Kurt Gödel

From Wikipedia, the free encyclopedia

Kurt Gödel
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.

• Collected Works: Oxford University Press: New York. Editor-in-chief: Solomon Feferman.
  • Volume I: Publications 1929–1936 ISBN 978-0-19-503964-1 / Paperback:ISBN 978-0-19-514720-9,
  • Volume II: Publications 1938–1974 ISBN 978-0-19-503972-6 / Paperback:ISBN 978-0-19-514721-6,
  • Volume III: Unpublished Essays and Lectures ISBN 978-0-19-507255-6 / Paperback:ISBN 978-0-19-514722-3,
  • Volume IV: Correspondence, A–G ISBN 978-0-19-850073-5,
  • Volume V: Correspondence, H–Z ISBN 978-0-19-850075-9.

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:

One type of non-traversable wormhole metric is the Schwarzschild solution (see the first diagram):

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.

John Archibald Wheeler

From Wikipedia, the free encyclopedia

 

John Archibald Wheeler
John Archibald Wheeler ( right ) together with Eckehard W. Mielke in front of lake in Holstein before the Hermann Weyl-Conference 1985 in Kiel, Germany
Born	July 9, 1911 Jacksonville, Florida, United States
Died	April 13, 2008 (aged 96) Hightstown, New Jersey, United States
Residence	United States
Nationality	American
Fields	Physics
Institutions	University of North Carolina Princeton University University of Texas at Austin
Alma mater	Johns Hopkins University (Ph.D)
Doctoral advisor	Karl Herzfeld
Doctoral students	Hugh Everett Richard Feynman Bahram Mashhoon James Griffin Demetrios Christodoulou Claudio Bunster Jacob Bekenstein Robert Geroch John R. Klauder Kenneth W. Ford Charles Misner Kip Thorne Arthur Wightman Bill Unruh Robert Wald Milton Plesset Benjamin Schumacher Dieter Brill Bei-lok Hu Warner A. Miller Yavuz Nutku Arkady Kheyfets Edward Fireman David Kerlick Harry King Ignazio Ciufolini
Known for	Breit–Wheeler process Popularizing the term "black hole" Nuclear fission Geometrodynamics General relativity Unified field theory Wheeler–Feynman absorber theory Wheeler's delayed choice experiment
Notable awards	Enrico Fermi Award (1968) Franklin Medal (1969) National Medal of Science (1970) Oersted Medal (1983) Albert Einstein Medal (1988) Matteucci Medal (1993) Wolf Prize (1997)

John Archibald Wheeler (July 9, 1911 – April 13, 2008) was an American theoretical physicist who was largely responsible for reviving interest in general relativity in the United States after World War II. Wheeler also worked with Niels Bohr in explaining the basic principles behind nuclear fission. One of the later collaborators of Albert Einstein, he tried to achieve Einstein's vision of a unified field theory. Together with G. Breit, Wheeler developed the concept of Breit–Wheeler process. He is also known for popularizing the term "black hole", for coining the terms "quantum foam", "wormhole", and "it from bit", and for hypothesizing the "one-electron universe". For most of his career, Wheeler was a professor at Princeton University, and was influential in mentoring a generation of physicists who made notable contributions to quantum mechanics and gravitation.

Biography

John Archibald Wheeler was born in Jacksonville, Florida to librarians Joseph Lewis Wheeler and Mabel Archibald Wheeler.^[1] He graduated from the Baltimore City College high school in 1926^[2] and earned his doctorate from Johns Hopkins University in 1933. His dissertation research work, carried out under the supervision of Karl Herzfeld, was on the theory of the dispersion and absorption of helium.

Wheeler started his academic career at the University of North Carolina at Chapel Hill in 1935 and in 1938 moved to Princeton University where he remained until 1976. He then became the director of the Center for Theoretical Physics at the University of Texas from 1976 to 1986, when he retired from academic work. At the time of his death, Wheeler had returned to Princeton University as a professor emeritus. Professor Wheeler's graduate students included Richard Feynman, Kip Thorne, Jacob Bekenstein, Charles Misner and Hugh Everett.^[3][4] Unlike some scholars, Wheeler gave a high priority to teaching. Even after he became a famous physicist, he continued to teach freshman and sophomore physics, saying that the young minds were the most important. Wheeler supervised more PhD as well as senior undergraduate theses than any other professor in the Princeton physics department.

Wheeler made important contributions to theoretical physics. In 1937, he introduced the S-matrix, which became an indispensable tool in particle physics. Wheeler was a pioneer in the theory of nuclear fission, along with Niels Bohr and Enrico Fermi. In 1939, Wheeler collaborated with Bohr on the liquid drop model of nuclear fission.

Together with many other leading physicists, during World War II, Wheeler interrupted his academic career to participate in the development of the atomic bomb during the Manhattan Project, working at the Hanford Site in Washington, where several large nuclear reactors were constructed to produce the element plutonium for atomic bombs. Even before the Hanford Site started up the "B-Pile" (the first of its three reactors), Wheeler had anticipated that the accumulation of "fission product poisons" would eventually impede the ongoing nuclear chain reaction by absorbing many of the thermal neutrons that were needed to continue a chain reaction. Wheeler deduced, by calculating its half-life in radioactive decay, that an isotope of the noble gas xenon (Xe¹³⁵) would be the one most responsible.^[5]

Some years later, Wheeler went on to work on the development of the more powerful hydrogen bomb under the nuclear weapons program.

After concluding his Manhattan Project work, Wheeler returned to Princeton University to resume his academic career. In 1957, while working on mathematical extensions to the Theory of General Relativity, Wheeler introduced the concept and the word wormhole to describe hypothetical "tunnels" in space-time.

During the 1950s, Wheeler formulated geometrodynamics, a program of physical and ontological reduction of every physical phenomenon, such as gravitation and electromagnetism, to the geometrical properties of a curved space-time. Aiming at a systematical identification of matter with space, geometrodynamics was often characterized as a continuation of the philosophy of nature as conceived by Descartes and Spinoza. Wheeler's geometrodynamics, however, failed to explain some important physical phenomena, such as the existence of fermions (electrons, muons, etc.) or that of gravitational singularities. Wheeler therefore abandoned his theory as somewhat fruitless during the early 1970s.

For a few decades, general relativity had not been considered a very respectable field of physics, being detached from experiment. Wheeler was a key figure in the revival of the subject, leading the school at Princeton, while Sciama and Zel'dovich developed the subject at Cambridge University and the University of Moscow. The work of Wheeler and his students made high contributions to the Golden Age of General Relativity.

His work in general relativity included the theory of gravitational collapse. He used the term black hole in 1967 during a talk he gave at the NASA Goddard Institute of Space Studies (GISS).^[6] He was also a pioneer in the field of quantum gravity with his development (with Bryce DeWitt) of the Wheeler–DeWitt equation, which is the equation governing the "wave function of the Universe", as he called it.

Recognizing Wheeler's colorful way with words, characterized by such confections as "mass without mass", the festschrift honoring his 60th birthday was fittingly entitled Magic Without Magic: John Archibald Wheeler: A collection of essays in honor of his sixtieth birthday, Ed: John R. Klauder, (W. H. Freeman, 1972, ISBN 0-7167-0337-8). That same writing style could also attract parodies, including one famous one by "John Archibald Wyler" that was affectionately published by a relativity journal.^[7][8]

Wheeler was the driving force behind the voluminous general relativity textbook Gravitation, co-written with Charles W. Misner and Kip Thorne. Its timely appearance during the golden age of general relativity and its comprehensiveness made it the most influential relativity textbook for a generation.

In 1979, Wheeler spoke to the American Association for the Advancement of Science (AAAS), asking it to expel parapsychology, which had been admitted ten years earlier at the request of Margaret Mead. He called it a pseudoscience,^[9] saying he did not oppose earnest research into the questions, but he thought the "air of legitimacy" of being an AAAS-Affiliate should be reserved until convincing tests of at least a few so-called psi effects could be demonstrated.^[10] His request was turned down, and the Parapsychological Association remained a member of the AAAS.

In 1990, Wheeler suggested that information is fundamental to the physics of the universe. According to this "it from bit" doctrine, all things physical are information-theoretic in origin.^[11]

Wheeler: It from bit. Otherwise put, every "it" — every particle, every field of force, even the space-time continuum itself — derives its function, its meaning, its very existence entirely — even if in some contexts indirectly — from the apparatus-elicited answers to yes-or-no questions, binary choices, bits. "It from bit" symbolizes the idea that every item of the physical world has at bottom — a very deep bottom, in most instances — an immaterial source and explanation; that which we call reality arises in the last analysis from the posing of yes-or-no questions and the registering of equipment-evoked responses; in short, that all things physical are information-theoretic in origin and that this is a participatory universe.

Wheeler was awarded the Wolf Prize in Physics in 1997.

Wheeler has speculated that reality is created by observers in the universe. "How does something arise from nothing?", he asks about the existence of space and time (Princeton Physics News, 2006). He also coined the term "Participatory Anthropic Principle" (PAP), a version of a Strong Anthropic Principle. From a transcript of a radio interview on "The anthropic universe":^[12]

Wheeler: We are participators in bringing into being not only the near and here but the far away and long ago. We are in this sense, participators in bringing about something of the universe in the distant past and if we have one explanation for what's happening in the distant past why should we need more?
Martin Redfern: Many don't agree with John Wheeler, but if he's right then we and presumably other conscious observers throughout the universe, are the creators — or at least the minds that make the universe manifest.

On April 13, 2008, Wheeler died of pneumonia at the age of 96 in Hightstown, New Jersey.^[13]

In April 2009, Wheeler was the focus of the monthly periodical Physics Today published by the American Institute of Physics. The articles contained reflection by prominent physicists, including many of those for whom he served as an academic advisor.

Books by Wheeler

• Wheeler, John Archibald (1962). Geometrodynamics. New York: Academic Press. OCLC 1317194.
• Misner, Charles W.; Kip S. Thorne; John Archibald Wheeler (September 1973). Gravitation. San Francisco: W. H. Freeman. ISBN 0-7167-0344-0.
• Some Men and Moments in the History of Nuclear Physics: The Interplay of Colleagues and Motivations. University of Minnesota Press. 1979.
• A Journey Into Gravity and Spacetime (1990). Scientific American Library. W.H. Freeman & Company 1999 reprint: ISBN 0-7167-6034-7
• Spacetime Physics: Introduction to Special Relativity (1992). W. H. Freeman, ISBN 0-7167-2327-1
• At Home in the Universe (1994). American Institute of Physics 1995 reprint: ISBN 1-56396-500-3
• Gravitation and Inertia (1995). Ignazio Ciufolini and John Archibald Wheeler. Princeton University Press. Princeton, New Jersey. ISBN 0-691-03323-4.
• Geons, Black Holes, and Quantum Foam: A Life in Physics (1998). New York: W.W. Norton & Co, hardcover: ISBN 0-393-04642-7, paperback: ISBN 0-393-31991-1 — autobiography and memoir.
• Exploring Black Holes: Introduction to General Relativity (2000). Addison Wesley, ISBN 0-201-38423-X

Bibliography

• "Update on John Archibald Wheeler", Princeton Physics News 2 (Winter 2006), Princeton University.