In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings. It describes how these strings propagate through interact with each other. On distance scales larger than the string scale, a string looks just like an ordinary particle, with its mass and other properties determined by the vibrational state of the string. In string theory, one of the many vibrational states of the string corresponds to the graviton, a quantum mechanical particle that carries gravitational force, thus string theory is a theory of quantum gravity. String theory is a broad and varied subject that attempts to address a number of deep questions of fundamental physics. String theory has been applied to a variety of problems in black hole physics, early universe cosmology, nuclear physics, condensed matter physics, it has stimulated a number of major developments in pure mathematics; because string theory provides a unified description of gravity and particle physics, it is a candidate for a theory of everything, a self-contained mathematical model that describes all fundamental forces and forms of matter.
Despite much work on these problems, it is not known to what extent string theory describes the real world or how much freedom the theory allows in the choice of its details. String theory was first studied in the late 1960s as a theory of the strong nuclear force, before being abandoned in favor of quantum chromodynamics. Subsequently, it was realized that the properties that made string theory unsuitable as a theory of nuclear physics made it a promising candidate for a quantum theory of gravity; the earliest version of string theory, bosonic string theory, incorporated only the class of particles known as bosons. It developed into superstring theory, which posits a connection called supersymmetry between bosons and the class of particles called fermions. Five consistent versions of superstring theory were developed before it was conjectured in the mid-1990s that they were all different limiting cases of a single theory in eleven dimensions known as M-theory. In late 1997, theorists discovered an important relationship called the AdS/CFT correspondence, which relates string theory to another type of physical theory called a quantum field theory.
One of the challenges of string theory is that the full theory does not have a satisfactory definition in all circumstances. Another issue is that the theory is thought to describe an enormous landscape of possible universes, this has complicated efforts to develop theories of particle physics based on string theory; these issues have led some in the community to criticize these approaches to physics and question the value of continued research on string theory unification. In the twentieth century, two theoretical frameworks emerged for formulating the laws of physics; the first is Albert Einstein's general theory of relativity, a theory that explains the force of gravity and the structure of space and time. The other is quantum mechanics, a different formulation to describe physical phenomena using the known probability principles. By the late 1970s, these two frameworks had proven to be sufficient to explain most of the observed features of the universe, from elementary particles to atoms to the evolution of stars and the universe as a whole.
In spite of these successes, there are still many problems. One of the deepest problems in modern physics is the problem of quantum gravity; the general theory of relativity is formulated within the framework of classical physics, whereas the other fundamental forces are described within the framework of quantum mechanics. A quantum theory of gravity is needed in order to reconcile general relativity with the principles of quantum mechanics, but difficulties arise when one attempts to apply the usual prescriptions of quantum theory to the force of gravity. In addition to the problem of developing a consistent theory of quantum gravity, there are many other fundamental problems in the physics of atomic nuclei, black holes, the early universe. String theory is a theoretical framework that attempts to address many others; the starting point for string theory is the idea that the point-like particles of particle physics can be modeled as one-dimensional objects called strings. String theory describes how strings propagate through interact with each other.
In a given version of string theory, there is only one kind of string, which may look like a small loop or segment of ordinary string, it can vibrate in different ways. On distance scales larger than the string scale, a string will look just like an ordinary particle, with its mass and other properties determined by the vibrational state of the string. In this way, all of the different elementary particles may be viewed as vibrating strings. In string theory, one of the vibrational states of the string gives rise to the graviton, a quantum mechanical particle that carries gravitational force, thus string theory is a theory of quantum gravity. One of the main developments of the past several decades in string theory was the discovery of certain "dualities", mathematical transformations that identify one physical theory with another. Physicists studying string theory have discovered a number of these dualities between different versions of string theory, this has led to the conjecture that all consistent versions of string theory are subsumed in a single framework known as M-theory.
Studies of string theory have yielded a number of results on the nature of black holes and the gravitational interaction. There are certain paradoxes that arise when one attempts to understand the quantum aspects of black holes, work on string theory
Quantum field theory
In theoretical physics, quantum field theory is a theoretical framework that combines classical field theory, special relativity, quantum mechanics and is used to construct physical models of subatomic particles and quasiparticles. QFT treats particles as excited states of their underlying fields, which are—in a sense—more fundamental than the basic particles. Interactions between particles are described by interaction terms in the Lagrangian involving their corresponding fields; each interaction can be visually represented by Feynman diagrams, which are formal computational tools, in the process of relativistic perturbation theory. As a successful theoretical framework today, quantum field theory emerged from the work of generations of theoretical physicists spanning much of the 20th century, its development began in the 1920s with the description of interactions between light and electrons, culminating in the first quantum field theory — quantum electrodynamics. A major theoretical obstacle soon followed with the appearance and persistence of various infinities in perturbative calculations, a problem only resolved in the 1950s with the invention of the renormalization procedure.
A second major barrier came with QFT's apparent inability to describe the weak and strong interactions, to the point where some theorists called for the abandonment of the field theoretic approach. The development of gauge theory and the completion of the Standard Model in the 1970s led to a renaissance of quantum field theory. Quantum field theory is the result of the combination of classical field theory, quantum mechanics, special relativity. A brief overview of these theoretical precursors is in order; the earliest successful classical field theory is one that emerged from Newton's law of universal gravitation, despite the complete absence of the concept of fields from his 1687 treatise Philosophiæ Naturalis Principia Mathematica. The force of gravity as described by Newton is an "action at a distance" — its effects on faraway objects are instantaneous, no matter the distance. In an exchange of letters with Richard Bentley, Newton stated that "it is inconceivable that inanimate brute matter should, without the mediation of something else, not material, operate upon and affect other matter without mutual contact."
It was not until the 18th century that mathematical physicists discovered a convenient description of gravity based on fields — a numerical quantity assigned to every point in space indicating the action of gravity on any particle at that point. However, this was considered a mathematical trick. Fields began to take on an existence of their own with the development of electromagnetism in the 19th century. Michael Faraday coined the English term "field" in 1845, he introduced fields as properties of space having physical effects. He argued against "action at a distance", proposed that interactions between objects occur via space-filling "lines of force"; this description of fields remains to this day. The theory of classical electromagnetism was completed in 1862 with Maxwell's equations, which described the relationship between the electric field, the magnetic field, electric current, electric charge. Maxwell's equations implied the existence of electromagnetic waves, a phenomenon whereby electric and magnetic fields propagate from one spatial point to another at a finite speed, which turns out to be the speed of light.
Action-at-a-distance was thus conclusively refuted. Despite the enormous success of classical electromagnetism, it was unable to account for the discrete lines in atomic spectra, nor for the distribution of blackbody radiation in different wavelengths. Max Planck's study of blackbody radiation marked the beginning of quantum mechanics, he treated atoms, which absorb and emit electromagnetic radiation, as tiny oscillators with the crucial property that their energies can only take on a series of discrete, rather than continuous, values. These are known as quantum harmonic oscillators; this process of restricting energies to discrete values is called quantization. Building on this idea, Albert Einstein proposed in 1905 an explanation for the photoelectric effect, that light is composed of individual packets of energy called photons; this implied that the electromagnetic radiation, while being waves in the classical electromagnetic field exists in the form of particles. In 1913, Niels Bohr introduced the Bohr model of atomic structure, wherein electrons within atoms can only take on a series of discrete, rather than continuous, energies.
This is another example of quantization. The Bohr model explained the discrete nature of atomic spectral lines. In 1924, Louis de Broglie proposed the hypothesis of wave-particle duality, that microscopic particles exhibit both wave-like and particle-like properties under different circumstances. Uniting these scattered ideas, a coherent discipline, quantum mechanics, was formulated between 1925 and 1926, with important contributions from de Broglie, Werner Heisenberg, Max Born, Erwin Schrödinger, Paul Dirac, Wolfgang Pauli.:22-23In the same year as his paper on the photoelectric effect, Einstein published his theory of special relativity, built on Maxwell's electromagnetism. New rules, called Lorentz transformation, were given for the way time and space coordinates of an event change under changes in the observer's velocity, the distinction between time and space was blurred.:19 It was proposed that all physical laws must be the same for observers at different velocities, i.e. that physical laws be invariant under Lorentz transformations.
Two difficulties remained. Observationally, the Schrödinger equation underlying q
In theoretical physics, the anti-de Sitter/conformal field theory correspondence, sometimes called Maldacena duality or gauge/gravity duality, is a conjectured relationship between two kinds of physical theories. On one side are anti-de Sitter spaces which are used in theories of quantum gravity, formulated in terms of string theory or M-theory. On the other side of the correspondence are conformal field theories which are quantum field theories, including theories similar to the Yang–Mills theories that describe elementary particles; the duality represents a major advance in our understanding of quantum gravity. This is because it provides a non-perturbative formulation of string theory with certain boundary conditions and because it is the most successful realization of the holographic principle, an idea in quantum gravity proposed by Gerard't Hooft and promoted by Leonard Susskind, it provides a powerful toolkit for studying coupled quantum field theories. Much of the usefulness of the duality results from the fact that it is a strong-weak duality: when the fields of the quantum field theory are interacting, the ones in the gravitational theory are weakly interacting and thus more mathematically tractable.
This fact has been used to study many aspects of nuclear and condensed matter physics by translating problems in those subjects into more mathematically tractable problems in string theory. The AdS/CFT correspondence was first proposed by Juan Maldacena in late 1997. Important aspects of the correspondence were elaborated in articles by Steven Gubser, Igor Klebanov, Alexander Polyakov, by Edward Witten. By 2015, Maldacena's article had over 10,000 citations, becoming the most cited article in the field of high energy physics. Current understanding of gravity is based on Albert Einstein's general theory of relativity. Formulated in 1915, general relativity explains gravity in terms of the geometry of space and time, or spacetime, it is formulated in the language of classical physics developed by physicists such as Isaac Newton and James Clerk Maxwell. The other nongravitational forces are explained in the framework of quantum mechanics. Developed in the first half of the twentieth century by a number of different physicists, quantum mechanics provides a radically different way of describing physical phenomena based on probability.
Quantum gravity is the branch of physics that seeks to describe gravity using the principles of quantum mechanics. The most popular approach to quantum gravity is string theory, which models elementary particles not as zero-dimensional points but as one-dimensional objects called strings. In the AdS/CFT correspondence, one considers theories of quantum gravity derived from string theory or its modern extension, M-theory. In everyday life, there are three familiar dimensions of space, there is one dimension of time. Thus, in the language of modern physics, one says. One peculiar feature of string theory and M-theory is that these theories require extra dimensions of spacetime for their mathematical consistency: in string theory spacetime is ten-dimensional, while in M-theory it is eleven-dimensional; the quantum gravity theories appearing in the AdS/CFT correspondence are obtained from string and M-theory by a process known as compactification. This produces a theory in which spacetime has a lower number of dimensions and the extra dimensions are "curled up" into circles.
A standard analogy for compactification is to consider a multidimensional object such as a garden hose. If the hose is viewed from a sufficient distance, it appears to have only one dimension, its length, but as one approaches the hose, one discovers that it contains a second dimension, its circumference. Thus, an ant crawling inside it would move in two dimensions; the application of quantum mechanics to physical objects such as the electromagnetic field, which are extended in space and time, is known as quantum field theory. In particle physics, quantum field theories form the basis for our understanding of elementary particles, which are modeled as excitations in the fundamental fields. Quantum field theories are used throughout condensed matter physics to model particle-like objects called quasiparticles. In the AdS/CFT correspondence, one considers, in addition to a theory of quantum gravity, a certain kind of quantum field theory called a conformal field theory; this is a symmetric and mathematically well behaved type of quantum field theory.
Such theories are studied in the context of string theory, where they are associated with the surface swept out by a string propagating through spacetime, in statistical mechanics, where they model systems at a thermodynamic critical point. In the AdS/CFT correspondence, one considers string theory or M-theory on an anti-de Sitter background; this means that the geometry of spacetime is described in terms of a certain vacuum solution of Einstein's equation called anti-de Sitter space. In elementary terms, anti-de Sitter space is a mathematical model of spacetime in which the notion of distance between points is different from the notion of distance in ordinary Euclidean geometry, it is related to hyperbolic space, which can be viewed as a disk as illustrated on the right. This image shows a tessellation of a disk by squares. One can define the distance between points of this disk in such a way that all the triangles and squares are the same size and the circular outer boundary is infinitely far from any point in the interior.
Now imagine a stack of hyperbolic disks where each disk represents the state of the universe at a given time. The resulting geometric object is three-dimensiona
Loop quantum gravity
Loop quantum gravity is a theory of quantum gravity, merging quantum mechanics and general relativity, making it a possible candidate for a theory of everything. Its goal is to unify gravity in a common theoretical framework with the other three fundamental forces of nature, beginning with relativity and adding quantum features, it competes with string theory that adds gravity. From the point of view of Einstein's theory, all attempts to treat gravity as another quantum force equal in importance to electromagnetism and the nuclear forces have failed. According to Einstein, gravity is not a force – it is a property of spacetime itself. Loop quantum gravity is an attempt to develop a quantum theory of gravity based directly on Einstein's geometric formulation. To do this, in LQG theory space and time are quantized, analogously to the way quantities like energy and momentum are quantized in quantum mechanics; the theory gives a physical picture of spacetime where space and time are granular and discrete directly because of quantization just like photons in the quantum theory of electromagnetism and the discrete energy levels of atoms.
Distance exists with a minimum. Space's structure prefers an fine fabric or network woven of finite loops; these networks of loops are called spin networks. The evolution of a spin network, or spin foam, has a scale on the order of a Planck length 10−35 metres, smaller scales do not exist. Not just matter, but space itself, prefers an atomic structure; the vast areas of research developed in several directions that involve about 30 research groups worldwide. They all share the basic physical assumptions and the mathematical description of quantum space. Research follows two directions: the more traditional canonical loop quantum gravity, the newer covariant loop quantum gravity, called spin foam theory. Physical consequences of the theory proceed in several directions; the most well-developed applies to cosmology, called loop quantum cosmology, the study of the early universe and the physics of the Big Bang. Its greatest consequence sees the evolution of the universe continuing beyond the Big Bang called the Big Bounce.
In 1986, Abhay Ashtekar reformulated Einstein's general relativity in a language closer to that of the rest of fundamental physics. Shortly after, Ted Jacobson and Lee Smolin realized that the formal equation of quantum gravity, called the Wheeler–DeWitt equation, admitted solutions labelled by loops when rewritten in the new Ashtekar variables. Carlo Rovelli and Lee Smolin defined a nonperturbative and background-independent quantum theory of gravity in terms of these loop solutions. Jorge Pullin and Jerzy Lewandowski understood that the intersections of the loops are essential for the consistency of the theory, the theory should be formulated in terms of intersecting loops, or graphs. In 1994, Rovelli and Smolin showed that the quantum operators of the theory associated to area and volume have a discrete spectrum; that is, geometry is quantized. This result defines an explicit basis of states of quantum geometry, which turned out to be labelled by Roger Penrose's spin networks, which are graphs labelled by spins.
The canonical version of the dynamics was put on firm ground by Thomas Thiemann, who defined an anomaly-free Hamiltonian operator, showing the existence of a mathematically consistent background-independent theory. The covariant or spin foam version of the dynamics developed during several decades, crystallized in 2008, from the joint work of research groups in France, Canada, UK, Germany, leading to the definition of a family of transition amplitudes, which in the classical limit can be shown to be related to a family of truncations of general relativity; the finiteness of these amplitudes was proven in 2011. It requires the existence of a positive cosmological constant, this is consistent with observed acceleration in the expansion of the Universe. In theoretical physics, general covariance is the invariance of the form of physical laws under arbitrary differentiable coordinate transformations; the essential idea is that coordinates are only artifices used in describing nature, hence should play no role in the formulation of fundamental physical laws.
A more significant requirement is the principle of general relativity that states that the laws of physics take the same form in all reference systems. This is a generalization of the principle of special relativity which states that the laws of physics take the same form in all inertial frames. In mathematics, a diffeomorphism is an isomorphism in the category of smooth manifolds, it is an invertible function that maps one differentiable manifold to another, such that both the function and its inverse are smooth. These are the defining symmetry transformations of General Relativity since the theory is formulated only in terms of a differentiable manifold. In general relativity, general covariance is intimately related to "diffeomorphism invariance"; this symmetry is one of the defining features of the theory. However, it is a common misunderstanding that "diffeomorphism invariance" refers to the invariance of the physical predictions of a theory under arbitrary coordinate transformations. Diffeomorphisms, as mathematicians define them, correspond to something much more radical.
Diffeomorphisms are the true symmetry transformations of general relativity, come about from the assertion that the formulation of the theory is based on a bare different
Edward Witten is an American theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, New Jersey. Witten is a researcher in string theory, quantum gravity, supersymmetric quantum field theories, other areas of mathematical physics. In addition to his contributions to physics, Witten's work has impacted pure mathematics. In 1990, he became the first physicist to be awarded a Fields Medal by the International Mathematical Union, awarded for his 1981 proof of the positive energy theorem in general relativity. Witten was born in Maryland, to a Jewish family, he is the son of Lorraine Witten and Louis Witten, a theoretical physicist specializing in gravitation and general relativity. Witten attended the Park School of Baltimore, received his Bachelor of Arts with a major in history and minor in linguistics from Brandeis University in 1971, he published articles in The Nation. He worked for George McGovern's presidential campaign. Witten attended the University of Wisconsin–Madison for one semester as an economics graduate student before dropping out.
He returned to academia, enrolling in applied mathematics at Princeton University in 1973 shifting departments and receiving a Ph. D. in physics in 1976 under David Gross, the 2004 Nobel laureate in Physics. He held a fellowship at Harvard University, visited Oxford University, was a junior fellow in the Harvard Society of Fellows, held a MacArthur Foundation fellowship. Witten was awarded the Fields Medal by the International Mathematical Union in 1990, becoming the first physicist to win the prize. In a written address to the ICM, Michael Atiyah said of Witten, Although he is a physicist his command of mathematics is rivaled by few mathematicians, his ability to interpret physical ideas in mathematical form is quite unique. Time and again he has surprised the mathematical community by a brilliant application of physical insight leading to new and deep mathematical theorems... E has made a profound impact on contemporary mathematics. In his hands physics is once again providing a rich source of inspiration and insight in mathematics.
As an example of Witten's work in pure mathematics, Atiyah cites his application of techniques from quantum field theory to the mathematical subject of low-dimensional topology. In the late 1980s, Witten coined the term topological quantum field theory for a certain type of physical theory in which the expectation values of observable quantities encode information about the topology of spacetime. In particular, Witten realized that a physical theory now called Chern–Simons theory could provide a framework for understanding the mathematical theory of knots and 3-manifolds. Although Witten's work was based on the mathematically ill-defined notion of a Feynman path integral and was therefore not mathematically rigorous, mathematicians were able to systematically develop Witten's ideas, leading to the theory of Reshetikhin–Turaev invariants. Another result for which Witten was awarded the Fields Medal was his proof in 1981 of the positive energy theorem in general relativity; this theorem asserts that the total energy of a gravitating system is always positive and can be zero only if the geometry of spacetime is that of flat Minkowski space.
It establishes Minkowski space as a stable ground state of the gravitational field. While the original proof of this result due to Richard Schoen and Shing-Tung Yau used variational methods, Witten's proof used ideas from supergravity theory to simplify the argument. A third area mentioned in Atiyah's address is Witten's work relating supersymmetry and Morse theory, a branch of mathematics that studies the topology of manifolds using the concept of a differentiable function. Witten's work gave a physical proof of a classical result, the Morse inequalities, by interpreting the theory in terms of supersymmetric quantum mechanics. By the mid 1990s, physicists working on string theory had developed five different consistent versions of the theory; these versions are known as type I, type IIA, type IIB, the two flavors of heterotic string theory. The thinking was that out of these five candidate theories, only one was the actual correct theory of everything, that theory was the one whose low-energy limit matched the physics observed in our world today.
Speaking at the string theory conference at University of Southern California in 1995, Witten made the surprising suggestion that these five string theories were in fact not distinct theories, but different limits of a single theory which he called M-theory. Witten's proposal was based on the observation that the five string theories can be mapped to one another by certain rules called dualities and are identified by these dualities. Witten's announcement led to a flurry of work now known as the second superstring revolution. Another of his contributions to physics was to the result of gauge/gravity duality. In 1997, Juan Maldacena formulated a result known as the AdS/CFT correspondence, which establishes a relationship between certain quantum field theories and theories of quantum gravity. Maldacena's discovery has dominated high energy theoretical physics for the past 15 years because of its applications to theoretical problems in quantum gravity and quantum field theory. Witten's foundational work following Maldacena's result has shed light on this relationship.
In collaboration with Nathan Seiberg, Witten established several powerful results in quantum field theories. In their paper on string theory and noncommutative geometry and Witten studied certain noncommutative quantum field theories that arise a
General relativity is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics. General relativity generalizes special relativity and Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time, or spacetime. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter and radiation are present; the relation is specified by the Einstein field equations, a system of partial differential equations. Some predictions of general relativity differ from those of classical physics concerning the passage of time, the geometry of space, the motion of bodies in free fall, the propagation of light. Examples of such differences include gravitational time dilation, gravitational lensing, the gravitational redshift of light, the gravitational time delay; the predictions of general relativity in relation to classical physics have been confirmed in all observations and experiments to date.
Although general relativity is not the only relativistic theory of gravity, it is the simplest theory, consistent with experimental data. However, unanswered questions remain, the most fundamental being how general relativity can be reconciled with the laws of quantum physics to produce a complete and self-consistent theory of quantum gravity. Einstein's theory has important astrophysical implications. For example, it implies the existence of black holes—regions of space in which space and time are distorted in such a way that nothing, not light, can escape—as an end-state for massive stars. There is ample evidence that the intense radiation emitted by certain kinds of astronomical objects is due to black holes; the bending of light by gravity can lead to the phenomenon of gravitational lensing, in which multiple images of the same distant astronomical object are visible in the sky. General relativity predicts the existence of gravitational waves, which have since been observed directly by the physics collaboration LIGO.
In addition, general relativity is the basis of current cosmological models of a expanding universe. Acknowledged as a theory of extraordinary beauty, general relativity has been described as the most beautiful of all existing physical theories. Soon after publishing the special theory of relativity in 1905, Einstein started thinking about how to incorporate gravity into his new relativistic framework. In 1907, beginning with a simple thought experiment involving an observer in free fall, he embarked on what would be an eight-year search for a relativistic theory of gravity. After numerous detours and false starts, his work culminated in the presentation to the Prussian Academy of Science in November 1915 of what are now known as the Einstein field equations; these equations specify how the geometry of space and time is influenced by whatever matter and radiation are present, form the core of Einstein's general theory of relativity. The Einstein field equations are nonlinear and difficult to solve.
Einstein used approximation methods in working out initial predictions of the theory. But as early as 1916, the astrophysicist Karl Schwarzschild found the first non-trivial exact solution to the Einstein field equations, the Schwarzschild metric; this solution laid the groundwork for the description of the final stages of gravitational collapse, the objects known today as black holes. In the same year, the first steps towards generalizing Schwarzschild's solution to electrically charged objects were taken, which resulted in the Reissner–Nordström solution, now associated with electrically charged black holes. In 1917, Einstein applied his theory to the universe as a whole, initiating the field of relativistic cosmology. In line with contemporary thinking, he assumed a static universe, adding a new parameter to his original field equations—the cosmological constant—to match that observational presumption. By 1929, the work of Hubble and others had shown that our universe is expanding; this is described by the expanding cosmological solutions found by Friedmann in 1922, which do not require a cosmological constant.
Lemaître used these solutions to formulate the earliest version of the Big Bang models, in which our universe has evolved from an hot and dense earlier state. Einstein declared the cosmological constant the biggest blunder of his life. During that period, general relativity remained something of a curiosity among physical theories, it was superior to Newtonian gravity, being consistent with special relativity and accounting for several effects unexplained by the Newtonian theory. Einstein himself had shown in 1915 how his theory explained the anomalous perihelion advance of the planet Mercury without any arbitrary parameters. A 1919 expedition led by Eddington confirmed general relativity's prediction for the deflection of starlight by the Sun during the total solar eclipse of May 29, 1919, making Einstein famous, yet the theory entered the mainstream of theoretical physics and astrophysics only with the developments between 1960 and 1975, now known as the golden age of general relativity. Physicists began to understand the concept of a black hole, to identify quasars as one of these objects' astrophysical manifestations.
More precise solar system tests confirmed the theory's predictive power, relativistic cosmology, became amenable to direct observational tests. Over the years, general relativity has acqui
In mathematics, topology is concerned with the properties of space that are preserved under continuous deformations, such as stretching, twisting and bending, but not tearing or gluing. An n-dimensional topological space is a space with certain properties of connectedness and compactness; the space discrete. It can be closed. Topology developed as a field of study out of geometry and set theory, through analysis of concepts such as space and transformation; such ideas go back to Gottfried Leibniz, who in the 17th century envisioned the geometria situs and analysis situs. Leonhard Euler's Seven Bridges of Königsberg Problem and Polyhedron Formula are arguably the field's first theorems; the term topology was introduced by Johann Benedict Listing in the 19th century, although it was not until the first decades of the 20th century that the idea of a topological space was developed. By the middle of the 20th century, topology had become a major branch of mathematics. Topology, as a well-defined mathematical discipline, originates in the early part of the twentieth century, but some isolated results can be traced back several centuries.
Among these are certain questions in geometry investigated by Leonhard Euler. His 1736 paper on the Seven Bridges of Königsberg is regarded as one of the first practical applications of topology. On 14 November 1750, Euler wrote to a friend that he had realised the importance of the edges of a polyhedron; this led to his polyhedron formula, V − E + F = 2. Some authorities regard this analysis as the first theorem. Further contributions were made by Augustin-Louis Cauchy, Ludwig Schläfli, Johann Benedict Listing, Bernhard Riemann and Enrico Betti. Listing introduced the term "Topologie" in Vorstudien zur Topologie, written in his native German, in 1847, having used the word for ten years in correspondence before its first appearance in print; the English form "topology" was used in 1883 in Listing's obituary in the journal Nature to distinguish "qualitative geometry from the ordinary geometry in which quantitative relations chiefly are treated". The term "topologist" in the sense of a specialist in topology was used in 1905 in the magazine Spectator.
Their work was corrected and extended by Henri Poincaré. In 1895, he published his ground-breaking paper on Analysis Situs, which introduced the concepts now known as homotopy and homology, which are now considered part of algebraic topology. Unifying the work on function spaces of Georg Cantor, Vito Volterra, Cesare Arzelà, Jacques Hadamard, Giulio Ascoli and others, Maurice Fréchet introduced the metric space in 1906. A metric space is now considered a special case of a general topological space, with any given topological space giving rise to many distinct metric spaces. In 1914, Felix Hausdorff coined the term "topological space" and gave the definition for what is now called a Hausdorff space. A topological space is a slight generalization of Hausdorff spaces, given in 1922 by Kazimierz Kuratowski. Modern topology depends on the ideas of set theory, developed by Georg Cantor in the part of the 19th century. In addition to establishing the basic ideas of set theory, Cantor considered point sets in Euclidean space as part of his study of Fourier series.
For further developments, see point-set topology and algebraic topology. Topology can be formally defined as "the study of qualitative properties of certain objects that are invariant under a certain kind of transformation those properties that are invariant under a certain kind of invertible transformation." Topology is used to refer to a structure imposed upon a set X, a structure that characterizes the set X as a topological space by taking proper care of properties such as convergence and continuity, upon transformation. Topological spaces show up in every branch of mathematics; this has made topology one of the great unifying ideas of mathematics. The motivating insight behind topology is that some geometric problems depend not on the exact shape of the objects involved, but rather on the way they are put together. For example, the square and the circle have many properties in common: they are both one dimensional objects and both separate the plane into two parts, the part inside and the part outside.
In one of the first papers in topology, Leonhard Euler demonstrated that it was impossible to find a route through the town of Königsberg that would cross each of its seven bridges once. This result did not depend on the lengths of the bridges or on their distance from one another, but only on connectivity properties: which bridges connect to which islands or riverbanks; this Seven Bridges of Königsberg problem led to the branch of mathematics known as graph theory. The hairy ball theorem of algebraic topology says that "one cannot comb the hair flat on a hairy ball without creating a cowlick." This fact is convincing to most people though they might not recognize the more formal statement of the theorem, that there is no nonvanishing continuous tangent vector field on the sphere. As with the Bridges of Königsberg, the result does not depend on the shape of t