infections.


MRgFUS is already approved for the treatment of uterine fibroids. It is in ongoing clinical trials for the treatment of breast, liver, prostate, and brain cancer and for the palliation of pain in bone metastasis. In addition to thermal ablation, FUS, with or without the use of microbubbles, can temporarily change vascular or cell membrane permeability and release or activate various compounds for targeted drug delivery or gene therapy. A disruptive technology, MRgFUS provides new therapeutic approaches and may cause major changes in patient management and several medical disciplines.





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Reporter and Curator: Dr. Sudipta Saha, Ph.D.


Babies born at or before 25 weeks have quite low survival outcomes, and in the US it is the leading cause of infant mortality and morbidity. Just a few weeks of extra ‘growing time’ can be the difference between severe health problems and a relatively healthy baby.


Researchers from The Children’s Hospital of Philadelphia (USA) Research Institute have shown it’s possible to nurture and protect a mammal in late stages of gestation inside an artificial womb; technology which could become a lifesaver for many premature human babies in just a few years.


The researchers took eight lambs between 105 to 120 days gestation (the physiological equivalent of 23 to 24 weeks in humans) and placed them inside the artificial womb. The artificial womb is a sealed and sterile bag filled with an electrolyte solution which acts like amniotic fluid in the uterus. The lamb’s own heart pumps the blood through the umbilical cord into a gas exchange machine outside the bag.


The artificial womb worked in this study and after just four weeks the lambs’ brains and lungs had matured like normal. They had also grown wool and could wiggle, open their eyes, and swallow. Although this study is looking incredibly promising but getting the research up to scratch for human babies still requires a big leap.


Nevertheless, if all goes well, the researchers hope to test the device on premature humans within three to five years. Potential therapeutic applications of this invention may include treatment of fetal growth retardation related to placental insufficiency or the salvage of preterm infants threatening to deliver after fetal intervention or fetal surgery.


The technology may also provide the opportunity to deliver infants affected by congenital malformations of the heart, lung and diaphragm for early correction or therapy before the institution of gas ventilation. Numerous applications related to fetal pharmacologic, stem cell or gene therapy could be facilitated by removing the possibility for maternal exposure and enabling direct delivery of therapeutic agents to the isolated fetus.





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Wave Theory Updated

Larry H. Bernstein, MD, FCAP, Curator




Mathematical Advance in Describing Waves Captures Essence of Modulational Instabilitywave theory

2/25/2016 – Charlotte Hsu, University at Buffalo

Researchers have shown, mathematically, that many different kinds of disturbances evolve to produce wave forms belonging to a single class, denoted by their identical asymptotic state.

Researchers have shown, mathematically, that many different kinds of disturbances evolve to produce wave forms belonging to a single class, denoted by their identical asymptotic state.

BUFFALO, NY — One of the great joys in mathematics is the ability to use it to describe phenomena seen in the physical world, says University at Buffalo mathematician Gino Biondini.

With UB postdoctoral researcher Dionyssios Mantzavinos, Biondini has published a new paper that advances the art — or shall we say, the math — of describing a wave. The findings, published January 27, 2016, in Physical Review Letters, are thought to apply to wave forms ranging from light waves in optical fibers to water waves in the sea.

The study explores what happens when a regular wave pattern has small irregularities, a question that scientists have been trying to answer for the last 50 years.

Researchers have long known that, in many cases, such minor imperfections grow and eventually completely distort the original wave as it travels over long distances, a phenomenon known as “modulational instability.” But the UB team has added to this story by showing, mathematically, that many different kinds of disturbances evolve to produce wave forms belonging to a single class, denoted by their identical asymptotic state.

“Ever since Isaac Newton used math to describe gravity, applied mathematicians have been inventing new mathematics or using existing forms to describe natural phenomena,” says Biondini, a professor of mathematics in the UB College of Arts and Sciences and an adjunct faculty member in the UB physics department. “Our research is, in a way, an extension of all the work that’s come before.”

He says the first great success in using math to represent waves came in the 1700s. The so-called wave equation, used to describe the propagation of waves such as light, sound and water waves, was discovered by Jean le Rond d’Alembert in the middle of that century. But the model has limitations.

“The wave equation is a great first approximation, but it breaks down when the waves are very large — or, in technical parlance — ‘nonlinear,’” Biondini said. “So, for example, in optical fibers, the wave equation is great for moderate distances, but if you send a laser pulse (which is an electromagnetic wave) through an optical fiber across the ocean or the continental U.S., the wave equation is not a good approximation anymore. “Similarly, when a water wave whitecaps and overturns, the wave equation is not a good description of the physics anymore.”

Over the next 250 years, scientists and mathematicians continued to develop new and better ways to describe waves. One of the models that researchers derived in the middle of the 20th century is the nonlinear Schrödinger equation, which helps to characterize wave trains in a variety of physical contexts, including in nonlinear optics and in deep water.

But many questions remained unanswered, including what happens when a wave has small imperfections at its origin.

This is the topic of Biondini and Mantzavinos’ new paper.

“Modulational instability has been known since the 1960s. When you have small perturbations at the input, you’ll have big changes at the output. But is there a way to describe precisely what happens?” Biondini said. “After laying out the foundations in two earlier papers, it took us a year of work to obtain a mathematical description of the solutions. We then used computers to test whether our math was correct, and the simulation results were pretty good — it appears that we have captured the essence of the phenomenon.”

The next step, Biondini said, is to partner with experimental researchers to see if the theoretical findings hold when applied to tangible, physical waves. He has started to collaborate with research groups in optics as well as water waves, and he hopes that it will soon be possible to test the theoretical predictions with real experiments.

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Discovery of Pi

Larry H. Bernstein, MD, FCAP, Curator



How a Farm Boy from Wales Gave the World Pi

3/14/2016 – Gareth Ffowc Roberts, Bangor University
Maths pi-oneer. William Hogarth/National Portrait Gallery
Maths pi-oneer. William Hogarth/National Portrait Gallery

One of the most important numbers in maths might today be named after the Greek letter π or “pi,” but the convention of representing it this way actually doesn’t come from Greece at all. It comes from the pen of an 18th century farmer’s son and largely self-taught mathematician from the small island of Anglesey in Wales. The Welsh Government has even renamed Pi Day (on March 14 or 3/14, which matches the first three digits of pi, 3.14) as “Pi Day Cymru.”

The importance of the number we now call pi has been known about since ancient Egyptian times. It allows you to calculate the circumference and area of a circle from its diameter (and vice versa). But it’s also a number that crops up across all scientific disciplines from cosmology to thermodynamics. Yet even after mathematicians worked out how to calculate pi accurately to over 100 decimal places at the start of the 18th century, we didn’t have an agreed symbol for the number.

From accountant to maths pioneer

This all changed thanks to William Jones who was born in 1674 in the parish of Llanfihangel Tre’r Beirdd. After attending a charity school, Jones landed a job as a merchant’s accountant and then as a maths teacher on a warship, before publishing A New Compendium of the Whole Art of Navigation, his first book in 1702 on the mathematics of navigation. On his return to Britain he began to teach maths in London, possibly starting by holding classes in coffee shops for a small fee.

Shortly afterwards he published Synopsis palmariorum matheseos, a summary of the current state of the art developments in mathematics which reflected his own particular interests. In it is the first recorded use of the symbol π as the number that gives the ratio of a circle’s circumference to its diameter.

We typically think of this number as being about 3.14, but Jones rightly suspected that the digits after its decimal point were infinite and non-repeating. This meant it could never be “expressed in numbers,” as he put it. That was why he recognised the number needed its own symbol. It is commonly thought that he chose pi either because it is the first letter of the word for periphery (περιφέρεια) or because it is the first letter of the word for perimeter (περίμετρος), or both.

Finding pi _Synopsis palmariorum matheseos_


In the pages of his Synopsis, Jones also showed his familiarity with the notion of an infinite series and how it could help calculate pi far more accurately than was possible just by drawing and measuring circles. An infinite series is the total of all the numbers in a sequence that goes on forever, for example ½ + ¼ + ⅛ + and so on. Adding an infinite sequence of ever-smaller fractions like this can bring you closer and closer to a number with an infinite number of digits after the decimal point — just like pi. So by defining the right sequence, mathematicians were able to calculate pi to an increasing number of decimal places.

Infinite series also assist our understanding of rational numbers, more commonly referred to as fractions. Irrational numbers are the ones, like pi, that can’t be written as a fraction, which is why Jones decided it needed its own symbol. What he wasn’t able to do was prove with maths that the digits of pi definitely were infinite and non-repeating and so that the number was truly irrational. This would eventually be achieved in 1768 by the French mathematician Johann Heinrich Lambert. Jones dipped his toes into the subject and showed an intuitive grasp of the complexity of pi but lacked the analytical tools to enable him to develop his ideas further.

Scientific success

Despite this — and his obscure background — Jones’s book was a success and led him to become an important and influential member of the scientific establishment. He was noticed and befriended by two of Britain’s foremost mathematicians — Edmund Halley and Sir Isaac Newton — and was elected a fellow of the Royal Society in 1711. He later became the editor and publisher of many of Newton’s manuscripts and built up an extraordinary library that was one of the greatest collections of books on science and mathematics ever known, and only recently fully dispersed.

Despite this success, the use of the symbol π spread slowly at first. It was popularised in 1737 by the Swiss mathematician Leonhard Euler (1707–83), one of the most eminent mathematicians of the 18th century, who likely came across Jones’ work while studying Newton at the University of Basel. His endorsement of the symbol in his own work ensured that it received wide publicity, yet even then the symbol wasn’t adopted universally until as late as 1934. Today π is instantly recognised worldwide but few know that its history can be traced back to a small village in the heart of Anglesey.

Gareth Ffowc Roberts, Emeritus Professor of Education, Bangor University. This article was originally published on The Conversation. Read the original article.


The Search for the Value of Pi

Tue, 03/15/2016 – 4:44pm
This “pi plate” shows some of the progress toward finding all the digits of pi. In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days. Piledhigheranddeeper, CC BY-SA

The number represented by pi (π) is used in calculations whenever something round (or nearly so) is involved, such as for circles, spheres, cylinders, cones and ellipses. Its value is necessary to compute many important quantities about these shapes, such as understanding the relationship between a circle’s radius and its circumference and area (circumference=2πr; area=πr2). Pi also appears in the calculations to determine the area of an ellipse and in finding the radius, surface area and volume of a sphere.

Our world contains many round and near-round objects; finding the exact value of pi helps us build, manufacture and work with them more accurately. Historically, people had only very coarse estimations of pi (such as 3, or 3.12, or 3.16), and while they knew these were estimates, they had no idea how far off they might be. The search for the accurate value of pi led not only to more accuracy, but also to the development of new concepts and techniques, such as limits and iterative algorithms, which then became fundamental to new areas of mathematics.

Finding the actual value of pi

Archimedes. André Thévet (1584)

Between 3,000 and 4,000 years ago, people used trial-and-error approximations of pi, without doing any math or considering potential errors. The earliest written approximations of pi are 3.125 in Babylon (1900-1600 B.C.) and 3.1605 in ancient Egypt (1650 B.C.). Both approximations start with 3.1 — pretty close to the actual value, but still relatively far off.

Archimedes’ method of calculating pi involved polygons with more and more sides. Leszek Krupinski, CC BY-SA

The first rigorous approach to finding the true value of pi was based on geometrical approximations. Around 250 B.C., the Greek mathematician Archimedes drew polygons both around the outside and within the interior of circles. Measuring the perimeters of those gave upper and lower bounds of the range containing pi. He started with hexagons; by using polygons with more and more sides, he ultimately calculated three accurate digits of pi: 3.14. Around A.D. 150, Greek-Roman scientist Ptolemy used this method to calculate a value of 3.1416.

Liu Hui’s method of calculating pi also used polygons, but in a slightly different way. Gisling and Pbroks13, CC BY-SA

Independently, around A.D. 265, Chinese mathematician Liu Hui created another simple polygon-based iterative algorithm. He proposed a very fast and efficient approximation method, which gave four accurate digits. Later, around A.D. 480, Zu Chongzhi adopted Liu Hui’s method and achieved seven digits of accuracy. This record held for another 800 years.

In 1630, Austrian astronomer Christoph Grienberger arrived at 38 digits, which is the most accurate approximation manually achieved using polygonal algorithms.

Moving beyond polygons

The development of infinite series techniques in the 16th and 17th centuries greatly enhanced people’s ability to approximate pi more efficiently. An infinite series is the sum (or much less commonly, product) of the terms of an infinite sequence, such as ½, ¼, 1/8, 1/16, … 1/(2n). The first written description of an infinite series that could be used to compute pi was laid out in Sanskrit verse by Indian astronomer Nilakantha Somayaji around 1500 A.D., the proof of which was presented around 1530 A.D.

Sir Isaac Newton Wellcome Trust, CC BY

In 1665, English mathematician and physicist Isaac Newton used infinite series to compute pi to 15 digits using calculus he and German mathematician Gottfried Wilhelm Leibniz discovered. After that, the record kept being broken. It reached 71 digits in 1699, 100 digits in 1706, and 620 digits in 1956 — the best approximation achieved without the aid of a calculator or computer.

Carl Louis Ferdinand von Lindemann

In tandem with these calculations, mathematicians were researching other characteristics of pi. Swiss mathematician Johann Heinrich Lambert (1728-1777) first proved that pi is an irrational number — it has an infinite number of digits that never enter a repeating pattern. In 1882, German mathematician Ferdinand von Lindemann proved that pi cannot be expressed in a rational algebraic equation (such as pi²=10 or 9pi4 – 240pi2 + 1492 = 0).

Toward even more digits of pi

Bursts of calculations of even more digits of pi followed the adoption of iterative algorithms, which repeatedly build an updated value by using a calculation performed on the previous value. A simple example of an iterative algorithm allows you to approximate the square root of 2 as follows, using the formula (x+2/x)/2:

  • (2+2/2)/2 = 1.5
  • (1.5+2/1.5)/2 = 1.4167
  • (1.4167+2/1.4167)/2 = 1.4142, which is a very close approximation already.

Advances toward more digits of pi came with the use of a Machin-like algorithm (a generalization of English mathematician John Machin’s formula developed in 1706) and the Gauss-Legendre algorithm (late 18th century) in electronic computers (invented mid-20th century). In 1946, ENIAC, the first electronic general-purpose computer, calculated 2,037 digits of pi in 70 hours. The most recent calculation found more than 13 trillion digits of pi in 208 days!

It has been widely accepted that for most numerical calculations involving pi, a dozen digits provides sufficient precision. According to mathematicians Jörg Arndt and Christoph Haenel, 39 digits are sufficient to perform most cosmological calculations, because that’s the accuracy necessary to calculate the circumference of the observable universe to within one atom’s diameter. Thereafter, more digits of pi are not of practical use in calculations; rather, today’s pursuit of more digits of pi is about testing supercomputers and numerical analysis algorithms.

Calculating pi by yourself

There are also fun and simple methods for estimating the value of pi. One of the best-known is a method called “Monte Carlo.”

A square with inscribed circle. Deweirdifier

The method is fairly simple. To try it at home, draw a circle and a square around it (as at left) on a piece of paper. Imagine the square’s sides are of length 2, so its area is 4; the circle’s diameter is therefore 2, and its area is pi. The ratio between their areas is pi/4, or about 0.7854.

Now pick up a pen, close your eyes and put dots on the square at random. If you do this enough times, and your efforts are truly random, eventually the percentage of times your dot landed inside the circle will approach 78.54 percent — or 0.7854.

Now you’ve joined the ranks of mathematicians who have calculated pi through the ages.

Xiaojing Ye, Assistant Professor of Mathematics and Statistics, Georgia State University. This article was originally published on The Conversation. Read the original article.


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History of Quantum Mechanics

Curator: Larry H. Bernstein, MD, FCAP




A history of Quantum Mechanics

It is hard to realise that the electron was only discovered a little over 100 years ago in 1897. That it was not expected is illustrated by a remark made by J J Thomson, the discoverer of the electron. He said

I was told long afterwards by a distinguished physicist who had been present at my lecture that he thought I had been pulling their leg.

The neutron was not discovered until 1932 so it is against this background that we trace the beginnings of quantum theory back to 1859.

In 1859 Gustav Kirchhoff proved a theorem about blackbody radiation. A blackbody is an object that absorbs all the energy that falls upon it and, because it reflects no light, it would appear black to an observer. A blackbody is also a perfect emitter and Kirchhoff proved that the energy emitted E depends only on the temperature T and the frequency v of the emitted energy, i.e.

E = J(T,v).

He challenged physicists to find the function J.

In 1879 Josef Stefan proposed, on experimental grounds, that the total energy emitted by a hot body was proportional to the fourth power of the temperature. In the generality stated by Stefan this is false. The same conclusion was reached in 1884 by Ludwig Boltzmann for blackbody radiation, this time from theoretical considerations using thermodynamics and Maxwell‘s electromagnetic theory. The result, now known as the StefanBoltzmann law, does not fully answer Kirchhoff‘s challenge since it does not answer the question for specific wavelengths.

In 1896 Wilhelm Wien proposed a solution to the Kirchhoff challenge. However although his solution matches experimental observations closely for small values of the wavelength, it was shown to break down in the far infrared by Rubens and Kurlbaum.

Kirchhoff, who had been at Heidelberg, moved to Berlin. Boltzmann was offered his chair in Heidelberg but turned it down. The chair was then offered to Hertz who also declined the offer, so it was offered again, this time to Planck and he accepted.

Rubens visited Planck in October 1900 and explained his results to him. Within a few hours of Rubens leaving Planck‘s house Planck had guessed the correct formula for Kirchhoff‘s J function. This guess fitted experimental evidence at all wavelengths very well but Planck was not satisfied with this and tried to give a theoretical derivation of the formula. To do this he made the unprecedented step of assuming that the total energy is made up of indistinguishable energy elements – quanta of energy. He wrote

Experience will prove whether this hypothesis is realised in nature

Planck himself gave credit to Boltzmann for his statistical method but Planck‘s approach was fundamentally different. However theory had now deviated from experiment and was based on a hypothesis with no experimental basis. Planck won the 1918 Nobel Prize for Physics for this work.

In 1901 Ricci and Levi-Civita published Absolute differential calculus. It had been Christoffel‘s discovery of ‘covariant differentiation’ in 1869 which let Ricci extend the theory of tensor analysis to Riemannian space of n dimensions. The Ricci and Levi-Civita definitions were thought to give the most general formulation of a tensor. This work was not done with quantum theory in mind but, as so often happens, the mathematics necessary to embody a physical theory had appeared at precisely the right moment.

In 1905 Einstein examined the photoelectric effect. The photoelectric effect is the release of electrons from certain metals or semiconductors by the action of light. The electromagnetic theory of light gives results at odds with experimental evidence. Einstein proposed a quantum theory of light to solve the difficulty and then he realised that Planck‘s theory made implicit use of the light quantum hypothesis. By 1906 Einstein had correctly guessed that energy changes occur in a quantum material oscillator in changes in jumps which are multiples of v where  is Planck‘s reduced constant and v is the frequency. Einstein received the 1921 Nobel Prize for Physics, in 1922, for this work on the photoelectric effect.

In 1913 Niels Bohr wrote a revolutionary paper on the hydrogen atom. He discovered the major laws of the spectral lines. This work earned Bohr the 1922 Nobel Prize for Physics. Arthur Compton derived relativistic kinematics for the scattering of a photon (a light quantum) off an electron at rest in 1923.

However there were concepts in the new quantum theory which gave major worries to many leading physicists. Einstein, in particular, worried about the element of ‘chance’ which had entered physics. In fact Rutherford had introduced spontaneous effect when discussing radio-active decay in 1900. In 1924 Einstein wrote:-

There are therefore now two theories of light, both indispensable, and – as one must admit today despite twenty years of tremendous effort on the part of theoretical physicists – without any logical connection.

In the same year, 1924, Bohr, Kramers and Slater made important theoretical proposals regarding the interaction of light and matter which rejected the photon. Although the proposals were the wrong way forward they stimulated important experimental work. Bohr addressed certain paradoxes in his work.

(i) How can energy be conserved when some energy changes are continuous and some are discontinuous, i.e. change by quantum amounts.
(ii) How does the electron know when to emit radiation.

Einstein had been puzzled by paradox (ii) and Pauli quickly told Bohr that he did not believe his theory. Further experimental work soon ended any resistance to belief in the electron. Other ways had to be found to resolve the paradoxes.

Up to this stage quantum theory was set up in Euclidean space and used Cartesian tensors of linear and angular momentum. However quantum theory was about to enter a new era.

The year 1924 saw the publication of another fundamental paper. It was written by Satyendra Nath Bose and rejected by a referee for publication. Bose then sent the manuscript to Einstein who immediately saw the importance of Bose‘s work and arranged for its publication. Bose proposed different states for the photon. He also proposed that there is no conservation of the number of photons. Instead of statistical independence of particles, Bose put particles into cells and talked about statistical independence of cells. Time has shown that Bose was right on all these points.

Work was going on at almost the same time as Bose‘s which was also of fundamental importance. The doctoral thesis of Louis de Broglie was presented which extended the particle-wave duality for light to all particles, in particular to electrons. Schrödinger in 1926 published a paper giving his equation for the hydrogen atom and heralded the birth of wave mechanics. Schrödingerintroduced operators associated with each dynamical variable.

The year 1926 saw the complete solution of the derivation of Planck‘s law after 26 years. It was solved by Dirac. Also in 1926 Born abandoned the causality of traditional physics. Speaking of collisions Born wrote

One does not get an answer to the question, What is the state after collision? but only to the question, How probable is a given effect of the collision? From the standpoint of our quantum mechanics, there is no quantity which causally fixes the effect of a collision in an individual event.

Heisenberg wrote his first paper on quantum mechanics in 1925 and 2 years later stated his uncertainty principle. It states that the process of measuring the position x of a particle disturbs the particle’s momentum p, so that

Dx Dp ≥  = h/2π

where Dx is the uncertainty of the position and Dp is the uncertainty of the momentum. Here h is Planck‘s constant and  is usually called the ‘reduced Planck‘s constant’. Heisenberg states that

the nonvalidity of rigorous causality is necessary and not just consistently possible.

Heisenberg‘s work used matrix methods made possible by the work of Cayley on matrices 50 years earlier. In fact ‘rival’ matrix mechanics deriving from Heisenberg‘s work and wave mechanics resulting from Schrödinger‘s work now entered the arena. These were not properly shown to be equivalent until the necessary mathematics was developed by Riesz about 25 years later.

Also in 1927 Bohr stated that space-time coordinates and causality are complementary. Pauli realised that spin, one of the states proposed by Bose, corresponded to a new kind of tensor, one not covered by the Ricci and Levi-Civita work of 1901. However the mathematics of this had been anticipated by Eli Cartan who introduced a ‘spinor’ as part of a much more general investigation in 1913.

Dirac, in 1928, gave the first solution of the problem of expressing quantum theory in a form which was invariant under the Lorentz group of transformations of special relativity. He expressedd’Alembert‘s wave equation in terms of operator algebra.

The uncertainty principle was not accepted by everyone. Its most outspoken opponent was Einstein. He devised a challenge to Niels Bohr which he made at a conference which they both attended in 1930. Einstein suggested a box filled with radiation with a clock fitted in one side. The clock is designed to open a shutter and allow one photon to escape. Weigh the box again some time later and the photon energy and its time of escape can both be measured with arbitrary accuracy. Of course this is not meant to be an actual experiment, only a ‘thought experiment’.

Niels Bohr is reported to have spent an unhappy evening, and Einstein a happy one, after this challenge by Einstein to the uncertainty principle. However Niels Bohr had the final triumph, for the next day he had the solution. The mass is measured by hanging a compensation weight under the box. This is turn imparts a momentum to the box and there is an error in measuring the position. Time, according to relativity, is not absolute and the error in the position of the box translates into an error in measuring the time.

Although Einstein was never happy with the uncertainty principle, he was forced, rather grudgingly, to accept it after Bohr‘s explanation.

In 1932 von Neumann put quantum theory on a firm theoretical basis. Some of the earlier work had lacked mathematical rigour, but von Neumann put the whole theory into the setting of operator algebra.

References (33 books/articles)

Article by: J J O’Connor and E F Robertson


A Brief History of Quantum Mechanics

Appendix A of
The Strange World of Quantum Mechanics

written by Dan StyerOberlin College Physics Department;
copyright © Daniel F. Styer 1999


One must understand not only the cleanest and most direct experimental evidence supporting our current theories (like the evidence presented in this book), but must understand also how those theories came to be accepted through a tightly interconnected web of many experiments, no one of which was completely convincing but which taken together presented an overwhelming argument

Thus a full history of quantum mechanics would have to discuss Schrödinger’s many mistresses, Ehrenfest’s suicide, and Heisenberg’s involvement with Nazism. It would have to treat the First World War’s effect on the development of science. It would need to mention “the Thomson model” of the atom, which was once the major competing theory to quantum mechanics. It would have to give appropriate weight to both theoretical and experimental developments.

Much of the work of science is done through informal conversations, and the resulting written record is often sanitized to avoid offending competing scientists. The invaluable oral record is passed down from professor to student repeatedly before anyone ever records it on paper. There is a tendency for the exciting stories to be repeated and the dull ones to be forgotten.

The fact is that scientific history, like the stock market and like everyday life, does not proceed in an orderly, coherent pattern. The story of quantum mechanics is a story full of serendipity, personal squabbles, opportunities missed and taken, and of luck both good and bad.

Status of physics: January 1900

In January 1900 the atomic hypothesis was widely but not universally accepted. Atoms were considered point particles, and it wasn’t clear how atoms of different elements differed. The electron had just been discovered (1897) and it wasn’t clear where (or even whether) electrons were located within atoms. One important outstanding problem concerned the colors emitted by atoms in a discharge tube (familiar today as the light from a fluorescent tube or from a neon sign). No one could understand why different gas atoms glowed in different colors. Another outstanding problem concerned the amount of heat required to change the temperature of a diatomic gas such as oxygen: the measured amounts were well below the value predicted by theory. Because quantum mechanics is important when applied to atomic phenomena, you might guess that investigations into questions like these would give rise to the discovery of quantum mechanics. Instead it came from a study of heat radiation.

Heat radiation

You know that the coals of a campfire, or the coils of an electric stove, glow red. You probably don’t know that even hotter objects glow white, but this fact is well known to blacksmiths. When objects are hotter still they glow blue. (This is why a gas stove should be adjusted to make a blue flame.) Indeed, objects at room temperature also glow (radiate), but the radiation they emit is infrared, which is not detectable by the eye. (The military has developed — for use in night warfare — special eye sets that convert infrared radiation to optical radiation.)

In the year 1900 several scientists were trying to turn these observations into a detailed explanation of and a quantitatively accurate formula for the color of heat radiation as a function of temperature. On 19 October 1900 the Berliner Max Planck (age 42) announced a formula that fit the experimental results perfectly, yet he had no explanation for the formula — it just happened to fit. He worked to find an explanation through the late fall and finally was able to derive his formula by assuming that the atomic jigglers could not take on any possible energy, but only certain special “allowed” values. He announced this result on 14 December 1900. We know this because the assumption of allowed energy values raises certain obvious questions. If a jiggling atom can only assume certain allowed values of energy, then there must also be restrictions on the positions and speeds that the atom can have. What are they?

Planck wrote (31 years after his discovery):

I had already fought for six years (since 1894) with the problem of equilibrium between radiation and matter without arriving at any successful result. I was aware that this problem was of fundamental importance in physics, and I knew the formula describing the energy distribution . . .

Here is another wonderful story, this one related by Werner Heisenberg:

In a period of most intensive work during the summer of 1900 [Planck] finally convinced himself that there was no way of escaping from this conclusion [of “allowed” energies]. It was told by Planck’s son that his father spoke to him about his new ideas on a long walk through the Grunewald, the wood in the suburbs of Berlin. On this walk he explained that he felt he had possibly made a discovery of the first rank, comparable perhaps only to the discoveries of Newton.

(the son would probably remember the nasty cold he caught better than any remarks his father made.)

The old quantum theory

Classical mechanics was assumed to hold, but with the additional assumption that only certain values of a physical quantity (the energy, say, or the projection of a magnetic arrow) were allowed. Any such quantity was said to be “quantized”. The trick seemed to be to guess the right quantization rules for the situation under study, or to find a general set of quantization rules that would work for all situations.

For example, in 1905 Albert Einstein (age 26) postulated that the total energy of a beam of light is quantized. Just one year later he used quantization ideas to explain the heat/temperature puzzle for diatomic gases. Five years after that, in 1911, Arnold Sommerfeld (age 43) at Munich began working on the implications of energy quantization for position and speed.

In the same year Ernest Rutherford (age 40), a New Zealander doing experiments in Manchester, England, discovered the atomic nucleus — only at this relatively late stage in the development of quantum mechanics did physicists have even a qualitatively correct picture of the atom! In 1913, Niels Bohr (age 28), a Dane who had recently worked in Rutherford’s laboratory, introduced quantization ideas for the hydrogen atom. His theory was remarkably successful in explaining the colors emitted by hydrogen glowing in a discharge tube, and it sparked enormous interest in developing and extending the old quantum theory.

During the WWI (in 1915) William Wilson (age 40, a native of Cumberland, England, working at King’s College in London) made progress on the implications of energy quantization for position and speed, and Sommerfeld also continued his work in that direction.

With the coming of the armistice in 1918, work in quantum mechanics expanded rapidly. Many theories were suggested and many experiments performed. To cite just one example, in 1922 Otto Stern and his graduate student Walther Gerlach (ages 34 and 23) performed their important experiment that is so essential to the way this book presents quantum mechanics. Jagdish Mehra and Helmut Rechenberg, in their monumental history of quantum mechanics, describe the situation at this juncture well:

At the turn of the year from 1922 to 1923, the physicists looked forward with enormous enthusiasm towards detailed solutions of the outstanding problems, such as the helium problem and the problem of the anomalous Zeeman effects. However, within less than a year, the investigation of these problems revealed an almost complete failure of Bohr’s atomic theory.

The matrix formulation of quantum mechanics

As more and more situations were encountered, more and more recipes for allowed values were required. This development took place mostly at Niels Bohr’s Institute for Theoretical Physics in Copenhagen, and at the University of Göttingen in northern Germany. The most important actors at Göttingen were Max Born (age 43, an established professor) and Werner Heisenberg (age 23, a freshly minted Ph.D. from Sommerfeld in Munich). According to Born “At Göttingen we also took part in the attempts to distill the unknown mechanics of the atom out of the experimental results. . . . The art of guessing correct formulas . . . was brought to considerable perfection.”

Heisenberg particularly was interested in general methods for making guesses. He began to develop systematic tables of allowed physical quantities, be they energies, or positions, or speeds. Born looked at these tables and saw that they could be interpreted as mathematical matrices. Fifty years later matrix mathematics would be taught even in high schools. But in 1925 it was an advanced and abstract technique, and Heisenberg struggled with it. His work was cut short in June 1925.

It was late spring in Göttingen, and Heisenberg suffered from an allergy attack so severe that he could hardly work. He asked his research director, Max Born, for a vacation, and spent it on the rocky North Sea island of Helgoland. At first he was so ill that could only stay in his rented room and admire the view of the sea. As his condition improved he began to take walks and to swim. With further improvement he began also to read Goethe and to work on physics. With nothing to distract him, he concentrated intensely on the problems that had faced him in Göttingen.

Heisenberg reproduced his earlier work, cleaning up the mathematics and simplifying the formulation. He worried that the mathematical scheme he invented might prove to be inconsistent, and in particular that it might violate the principle of the conservation of energy. In Heisenberg’s own words:

The energy principle had held for all the terms, and I could no longer doubt the mathematical consistency and coherence of the kind of quantum mechanics to which my calculations pointed. At first, I was deeply alarmed. I had the feeling that, through the surface of atomic phenomena, I was looking at a strangely beautiful interior, and felt almost giddy at the thought that I now had to probe this wealth of mathematical structures nature had so generously spread out before me.

By the end of the summer Heisenberg, Born, and Pascual Jordan (age 22) had developed a complete and consistent theory of quantum mechanics. (Jordan had entered the collaboration when he overheard Born discussing quantum mechanics with a colleague on a train.)

This theory, called “matrix mechanics” or “the matrix formulation of quantum mechanics”, is not the theory I have presented in this book. It is extremely and intrinsically mathematical, and even for master mathematicians it was difficult to work with. Although we now know it to be complete and consistent, this wasn’t clear until much later. Heisenberg had been keeping Wolfgang Pauli apprised of his progress. (Pauli, age 25, was Heisenberg’s friend from graduate student days, when they studied together under Sommerfeld.) Pauli found the work too mathematical for his tastes, and called it “Göttingen’s deluge of formal learning”. On 12 October 1925 Heisenberg could stand Pauli’s biting criticism no longer. He wrote to Pauli:

With respect to both of your last letters I must preach you a sermon, and beg your pardon… When you reproach us that we are such big donkeys that we have never produced anything new in physics, it may well be true. But then, you are also an equally big jackass because you have not accomplished it either . . . . . . (The dots denote a curse of about two-minute duration!) Do not think badly of me and many greetings.

The wavefunction formulation of quantum mechanics

While this work was going on at Göttingen and Helgoland, others were busy as well. In 1923 Louis de Broglie (age 31), associated an “internal periodic phenomenon” — a wave — with a particle. He was never very precise about just what that meant. (De Broglie is sometimes called “Prince de Broglie” because his family descended from the French nobility. To be strictly correct, however, only his eldest brother could claim the title.)

It fell to Erwin Schroedinger, an Austrian working in Zürich, to build this vague idea into a theory of wave mechanics. He did so during the Christmas season of 1925 (at age 38), at the alpine resort of Arosa, Switzerland, in the company of “an old girlfriend [from] Vienna”, while his wife stayed home in Zürich.

In short, just twenty-five years after Planck glimpsed the first sight of a new physics, there was not one, but two competing versions of that new physics!

The two versions seemed utterly different and there was an acrimonious debate over which one was correct. In a footnote to a 1926 paper Schrödinger claimed to be “discouraged, if not repelled” by matrix mechanics. Meanwhile, Heisenberg wrote to Pauli (8 June 1926) that

The more I think of the physical part of the Schrödinger theory, the more detestable I find it. What Schrödinger writes about visualization makes scarcely any sense, in other words I think it is shit. The greatest result of his theory is the calculation of matrix elements.

Fortunately the debate was soon stilled: in 1926 Schrödinger and, independently, Carl Eckert (age 24) of Caltech proved that the two new mechanics, although very different in superficial appearance, were equivalent to each other. [Very much as the process of adding arabic numerals is quite different from the process of adding roman numerals, but the two processes nevertheless always give the same result.] (Pauli also proved this, but never published the result.)


With not just one, but two complete formulations of quantum mechanics in hand, the quantum theory grew explosively. It was applied to atoms, molecules, and solids. It solved with ease the problem of helium that had defeated the old quantum theory. It resolved questions concerning the structure of stars, the nature of superconductors, and the properties of magnets. One particularly important contributor was P.A.M. Dirac, who in 1926 (at age 22) extended the theory to relativistic and field-theoretic situations. Another was Linus Pauling, who in 1931 (at age 30) developed quantum mechanical ideas to explain chemical bonding, which previously had been understood only on empirical grounds. Even today quantum mechanics is being applied to new problems and new situations. It would be impossible to mention all of them. All I can say is that quantum mechanics, strange though it may be, has been tremendously successful.

The Bohr-Einstein debate

The extraordinary success of quantum mechanics in applications did not overwhelm everyone. A number of scientists, including Schrödinger, de Broglie, and — most prominently — Einstein, remained unhappy with the standard probabilistic interpretation of quantum mechanics. In a letter to Max Born (4 December 1926), Einstein made his famous statement that

Quantum mechanics is very impressive. But an inner voice tells me that it is not yet the real thing. The theory produces a good deal but hardly brings us closer to the secret of the Old One. I am at all events convinced that He does not play dice.

In concrete terms, Einstein’s “inner voice” led him, until his death, to issue occasional detailed critiques of quantum mechanics and its probabilistic interpretation. Niels Bohr undertook to reply to these critiques, and the resulting exchange is now called the “Bohr-Einstein debate”. At one memorable stage of the debate (Fifth Solvay Congress, 1927), Einstein made an objection similar to the one quoted above and Bohr

replied by pointing out the great caution, already called for by ancient thinkers, in ascribing attributes to Providence in every-day language.

These two statements are often paraphrased as, Einstein to Bohr: “God does not play dice with the universe.” Bohr to Einstein: “Stop telling God how to behave!” While the actual exchange was not quite so dramatic and quick as the paraphrase would have it, there was nevertheless a wonderful rejoinder from what must have been a severely exasperated Bohr.

The Bohr-Einstein debate had the benefit of forcing the creators of quantum mechanics to sharpen their reasoning and face the consequences of their theory in its most starkly non-intuitive situations. It also had (in my opinion) one disastrous consequence: because Einstein phrased his objections in purely classical terms, Bohr was compelled to reply in nearly classical terms, giving the impression that in quantum mechanics, an electron is “really classical” but that somehow nature puts limits on how well we can determine those classical properties. … this is a misconception: the reason we cannot measure simultaneously the exact position and speed of an electron is because an electron does not have simultaneously an exact position and speed — an electron is not just a smaller, harder edition of a marble. This misconception — this picture of a classical world underlying the quantum world … avoid it.

On the other hand, the Bohr-Einstein debate also had at least one salutary product. In 1935 Einstein, in collaboration with Boris Podolsky and Nathan Rosen, invented a situation in which the results of quantum mechanics seemed completely at odds with common sense, a situation in which the measurement of a particle at one location could reveal instantly information about a second particle far away. The three scientists published a paper which claimed that “No reasonable definition of reality could be expected to permit this.” Bohr produced a recondite response and the issue was forgotten by most physicists, who were justifiably busy with the applications of rather than the foundations of quantum mechanics. But the ideas did not vanish entirely, and they eventually raised the interest of John Bell. In 1964 Bell used the Einstein-Podolsky-Rosen situation to produce a theorem about the results from certain distant measurements for any deterministic scheme, not just classical mechanics. In 1982 Alain Aspect and his collaborators put Bell’s theorem to the test and found that nature did indeed behave in the manner that Einstein (and others!) found so counterintuitive.

The amplitude formulation of quantum mechanics

The version of quantum mechanics presented in this book is neither matrix nor wave mechanics. It is yet another formulation, different in approach and outlook, but fundamentally equivalent to the two formulations already mentioned. It is called amplitude mechanics (or “the sum over histories technique”, or “the many paths approach”, or “the path integral formulation”, or “the Lagrangian approach”, or “the method of least action”), and it was developed by Richard Feynman in 1941 while he was a graduate student (age 23) at Princeton. Its discovery is well described by Feynman himself in his Nobel lecture:

I went to a beer party in the Nassau Tavern in Princeton. There was a gentleman, newly arrived from Europe (Herbert Jehle) who came and sat next to me. Europeans are much more serious than we are in America because they think a good place to discuss intellectual matters is a beer party. So he sat by me and asked, “What are you doing” and so on, and I said, “I’m drinking beer.” Then I realized that he wanted to know what work I was doing and I told him I was struggling with this problem, and I simply turned to him and said “Listen, do you know any way of doing quantum mechanics starting with action — where the action integral comes into the quantum mechanics?” “No,” he said, “but Dirac has a paper in which the Lagrangian, at least, comes into quantum mechanics. I will show it to you tomorrow.”

Next day we went to the Princeton Library (they have little rooms on the side to discuss things) and he showed me this paper.

Dirac’s short paper in the Physikalische Zeitschrift der Sowjetunion claimed that a mathematical tool which governs the time development of a quantal system was “analogous” to the classical Lagrangian.

Professor Jehle showed me this; I read it; he explained it to me, and I said, “What does he mean, they are analogous; what does that mean,analogous? What is the use of that?” He said, “You Americans! You always want to find a use for everything!” I said that I thought that Dirac must mean that they were equal. “No,” he explained, “he doesn’t mean they are equal.” “Well,” I said, “let’s see what happens if we make them equal.”

So, I simply put them equal, taking the simplest example . . . but soon found that I had to put a constant of proportionality A in, suitably adjusted. When I substituted . . . and just calculated things out by Taylor-series expansion, out came the Schrödinger equation. So I turned to Professor Jehle, not really understanding, and said, “Well you see Professor Dirac meant that they were proportional.” Professor Jehle’s eyes were bugging out — he had taken out a little notebook and was rapidly copying it down from the blackboard and said, “No, no, this is an important discovery.”

Feynman’s thesis advisor, John Archibald Wheeler (age 30), was equally impressed. He believed that the amplitude formulation of quantum mechanics — although mathematically equivalent to the matrix and wave formulations — was so much more natural than the previous formulations that it had a chance of convincing quantum mechanics’s most determined critic. Wheeler writes:

Visiting Einstein one day, I could not resist telling him about Feynman’s new way to express quantum theory. “Feynman has found a beautiful picture to understand the probability amplitude for a dynamical system to go from one specified configuration at one time to another specified configuration at a later time. He treats on a footing of absolute equality every conceivable history that leads from the initial state to the final one, no matter how crazy the motion in between. The contributions of these histories differ not at all in amplitude, only in phase. . . . This prescription reproduces all of standard quantum theory. How could one ever want a simpler way to see what quantum theory is all about! Doesn’t this marvelous discovery make you willing to accept the quantum theory, Professor Einstein?” He replied in a serious voice, “I still cannot believe that God plays dice. But maybe”, he smiled, “I have earned the right to make my mistakes.”








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Gravitational waves detected

Reporter: Danut Dragoi, PhD

Many physicists remember the general relativity classes and the Einstein equations, see link in here, whose solution gives us the wave vector and the wavelength. More on this topic can be found in here.

A famous physicist, Serban Titeica,  said on his classes, link in here, that a theory is good if it is supported by experiment, otherwise is null. Since the ripples of a gravitational perturbation were recently proved experimentally, see link in here, we are thinking like Titeica, asking ourselves, what is the benefit of General Relativity. Some business people will ask for how much profit they can get, of course they talk about monetary profit. The others, it depends were in the social field they work, will say the gravitational waves detected is a great result for education, for humanity knowledge, and for intellectual enlightenment.

In a public discussion, see the video link, a participant said the benefits are in the new technology that can be transferred to the industry. Now we cannot forget the benefits from space exploration by NASA that brought innovative solutions to many industries including computers and communications. The picture below taken from the previous video,. shows the essence of actual activity on gravitation research. On the lower right corner of the picture is the LIGO, Laser Interferometer Gravitational-Wave Observatory, which is a Michelson type of interferometer with two perpendicular arms, each having a length of about 4 Km. Two gravitational interferometers are operating on two different locations on US, one in Livingston, Louisiana State, the other in Hanford, Washington State. Both stations detected same signal with same characteristics.


Watch these videos

1st video is from youtube

and 2nd video from Caltech, California.

Stephen Hawking, a well known author and scientist on Time and Gravitation Theory and Black Holes,  congratulated LIGO team, watch video at this link in here.

Regarding the applications in medicine, I think all physical fields, including gravitation, have an influence on our life, on all living cells. It is well established that the electric field has a major effect on living cells, since all processes in the living cells are based on electric charge transfer from one molecule to another in a complex collective material interaction. Other physical fields, like electromagnetic fields have many applications in medicine starting with RF therapeutics, laser, and optical communication. We can include here NASA’s research tremendous contribution on non-medical industries and new technologies, but this in another posting.

Now, not only the physicists can enjoy the knowledge about recent results on gravitation, but also the engineers and anybody who is interested on the philosophy of the universe asking themselves what is the answer to crucial questions like why we live in this Universe, how the Universes influence us, etc.





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Einstein and General Theory of Relativity

Larry H. Bernstein, MD, FCAP, Curator




General Relativity And The ‘Lone Genius’ Model Of Science

Chad Orzel


(Credit: AP)


One hundred years ago this Wednesday, Albert Einstein gave the last of a series of presentations to the Prussian Academy of Sciences, which marks the official completion of his General Theory of Relativity. This anniversary is generating a good deal of press and various celebratory events, such as the premiere of a new documentary special. If you prefer your physics explanations in the plainest language possible, there’s even an “Up Goer Five” version (personally, I don’t find these all that illuminating, but lots of people seem to love it).

Einstein is, of course, the most iconic scientist in history, and much of the attention to this week’s centennial will center on the idea of his singular genius. Honestly, general relativity is esoteric enough that were it not for Einstein’s personal fame, there probably wouldn’t be all that much attention paid to this outside of the specialist science audience.

But, of course, while the notion of Einstein as a lone, unrecognized genius is a big part of his myth, he didn’t create relativity entirely on his own, asthis article in Nature News makes clear. The genesis of relativity is a single simple idea, but even in the early stages, when he developed Special Relativity while working as a patent clerk, he honed his ideas through frequent discussions with friends and colleagues. Most notable among these was probably Michele Besso, who Einstein later referred to as “the best sounding board in Europe.”

And most of the work on General Relativity came not when Einstein was toiling in obscurity, but after he had begun to climb the academic ladder in Europe. In the ten years between the Special and General theories, he went through a series of faculty jobs of increasing prestige. He also laboriously learned a great deal of mathematics in order to reach the final form of the theory, largely with the assistance of his friend Marcel Grossmann. The path to General Relativity was neither simple nor solitary, and the Nature piece documents both the mis-steps along the way and the various people who helped out.

While Einstein wasn’t working alone, though, the Nature piece also makes an indirect case for his status as a genius worth celebrating. Not because of the way he solved the problem, but through the choice of problem to solve. Einstein pursued a theory that would incorporate gravitation into relativity with dogged determination through those years, but he was one of a very few people working on it. There were a couple of other theories kicking around, particularly Gunnar Nordström’s, but these didn’t generate all that much attention. The mathematician David Hilbert nearly scooped Einstein with the final form of the field equations in November of 1915 (some say he did get there first), but Hilbert was a latecomer who only got interested in the problem of gravitation after hearing about it from Einstein, and his success was a matter of greater familiarity with the necessary math. One of the books I used when I taught a relativity class last year quoted Hilbert as saying that “every child in the streets of Göttingen knows more about four-dimensional geometry than Einstein,” but that Einstein’s physical insight got him to the theory before superior mathematicians.


History: Einstein was no lone genius

Michel Janssen & Jürgen Renn   

16 November 2015 Corrected:   17 November 2015    Nature Nov 2015; 527(7578)

Lesser-known and junior colleagues helped the great physicist to piece together his general theory of relativity, explain Michel Janssen and Jürgen Renn.!/image/Comment2.jpg_gen/derivatives/landscape_630/Comment2.jpg

Marcel Grossmann (left) and Michele Besso (right), university friends of Albert Einstein (centre), both made important contributions to general relativity.


A century ago, in November 1915, Albert Einstein published his general theory of relativity in four short papers in the proceedings of the Prussian Academy of Sciences in Berlin1. The landmark theory is often presented as the work of a lone genius. In fact, the physicist received a great deal of help from friends and colleagues, most of whom never rose to prominence and have been forgotten2, 3, 4, 5. (For full reference details of all Einstein texts mentioned in this piece, seeSupplementary Information.)

Here we tell the story of how their insights were woven into the final version of the theory. Two friends from Einstein’s student days — Marcel Grossmann and Michele Besso — were particularly important. Grossmann was a gifted mathematician and organized student who helped the more visionary and fanciful Einstein at crucial moments. Besso was an engineer, imaginative and somewhat disorganized, and a caring and lifelong friend to Einstein. A cast of others contributed too.

Einstein met Grossmann and Besso at the Swiss Federal Polytechnical School in Zurich6 — later renamed the Swiss Federal Institute of Technology (Eidgenössische Technische Hochschule; ETH) — where, between 1896 and 1900, he studied to become a school teacher in physics and mathematics. Einstein also met his future wife at the ETH, classmate Mileva Marić. Legend has it that Einstein often skipped class and relied on Grossmann’s notes to pass exams.!/image/entanglement.jpg_gen/derivatives/fullsize/entanglement.jpg


Grossmann’s father helped Einstein to secure a position at the patent office in Berne in 1902, where Besso joined him two years later. Discussions between Besso and Einstein earned the former the sole acknowledgment in the most famous of Einstein’s 1905 papers, the one introducing the special theory of relativity. As well as publishing the papers that made 1905 his annus mirabilis, Einstein completed his dissertation that year to earn a PhD in physics from the University of Zurich.

In 1907, while still at the patent office, he started to think about extending the principle of relativity from uniform to arbitrary motion through a new theory of gravity. Presciently, Einstein wrote to his friend Conrad Habicht — whom he knew from a reading group in Berne mockingly called the Olympia Academy by its three members — saying that he hoped that this new theory would account for a discrepancy of about 43˝ (seconds of arc) per century between Newtonian predictions and observations of the motion of Mercury’s perihelion, the point of its orbit closest to the Sun.

Einstein started to work in earnest on this new theory only after he left the patent office in 1909, to take up professorships first at the University of Zurich and two years later at the Charles University in Prague. He realized that gravity must be incorporated into the structure of space-time, such that a particle subject to no other force would follow the straightest possible trajectory through a curved space-time.

In 1912, Einstein returned to Zurich and was reunited with Grossmann at the ETH. The pair joined forces to generate a fully fledged theory. The relevant mathematics was Gauss’s theory of curved surfaces, which Einstein probably learned from Grossmann’s notes. As we know from recollected conversations, Einstein told Grossmann7: “You must help me, or else I’ll go crazy.”

Their collaboration, recorded in Einstein’s ‘Zurich notebook‘, resulted in a joint paper published in June 1913, known as the Entwurf (‘outline’) paper. The main advance between this 1913 Entwurf theory and the general relativity theory of November 1915 are the field equations, which determine how matter curves space-time. The final field equations are ‘generally covariant’: they retain their form no matter what system of coordinates is chosen to express them. The covariance of the Entwurf field equations, by contrast, was severely limited.!/image/einstein_lost.jpg_gen/derivatives/fullsize/einstein_lost.jpg

Einstein’s lost theory uncovered


Two Theories

In May 1913, as he and Grossmann put the finishing touches to their Entwurf paper, Einstein was asked to lecture at the annual meeting of the Society of German Natural Scientists and Physicians to be held that September in Vienna, an invitation that reflects the high esteem in which the 34-year-old was held by his peers.

In July 1913, Max Planck and Walther Nernst, two leading physicists from Berlin, came to Zurich to offer Einstein a well-paid and teaching-free position at the Prussian Academy of Sciences in Berlin, which he swiftly accepted and took up in March 1914. Gravity was not a pressing problem for Planck and Nernst; they were mainly interested in what Einstein could do for quantum physics.  (It was Walther Nernst who advised that Germany could not engage in WWI and win unless it was a short war).

Several new theories had been proposed in which gravity, like electromagnetism, was represented by a field in the flat space-time of special relativity. A particularly promising one came from the young Finnish physicist Gunnar Nordström. In his Vienna lecture, Einstein compared his own Entwurf theory to Nordström’s theory. Einstein worked on both theories between May and late August 1913, when he submitted the text of his lecture for publication in the proceedings of the 1913 Vienna meeting.

In the summer of 1913, Nordström visited Einstein in Zurich. Einstein convinced him that the source of the gravitational field in both their theories should be constructed out of the ‘energy–momentum tensor’: in pre-relativistic theories, the density and the flow of energy and momentum were represented by separate quantities; in relativity theory, they are combined into one quantity with ten different components.!/image/Comment4.jpg_gen/derivatives/landscape_630/Comment4.jpg

ETH-Bibliothek Zürich, Bildarchiv

ETH Zurich, where Einstein met friends with whom he worked on general relativity.


This energy–momentum tensor made its first appearance in 1907–8 in the special-relativistic reformulation of the theory of electrodynamics of James Clerk Maxwell and Hendrik Antoon Lorentz by Hermann Minkowski. It soon became clear that an energy–momentum tensor could be defined for physical systems other than electromagnetic fields. The tensor took centre stage in the new relativistic mechanics presented in the first textbook on special relativity, Das Relativitätsprinzip, written by Max Laue in 1911. In 1912, a young Viennese physicist, Friedrich Kottler, generalized Laue’s formalism from flat to curved space-time. Einstein and Grossmann relied on this generalization in their formulation of the Entwurf theory. During his Vienna lecture, Einstein called for Kottler to stand up and be recognized for this work8.

Einstein also worked with Besso that summer to investigate whether the Entwurf theory could account for the missing 43˝ per century for Mercury’s perihelion. Unfortunately, they found that it could only explain 18˝. Nordström’s theory, Besso checked later, gave 7˝ in the wrong direction. These calculations are preserved in the ‘Einstein–Besso manuscript‘ of 1913.

Besso contributed significantly to the calculations and raised interesting questions. He wondered, for instance, whether the Entwurf field equations have an unambiguous solution that uniquely determines the gravitational field of the Sun. Historical analysis of extant manuscripts suggests that this query gave Einstein the idea for an argument that reconciled him with the restricted covariance of the Entwurf equations. This ‘hole argument’ seemed to show that generally covariant field equations cannot uniquely determine the gravitational field and are therefore inadmissible9.

Einstein and Besso also checked whether the Entwurf equations hold in a rotating coordinate system. In that case the inertial forces of rotation, such as the centrifugal force we experience on a merry-go-round, can be interpreted as gravitational forces. The theory seemed to pass this test. In August 1913, however, Besso warned him that it did not. Einstein did not heed the warning, which would come back to haunt him.!/image/integrity.jpg_gen/derivatives/fullsize/integrity.jpg

Scientific method: Defend the integrity of physics


In his lecture in Vienna in September 1913, Einstein concluded his comparison of the two theories with a call for experiment to decide. The Entwurf theory predicts that gravity bends light, whereas Nordström’s does not. It would take another five years to find out. Erwin Finlay Freundlich, a junior astronomer in Berlin with whom Einstein had been in touch since his days in Prague, travelled to Crimea for the solar eclipse of August 1914 to determine whether gravity bends light but was interned by the Russians just as the First World War broke out. Finally, in 1919, English astronomer Arthur Eddington confirmed Einstein’s prediction of light bending by observing the deflection of distant stars seen close to the Sun’s edge during another eclipse, making Einstein a household name10.

Back in Zurich, after the Vienna lecture, Einstein teamed up with another young physicist, Adriaan Fokker, a student of Lorentz, to reformulate the Nordström theory using the same kind of mathematics that he and Grossmann had used to formulate the Entwurf theory. Einstein and Fokker showed that in both theories the gravitational field can be incorporated into the structure of a curved space-time. This work also gave Einstein a clearer picture of the structure of the Entwurf theory, which helped him and Grossmann in a second joint paper on the theory. By the time it was published in May 1914, Einstein had left for Berlin.!/image/Einstein_frontal_small.jpg_gen/derivatives/fullsize/Einstein_frontal_small.jpg

Snapshots explore Einstein’s unusual brain


The Breakup

Turmoil erupted soon after the move. Einstein’s marriage fell apart and Mileva moved back to Zurich with their two young sons. Albert renewed the affair he had started and broken off two years before with his cousin Elsa Löwenthal (née Einstein). The First World War began. Berlin’s scientific elite showed no interest in the Entwurf theory, although renowned colleagues elsewhere did, such as Lorentz and Paul Ehrenfest in Leiden, the Netherlands. Einstein soldiered on.

By the end of 1914, his confidence had grown enough to write a long exposition of the theory. But in the summer of 1915, after a series of his lectures in Göttingen had piqued the interest of the great mathematician David Hilbert, Einstein started to have serious doubts. He discovered to his dismay that the Entwurf theory does not make rotational motion relative. Besso was right. Einstein wrote to Freundlich for help: his “mind was in a deep rut”, so he hoped that the young astronomer as “a fellow human being with unspoiled brain matter” could tell him what he was doing wrong. Freundlich could not help him.

“Worried that Hilbert might beat him to the punch, Einstein rushed new equations into print.”

The problem, Einstein soon realized, lay with the Entwurf field equations. Worried that Hilbert might beat him to the punch, Einstein rushed new equations into print in early November 1915, modifying them the following week and again two weeks later in subsequent papers submitted to the Prussian Academy. The field equations were generally covariant at last.

In the first November paper, Einstein wrote that the theory was “a real triumph” of the mathematics of Carl Friedrich Gauss and Bernhard Riemann. He recalled in this paper that he and Grossmann had considered the same equations before, and suggested that if only they had allowed themselves to be guided by pure mathematics rather than physics, they would never have accepted equations of limited covariance in the first place.

Other passages in the first November paper, however, as well as his other papers and correspondence in 1913–15, tell a different story. It was thanks to the elaboration of the Entwurf theory, with the help of Grossmann, Besso, Nordström and Fokker, that Einstein saw how to solve the problems with the physical interpretation of these equations that had previously defeated him.

In setting out the generally covariant field equations in the second and fourth papers, he made no mention of the hole argument. Only when Besso and Ehrenfest pressed him a few weeks after the final paper, dated 25 November, did Einstein find a way out of this bind — by realizing that only coincident events and not coordinates have physical meaning. Besso had suggested a similar escape two years earlier, which Einstein had brusquely rejected2.

In his third November paper, Einstein returned to the perihelion motion of Mercury. Inserting the astronomical data supplied by Freundlich into the formula he derived using his new theory, Einstein arrived at the result of 43″ per century and could thus fully account for the difference between Newtonian theory and observation. “Congratulations on conquering the perihelion motion,” Hilbert wrote to him on 19 November. “If I could calculate as fast as you can,” he quipped, “the hydrogen atom would have to bring a note from home to be excused for not radiating.”

Einstein kept quiet on why he had been able to do the calculations so fast. They were minor variations on the ones he had done with Besso in 1913. He probably enjoyed giving Hilbert a taste of his own medicine: in a letter to Ehrenfest written in May 1916, Einstein characterized Hilbert’s style as “creating the impression of being superhuman by obfuscating one’s methods”.

Einstein emphasized that his general theory of relativity built on the work of Gauss and Riemann, giants of the mathematical world. But it also built on the work of towering figures in physics, such as Maxwell and Lorentz, and on the work of researchers of lesser stature, notably Grossmann, Besso, Freundlich, Kottler, Nordström and Fokker. As with many other major breakthroughs in the history of science, Einstein was standing on the shoulders of many scientists, not just the proverbial giants4.!/image/cartoon.jpg_gen/derivatives/landscape_630/cartoon.jpg

Berlin’s physics elite (Fritz Haber, Walther Nernst, Heinrich Rubens, Max Planck) and Einstein’s old and new family (Mileva Einstein-Marić and heir sons Eduard and Hans Albert; Elsa Einstein-Löwenthal and her daughters Ilse and Margot) are watching as Einstein is pursuing his new theory of gravity and his idée fixeof generalizing the relativity principle while carried by giants of both physics and mathematics (Isaac Newton, James Clerk Maxwell, Carl Friedrich Gauss, Bernhard Riemann) and scientists of lesser stature (Marcel Grossmann, Gunnar Nordström, Erwin Finlay Freundlich, Michele Besso).

Nature 527, 298–300 (19 Nov 2015)



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