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Sunday, September 21, 2014

Roger Penrose

Roger Penrose

From Wikipedia, the free encyclopedia

Sir Roger Penrose
Roger Penrose-6Nov2005.jpg
Roger Penrose, 2005
Born 8 August 1931 (age 83)
Colchester, Essex, England
Residence United Kingdom
Nationality British
Fields Mathematical physics
Institutions
Alma mater
Doctoral advisor John A. Todd
Other academic advisors W. V. D. Hodge
Doctoral students
Known for
Influenced
Notable awards
Notes
He is the brother of Jonathan Penrose, Oliver Penrose and Shirley Hodgson; son of Lionel Penrose; nephew of Roland Penrose.

Sir Roger Penrose OM FRS (born 8 August 1931), is an English mathematical physicist, mathematician and philosopher of science. He is the Emeritus Rouse Ball Professor of Mathematics at the Mathematical Institute of the University of Oxford, as well as an Emeritus Fellow of Wadham College.

Penrose is known for his work in mathematical physics, in particular for his contributions to general relativity and cosmology. He has received a number of prizes and awards, including the 1988 Wolf Prize for physics, which he shared with Stephen Hawking for their contribution to our understanding of the universe.[1]

Early life and academia

Born in Colchester, Essex, England, Roger Penrose is a son of psychiatrist and mathematician Lionel Penrose and Margaret Leathes,[2] and the grandson of the physiologist John Beresford Leathes. His uncle was artist Roland Penrose, whose son with photographer Lee Miller is Antony Penrose. Penrose is the brother of mathematician Oliver Penrose and of chess Grandmaster Jonathan Penrose. Penrose attended University College School and University College, London, where he graduated with a first class degree in mathematics. In 1955, while still a student, Penrose reintroduced the E. H. Moore generalised matrix inverse, also known as the Moore–Penrose inverse,[3] after it had been reinvented by Arne Bjerhammar (1951). Penrose earned his PhD at Cambridge (St John's College) in 1958, writing a thesis on "tensor methods in algebraic geometry" under algebraist and geometer John A. Todd. He devised and popularised the Penrose triangle in the 1950s, describing it as "impossibility in its purest form" and exchanged material with the artist M. C. Escher, whose earlier depictions of impossible objects partly inspired it. Escher's Waterfall, and Ascending and Descending were in turn inspired by Penrose. As reviewer Manjit Kumar puts it:
As a student in 1954, Penrose was attending a conference in Amsterdam when by chance he came across an exhibition of Escher's work. Soon he was trying to conjure up impossible figures of his own and discovered the tri-bar – a triangle that looks like a real, solid three-dimensional object, but isn't. Together with his father, a physicist and mathematician, Penrose went on to design a staircase that simultaneously loops up and down. An article followed and a copy was sent to Escher. Completing a cyclical flow of creativity, the Dutch master of geometrical illusions was inspired to produce his two masterpieces.[4]
In 1965, at Cambridge, Penrose proved that singularities (such as black holes) could be formed from the gravitational collapse of immense, dying stars.[5] This work was extended by Hawking to prove the Penrose–Hawking singularity theorems.
Oil painting by Urs Schmid (1995) of a Penrose tiling using fat and thin rhombi.

In 1967, Penrose invented the twistor theory which maps geometric objects in Minkowski space into the 4-dimensional complex space with the metric signature (2,2). In 1969, he conjectured the cosmic censorship hypothesis. This proposes (rather informally) that the universe protects us from the inherent unpredictability of singularities (such as the one in the centre of a black hole) by hiding them from our view behind an event horizon. This form is now known as the "weak censorship hypothesis"; in 1979, Penrose formulated a stronger version called the "strong censorship hypothesis". Together with the BKL conjecture and issues of nonlinear stability, settling the censorship conjectures is one of the most important outstanding problems in general relativity. Also from 1979 dates Penrose's influential Weyl curvature hypothesis on the initial conditions of the observable part of the Universe and the origin of the second law of thermodynamics.[6] Penrose and James Terrell independently realised that objects travelling near the speed of light will appear to undergo a peculiar skewing or rotation. This effect has come to be called the Terrell rotation or Penrose–Terrell rotation.[7][8]
A Penrose tiling

Penrose is well known for his 1974 discovery of Penrose tilings, which are formed from two tiles that can only tile the plane nonperiodically, and are the first tilings to exhibit fivefold rotational symmetry. Penrose developed these ideas based on the article Deux types fondamentaux de distribution statistique[9] (1938; an English translation Two Basic Types of Statistical Distribution) by Czech geographer, demographer and statistician Jaromír Korčák. In 1984, such patterns were observed in the arrangement of atoms in quasicrystals.[10] Another noteworthy contribution is his 1971 invention of spin networks, which later came to form the geometry of spacetime in loop quantum gravity. He was influential in popularising what are commonly known as Penrose diagrams (causal diagrams).

In 1983, Penrose was invited to teach at Rice University in Houston, by the then provost Bill Grodon. Roger Penrose worked at Rice University form 1983 to 1987.[11]

Later activity

In 2004 Penrose released The Road to Reality: A Complete Guide to the Laws of the Universe, a 1,099-page book aimed at giving a comprehensive guide to the laws of physics. He has proposed a novel interpretation of quantum mechanics.[12]

Penrose is the Francis and Helen Pentz Distinguished (visiting) Professor of Physics and Mathematics at Pennsylvania State University.[13] He is also a member of the Astronomical Review Editorial Board.

An earlier universe

WMAP image of the (extremely tiny) anisotropies in the cosmic background radiation

In 2010, Penrose reported possible evidence, based on concentric circles found in WMAP data of the CMB sky, of an earlier universe existing before the Big Bang of our own present universe.[14] He mentions this evidence in the epilogue of his 2010 book Cycles of Time,[15] a book in which he presents his reasons, to do with Einstein's field equations, the Weyl curvature C, and the Weyl curvature hypothesis, that the transition at the Big Bang could have been smooth enough for a previous universe to survive it. He made several conjectures about C and the WCH, some of which were subsequently proved by others, and the smoothness is real. In simple terms, he believes that the singularity in Einstein's field equation at the Big Bang is only an apparent singularity, similar to the well-known apparent singularity at the event horizon of a black hole. The latter singularity can be removed by a change of coordinate system, and Penrose proposes a different change of coordinate system that will remove the singularity at the big bang. This was a daring step, relying on certain conjectures being proved, but these have subsequently been proved.[citation needed] One implication of this is that the major events at the Big Bang can be understood without unifying general relativity and quantum mechanics, and therefore we are not necessarily constrained by the Wheeler–DeWitt equation, which disrupts time.

Physics and consciousness

Prof. Penrose at a conference.

Penrose has written books on the connection between fundamental physics and human (or animal) consciousness. In The Emperor's New Mind (1989), he argues that known laws of physics are inadequate to explain the phenomenon of consciousness. Penrose proposes the characteristics this new physics may have and specifies the requirements for a bridge between classical and quantum mechanics (what he calls correct quantum gravity). Penrose uses a variant of Turing's halting theorem to demonstrate that a system can be deterministic without being algorithmic. (For example, imagine a system with only two states, ON and OFF. If the system's state is ON when a given Turing machine halts and OFF when the Turing machine does not halt, then the system's state is completely determined by the machine; nevertheless, there is no algorithmic way to determine whether the Turing machine stops.)

Penrose believes that such deterministic yet non-algorithmic processes may come into play in the quantum mechanical wave function reduction, and may be harnessed by the brain. He argues that the present computer is unable to have intelligence because it is an algorithmically deterministic system. He argues against the viewpoint that the rational processes of the mind are completely algorithmic and can thus be duplicated by a sufficiently complex computer. This contrasts with supporters of strong artificial intelligence, who contend that thought can be simulated algorithmically. He bases this on claims that consciousness transcends formal logic because things such as the insolubility of the halting problem and Gödel's incompleteness theorem prevent an algorithmically based system of logic from reproducing such traits of human intelligence as mathematical insight. These claims were originally espoused by the philosopher John Lucas of Merton College, Oxford.

The Penrose/Lucas argument about the implications of Gödel's incompleteness theorem for computational theories of human intelligence has been widely criticised by mathematicians, computer scientists and philosophers, and the consensus among experts in these fields seems to be that the argument fails, though different authors may choose different aspects of the argument to attack.[16] Marvin Minsky, a leading proponent of artificial intelligence, was particularly critical, stating that Penrose "tries to show, in chapter after chapter, that human thought cannot be based on any known scientific principle." Minsky's position is exactly the opposite – he believes that humans are, in fact, machines, whose functioning, although complex, is fully explainable by current physics. Minsky maintains that "one can carry that quest [for scientific explanation] too far by only seeking new basic principles instead of attacking the real detail. This is what I see in Penrose's quest for a new basic principle of physics that will account for consciousness."[17]

Penrose responded to criticism of The Emperor's New Mind with his follow up 1994 book Shadows of the Mind, and in 1997 with The Large, the Small and the Human Mind. In those works, he also combined his observations with that of anesthesiologist Stuart Hameroff.

Penrose and Hameroff have argued that consciousness is the result of quantum gravity effects in microtubules, which they dubbed Orch-OR (orchestrated objective reduction). Max Tegmark, in a paper in Physical Review E,[18] calculated that the time scale of neuron firing and excitations in microtubules is slower than the decoherence time by a factor of at least 10,000,000,000. The reception of the paper is summed up by this statement in Tegmark's support: "Physicists outside the fray, such as IBM's John A. Smolin, say the calculations confirm what they had suspected all along. 'We're not working with a brain that's near absolute zero. It's reasonably unlikely that the brain evolved quantum behavior'".[19] Tegmark's paper has been widely cited by critics of the Penrose–Hameroff position.

In their reply to Tegmark's paper, also published in Physical Review E, the physicists Scott Hagan, Jack Tuszynski and Hameroff[20][21] claimed that Tegmark did not address the Orch-OR model, but instead a model of his own construction. This involved superpositions of quanta separated by 24 nm rather than the much smaller separations stipulated for Orch-OR. As a result, Hameroff's group claimed a decoherence time seven orders of magnitude greater than Tegmark's, but still well short of the 25 ms required if the quantum processing in the theory was to be linked to the 40 Hz gamma synchrony, as Orch-OR suggested. To bridge this gap, the group made a series of proposals. It was supposed that the interiors of neurons could alternate between liquid and gel states. In the gel state, it was further hypothesized that the water electrical dipoles are oriented in the same direction, along the outer edge of the microtubule tubulin subunits. Hameroff et al. proposed that this ordered water could screen any quantum coherence within the tubulin of the microtubules from the environment of the rest of the brain. Each tubulin also has a tail extending out from the microtubules, which is negatively charged, and therefore attracts positively charged ions. It is suggested that this could provide further screening. Further to this, there was a suggestion that the microtubules could be pumped into a coherent state by biochemical energy.
Roger Penrose in the University of Santiago de Compostela to receive the Fonseca Prize.

Finally, it is suggested that the configuration of the microtubule lattice might be suitable for quantum error correction, a means of holding together quantum coherence in the face of environmental interaction. In the last decade, some researchers who are sympathetic to Penrose's ideas have proposed an alternative scheme for quantum processing in microtubules based on the interaction of tubulin tails with microtubule-associated proteins, motor proteins and presynaptic scaffold proteins. These proposed alternative processes have the advantage of taking place within Tegmark's time to decoherence.

Hameroff, in a lecture in part of a Google Tech talks series exploring quantum biology, gave an overview of current research in the area, and responded to subsequent criticisms of the Orch-OR model.[22] In addition to this, a recent 2011 paper by Roger Penrose and Stuart Hameroff gives an updated model of their Orch-OR theory, in light of criticisms, and discusses the place of consciousness within the universe.[23]

Phillip Tetlow, although himself supportive of Penrose's views, acknowledges that Penrose's ideas about the human thought process are at present a minority view in scientific circles, citing Minsky's criticisms and quoting science journalist Charles Seife's description of Penrose as "one of a handful of scientists" who believe that the nature of consciousness suggests a quantum process.[19]

In January 2014 Hameroff and Penrose announced that a discovery of quantum vibrations in microtubules by Anirban Bandyopadhyay of the National Institute for Materials Science in Japan[24] confirms the hypothesis of Orch-OR theory.[25] A reviewed and updated version of the theory was published along with critical commentary and debate in the March 2014 issue of Physics of Life Reviews.[26]

Personal life

Family life

Penrose is married to Vanessa Thomas, head of mathematics at Abingdon School,[27][28] with whom he has one son.[27] He has three sons from a previous marriage to American Joan Isabel Wedge, whom he married in 1959. He is the elder brother of Jonathan Penrose, the chessplayer.

Religious views

Penrose does not hold to any religious doctrine,[29] and refers to himself as an atheist.[30] In the film A Brief History of Time, he said, "I think I would say that the universe has a purpose, it's not somehow just there by chance ... some people, I think, take the view that the universe is just there and it runs along – it's a bit like it just sort of computes, and we happen somehow by accident to find ourselves in this thing. But I don't think that's a very fruitful or helpful way of looking at the universe, I think that there is something much deeper about it."[31] Penrose is a Distinguished Supporter of the British Humanist Association.

Awards and honours

Roger Penrose during a lecture

Penrose has been awarded many prizes for his contributions to science. He was elected a Fellow of the Royal Society of London in 1972. In 1975, Stephen Hawking and Penrose were jointly awarded the Eddington Medal of the Royal Astronomical Society. In 1985, he was awarded the Royal Society Royal Medal. Along with Stephen Hawking, he was awarded the prestigious Wolf Foundation Prize for Physics in 1988. In 1989 he was awarded the Dirac Medal and Prize of the British Institute of Physics. In 1990 Penrose was awarded the Albert Einstein Medal for outstanding work related to the work of Albert Einstein by the Albert Einstein Society. In 1991, he was awarded the Naylor Prize of the London Mathematical Society. From 1992 to 1995 he served as President of the International Society on General Relativity and Gravitation. In 1994, Penrose was knighted for services to science.[32] In the same year he was also awarded an Honorary Degree (Doctor of Science) by the University of Bath.[33] In 1998, he was elected Foreign Associate of the United States National Academy of Sciences. In 2000 he was appointed to the Order of Merit. In 2004 he was awarded the De Morgan Medal for his wide and original contributions to mathematical physics. To quote the citation from the London Mathematical Society:
His deep work on General Relativity has been a major factor in our understanding of black holes. His development of Twistor Theory has produced a beautiful and productive approach to the classical equations of mathematical physics. His tilings of the plane underlie the newly discovered quasi-crystals.[34]
In 2005 Penrose was awarded an honorary doctorate by Warsaw University and Katholieke Universiteit Leuven (Belgium), and in 2006 by the University of York. In 2008 Penrose was awarded the Copley Medal. He is also a Distinguished Supporter of the British Humanist Association and one of the patrons of the Oxford University Scientific Society. In 2011, Penrose was awarded the Fonseca Prize by the University of Santiago de Compostela. In 2012, Penrose was awarded the Richard R. Ernst Medal by ETH Zürich for his contributions to science and strengthening the connection between science and society.

Works

Forewords to Beating the Odds: The Life and Times of E. A. Milne, written by Meg Weston Smith. Published by World Scientific Publishing Co in June 2013.
Penrose also wrote forewords to Quantum Aspects of Life and Anthony Zee's book Fearful Symmetry (foreword).
at

Kurt Gödel

Kurt Gödel

From Wikipedia, the free encyclopedia

Kurt Gödel
Kurt gödel.jpg
Born Kurt Friedrich Gödel
April 28, 1906
Brünn, Austria-Hungary (now Brno, Czech Republic)
Died January 14, 1978 (aged 71)
Princeton, New Jersey, United States
Residence United States
Citizenship Austria, USA
Fields Mathematics, Mathematical logic
Institutions Institute for Advanced Study
Alma mater University of Vienna
Doctoral advisor Hans Hahn
Known for Gödel's incompleteness theorems, Gödel's completeness theorem, the consistency of the Continuum hypothesis with ZFC, Gödel metric, Gödel's ontological proof
Notable awards Albert Einstein Award (1951); National Medal of Science (USA) in Mathematical, Statistical, and Computational Sciences (1974)
Fellow of the British Academy
Signature

Kurt Friedrich Gödel (/ˈkɜrt ɡɜrdəl/; German: [ˈkʊʁt ˈɡøːdəl] ( listen); April 28, 1906 – January 14, 1978) was an Austrian, and later American, logician, mathematician, and philosopher. Considered with Aristotle and Gottlob Frege to be one of the most significant logicians in history, Gödel made an immense impact upon scientific and philosophical thinking in the 20th century, a time when others such as Bertrand Russell,[1] A. N. Whitehead,[1] and David Hilbert were pioneering the use of logic and set theory to understand the foundations of mathematics.

Gödel published his two incompleteness theorems in 1931 when he was 25 years old, one year after finishing his doctorate at the University of Vienna. The first incompleteness theorem states that for any self-consistent recursive axiomatic system powerful enough to describe the arithmetic of the natural numbers (for example Peano arithmetic), there are true propositions about the naturals that cannot be proved from the axioms. To prove this theorem, Gödel developed a technique now known
as Gödel numbering, which codes formal expressions as natural numbers.

He also showed that neither the axiom of choice nor the continuum hypothesis can be disproved from the accepted axioms of set theory, assuming these axioms are consistent. The former result opened the door for mathematicians to assume the axiom of choice in their proofs. He also made important contributions to proof theory by clarifying the connections between classical logic, intuitionistic logic, and modal logic.

Childhood

Gödel was born April 28, 1906, in Brünn, Austria-Hungary (now Brno, Czech Republic) into the ethnic German family of Rudolf Gödel, the manager of a textile factory, and Marianne Gödel (born Handschuh).[2] At the time of his birth the city had a German-speaking majority,[3] and this was the language of his parents.[4] The marriage of his parents was confessionally mixed, the father being a Catholic and the mother a Protestant. The children were raised in the Protestant confession. The ancestors of Kurt Gödel were often active in Brünn's cultural life. For example, his grandfather Joseph Gödel was a famous singer of that time and for some years a member of the "Brünner Männergesangverein".[5]

Gödel automatically became a Czechoslovak citizen at age 12 when the Austro-Hungarian Empire broke up at the end of World War I. According to his classmate Klepetař, like many residents of the predominantly German Sudetenländer, "Gödel considered himself always Austrian and an exile in Czechoslovakia".[6] He chose to become an Austrian citizen at age 23[citation needed]. When Germany annexed Austria in 1938, Gödel automatically became a German citizen at age 32. After World War II, at the age of 42, he became an American citizen.

In his family, young Kurt was known as Herr Warum ("Mr. Why") because of his insatiable curiosity. According to his brother Rudolf, at the age of six or seven Kurt suffered from rheumatic fever; he completely recovered, but for the rest of his life he remained convinced that his heart had suffered permanent damage.

Gödel attended the Evangelische Volksschule, a Lutheran school in Brünn from 1912 to 1916, and was enrolled in the Deutsches Staats-Realgymnasium from 1916 to 1924, excelling with honors in all his subjects, particularly in mathematics, languages and religion. Although Kurt had first excelled in languages, he later became more interested in history and mathematics. His interest in mathematics increased when in 1920 his older brother Rudolf (born 1902) left for Vienna to go to medical school at the University of Vienna. During his teens, Kurt studied Gabelsberger shorthand, Goethe's Theory of Colours and criticisms of Isaac Newton, and the writings of Immanuel Kant.

Studying in Vienna

At the age of 18, Gödel joined his brother in Vienna and entered the University of Vienna. By that time, he had already mastered university-level mathematics.[7] Although initially intending to study theoretical physics, he also attended courses on mathematics and philosophy. During this time, he adopted ideas of mathematical realism. He read Kant's Metaphysische Anfangsgründe der Naturwissenschaft, and participated in the Vienna Circle with Moritz Schlick, Hans Hahn, and Rudolf Carnap. Gödel then studied number theory, but when he took part in a seminar run by Moritz Schlick which studied Bertrand Russell's book Introduction to Mathematical Philosophy, he became interested in mathematical logic. According to Gödel mathematical logic was "a science prior to all others, which contains the ideas and principles underlying all sciences."[8]

Attending a lecture by David Hilbert in Bologna on completeness and consistency of mathematical systems may have set Gödel's life course. In 1928, Hilbert and Wilhelm Ackermann published Grundzüge der theoretischen Logik (Principles of Mathematical Logic), an introduction to first-order logic in which the problem of completeness was posed: Are the axioms of a formal system sufficient to derive every statement that is true in all models of the system?

This was the topic chosen by Gödel for his doctoral work. In 1929, at the age of 23, he completed his doctoral dissertation under Hans Hahn's supervision. In it, he established the completeness of the first-order predicate calculus (Gödel's completeness theorem). He was awarded his doctorate in 1930. His thesis, along with some additional work, was published by the Vienna Academy of Science.

The Incompleteness Theorem

In 1931 and while still in Vienna, Gödel published his incompleteness theorems in Über formal unentscheidbare Sätze der "Principia Mathematica" und verwandter Systeme (called in English "On Formally Undecidable Propositions of "Principia Mathematica" and Related Systems"). In that article, he proved for any computable axiomatic system that is powerful enough to describe the arithmetic of the natural numbers (e.g. the Peano axioms or Zermelo–Fraenkel set theory with the axiom of choice), that:
  1. If the system is consistent, it cannot be complete.
  2. The consistency of the axioms cannot be proven within the system.
These theorems ended a half-century of attempts, beginning with the work of Frege and culminating in Principia Mathematica and Hilbert's formalism, to find a set of axioms sufficient for all mathematics.

In hindsight, the basic idea at the heart of the incompleteness theorem is rather simple. Gödel essentially constructed a formula that claims that it is unprovable in a given formal system. If it were provable, it would be false, which contradicts the idea that in a consistent system, provable statements are always true. Thus there will always be at least one true but unprovable statement. That is, for any computably enumerable set of axioms for arithmetic (that is, a set that can in principle be printed out by an idealized computer with unlimited resources), there is a formula that obtains in arithmetic, but which is not provable in that system. To make this precise, however, Gödel needed to produce a method to encode statements, proofs, and the concept of provability as natural numbers. He did this using a process known as Gödel numbering.

In his two-page paper Zum intuitionistischen Aussagenkalkül (1932) Gödel refuted the finite-valuedness of intuitionistic logic. In the proof he implicitly used what has later become known as Gödel–Dummett intermediate logic (or Gödel fuzzy logic).

The mid-1930s: further work and visits to the US

Gödel earned his habilitation at Vienna in 1932, and in 1933 he became a Privatdozent (unpaid lecturer) there. In 1933 Adolf Hitler came to power in Germany and over the following years the Nazis rose in influence in Austria, and among Vienna's mathematicians. In June 1936, Moritz Schlick, whose seminar had aroused Gödel's interest in logic, was assassinated by a pro-Nazi student. This triggered "a severe nervous crisis" in Gödel.[10] He developed paranoid symptoms, including a fear of being poisoned, and spent several months in a sanitarium for nervous diseases.[11]

In 1933, Gödel first traveled to the U.S., where he met Albert Einstein, who became a good friend.[12] He delivered an address to the annual meeting of the American Mathematical Society. During this year, Gödel also developed the ideas of computability and recursive functions to the point where he delivered a lecture on general recursive functions and the concept of truth. This work was developed in number theory, using Gödel numbering.

In 1934 Gödel gave a series of lectures at the Institute for Advanced Study (IAS) in Princeton, New Jersey, entitled On undecidable propositions of formal mathematical systems. Stephen Kleene, who had just completed his PhD at Princeton, took notes of these lectures that have been subsequently published.

Gödel would visit the IAS again in the autumn of 1935. The traveling and the hard work had exhausted him, and the next year he took a break to recover from a depressive episode. He returned to teaching in 1937. During this time, he worked on the proof of consistency of the axiom of choice and of the continuum hypothesis; he would go on to show that these hypotheses cannot be disproved from the common system of axioms of set theory.

He married Adele Nimbursky (née Porkert, 1899–1981), whom he had known for over 10 years, on September 20, 1938. Their relationship had been opposed by his parents on the grounds that she was a divorced dancer, six years older than he was.

Subsequently, he left for another visit to the USA, spending the autumn of 1938 at the IAS and the spring of 1939 at the University of Notre Dame.

Relocation to Princeton, Einstein and US citizenship

After the Anschluss in 1938, Austria had become a part of Nazi Germany. Germany abolished the title of Privatdozent, so Gödel had to apply for a different position under the new order. His former association with Jewish members of the Vienna Circle, especially with Hahn, weighed against him.
The University of Vienna turned his application down.

His predicament intensified when the German army found him fit for conscription. World War II started in September 1939. Before the year was up, Gödel and his wife left Vienna for Princeton. To avoid the difficulty of an Atlantic crossing, the Gödels took the trans-Siberian railway to the Pacific, sailed from Japan to San Francisco (which they reached on March 4, 1940), then crossed the U.S. by train to Princeton, where Gödel would accept a position at the Institute for Advanced Study (IAS).
Gödel very quickly resumed his mathematical work. In 1940, he published his work Consistency of the axiom of choice and of the generalized continuum-hypothesis with the axioms of set theory, which is a classic of modern mathematics.[citation needed] In that work he introduced the constructible universe, a model of set theory in which the only sets that exist are those that can be constructed from simpler sets. Gödel showed that both the axiom of choice (AC) and the generalized continuum hypothesis (GCH) are true in the constructible universe, and therefore must be consistent with the Zermelo–Fraenkel axioms for set theory (ZF). This result has had considerable consequences for working mathematicians, as it means that they can assume the axiom of choice when proving the Hahn-Banach theorem. Paul Cohen later constructed a model of ZF in which AC and GCH are false; together these proofs mean that AC and GCH are independent of the ZF axioms for set theory.
Albert Einstein was also living at Princeton during this time. Gödel and Einstein developed a strong friendship, and were known to take long walks together to and from the Institute for Advanced Study. The nature of their conversations was a mystery to the other Institute members. Economist Oskar Morgenstern recounts that toward the end of his life Einstein confided that his "own work no longer meant much, that he came to the Institute merely ... to have the privilege of walking home with Gödel".[13]

Gödel and his wife Adele spent the summer of 1942 in Blue Hill, Maine, at the Blue Hill Inn at the top of the bay. Gödel was not merely vacationing but had a very productive summer of work. Using Heft 15 [volume 15] of Gödel's still-unpublished Arbeitshefte [working notebooks], John W. Dawson, Jr. conjectures that Gödel discovered a proof for the independence of the axiom of choice from finite type theory, a weakened form of set theory, while in Blue Hill in 1942. Gödel's close friend Hao Wang supports this conjecture, noting that Gödel's Blue Hill notebooks contain his most extensive treatment of the problem.

On December 5, 1947, Einstein and Morgenstern accompanied Gödel to his U.S. citizenship exam, where they acted as witnesses. Gödel had confided in them that he had discovered an inconsistency in the U.S. Constitution that would allow the U.S. to become a dictatorship. Einstein and Morgenstern were concerned that their friend's unpredictable behavior might jeopardize his application. Fortunately, the judge turned out to be Phillip Forman, who knew Einstein and had administered the oath at Einstein's own citizenship hearing. Everything went smoothly until Forman happened to ask Gödel if he thought a dictatorship like the Nazi regime could happen in the U.S. Gödel then started to explain his discovery to Forman. Forman understood what was going on, cut Gödel off, and moved the hearing on to other questions and a routine conclusion.[14][15]

Later years and death

Gödel became a permanent member of the Institute for Advanced Study at Princeton in 1946. Around this time he stopped publishing, though he continued to work. He became a full professor at the Institute in 1953 and an emeritus professor in 1976.[citation needed]

During his many years at the Institute, Gödel's interests turned to philosophy and physics. In 1949, he demonstrated the existence of solutions involving closed time-like curves, to Albert Einstein's field equations in general relativity.[16] He is said to have given this elaboration to Einstein as a present for his 70th birthday.[17] His "rotating universes" would allow time travel to the past and caused Einstein to have doubts about his own theory. His solutions are known as the Gödel metric (an exact solution of the Einstein field equation).

He studied and admired the works of Gottfried Leibniz, but came to believe that a hostile conspiracy had caused some of Leibniz's works to be suppressed.[18] To a lesser extent he studied Immanuel Kant and Edmund Husserl. In the early 1970s, Gödel circulated among his friends an elaboration of Leibniz's version of Anselm of Canterbury's ontological proof of God's existence. This is now known as Gödel's ontological proof. Gödel was awarded (with Julian Schwinger) the first Albert Einstein Award in 1951, and was also awarded the National Medal of Science, in 1974.[citation needed]
Gravestone of Kurt and Adele Gödel in the Princeton, N.J., cemetery

In later life, Gödel suffered periods of mental instability and illness. He had an obsessive fear of being poisoned; he would eat only food that his wife, Adele, prepared for him. Late in 1977, she was hospitalized for six months and could no longer prepare her husband's food. In her absence, he refused to eat, eventually starving to death.[19] He weighed 65 pounds (approximately 30 kg) when he died. His death certificate reported that he died of "malnutrition and inanition caused by personality disturbance" in Princeton Hospital on January 14, 1978.[20] Adele's death followed in 1981.

Religious views

Gödel was a convinced theist.[21] He held the notion that God was personal, which differed from the religious views of his friend Albert Einstein.

He believed firmly in an afterlife, stating: "Of course this supposes that there are many relationships which today's science and received wisdom haven't any inkling of. But I am convinced of this [the afterlife], independently of any theology." It is "possible today to perceive, by pure reasoning" that it "is entirely consistent with known facts." "If the world is rationally constructed and has meaning, then there must be such a thing [as an afterlife]."[22]

In an unmailed answer to a questionnaire, Gödel described his religion as "baptized Lutheran (but not member of any religious congregation). My belief is theistic, not pantheistic, following Leibniz rather than Spinoza."[23] Describing religion(s) in general, Gödel said: "Religions are, for the most part, bad—but religion is not".[24] About Islam he said: "I like Islam, it is a consistent [or consequential] idea of religion and open-minded."[25]

Legacy

The Kurt Gödel Society, founded in 1987, was named in his honor. It is an international organization for the promotion of research in the areas of logic, philosophy, and the history of mathematics. The University of Vienna hosts the Kurt Gödel Research Center for Mathematical Logic. The Association for Symbolic Logic has invited an annual Kurt Gödel lecturer each year since 1990.

Five volumes of Gödel's collected works have been published. The first two include Gödel's publications; the third includes unpublished manuscripts from Gödel's Nachlass, and the final two include correspondence.

A biography of Gödel was published by John Dawson in 2005. Gödel was also one of four mathematicians examined in the 2008 BBC documentary entitled Dangerous Knowledge by David Malone.[26]

Douglas Hofstadter wrote a popular book in 1979 called Gödel, Escher, Bach to celebrate the work and ideas of Gödel, along with those of artist M. C. Escher and composer Johann Sebastian Bach. The book partly explores the ramifications of the fact that Gödel's incompleteness theorem can be applied to any Turing-complete computational system, which may include the human brain.

Important publications

In German:
  • 1930, "Die Vollständigkeit der Axiome des logischen Funktionenkalküls." Monatshefte für Mathematik und Physik 37: 349–60.
  • 1931, "Über formal unentscheidbare Sätze der Principia Mathematica und verwandter Systeme, I." Monatshefte für Mathematik und Physik 38: 173–98.
  • 1932, "Zum intuitionistischen Aussagenkalkül", Anzeiger Akademie der Wissenschaften Wien 69: 65–66.
In English:
  • 1940. The Consistency of the Axiom of Choice and of the Generalized Continuum Hypothesis with the Axioms of Set Theory. Princeton University Press.
  • 1947. "What is Cantor's continuum problem?" The American Mathematical Monthly 54: 515–25. Revised version in Paul Benacerraf and Hilary Putnam, eds., 1984 (1964). Philosophy of Mathematics: Selected Readings. Cambridge Univ. Press: 470–85.
  • 1950, "Rotating Universes in General Relativity Theory." Proceedings of the international Congress of Mathematicians in Cambridge, 1: 175–81
In English translation:
  • Kurt Godel, 1992. On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. B. Meltzer, with a comprehensive introduction by Richard Braithwaite. Dover reprint of the 1962 Basic Books edition.
  • Kurt Godel, 2000.[27] On Formally Undecidable Propositions Of Principia Mathematica And Related Systems, tr. Martin Hirzel
  • Jean van Heijenoort, 1967. A Source Book in Mathematical Logic, 1879–1931. Harvard Univ. Press.
    • 1930. "The completeness of the axioms of the functional calculus of logic," 582–91.
    • 1930. "Some metamathematical results on completeness and consistency," 595–96. Abstract to (1931).
    • 1931. "On formally undecidable propositions of Principia Mathematica and related systems," 596–616.
    • 1931a. "On completeness and consistency," 616–17.
  • "My philosophical viewpoint", c. 1960, unpublished.
  • "The modern development of the foundations of mathematics in the light of philosophy", 1961, unpublished.
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Wormhole

Wormhole

From Wikipedia, the free encyclopedia

A wormhole, also known as an Einstein–Rosen bridge, is a hypothetical topological feature of spacetime that would fundamentally be a "shortcut" through spacetime. A wormhole is much like a tunnel with two ends each in separate points in spacetime.

For a simplified notion of a wormhole, visualize space as a two-dimensional (2D) surface. In this case, a wormhole can be pictured as a hole in that surface that leads into a 3D tube (the inside surface of a cylinder). This tube then re-emerges at another location on the 2D surface with a similar hole as the entrance. An actual wormhole would be analogous to this but with the spatial dimensions raised by one. For example, instead of circular holes on a 2D plane, a real wormhole's mouths could be spheres in 3D space.

Researchers have no observational evidence for wormholes, but the equations of the theory of general relativity have valid solutions that contain wormholes. Because of its robust theoretical strength, a wormhole is one of the great physics metaphors for teaching general relativity. The first type of wormhole solution discovered was the Schwarzschild wormhole, which would be present in the Schwarzschild metric describing an eternal black hole, but it was found that this type of wormhole would collapse too quickly for anything to cross from one end to the other. Wormholes which could actually be crossed in both directions, known as traversable wormholes, would only be possible if exotic matter with negative energy density could be used to stabilize them.

The Casimir effect shows that quantum field theory allows the energy density in certain regions of space to be negative relative to the ordinary vacuum energy, and it has been shown theoretically that quantum field theory allows states where the energy can be arbitrarily negative at a given point.[1]
Many physicists such as Stephen Hawking,[2] Kip Thorne,[3] and others[4][5][6] therefore argue that such effects might make it possible to stabilize a traversable wormhole. Physicists have not found any natural process that would be predicted to form a wormhole naturally in the context of general relativity, although the quantum foam hypothesis is sometimes used to suggest that tiny wormholes might appear and disappear spontaneously at the Planck scale,[7][8] and stable versions of such wormholes have been suggested as dark matter candidates.[9][10] It has also been proposed that if a tiny wormhole held open by a negative-mass cosmic string had appeared around the time of the Big Bang, it could have been inflated to macroscopic size by cosmic inflation.[11]

The American theoretical physicist John Archibald Wheeler coined the term wormhole in 1957; however, in 1921, the German mathematician Hermann Weyl already had proposed the wormhole theory, in connection with mass analysis of electromagnetic field energy.[12]
This analysis forces one to consider situations...where there is a net flux of lines of force, through what topologists would call "a handle" of the multiply-connected space, and what physicists might perhaps be excused for more vividly terming a "wormhole".
—John Wheeler in Annals of Physics

Definition

The basic notion of an intra-universe wormhole is that it is a compact region of spacetime whose boundary is topologically trivial but whose interior is not simply connected. Formalizing this idea leads to definitions such as the following, taken from Matt Visser's Lorentzian Wormholes.
If a Minkowski spacetime contains a compact region Ω, and if the topology of Ω is of the form Ω ~ R x Σ, where Σ is a three-manifold of the nontrivial topology, whose boundary has topology of the form ∂Σ ~ S2, and if, furthermore, the hypersurfaces Σ are all spacelike, then the region Ω contains a quasipermanent intra-universe wormhole.
Characterizing inter-universe wormholes is more difficult. For example, one can imagine a 'baby' universe connected to its 'parent' by a narrow 'umbilicus'. One might like to regard the umbilicus as the throat of a wormhole, but the spacetime is simply connected. For this reason wormholes have been defined geometrically, as opposed to topologically, as regions of spacetime that constrain the incremental deformation of closed surfaces. For example, in Enrico Rodrigo’s The Physics of Stargates a wormhole is defined informally as
a region of spacetime containing a "world tube" (the time evolution of a closed surface) that cannot be continuously deformed (shrunk) to a world line [(the time evolution of a point)].

Schwarzschild wormholes

An artist's impression of a wormhole from an observer's perspective, crossing the event horizon of a Schwarzschild wormhole which bridges two different universes. The observer originates from the right, and another universe becomes visible in the center of the wormhole’s shadow once the horizon is crossed, the observer seeing light that has fallen into the black hole interior region from the other universe; however, this other universe is unreachable in the case of a Schwarzschild wormhole, as the bridge always collapses before the observer has time to cross it, and everything that has fallen through the event horizon of either universe is inevitably crushed in the singularity.

Lorentzian wormholes known as Schwarzschild wormholes or Einstein–Rosen bridges are connections between areas of space that can be modeled as vacuum solutions to the Einstein field equations, and which are now understood to be intrinsic parts of the maximally extended version of the Schwarzschild metric describing an eternal black hole with no charge and no rotation. Here, "maximally extended" refers to the idea that the spacetime should not have any "edges": for any possible trajectory of a free-falling particle (following a geodesic) in the spacetime, it should be possible to continue this path arbitrarily far into the particle's future or past, unless the trajectory hits a gravitational singularity like the one at the center of the black hole's interior. In order to satisfy this requirement, it turns out that in addition to the black hole interior region which particles enter when they fall through the event horizon from the outside, there must be a separate white hole interior region which allows us to extrapolate the trajectories of particles which an outside observer sees rising up away from the event horizon. And just as there are two separate interior regions of the maximally extended spacetime, there are also two separate exterior regions, sometimes called two different "universes", with the second universe allowing us to extrapolate some possible particle trajectories in the two interior regions. This means that the interior black hole region can contain a mix of particles that fell in from either universe (and thus an observer who fell in from one universe might be able to see light that fell in from the other one), and likewise particles from the interior white hole region can escape into either universe. All four regions can be seen in a spacetime diagram which uses Kruskal–Szekeres coordinates.

In this spacetime, it is possible to come up with coordinate systems such that if you pick a hypersurface of constant time (a set of points that all have the same time coordinate, such that every point on the surface has a space-like separation, giving what is called a 'space-like surface') and draw an "embedding diagram" depicting the curvature of space at that time, the embedding diagram will look like a tube connecting the two exterior regions, known as an "Einstein–Rosen bridge". Note that the Schwarzschild metric describes an idealized black hole that exists eternally from the perspective of external observers; a more realistic black hole that forms at some particular time from a collapsing star would require a different metric. When the infalling stellar matter is added to a diagram of a black hole's history, it removes the part of the diagram corresponding to the white hole interior region, along with the part of the diagram corresponding to the other universe.[13]

The Einstein–Rosen bridge was discovered by Albert Einstein and his colleague Nathan Rosen, who first published the result in 1935. However, in 1962 John A. Wheeler and Robert W. Fuller published a paper showing that this type of wormhole is unstable if it connects two parts of the same universe, and that it will pinch off too quickly for light (or any particle moving slower than light) that falls in from one exterior region to make it to the other exterior region.

The motion through a Schwarzschild wormhole connecting two universes is possible in only one direction. The analysis of the radial geodesic motion of a massive particle into an Einstein–Rosen bridge shows that the proper time of the particle extends to infinity. Timelike and null geodesics in the gravitational field of a Schwarzschild wormhole are complete because the expansion scalar in the Raychaudhuri equation has a discontinuity at the event horizon, and because an Einstein–Rosen bridge is represented by the Kruskal diagram in which the two antipodal future event horizons are identified. Schwarzschild wormholes and Schwarzschild black holes are different, mathematical solutions of general relativity and Einstein–Cartan–Sciama–Kibble theory of gravity. Yet for distant observers, both solutions with the same mass are indistinguishable. These results suggest that all observed astrophysical black holes may be Einstein–Rosen bridges, each with a new universe inside that formed simultaneously with the black hole. Accordingly, our own Universe may be the interior of a black hole existing inside another universe.[14]

According to general relativity, the gravitational collapse of a sufficiently compact mass forms a singular Schwarzschild black hole. In the Einstein–Cartan–Sciama–Kibble theory of gravity, however, it forms a regular Einstein–Rosen bridge. This theory extends general relativity by removing a constraint of the symmetry of the affine connection and regarding its antisymmetric part, the torsion tensor, as a dynamical variable. Torsion naturally accounts for the quantum-mechanical, intrinsic angular momentum (spin) of matter. The minimal coupling between torsion and Dirac spinors generates a repulsive spin–spin interaction which is significant in fermionic matter at extremely high densities. Such an interaction prevents the formation of a gravitational singularity. Instead, the collapsing matter reaches an enormous but finite density and rebounds, forming the other side of the bridge.[15]

Before the stability problems of Schwarzschild wormholes were apparent, it was proposed that quasars were white holes forming the ends of wormholes of this type.[citation needed]

While Schwarzschild wormholes are not traversable in both directions, their existence inspired Kip Thorne to imagine traversable wormholes created by holding the 'throat' of a Schwarzschild wormhole open with exotic matter (material that has negative mass/energy).

Traversable wormholes

Image of a traversable wormhole that connects the square in front of the physical institutes of Tübingen University with the sand dunes near Boulogne sur Mer in the north of France. The image is calculated with 4D raytracing in a Morris–Thorne wormhole metric, but the gravitational effects on the wavelength of light have not been simulated.[16]

Lorentzian traversable wormholes would allow travel in both directions from one part of the universe to another part of that same universe very quickly or would allow travel from one universe to another. The possibility of traversable wormholes in general relativity was first demonstrated by Kip Thorne and his graduate student Mike Morris in a 1988 paper. For this reason, the type of traversable wormhole they proposed, held open by a spherical shell of exotic matter, is referred to as a Morris–Thorne wormhole. Later, other types of traversable wormholes were discovered as allowable solutions to the equations of general relativity, including a variety analyzed in a 1989 paper by Matt Visser, in which a path through the wormhole can be made where the traversing path does not pass through a region of exotic matter. However, in the pure Gauss–Bonnet gravity (a modification to general relativity involving extra spatial dimensions which is sometimes studied in the context of brane cosmology) exotic matter is not needed in order for wormholes to exist—they can exist even with no matter.[17] A type held open by negative mass cosmic strings was put forth by Visser in collaboration with Cramer et al.,[11] in which it was proposed that such wormholes could have been naturally created in the early universe.

Wormholes connect two points in spacetime, which means that they would in principle allow travel in time, as well as in space. In 1988, Morris, Thorne and Yurtsever worked out explicitly how to convert a wormhole traversing space into one traversing time.[3] However, according to general relativity it would not be possible to use a wormhole to travel back to a time earlier than when the wormhole was first converted into a time machine by accelerating one of its two mouths.[18]

Raychaudhuri's theorem and exotic matter

To see why exotic matter is required, consider an incoming light front traveling along geodesics, which then crosses the wormhole and re-expands on the other side. The expansion goes from negative to positive. As the wormhole neck is of finite size, we would not expect caustics to develop, at least within the vicinity of the neck. According to the optical Raychaudhuri's theorem, this requires a violation of the averaged null energy condition. Quantum effects such as the Casimir effect cannot violate the averaged null energy condition in any neighborhood of space with zero curvature,[19] but calculations in semiclassical gravity suggest that quantum effects may be able to violate this condition in curved spacetime.[20] Although it was hoped recently that quantum effects could not violate an achronal version of the averaged null energy condition,[21] violations have nevertheless been found,[22] so it remains an open possibility that quantum effects might be used to support a wormhole.

Faster-than-light travel

The impossibility of faster-than-light relative speed only applies locally. Wormholes might allow superluminal (faster-than-light) travel by ensuring that the speed of light is not exceeded locally at any time. While traveling through a wormhole, subluminal (slower-than-light) speeds are used. If two points are connected by a wormhole whose length is shorter than the distance between them outside the wormhole, the time taken to traverse it could be less than the time it would take a light beam to make the journey if it took a path through the space outside the wormhole. However, a light beam traveling through the wormhole would always beat the traveler. As an analogy, sprinting around to the opposite side of a mountain at maximum speed may take longer than walking through a tunnel crossing it.

Time travel

The theory of general relativity predicts that if traversable wormholes exist, they could allow time travel.[3] This would be accomplished by accelerating one end of the wormhole to a high velocity relative to the other, and then sometime later bringing it back; relativistic time dilation would result in the accelerated wormhole mouth aging less than the stationary one as seen by an external observer, similar to what is seen in the twin paradox. However, time connects differently through the wormhole than outside it, so that synchronized clocks at each mouth will remain synchronized to someone traveling through the wormhole itself, no matter how the mouths move around.[23] This means that anything which entered the accelerated wormhole mouth would exit the stationary one at a point in time prior to its entry.
For example, consider two clocks at both mouths both showing the date as 2000. After being taken on a trip at relativistic velocities, the accelerated mouth is brought back to the same region as the stationary mouth with the accelerated mouth's clock reading 2004 while the stationary mouth's clock read 2012. A traveler who entered the accelerated mouth at this moment would exit the stationary mouth when its clock also read 2004, in the same region but now eight years in the past. Such a configuration of wormholes would allow for a particle's world line to form a closed loop in spacetime, known as a closed timelike curve. An object traveling through a wormhole could carry energy or charge from one time to another, but this would not violate conservation of energy or charge in each time, because the energy/charge of the wormhole mouth itself would change to compensate for the object that fell into it or emerged from it.[24][25]

It is thought that it may not be possible to convert a wormhole into a time machine in this manner; the predictions are made in the context of general relativity, but general relativity does not include quantum effects. Analyses using the semiclassical approach to incorporating quantum effects into general relativity have sometimes indicated that a feedback loop of virtual particles would circulate through the wormhole and pile up on themselves, driving the energy density in the region very high and possibly destroying it before any information could be passed through it, in keeping with the chronology protection conjecture. The debate on this matter is described by Kip S. Thorne in the book Black Holes and Time Warps, and a more technical discussion can be found in The quantum physics of chronology protection by Matt Visser.[26] There is also the Roman ring, which is a configuration of more than one wormhole. This ring seems to allow a closed time loop with stable wormholes when analyzed using semiclassical gravity, although without a full theory of quantum gravity it is uncertain whether the semiclassical approach is reliable in this case.

Inter-universe travel

A possible resolution to the paradoxes resulting from wormhole-enabled time travel rests on the many-worlds interpretation of quantum mechanics. In 1991 David Deutsch showed that quantum theory is fully consistent (in the sense that the so-called density matrix can be made free of discontinuities) in spacetimes with closed timelike curves.[27] However, later it was shown that such model of closed timelike curve can have internal inconsistencies as it will lead to strange phenomena like distinguishing non orthogonal quantum states and distinguishing proper and improper mixture.[28][29] Accordingly, the destructive positive feedback loop of virtual particles circulating through a wormhole time machine, a result indicated by semi-classical calculations, is averted. A particle returning from the future does not return to its universe of origination but to a parallel universe. This suggests that a wormhole time machine with an exceedingly short time jump is a theoretical bridge between contemporaneous parallel universes.[30] Because a wormhole time-machine introduces a type of nonlinearity into quantum theory, this sort of communication between parallel universes is consistent with Joseph Polchinski’s discovery of an “Everett phone” in Steven Weinberg’s formulation of nonlinear quantum mechanics.[31]

Metrics

Theories of wormhole metrics describe the spacetime geometry of a wormhole and serve as theoretical models for time travel. An example of a (traversable) wormhole metric is the following:
ds^2= - c^2 dt^2 + dl^2 + (k^2 + l^2)(d \theta^2 + \sin^2 \theta \, d\phi^2).
One type of non-traversable wormhole metric is the Schwarzschild solution (see the first diagram):
ds^2= - c^2 \left(1 - \frac{2GM}{rc^2}\right)dt^2 + \frac{dr^2}{1 - \frac{2GM}{rc^2}} + r^2(d \theta^2 + \sin^2 \theta \, d\phi^2).

In fiction

Wormholes are a common element in science fiction as they allow interstellar, intergalactic, and sometimes interuniversal travel within human timescales. They have also served as a method for time travel.
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