What is Perspicacia (Part 2)?

Some of the greatest insights in physics have come out of paradoxes and incompatibilities between well-established theories of nature and newly observed phenomena. In Part 1 of this blog, we saw the development of classical physics up to the 19th century. At the dawn of the 20th century, it was widely believed that the end of physics was near. Lord Kelvin, a pre-eminent physicist of the day is said to have famously pronounced (in a possibly apocryphal event) "There is nothing new to be discovered in physics now, all that remains is more and more precise measurement". Newton's laws were spectacularly successful in describing the motion of bodies, both earthly and celestial. The macroscopic and thermodynamic properties of matter could be explained in terms of the statistical behavior of atoms that obeyed Newtonian mechanics. Maxwell's equations gave a complete mathematical description of electromagnetic phenomena. While Faraday's original formulation of electromagnetic theory did not postulate any medium for the transmission of electromagnetic fields, it was widely believed that an all-pervading material called the aether was the medium for electromagnetic phenomena. Even Maxwell is said to have believed that electromagnetic waves traveled as oscillations of the aether.  Also, classical physics at the turn of the century rested on the twin foundations of causality and determinism. These principles demanded that given initial conditions, the dynamic evolution of any system is governed by physical laws that can be expressed by differential equations, whose solutions allow precise calculation of positions and velocities of particles involved in the dynamics of the system.

However, there were some ominous clouds gathering over the prevailing sense of completeness and finality in the subject. Attempts to fit the Faraday-Maxwell theory of electromagnetism into the mechanistic framework of the aether were ad hoc at best. No experiment could detect the all-pervading aether. Precise experiments such as those of Michelson and Morley failed to detect any changes in the speed of light due to the earth's motion through the aether (unlike for example the change in the speed of sound due to the motion of the sound source through the air). There were other vexing astronomical anomalies such as the perihelion motion of Mercury's orbit. On the earthly front, the existing wave theory of light could not explain the photoelectric effect, where light shining on a metal resulted in the production of an electric current whose energy depended only on the frequency of the light as opposed to the intensity of the light. There were still many questions about the existence of atoms, their structure, and their relationship to radiation phenomena. But what triggered the quantum revolution was the problem of "black body" radiation.

The Quantum Revolution

A black body is one that absorbs all the light falling on it. When a black body is heated it starts to radiate and a major problem in theoretical physics was to derive the observed spectrum of the radiation using known laws of physics. A theoretical model created by Rayleigh and Jeans based on the classical theory of electromagnetism produced the paradoxical result that the energy emitted by the body would grow indefinitely as a function of frequency. This divergence was called the 'ultraviolet catastrophe' by the physicist Paul Ehrenfest. Max Planck overcame this paradox in 1900 by postulating that energy could only be exchanged in discrete units called "quanta". A quantum of energy is related to the frequency of the emitted radiation via the famous Planck's constant h. Using his equation Planck was able to produce the observed distribution of radiation energy as a function of radiation frequency and temperature. Planck was of a conservative bent of mind and by his own admission hardly a revolutionary. But he had the courage to go where the problem took him and make a counter-intuitive postulate in order to resolve a paradox. But Albert Einstein in a fashion typical of him took what was an ad hoc postulate and generalized it to a universal property of light itself (and all electromagnetic radiation). He postulated that light consists of packets of energy called photons, which carry momenta and energy that can knock electrons off certain metals resulting in the photoelectric effect. This was a dramatic development since Thomas Young had disproved in 1800 Newton's assertion that light consisted of corpuscles by demonstrating the interference of light using the double-slit experiment. The wave theory of light had been an established fact for more than half a century and had gotten an additional boost from Maxwell's theory of electromagnetism. But here was Einstein postulating that light consisted of particles. Einstein of course knew that this implied the paradoxical statement that light was both a wave and a particle. He hoped that a deeper theory of nature would resolve this paradox. Little did he know that the theory that would eventually emerge would go against his deepest beliefs about nature.  Quantum mechanics got another early boost from Niels Bohr's work on atomic theory. Early models of the atom such as Rutherford's planetary model suffered from the uncomfortable fact that they violated known laws of electromagnetism. Unlike planets revolving around the sun, an electron revolving around a nucleus should radiate energy and fall into the nucleus. This led Bohr to conclude that the stability and spectrum of atoms could best be explained by postulating that electrons in an atom can assume only certain discrete energy states. The observed line spectra of gases such as Hydrogen could then also be explained as the result of transitions between these discrete energy states (quantum jumps). The hallmark of early quantum theory was the realization that when it comes to atomic and radiation phenomena, energy is not a continuous entity (as in classical physics) but comes in discrete "lumps" (terminology used by Richard Feynman - see Feynman Lectures in Physics Vol. 3). 

Numerous attempts to fit this realization into known classical models of electromagnetic theory and Newtonian mechanics met with failure. In the early 1920s, de Broglie and Schrödinger developed a wave theory of matter that carried a tremendous intuitive and visual appeal to classical physicists. Schrödinger's famous equation was also a major milestone that allowed one to describe the time evolution of a system using a "state vector" known also as the wave function. Schrödinger's interpretation of the wave function was in terms of physical "matter waves". But it became clear over the next 2 decades largely due to the work of Heisenberg, Pauli, Max Born, and most importantly Niels Bohr, that all visual and classical ideas of nature had to be abandoned when it came to atomic phenomena. The consensus that emerged (known as the Copenhagen Interpretation) was that wave and particle were complementary aspects of nature that manifested themselves depending upon the kind of experiment that was being performed. All measurements carry an inherent uncertainty where dual observables such as position and momenta cannot be simultaneously measured with unlimited precision. The wave function itself is an "infinite-dimensional" mathematical entity, whose only physical interpretation is that its "modulus square" represents the probability of finding a particle at a location during a measurement. Interestingly the author of this probabilistic interpretation of quantum mechanics (Max Born) attributed the idea originally to Einstein, who had suggested in an unpublished manuscript that the wave nature of light actually represented a "wave of probabilities" of photons. However, Einstein believed that these probabilities applied to an ensemble of photons and not to an individual photon. When it came to individual particles, Einstein firmly believed that deterministic laws would apply. On the other hand, Max Born's interpretation of the wave function and the later Copenhagen interpretation of quantum mechanics asserted that determinism did not apply to atomic phenomena. When it came to atomic measurements one could only speak in terms of probabilities. Einstein never accepted this "probabilistic" interpretation of quantum mechanics and maintained till the end of his life that "God does not play dice with the universe". The debates that raged between Bohr, Heisenberg, Schrödinger, and Einstein could be counted as one of the greatest philosophical debates of all time. The Copenhagen interpretation of quantum mechanics is widely accepted by physicists today with some notable exceptions. 

As Feynman stated in his numerous lectures, it is safe to say that nobody really understands quantum mechanics. Niels Bohr, one of the founding fathers of quantum theory stated that anyone who is not shocked by quantum mechanics has not really understood it. But it is indeed one of the most successful theories of nature and is the foundation of much of modern physics and technology. Quantum mechanics plays a foundational role in solid-state physics, condensed matter physics, the theory of superconductivity, nuclear physics, stellar evolution, particle physics, modern chemistry, and even modern biology. Today's technology including semiconductors, lasers, MRI machines, electronics, etc. would not be possible without quantum mechanics. 

Even when he was wrong, Albert Einstein was remarkably prescient and insightful. His ingenious thought experiments aimed at showing that Quantum Mechanics was incomplete, culminated in the famous EPR (Einstein-Podolski-Rosen) paper that highlighted the "spooky non-local" nature of quantum entanglement of particles separated by large distances. However, entanglement is a fundamental and experimentally verified aspect of quantum mechanics. It is an active area of research in both physics and quantum information theory today. Quantum mechanics is also penetrating other areas of knowledge such as information science and computing. Research labs around the world are racing to build the first quantum computer. Such a development would transform computing and communication as we know it. 

Relativity

At the turn of the 20th century, it was realized that Maxwell's equations had the strange property that the speed of an electromagnetic wave (such as light for example) seemed to be the same in all frames of reference. This was clearly incompatible with the intuitively obvious concept of relative motion. For instance, two trains moving at the same speed in the same direction appear to be at rest relative to each other. Not so with light, which seemed to travel at a fixed constant speed of 186,000 miles/sec irrespective of the speed of the observer. The speed of light (c) shows up in Maxwell's equations as a constant number that can be computed from the electrical permittivity and magnetic permeability of empty space. Maxwell himself assumed that this speed must be relative to an absolute frame of reference namely the aether. The Michelson-Morley experiment of 1887 failed to measure any movement of the earth relative to the aether. Their experiment showed that light traveled at exactly the same speed both parallel and perpendicular to the motion of the earth (which contradicted the hypothesis of a drag in the aether caused by the earth's motion). Nobody appreciated the constancy of the speed of light more deeply than Albert Einstein. After all, as a teenager, he had performed a Gedankenexperiment (thought experiment) of racing behind a ray of light and wondering what he would see. Common sense indicated that he should see a standing light wave, but that seemed to violate Maxwell's equations. Einstein was also aware of Galileo's discovery of "inertial frames of reference" and the "principle of relativity". Galileo had shown that there was no fundamental difference between a frame of reference at rest and one at uniform motion relative to it. In other words, no experiment conducted within a uniformly moving frame of reference can tell whether it is at rest or at motion. So all uniform motion seems to be relative and all laws of physics should be the same in all inertial frames of reference. Einstein in 1905, resolved the paradox of the constancy of the speed of light in all frames of reference by developing a new theory of space and time called the "Special Theory of Relativity". By elevating the "Principle of Relativity" and "the Constancy of Speed of Light" to the status of fundamental postulates, Einstein derived a host of consequences that transformed our understanding of simultaneity of events, lengths of objects, and time intervals. In Einstein's theory, space and time lost their absolute and independent nature and blended into a 4-dimensional continuum called spacetime whose geometry was such that no signal could travel faster than the speed of light. The mathematician Hermann Minkowski announced rather dramatically in a 1908 speech, "henceforth space by itself and time by itself are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality".  Some of the remarkable consequences of this theory were length contraction of moving objects, time dilation (slowing down of moving clocks), and the famous equivalence of mass and energy given by the equation E = mc^2. 

Revolutionary as the special theory of relativity was, there were some unsatisfactory aspects to it, at least as far as Einstein was concerned. The circular nature of the definition of an inertial frame had not escaped his attention. Clearly, the notion of an inertial frame was a kind of idealization or approximation. If you were far away from all other objects and if there were no measurable forces then you could treat your frame of reference as an inertial one. The special status given to inertial frames irked him and made him ask the question as to why the laws of physics would not be the same in ALL frames of reference (and not just inertial frames of reference). In this, he was influenced by the philosophical ideas of Ernst Mach. Further, he puzzled over the nature of gravity. Newton's law of gravity suggests an "action at a distance" that is transmitted instantaneously across distances between two masses. This clearly violates the special theory of relativity where nothing can travel faster than the speed of light. There was also the mystery of the equivalence of inertial and gravitational mass as discovered by Galileo (which is equivalent to the statement that all objects of any mass accelerate at the same rate near the earth). All of this set the stage for arguably one of the greatest leaps of human intuition since the time of Newton, namely the discovery of the General Theory of Relativity.  Based not just on empirically observed facts, but on phenomenological and philosophical considerations involving frames of reference, Einstein developed a dynamic theory of space, time, and gravitation that completely revolutionized our understanding of the universe. In the General Theory of Relativity, Einstein declared that the laws of physics were the same in all frames of reference (Principle of General Covariance). Moreover, spacetime in general relativity is a dynamic entity that is curved by the presence of matter. The curvature of spacetime manifests itself as the force of gravity. Bodies in motion follow geodesics (paths of least "distance") in curved spacetime. The geometry of spacetime in the vicinity of matter can be best understood using the mathematics of non-Euclidean geometry developed by Bernhard Riemann in 1854. The language and formalism of tensor analysis were used by Einstein to derive his famous field equations of gravitation. These equations provide a precise relationship between the geometry of spacetime and the distribution of matter and energy in a region. Einstein was able to show that Newton's Law of Gravitation appeared as a limiting case when gravitational fields were weak.  Einstein was able to precisely account for the observed perihelion motion of Mercury thus solving a longstanding mystery in astronomy. More remarkably he was also able to predict effects such as the bending of light due to gravity (gravitational lensing) and gravitational waves (recently detected by the LIGO observatories). 

General relativity has withstood the test of time and continues to be tested by precise experiments conducted on earth and space. It is one of the two pillars of modern physics today, the other being quantum mechanics. Einstein's theory has paved the way for much of the dramatic developments in modern cosmology and astrophysics including the theory of black holes, the Big Bang theory, and the theory of the expanding universe. 

However, lest one should think of Einstein's theory as an esoteric theory with no practical applications, it should be pointed out that both special relativity and general relativity play a critical role in modern GPS technology. The time dilation of atomic clocks due to the motion of satellites (a special relativistic effect) and the speedup of the clocks due to the location of the satellites above the earth (a general relativistic effect) impact the precision of calculation of the location of objects on earth. Modern GPS software accounts for the relativistic effects of geolocation. Anyone who questions the practical value of pursuing theoretical physics must stop to think about how the smartphone is able to provide precise turn-by-turn directions!

Remarkable applications of relativity and quantum mechanics can also be found in the health sciences - especially in the science behind MRI (Magnetic Resonance Imaging) and PET (Positron Emission Tomography). Amazing things happen when special relativity and quantum mechanics meet each other. The first one to discover this was Paul Dirac. Special relativity demands that equations of physics be written in such a way that they maintain their form in all inertial frames of reference (assuming that the gravitational effects are weak). The Schrödinger equation did not meet this requirement.  Many attempts to address this problem failed until Dirac produced his relativistic quantum equation in 1928. Two remarkable consequences of the Dirac equation were the discovery of spin angular momentum of a particle and the discovery of the positron. The former is the source of magnetic phenomena such as those used in MRI. The latter is a remarkable example of a theory predicting the existence of a completely unknown and new type of matter known as an anti-particle. A positron is an "anti-electron", which would annihilate an electron if it came into contact with it resulting in radiation energy. Positrons are used in the health sciences in PET scans (Positron Emission Tomography) using radioactive materials called tracers.  These are again examples of fundamental science having a profound impact on modern society. 

Physics Today

The last century has seen remarkable progress in the development of the physical sciences. The discovery of an expanding universe by Hubble and detection of the cosmic microwave background radiation (CMB) have allowed us to create an effective model of the universe and its evolution from the time of the Big Bang. Precision cosmology has also provided strong evidence for the inflationary expansion of the universe right after the Big Bang when the universe supposedly underwent a very rapid expansion and slowed down to allow for the creation of galaxies. X-Ray, Gamma Ray, and Infrared astronomy have provided strong evidence for the existence of extraordinarily compact astrophysical objects such as Neutron Stars and Black Holes. The recent dramatic discovery of gravitational waves by the LIGO observatories in Washington and Louisiana have provided us a window into the strange and violent nature of the universe. However, in many ways, we are in the same situation as the Greeks. Our model of the universe is incomplete and many problems remain unsolved. Two great puzzles of cosmology are the apparent presence of dark matter and dark energy. Dark matter refers to invisible matter whose presence is needed to explain the stability of spinning galaxies. Dark energy refers to a mysterious force that seems to cause the universe to expand at an increasingly rapid pace (as observed by the Hubble space telescope). It is estimated that roughly 5% of the universe is visible matter, 27% is dark matter and the rest (68%) is dark energy. However, we don't know what dark matter is and nor do we have a clear framework to describe dark energy. There are theoretical proposals such as WIMPs (Weakly Interacting Massive Particles) designed to explain dark matter.  But they have not been detected in any experiment. Dark energy is supposed to be the intrinsic energy of vacuum (represented by Einstein's cosmological constant) that causes a negative pressure leading to the expansion of the universe. However, most quantum field theories predict that the energy of the vacuum has to be 100 orders of magnitude larger than what is required to explain the observed rate of expansion. Again many alternative theories have been proposed but none are conclusive.

Quantum mechanics and special relativity have been remarkably successful in explaining the properties of matter and its internal constituents. A central problem of theoretical physics is Einstein's dream of unification of the forces of nature. The Standard Model of Particle Physics has been successful in unifying the electromagnetic, strong, and weak nuclear forces into a common theoretical framework. Experiments at the LHC (Large Hadron Collider) have verified the Standard Model by detecting the Higgs Boson. However, the force of gravity has resisted attempts at unification with the rest of the forces. String theory is a potential approach to unification, but many mysteries remain and string theory is far from being testable (let alone tested). Any such unification would require a quantum theory of gravity.  When gravitation and quantum mechanics meet in the strong field regime of a black hole, very interesting puzzles emerge. The most interesting of these is the apparent loss of information due to the evaporation of a black hole due to Hawking radiation. All of these puzzles, the proliferation of proposed theories to resolve them and the experimental attempts to probe higher energies and deeper realms of the cosmos suggest that we are at the cusp of another paradigm shift in our understanding of the universe.

Perspikacia

From ancient times to the present day, deep contemplation, precise reasoning, mathematical analysis, careful observation, and sophisticated experimentation have shown us that the world is not as it seems to be nor is it as one may imagine or wish it to be. Time and again visionary individuals such as Galileo, Newton, Faraday, Maxwell, Einstein, Bohr, Dirac, Feynman, Hawking, and many others have used their perspicacity to give us dramatically new perspectives on the inner workings of nature. In Science as in great art, it is individual perspicacity that produces shifts in human consciousness that deepens our understanding of the universe. While in some cases these individuals are driven by just "the pleasure of finding things out" (Feynman), in other cases they are propelled by the desire to "know the mind of God" (Einstein). There is something in human consciousness that takes pleasure in understanding the order of things. 

"The scientists’ religious feeling takes the form of a rapturous amazement at the harmony of natural law, which reveals an intelligence of such superiority that, compared with it, all the systematic thinking and acting of human beings is an utterly insignificant reflection."  - Albert Einstein

                                                         

What is Perspikacia (Part 1)?

                                                                            

There is no logical way to the discovery of elemental laws. There is only the way of intuition, which is helped by a feeling for the order lying behind the appearance.

- Albert Einstein

                                                 

The name Perspikacia is inspired by the Spanish word Perspicacia, which means insight. According to the 16th century French philosopher Rene' Descartes, human intelligence consists of two faculties: sagacity and perspicacity. The former enables reasoning about details to make deductions, and the latter concerns intuition and the ability to discern things directly. In his book "Rules for the Direction of the Mind", Descartes describes insight as follows: "one must focus the vision of the natural intelligence on the smallest and easiest things, and dwell on them for a long time, so long, until we have become accustomed to intuiting the truth distinctly and perspicuously" (Rule 9 - "On The Perspicacity of Intuition"). Sagacity is undoubtedly a very important quality to have in both human and scientific affairs. Science would not be possible without the painstaking work of measurement, tabulation, calculation, and deduction. However, revolutionary discoveries in science happen mainly through the process of direct insight. This series of blog posts try to give a flavor of the kinds of insights that have brought about major paradigm shifts in the physical sciences.

A Geocentric Universe

As far as we know, homo sapiens is the only species that contemplates its immediate surroundings and tries to comprehend the universe of experience and beyond. The ancients gazed at the flickering stars in the heavens and wondered about their true nature. They perceived the cycles of nature such as the periodic movement of heavenly bodies and the changing of the seasons. They saw violent and fearsome forces of nature at work in phenomena such as lightning, storms and fire. They tried to explain natural phenomena using extra-natural agents such as spirits and gods. Even until a few hundred years ago many people believed that there were invisible angels that kept the planets moving around the earth (see Feynman Lectures in Physics Vol. 1, Chapter 7). However, the development of civilization in many parts of the world (such as India, China, Persia, Arabia, and later Rome and Greece) saw a parallel development in science, astronomy, philosophy, and mathematics, where superstition and speculation were replaced by observation and rational thought. 

To the unaided and uncritical eye, the earth seems to be a flat surface that is at absolute rest (barring local features such as hills, mountains, ocean tides, etc). The heavenly bodies such as the sun, moon, and stars seem to move about the earth on a firmament of space that is inert and infinite. Time seems to flow in a definite and absolute manner and history is created by a universal and relentless march of time. The ancient civilizations and later the Greeks contemplated the universe around them and came up with theories to explain various phenomena. There were endless debates among the philosophers about the nature of the universe. Greek philosophers contemplated even abstract concepts such as motion, rest, and the constituents of matter. The philosopher Democritus believed that all matter was composed of indivisible units called atoms. Aristotle (a disciple of Plato) believed that all objects by their very nature tended to stay at rest and needed a force to keep them moving. Zeno of Elea worried about the very possibility of motion since his analysis seemed to show that a person would have to take infinitely many steps to get anywhere (Zeno's paradox). While most ancient cosmologies assumed that the Earth was a flat disc, the Pythagoreans had concluded by the 6th century BC (using observations of eclipses) that the Earth was a sphere. This idea of spherical earth at the center of the universe was further solidified by Plato and Aristotle in the 4 century BC. Plato and Aristotle had a strong belief in perfect circles and spheres as the models for celestial mechanics. But their perfect scheme ran into trouble with precise observations of planetary motion and, in the 2nd century AD, the astronomer Ptolemy introduced the concept of epicycles to resolve the conflict.  In the Ptolemaic universe the earth was still inert and at the center, but the planets moved in perfectly circular orbits called epicycles, that themselves orbited around the earth in perfect circles. This model managed to explain and predict movements of planets to a relatively high degree of accuracy and along with Aristotelian philosophy, became the chief dogma for several centuries. 

Heliocentrism and the Copernican Revolution

For a theory to survive, it should match known observed phenomena, it should have internal logical consistency and it should meet the criteria of "economy" (least number of assumptions). Science progresses when a dogma that does not meet these criteria is questioned and alternative theories are proposed. Social, religious, and political factors may prevent alternative ideas from being accepted and even lead to individual persecution (as in the case of Galileo). But eventually, the truth prevails and better models are embraced. Interestingly, the earliest challenges to the Ptolemaic and Aristotelian dogma came from Muslim astronomers in the 10th century AD.  They accepted the Geocentric view of the universe, but many questioned the static nature of the earth and the circular orbits of the epicycles. Instead, they advocated for the rotation of the earth and proposed elliptical orbits for certain planets. But the first known major challenge to Ptolemaic cosmology came from Copernicus in 1542. While Heliocentric theories had been proposed as early as the 3rd century BC by Aristarchus, it was Copernicus who in his book "On the Revolutions of the Heavenly Spheres", presented a geometric model that unambiguously posited the Earth and the planets moving around the Sun. What is interesting is that the Copernican system did not improve upon known observational data nor did it make any predictions that were better than the Ptolemaic Geocentric system. Copernicus had used circular orbits for the planets just like his predecessors and was in fact forced to introduce epicycles in order to explain certain observed planetary movements. It appears that Copernicus was primarily motivated by aesthetic and mathematical considerations more than anything else. He was applying the principle of Occam's Razor (using the least number of assumptions) to explain the observed motion of planets.  He was also the first person in history to create a complete and general system, combining mathematics, physics, and cosmology. The Heliocentric model collided with the prevailing Aristotelian world view. But apparently, it did not get Copernicus into direct trouble with the authorities. It seems that many people including members of the clergy encouraged him to publish his ideas, but Copernicus was skittish about publicizing his ideas out of fear of angering the establishment. His book was eventually published only around the time of his death at the age of 70. A century later the Italian astronomer Galileo Galilei was not as cautious and got into serious trouble for promoting Copernicus' radical ideas.

Galileo's "Heresy"

The true scientific method began with Galileo, who combined experimentation, careful observation using telescopes, deduction, and insight to uncover laws of nature that were not based on philosophical speculation or established conventions. By dropping objects of different sizes from the tower of Pisa, he showed that all objects regardless of their size fell at the same rate of acceleration (this is known as the "principle of equivalence" and served as a powerful motivation for Einstein's general theory of relativity). Galileo's insights showed that objects have something called inertia that keeps them at rest or in uniform motion in the absence of forces (in direct contrast to Aristotle's idea that everything tended to stay at rest and needed a force to keep it moving). Moreover, he discovered that a frame of reference that is in uniform motion is indistinguishable from a frame of reference at rest (a concept known as "the principle of relativity"). Galileo's main target was the prevailing Aristotelian and Ptolemaic dogma about the Geocentric model of the universe. In his book "Dialogue Concerning the Two Chief World Systems" Galileo advocated in favor of the Copernican Heliocentric model as opposed to Ptolemy's Geocentric model. In his book, Galileo presented a debate between a fictitious philosopher Salviati (representing Galileo himself), and the simple-minded and dogmatic character Simplicius (representing the philosophical establishment). In the debate, Salviati explains to Simplicius that the absence of speed sensations on the earth is not a justification for asserting that the earth is at absolute rest. Instead, he gives an evocative account of a man in a uniformly moving ship who observes water dripping from a bottle, fish swimming in a tank, and butterflies flying in a manner identical to when the ship is at rest. In modern language, Galileo was describing an "inertial frame of reference", a notion that played a central role in Einstein's special theory of relativity more than 200 years later. Galileo's book is a remarkable account of the fallacy of believing in what appears obvious to the senses without critical examination. Galileo was also the first to use a telescope to observe planets and their moons. Using his telescope he observed dark lunar spots (craters) and discovered the moons of Jupiter. The former implied that the moon was not a perfectly spherical body, something that went against the prevailing religious belief in the perfect spherical nature of all celestial objects. Clearly, Galileo's pronouncements did not sit well with his contemporaries. One of Galileo's influential and dogmatic contemporaries (a real-life Simplicius) simply refused to look through the telescope. Eventually, Galileo's views fell afoul of the Papal authorities, and Galileo was tried for heresy and sentenced to house arrest in 1633. Tragically Galileo was kept under house arrest until his death in 1642. His masterpiece was banned by the Catholic Church and remained on its list of banned books until 1835. 

Newton's Universal Laws

The Pythagoreans may have discovered that the earth was not flat, but it took the profound insights of Copernicus, Galileo, Kepler, and Newton to decisively establish a new cosmology in which spherical earth and other planets moved around the Sun in elliptical orbits following definite mathematical laws. Kepler performed a painstaking mathematical analysis of astronomical data observed and recorded by Tycho Brahe and in the process discovered his beautiful laws of planetary motion. Kepler's laws stated that a) planets follow elliptical orbits around the sun, b) a planet sweeps equal areas in equal intervals of time while orbiting the sun, and c) the square of a planet's orbital period is proportional to the cube of the semi-major axis. Newton more than anyone else established mathematics as a powerful tool and a universal language for the physical sciences. In his landmark tome "Philosophiae Naturalis Principia Mathematica", Newton revolutionized man's conception of the universe by writing down his famous Universal Laws of Motion and the Universal Law of Gravitation in a mathematical format. By postulating that the force that kept us wedded to the ground (and made apples fall from trees) was the same force that kept the planets and the moon in their orbits, Newton made arguably the greatest conceptual leap known to man. 

In Newton's Heliocentric scheme planets do not need an agent to keep them moving. They are just moving due to their inertia, having been set in motion at some original time (possibly at the time of the formation of the solar system). What keeps them from flying off is the centripetal force of gravitation. In fact, the moon is simply falling towards the earth in the same way that a baseball falls towards the earth after being thrown in the air. The only difference is that the moon is much farther away and is traveling much faster. As the moon falls towards the earth, the earth curves away thus keeping the moon in orbit around the earth (see Feynman Lectures in Physics Vol. 2). Newton explained that the ocean tides were caused by the gravitation pull of the moon as well as the centrifugal force of the rotation of the earth-moon system. With his newly developed Calculus, he was able to derive Kepler's laws of planetary motion. The mathematics and the science of the Principia were put to spectacular use by Edmund Halley when he accurately predicted the timing of the arrival of Halley's Comet (25 December 1758). Another such application of Newton's laws was the discovery of the planet Neptune. The French astronomer Le Verrier in 1846 tried to explain the irregular motion of Uranus using Newtonian mechanics by postulating the presence of another planet nearby. Using Newton's laws, Le Verrier (and earlier John Adams) made a precise prediction of the mass, orbit, and location of the body that they believed was responsible for Uranus' eccentric motion. The presence of the new planet was confirmed the same year by a German astronomer Johann Gottfried Galle, thus providing a powerful validation of Newton's laws. 

As an interesting aside, Le Verrier tried unsuccessfully to explain the anomalous motion of Mercury in a similar manner. Careful measurements of Mercury's orbit over a period of a few centuries showed that Mercury's perihelion (closest point of approach to the sun) precessed by 43 arc seconds per century. This did not match the calculations made by Le Verrier using Newtonian mechanics. One of the explanations that he offered for the aberration was the presence of a mysterious planet named Vulcan. No such planet was discovered and the precession of Mercury's perihelion remained a mystery until Albert Einstein explained it precisely using his General Theory of Relativity in 1915. 

The Kinetic Theory of Gases and the Atomic Hypothesis

Newton's laws reigned supreme for 300 years and met with spectacular success (barring little exceptions such as the motion of Mercury). While Newtonian mechanics dealt with dynamics of moving bodies, the theory was extended to the study of the motion of bulk matter such as fluids by people like Bernoulli, d'Alembert, and Leonhard Euler. The Industrial Revolution brought steam engines and factories into the scene. People became interested in macroscopic properties of matter such as heat, temperature, pressure, and their relationship to energy and work. Through the work of Maxwell and Boltzmann thermodynamics was shown to be a manifestation of the statistical properties of atoms that they believed were the microscopic constituents of matter. Maxwell and Boltzmann developed the kinetic theory of gases and provided a statistical distribution of velocities of atoms in an idealized gas as a function of the temperature of the gas. Boltzmann's famous equation named after him relates the entropy of a system (a measure of the macroscopic disorder of the system) to the logarithm of the number of possible "micro-states" of the system. The second law of thermodynamics postulates that entropy of a system almost always increases, meaning the disorder of a system tends to increase. A subject of great controversy during Boltzmann's time was the atomic theory of matter. While Maxwell and Boltzmann were firm believers in the atomic hypothesis and used it to explain macroscopic phenomena, some influential physicists and philosophers of the day (such as Ernst Mach and even initially Max Planck) were opposed to such a concept because atoms were not observable at that time. So much so that it drove the depressive Boltzmann to commit suicide in 1906. It was only when Einstein in 1905 and Jean Perrin in 1908 demonstrated theoretically and experimentally that the phenomena of Brownian motion could be used to compute the sizes of atoms that the atomic theory became widely accepted. Boltzmann's entropy equation is carved on his gravestone in Vienna.

The Faraday-Maxwell Theory of Electromagnetism

Statistical mechanics showed that the thermodynamic behavior of matter was fully compatible with Newtonian mechanics since it ultimately involved the statistical behavior of atoms moving according to Newton's laws. However, in the latter half of the 19th century, largely due to the experiments of Michael Faraday, the phenomena of electricity and magnetism were found to not fit Newton's perfect mechanistic scheme. Through a remarkable series of experiments, Faraday provided an almost complete albeit non-mathematical description of electricity, magnetism, and their relationship to each other.  More importantly, the subject saw the establishment of the abstract concept of a "field" in physics. The electromagnetic "field" is an abstract entity that permeates the space around an electrical or magnetic material. One can draw lines of force to visualize them, but they do not represent any material substance or a medium (something that bothered many scientists of the day including Lord Kelvin who accused Maxwell of resorting to mysticism). The application of mathematics (especially vector calculus ) to Faraday's experimental results, resulted in Maxwell's beautiful set of equations that completely described electromagnetic phenomena. Maxwell's equations describe the spatial and dynamic properties of the electric and magnetic fields as functions of each other and as functions of charges and currents. It is the first known instance of unification of two fundamental forces of nature (a theme that continues to occupy physicists to this day). The most spectacular outcome of Maxwell's field theory of electromagnetism was his startling insight that light is an electromagnetic wave. Light was known to be a wave since the early 1800s due to the work of Thomas Young on light interference and diffraction. Maxwell noticed that his mathematics produced a set of wave equations whose velocity factor (computed from the constants of electrical permittivity and magnetic permeability) matched exactly the empirically known speed of light. He then made a dramatic conceptual leap, when he postulated that in fact, light is nothing but an electromagnetic wave. This deep insight was later experimentally verified by Heinrich Hertz and led to the development of much of modern radio and wireless technology. It also set the stage for the twin revolutions of Relativity and Quantum Mechanics in the early part of the 20th century. It is remarkable that what was once simply an object of curiosity at the time of the Greeks, namely the behavior of materials like amber and lodestone eventually turned out to be one of the most fundamental and ubiquitous forces of nature. The solidity of materials is due to the perfect match between the positive and negative charges in the atom and the quantum effects that prevent them from collapsing into each other. All chemical processes can be described in terms of electrostatic phenomena. The neurons in the brain send messages through electrical signals and the human heartbeats due to the electrical impulses sent by the sinus node (a cluster of cells in the upper right chamber of the heart). Most of the modern technology including electronics and electrical systems are the outcome of Maxwell's equations. Most of life and human experience can arguably be described in terms of electromagnetic phenomena. Richard Feynman had this to say in his famous lectures: "From a long view of the history of mankind—seen from, say, ten thousand years from now—there can be little doubt that the most significant event of the 19th century will be judged as Maxwell’s discovery of the laws of electrodynamics. The American Civil War will pale into provincial insignificance in comparison with this important scientific event of the same decade."

                                                                                    

                                                            

The Einstein centenary

November 2015 was the centennial of a landmark event in the history of science. In this month in 1915, Albert Einstein presented his General Theory of Relativity (GR for short) to the Prussian Academy of Sciences. In a series of four papers Einstein presented a new vision that completely transformed our understanding of the universe. Arguably an event of such a significance had not occurred since the publication of Newton's "Philosophiae Naturalis Principia Mathematica".  GR was the culmination of a decade long meditation by Einstein on the relationship between space, time and gravitation. Einstein's theory was a substantial generalization of Newton's theory of gravitation. Already in 1905, Einstein had abolished the Newtonian concepts of absolute space and absolute time and had replaced them with a unified notion of "spacetime" (the geometrical formulation of this was due to Minkowski). In the Special Theory of Relativity (SR for short), Einstein made two fundamental postulates -  that the laws of physics are the same in all inertial frames of reference; and that the speed of light is the same constant in all inertial frames of reference.  As a consequence, he had demonstrated that the spacetime has some unique properties, one of which is that no signal could travel faster than the speed of light. In addition, the lengths of moving bodies, the passage of time and the mass of an object are no longer absolute quantities, but are affected by the relative motion of the observer's frame of reference. Thus in SR spacetime had an effect on moving bodies. However, the spacetime of SR was itself unaffected by the presence of matter. In GR, Einstein replaced the static and "flat" spacetime of SR with a new dynamic spacetime that is curved by the presence of matter.  The gravitational field was shown to be simply a manifestation of the curvature of spacetime. Einstein postulated in GR, that the laws of physics are the same in all frames of reference including accelerated frames (this is known as the Principle of Covariance). In the language of modern mathematics this can be stated as "the equations of physics should be invariant under all diffeomorphisms of the spacetime manifold". The Principle of Equivalence is the empirically observed fact that all frames of reference in a uniform gravitational field behave in the same way as a uniformly accelerated frame of reference. Einstein had the insight that this implied a deep connection between spacetime geometry and gravitation (in fact the Principle of Covariance is a stronger form of the Principle of Equivalence). Using this principle and the effects of SR on lengths and time intervals, Einstein argued that the geometry of spacetime in a gravitational field had to be non-Euclidean. This led him to conclude that the metric properties of spacetime represent the gravitational field near a massive object. Using the mathematics of Riemannian Geometry developed by Bernhard Riemann, Tullio Levi Civita, and others, Einstein developed a geometrical theory of space, time and gravitation culminating in his famous Field Equations of Gravitation (shown in the picture above). The equations relate a geometrical quantity (the Einstein tensor) to the matter "density" (the Energy-Momentum tensor). Einstein also noticed that objects tend to move in optimal paths known as geodesics in a uniform gravitational field and postulated that the same was true in all gravitational fields. The dynamics of moving bodies in a gravitational field could be predicted based on equations of geodesics. In summary, "spacetime tells matter how to move; matter tells spacetime how to curve" (a quote by John Wheeler). The bending of starlight in the presence of a massive object (like a star or a galaxy) was one of the remarkable predictions of this theory. This prediction was famously verified by Arthur Eddington in 1919, which made Einstein an overnight sensation and a household name. 

To celebrate this occasion, many conferences were held around the world. I had the privilege to attend the Einstein Conference in Berlin between November 30 and December 5.  The conference was actually a combination of two conferences jointly organized by the Max Planck Institute for Gravitational Physics (also known as the Albert Einstein Institute) and the Max Planck Institute for the History of Science. It was held in the Harnack Haus, a place where Einstein used to lecture regularly when he was a professor in Berlin. 

There were two parts to the conference: a technical part and a historical part. The first half (Nov 30 - Dec 2) was devoted to the latest technical developments in the fields related to GR. The second half (Dec 2 - Dec 5) was devoted to various historical aspects of the development of GR in the last century. The conference was a gathering of a variety of individuals interested in the field including theorists, mathematicians, experimentalists, astrophysicists, cosmologists and historians of science. There were a few luminaries present - including the famous physicist and author Roger Penrose, the Nobel Laureate (and String Theorist) David Gross and the veteran cosmologist Jim Peebles. The range of topics were broad and the talks were mostly accessible to a semi-technical audience. There were talks on the attempts to detect Gravitational Waves, on the astrophysics of Black Holes, on the latest developments in Cosmology, experimental tests of Gravity,  the mathematical theory of Black Holes, the Black Hole Information Paradox, String Theory and Quantum Gravity. The historical talks focused on the early decline of GR during the post-war period and the revival (renaissance?) of GR as an active area of research in the 60s and 70s. There were also talks on some of the controversies surrounding the Black Hole Information Paradox and String Theory. There were panel discussions and some debates on some of the recent attempts to quantize gravity. Given the august nature of the conference (celebration of Einstein's work) the debates were of a civil nature. During the coffee breaks there was a booth hosted by Springer Verlag and IOPscience (publisher of the journal "Classical and Quantum Gravity"). The latter was giving away volumes of latest issues of CQG, which are also available online. Springer had some of their latest books on GR on display. Needless to say I came back with quite a stash of books and journals from this conference. 

One of the best parts of the conference was an exhibit at the basement of the Harnack Haus entitled "Einstein's Road to General Relativity". The exhibition contained original manuscripts of Einstein, including a notebook containing his notes on tensor analysis, his letters on his formulation of the so called "Entwurf Theory" (Outline theory) of GR, some of the mistakes he committed in his calculations, which were corrected by his friend Michele Besso and a magnificent manuscript containing his full exposition of GR. 

The manuscripts are owned by the Hebrew University of Jerusalem and were on loan to the Max Planck Institute for the conference. A facsimile of the manuscripts were launched on a Space X Dragon spacecraft to the International Space Station in 2013 and were signed by the Italian astronaut Luca Parmitano before being returned to earth.

I have compiled here a list of lectures (organized by broad subject areas) along with a summary of some of the talks with some pictures from my photo album of this visit.

Gravitational Waves

09:00 - 10:00 Rai Weiss (Massachusetts Institute of Technology, Cambridge):

"Gravitational waves: Theoretical insight to measurement"

10:00 - 10:30 Daniel Kennefick (University of Arkansas):

"Waves without Energy - Einstein and the enigma of gravitational waves: Do they actually transport energy?"

11:30 - 12:30 Abhay Ashtekar (Penn State University, University Park):

"Even a tiny cosmological constant casts a long shadow"

Rai Weiss is a leading figure in physical cosmology (especially Cosmic Microwave Background radiation) and gravitational wave detection. He has been involved with the LIGO (Laser Interferometer Gravitational Observatory) experiments from their very inception. Rai gave an excellent talk on the history of gravitational waves, starting with Einstein's paper in 1918 in which he calculated the effects of gravitational waves resulting in the famous "quadruple formula". The formula computed the change in the metric as a function of changes in the quadruple moment of the mass.  Einstein arrived at this in analogy with electrodynamics where accelerating charges produce electromagnetic waves and are a function of the dipole moment of the charges. However, Einstein found that energy of the waves was of the order of (v/c)^5, which is an incredibly small number. The weakness of the gravitational waves led Einstein to assume that they would never be detected. In fact Einstein and Rosen even came to the conclusion in 1936, that gravitational waves simply do not exist. Einstein and Rosen's paper was rejected by the Physical Review and they published the paper in another journal. There followed a period of considerable confusion after this. It was only much later that the work of Felix Pirani, Hermann Bondi and other firmly established the possibility that gravity waves do in fact exist. On the experimental side there was the pioneering work of Joe Weber, which was unfortunately discredited due to false claims of having detected gravitational waves. The gold standard in this field according Rai Weiss was the discovery in 1975 of the binary pulsar "PSR 1913 + 16" by Hulse and Taylor, who showed remarkably that the period of revolution of the pulsar decreased due to energy loss in gravitational waves precisely as predicted by the quadruple formula. This was a sensational discovery and provided the impetus for attempts to directly observe gravitational waves. Rai Weiss gave an overview of the LIGO experiments, the amazing degrees of sensitivity achieved by Advanced LIGO and the remarkable LISA Pathfinder mission to detect gravitational waves using interferometers in space.  Daniel Kennefick's talk added additional color to the history of the theory of gravitational waves. 

Abhay Ashtekar is a leading relativist from U Penn and is most famous for his "Loop Quantum Gravity" approach to quantizing GR. He did not spend any time on quantum gravity a controversial topic needless to say), but instead focused on the problem of calculating the effect of the cosmological constant on the energy of the gravitational waves. There are numerous conceptual issues that crop up and the problem is non-trivial.

Pictures: Weiss and Kennefick

Pictures: Ashtekar

Relativistic Astrophysics

11:30 - 12:30 Andrea Ghez (University of California, Los Angeles): 

"Our Galactic Center: A Laboratory for Exploring the Physics & Astrophysics of Black Holes"

15:00 - 16:00 Reinhard Genzel (Max-Planck-Institut für extraterrestrische Physik, Garching):

"Testing the Massive Black Hole Paradigm in the Center of the Milky Way"
16:00 - 16:30 Luisa Bonolis (Max Planck Institute for the History of Science, Berlin):
"From “Dark Stars” to Gravitational Collapse within Einstein’s Theory: The Emergence of Relativistic Astrophysics"

One of my favorite talks of the conference was the one by the UCLA astronomer Andrea Ghez. Andrea is a famous for her TED talks (and other presentations on BBC, Discovery and Nova) on the hunt for a supermassive black hole at the center of our galaxy. She uses advanced advanced imaging techniques known as Adaptive Optics at the Keck Telescopes (in Hawaii) to study the center of our galaxy known as Saggitarius A*. In particular, she studies the motion of stars near the dark center of the galaxy to estimate the density of mass at the center, which turns out to be 10 million times that of the Sun. This is the strongest evidence we have to date of the existence of a supermassive black hole. Andrea's results confirmed and extended the work done by Reinhard Genzel's group at the Max Planck Institute in Garching. Genzel also gave a talk on how these independent groups are reinforcing each other's work. A fascinating aspect of Andrea's talk were some of the puzzles emerging from the observation of the center of our galaxy. In particular, a puzzle is the proliferation of young stars and the paucity of old red giants in the vicinity of the black hole. Another is the behavior of the gas cloud G2, which is showing accretion phenomena, but also survived periapse (closest approach) to the black hole. The theory is that the gas cloud is actually a binary star merger that is driven by the black hole!

Pictures: Andrea Ghez, Reinhard Genzel

Pictures: Luisa Bonolis

Mathematics of Black Holes

14:00 - 15:00 Sergiu Klainerman (Princeton University, Princeton): 

"Are black holes real? - A Mathematics perspective"
14:30- 15:00 Yvonne Choquet-Bruhat (Université Pierre et Marie Curie/Sorbonne): 
"Some memories from meeting Einstein, 1951-1952"

Mathematical general relativity started with the work of Yvonne Choquet-Bruhat, who was a young French mathematician studying under Jean Leray at the Institute for Advanced Study at Princeton at the time that Einstein was at the Institute (1951-52). Leray suggested to Yvonne that she study the so-called "Cauchy Problem" for Einstein field equations. A Cauchy problem in mathematics asks for the solution of a partial differential equation that satisfies certain conditions that are given on a hypersurface in the domain. Einstein's equations are non-linear hyperbolic equations. Yvonne showed the local existence and uniqueness of solutions to the vacuum Einstein equations. Yvonne is 92 years of age today, but nevertheless is mentally sharp. She talked about her meetings with Einstein and the experiences that she had. She described Einstein as a kind old man who was disengaged from the prevailing scientific community, but took an interest in the work of the young mathematician. Yvonne showed her results to Einstein, who was very patient, kind and gracious towards her and gave her encouragement to continue her research in the field. Yvonne painted a picture of a man who had withdrawn himself from the community of physicists. He never attended any physics lectures or seminars. He had a fixed routine and stuck to that routine. His whiteboard was always covered with equations that were attempts at unifying gravity with electromagnetism. He did not actively follow the latest developments in nuclear and particle physics. He tried gently to get Yvonne engaged in his attempts at unification, but as a young graduate Yvonne did not want to get into an area that had limited potential for success. Eventually Yvonne left IAS to join the teaching faculty at the University of Marseilles. Einstein wished her all the best in her teaching career and Yvonne was a little disappointed that he did not wish her the best in her research. Overall, Yvonne remembers Einstein as a kindly, old and somewhat lonely individual who was willing to make time for a young student like her. It was a very moving talk.

Mathematical GR is now a very active area of research, especially in the differential geometry and PDE community. Famous mathematicians such as Shing Tung-Yau and Rick Schoen have made substantial contributions to the field including the proof of the "Positive Mass Theorem" (also known as the "Positive Energy Theorem"), which states that the "energy" of an asymptotically flat spacetime is non-negative and is zero only for flat Minkowski spacetime. Sergiu Klainerman is an expert in the theory of non-linear hyperbolic PDEs and mathematical GR. He described some of the hard mathematical problems in the mathematical theory of black holes. Specifically, he talked about the problems of rigidity  and stability of Kerr black hole solutions of Einstein equations (solutions for a spherically symmetric rotating mass). Essentially these problems deal with whether the known solutions exhaust all possible vacuum blackholes and whether the solutions are stable under arbitrarily small perturbations of the metric. The difficulty of such problems is indicated by the fact that in a landmark paper Klainerman and the Greek mathematician Christodoulou proved the stability of flat Minkowski space in 1993. Klainerman also talked about the mathematical problem of Collapse, whether certain initial conditions can indeed result in the creation of a stable black hole. It was some pretty heavy stuff, but fascinating nevertheless.

Pictures: Klainerman

Experimental Tests of GR

10:00 - 11:00 Eric Adelberger (University of Washington, Seattle): 

"Tests of Einstein's equivalence principle and Newton's inverse-square law"

Progress in science is impossible without precise experiments and measurements. Henry Cavendish famously demonstrated the absence of any force due to gravity inside a hollow shell of matter. This was an indirect evidence for Newton's inverse square law of gravitation. However, the Hungarian experimentalist Loránd Eötvös was the first to use a torsion balance to make direct measurements of Newton's inverse square law of gravitation in the early 1900s. Inspired by Eötvös, a unique group of experimentalists at the University of Washington that call themselves Eöt-Wash are pushing the boundaries of precision measurements of the inverse square law, Newton's gravitational constant G and the principle of equivalence. The motivation behind these experiments is to attempt to detect quantum gravity effects or other types of forces in hitherto unexplored length scales. So far no such effects have been discovered at length scales of the order of 10^-15 cms and the equivalence principle (and therefore GR) has withstood any attempts to detect violations. Given the enormous focus on string theory, quantum gravity, LHC and other high energy experiments it is refreshing to see a group that is using simple mechanical approaches to make high precision measurements of gravity. The LIGO experiments (attempting to detect gravity waves) and the Eöt-Wash experiments are part of a fascinating trend in experimental physics attempting to make measurements at a mind-boggling level of precision (length scales of a fraction of the diameter of a proton!). The technology used to cancel out the noise due to seismic and human activity and even accounting for quantum noise at such length scales will surely have applications far beyond gravity experiments.

Pictures: Adelberger

Black Hole Information Paradox

14:00 - 15:00 Ted Jacobson (University of Maryland, College Park):

"Einstein's equation from maximal entropy of vacuum entanglement"

12:15 - 13:00 Jeroen van Dongen (University of Amsterdam and Utrecht University):
"Can we understand the Black Hole Information Paradox by studying its history?"

One of the most exciting areas of modern theoretical physics concerns the physics of black holes. Black holes arise out of solutions to Einstein's field equations and were considered by most physicists (including Einstein) to be a mathematical curiosity that had no physical meaning. Through the remarkable developments in both theoretical and experimental astrophysics in the latter half of the 20th century their reality became widely accepted. Hawking and Penrose are two of the leading lights in the theory of black holes. Their extensive work on singularities established that black holes are an essential aspect of general relativity. What is interesting is that a black hole is at the same time one of the simplest objects in the universe and also one of the most complex objects. The former became evident when Hawking and others showed that a black hole can be completely characterized by its mass, angular momentum and charge ("the no hair conjecture"). The latter was the subject of the talk. It starts with the discovery by Jakob Bekenstein and later Hawking that black holes must carry an entropy that is proportional to the area of the event horizon of the black hole. This was an amazing application of the 2nd law of thermodynamics to the subject of black holes. However, this then begs the question as to source of the enormously large entropy of black hole, which on the face of it seems to be an exceedingly simple object. There must some kind of ensemble that leads to the entropy. Recent work of Strominger and Vafa have provided some tantalizing clues, indicating that the so called "branes" of string theory account for the entropy. In any case, an even more remarkable development in the theory of black holes occurred when Hawking declared that black holes radiate and all information that went into the creation of the black hole is lost in the thermal radiation. This caused a stir in the physics community and has been the subject of numerous articles and books including a popular one by Leonard Susskind. The subject is an active area of research and seems to be still unresolved (notwithstanding recent announcements by Hawking and others). These were fascinating talks on an esoteric topic, which has captured the popular imagination. My favorite part of Ted Jacobsen's talk was when he talked about the recent experiments on "analog black holes". These are black hole like objects created in fluids (such as Bose-Einstein condensates). The second talk focused on some of the recent controversies involving string theory and the black hole information paradox. 

Pictures: Black Hole Information Paradox

Einstein's Legacy

11:30 - 12:30 David Gross (Kavli Institute for Theoretical Physics, Santa Barbara):

"The Enduring Legacy of Albert Einstein"

16:30 - 17:30 Joseph Polchinski (Kavli Institute for Theoretical Physics, Santa Barbara): 

"Quantum Gravity and Strings"

David Gross is a Nobel Laureate in Physics and one of the leaders in the field of String Theory. Edward Witten, who is considered one of the greatest physicists today was a student of Gross. Joe Polchinski is also one of the leading practitioners of string theory, author of an authoritative textbook on string theory and a central figure in the subject of black hole information paradox. Both are strong and influential proponents and defenders of string theory and have been at the center of some of the controversy surrounding string theory. They both were coming off of another conference in Germany on the question of whether string theory is real science. David's lecture on Einstein's legacy was quite masterful and well presented although he did make a plug for string theory at the end. Polchinski gave an overview of string theory, the successes and the challenges ahead. Not everyone in the audience bought into the vision. In particular, there were some who were actively pursuing alternative models of quantum gravity such as "Loop Quantum Gravity". During lunch, one of the old relativists told me sadly that "the particle physicists have taken over relativity". String theory has had some amazing recent successes, but many challenges remain. It remains to be seen how the subject evolves. The most important takeaway from Gross lecture was Einstein's role in changing how physicists thought about symmetry. Symmetry has played a central role in particle physics and in theories like the Standard Model.

Pictures: David Gross

Penrose Diagrams and Conformal Geometry

9:30 - 10:15 Roger Penrose (Oxford University): 

"Conformal Geometry"

10:15 - 11:00 Aaron Wright (Harvard University): 

"New Ways of Seeing in the Renaissance of General Relativity: Penrose Diagrams as Paper Tools"

Roger Penrose is a legendary physicist, fellow graduate student and collaborator of Hawking and a prolific author of books such as "The Road To Reality" and "The Emperor's New Mind". It was great to see him speak although I understood very little of his lecture on Conformal Geometry. He has been thinking a lot about thermodynamics and cosmology and has a specific model on cosmic evolution that attempts to resolve some of the questions related to very low entropy at the beginning of the universe. His model seems to propose a cosmic cycle involving the big bang and big crunch. My favorite part of both of these talks was the discussion of Penrose diagrams. These diagrams allows one to represent the infinite space around a black hole on a piece of paper! It is essentially a pictorial "compactification" of infinite spacetime that makes it easy to reason about behavior of particles near a black hole. The second talk suggested that Penrose's discovery and presentation of these diagrams played a central role in the revival of general relativity as an active area if research in the latter half of the 20th century.

Pictures: Penrose and Wright

Effective Field Theories, Matter from space, etc.

9:00 - 10:00 Thibault Damour, (Institut des Hautes Études Scientifique) The Problem of Motion in General Relativity: A Centenary Assessment

12:15 – 13:00 Dennis Lehmkuhl (California Institute of Technology) On Different Approaches to the Problem of Motion in General Relativity

14:00 – 14:45 Domenico Giulini (University of Hannover) Matter from Space

The main gist of these talks were early attempts to understand the role of matter in general relativity. As one speaker put it Einstein's field equation involves the "beautiful and clean" left hand side (the Einstein tensor expressing the geometry of spacetime) and "messy and ugly" right hand side (the energy momentum tensor) representing the density of matter-energy, the flux of matter-energy and the pressure due to matter-energy. The right hand side is where quantum properties of matter start becoming relevant. I walked into one of the talks after visiting the Einstein exhibition, where the speaker was talking about Feynman's attempts to derive a quantum theory of gravity. His approach was to create a quantum field theory involving spin 2 particles (gravitons) and in Minkowski spacetime and derive general covariance based on certain constraints that were imposed by this approach. His approach which he captured in a series of lectures suppressed the geometrical aspects of the theory. Later Steven Weinberg apparently gave a similar but more effective treatment of this approach. These approaches treat gravity as just another field that has a quantum field theory like any other force. All this left me in a state of shock, since Einstein's approach started with general covariance and naturally led to a geometrical theory of spacetime. It was somewhat disconcerting to see that being swept aside by a standard quantum mechanical treatment of gravity (known as "gravity as an effective field theory") based on spin 2 gravitons. In any case I understood very little of it and resolved to read more on the subject when I had the time.

Pictures