1.
General relativity
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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 Newtons law of gravitation, providing a unified description of gravity as a geometric property of space and time. In particular, the curvature of spacetime is directly related to the energy and momentum of whatever matter, the relation is specified by the Einstein field equations, a system of partial differential equations. Examples of such differences include gravitational time dilation, gravitational lensing, the redshift of light. The predictions of relativity have been confirmed in all observations. Although general relativity is not the only theory of gravity. Einsteins 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 even light, can escape—as an end-state for massive stars. The bending of light by gravity can lead to the phenomenon of gravitational lensing, General relativity also predicts the existence of gravitational waves, which have since been observed directly by physics collaboration LIGO. In addition, general relativity is the basis of current cosmological models of an expanding universe. 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 thought experiment involving an observer in free fall. 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, the Einstein field equations are nonlinear and very 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 stages of gravitational collapse. In 1917, Einstein applied his theory to the universe as a whole, 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, however, the work of Hubble and others had shown that our universe is expanding and this is readily 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 extremely hot, Einstein later declared the cosmological constant the biggest blunder of his life

2.
Anti-de Sitter space
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In mathematics and physics, n-dimensional anti-de Sitter space is a maximally symmetric Lorentzian manifold with constant negative scalar curvature. The anti-de Sitter space and de Sitter space are named after Willem de Sitter, professor of astronomy at Leiden University, Willem de Sitter and Albert Einstein worked together closely in the 1920s in Leiden on the spacetime structure of the universe. Einsteins theory of relativity places space and time on equal footing, so that one considers the geometry of a unified spacetime instead of considering space, the cases of spacetime of constant curvature are de Sitter space, Minkowski space, and anti-de Sitter space. As such, they are solutions of Einsteins field equations for an empty universe with a positive, zero, or negative cosmological constant. Anti-de Sitter space generalises to any number of space dimensions and this non-technical explanation first defines the terms used in the introductory material of this entry. Then, it sets forth the underlying idea of a general relativity-like spacetime. It also explains that Minkowski space, de Sitter space and anti-de Sitter space, as applied to general relativity, finally, it offers some caveats that describe in general terms how this non-technical explanation fails to capture the full detail of the mathematical concept. The space of special relativity is an example, negative curvature means curved hyperbolically, like a saddle surface or the Gabriels Horn surface, similar to that of a trumpet bell. It might be described as being the opposite of the surface of a sphere, general relativity is a theory of the nature of time, space and gravity in which gravity is a curvature of space and time that results from the presence of matter or energy. Energy and matter are equivalent, and space and time can be translated into equivalent units based on the speed of light, of course, in general relativity, both the small and large objects mutually influence the curvature of spacetime. The attractive force of gravity created by matter is due to a curvature of spacetime. As a result, in relativity, the familiar Newtonian equation of gravity F = G m 1 m 2 r 2 is merely an approximation of the gravity-like effects seen in general relativity. However this approximation becomes inaccurate in extreme physical situations, for example, in general relativity, objects in motion have a slightly different gravitation effect than objects at rest. In normal circumstances, gravity bends time so slightly that the differences between Newtonian gravity and general relativity are detectable only with precise instruments, de Sitter space involves a variation of general relativity in which spacetime is slightly curved in the absence of matter or energy. This is analogous to the relationship between Euclidean geometry and non-Euclidean geometry, an intrinsic curvature of spacetime in the absence of matter or energy is modeled by the cosmological constant in general relativity. This corresponds to the vacuum having a density and pressure. This spacetime geometry results in initially parallel timelike geodesics diverging, with spacelike sections having positive curvature, an anti-de Sitter space in general relativity is similar to a de Sitter space, except with the sign of the curvature changed. This corresponds to a cosmological constant

3.
Cosmological constant
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In cosmology, the cosmological constant is the value of the energy density of the vacuum of space. It was originally introduced by Albert Einstein in 1917 as an addition to his theory of relativity to hold back gravity and achieve a static universe. Einstein abandoned the concept after Hubbles 1929 discovery that all galaxies outside the Local Group are moving away from each other, from 1929 until the early 1990s, most cosmology researchers assumed the cosmological constant to be zero. When Λ is zero, this reduces to the field equation of general relativity. When T is zero, the equation describes empty space. The cosmological constant has the effect as an intrinsic energy density of the vacuum. In this context, it is moved onto the right-hand side of the equation, and defined with a proportionality factor of 8π, Λ = 8πρvac. It is common to quote values of energy density directly, though using the name cosmological constant. A positive vacuum energy density resulting from a cosmological constant implies a negative pressure, if the energy density is positive, the associated negative pressure will drive an accelerated expansion of the universe, as observed. This ratio is usually denoted ΩΛ, and is estimated to be 0. 6911±0.0062, according to results published by the Planck Collaboration in 2015. In a flat universe ΩΛ is the fraction of the energy of the due to the cosmological constant. Another ratio that is used by scientists is the equation of state, usually denoted w and this ratio is w = −1 for a true cosmological constant, and is generally different for alternative time-varying forms of vacuum energy such as quintessence. To counteract this possibility, Einstein added the cosmological constant, likewise, a universe that contracts slightly will continue contracting. However, the cosmological constant remained a subject of theoretical and empirical interest, empirically, the onslaught of cosmological data in the past decades strongly suggests that our universe has a positive cosmological constant. The explanation of this small but positive value is a theoretical challenge. Observations announced in 1998 of distance–redshift relation for Type Ia supernovae indicated that the expansion of the universe is accelerating. When combined with measurements of the microwave background radiation these implied a value of ΩΛ ≈0.7. There are other causes of an accelerating universe, such as quintessence

4.
Topological quantum field theory
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A topological quantum field theory is a quantum field theory which computes topological invariants. Donaldson, Jones, Witten, and Kontsevich have all won Fields Medals for work related to field theory. In a topological field theory, the correlation functions do not depend on the metric of spacetime and this means that the theory is not sensitive to changes in the shape of spacetime, if the spacetime warps or contracts, the correlation functions do not change. Topological field theories are not very interesting on the flat Minkowski spacetime used in particle physics, Minkowski space can be contracted to a point, so a TQFT on Minkowski space computes only trivial topological invariants. Consequently, TQFTs are usually studied on curved spacetimes, such as, for example, most of the known topological field theories are defined on spacetimes of dimension less than five. It seems that a few theories exist, but they are not very well understood. Quantum gravity is believed to be background-independent, and TQFTs provide examples of independent quantum field theories. This has prompted ongoing theoretical investigation of this class of models, the known topological field theories fall into two general classes, Schwarz-type TQFTs and Witten-type TQFTs. Witten TQFTs are also referred to as cohomological field theories. In Schwarz-type TQFTs, the functions or partition functions of the system are computed by the path integral of metric independent action functionals. For instance, in the BF model, the spacetime is a two-dimensional manifold M, the observables are constructed from a two-form F, an auxiliary scalar B, and their derivatives. The action is S = ∫ M B F The spacetime metric does not appear anywhere in the theory, the first example appeared in 1977 and is due to A. Schwarz, its action functional is, ∫ M A ∧ d A. Another more famous example is Chern–Simons theory, which can be used to compute knot invariants, in general partition functions depend on a metric but the above examples are shown to be metric-independent. The first example of the field theories of Witten-type appeared in Wittens paper in 1988. Though its action functional contains the spacetime metric gαβ, after a topological twist it turns out to be metric independent, the independence of the stress-energy tensor Tαβ of the system from the metric depends on whether BRST-operator is closed. Following Wittens example a lot of examples are found in string theory,4. The stress-energy-tensor is of the form T α β = δ G α β for an arbitrary tensor G α β. As an example given a 2-form field B with the differential operator δ which satisfies δ2 =0, the expression δ δ B α β S is proportional to δ G with another 2-form G. In the third equality it was used the fact that δ O i = δ S =0, since ∫ d μ O i G e i S is only a number, the Lie derivative applied on it vanishes

5.
Edward Witten
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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, in addition to his contributions to physics, Wittens work has significantly impacted pure mathematics. In 1990 he became the first and so far the only physicist to be awarded a Fields Medal by the International Mathematical Union, in 2004, Time magazine stated that Witten is widely thought to be the worlds smartest living theoretical physicist. Witten was born in Baltimore, Maryland, to a Jewish family and 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, and received his Bachelor of Arts with a major in history and he published articles in The New Republic and The Nation. In 1968, Witten published an article in The Nation arguing that the New Left had no strategy and he worked briefly for George McGoverns presidential campaign. Witten attended the University of Wisconsin–Madison for one semester as a graduate student before dropping out. He held a fellowship at Harvard University, visited Oxford University, was a fellow in the Harvard Society of Fellows. Witten was awarded the Fields Medal by the International Mathematical Union in 1990, 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 an impact on contemporary mathematics. In his hands physics is once again providing a source of inspiration. As an example of Wittens work in mathematics, Atiyah cites his application of techniques from quantum field theory to the mathematical subject of low-dimensional topology. In particular, Witten realized that a theory now called Chern–Simons theory could provide a framework for understanding the mathematical theory of knots. Another result for which Witten was awarded the Fields Medal was his proof in 1981 of the energy theorem in general relativity. This theorem asserts that the energy of a gravitating system is always positive. It establishes Minkowski space as a ground state of the gravitational field. While the original proof of this due to Richard Schoen and Shing-Tung Yau used variational methods. Wittens work gave a proof of a classical result, the Morse inequalities

6.
Gauge theory
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In physics, a gauge theory is a type of field theory in which the Lagrangian is invariant under a continuous group of local transformations. An invariant is a model that holds no matter the mathematical procedure applied to it. This is the concept behind gauge invariance, the idea of fields as described by Michael Faraday in his study of electromagnetism led to the postulate that fields could be described mathematically as scalars and vectors. When a field is transformed, but the result is not, applying gauge theory creates a unification which describes mathematical formulas or models that hold good for all fields of the same class. The term gauge refers to any specific mathematical formalism to regulate redundant degrees of freedom in the Lagrangian, the transformations between possible gauges, called gauge transformations, form a Lie group—referred to as the symmetry group or the gauge group of the theory. Associated with any Lie group is the Lie algebra of group generators, for each group generator there necessarily arises a corresponding field called the gauge field. Gauge fields are included in the Lagrangian to ensure its invariance under the local group transformations, when such a theory is quantized, the quanta of the gauge fields are called gauge bosons. If the symmetry group is non-commutative, the theory is referred to as non-abelian. Many powerful theories in physics are described by Lagrangians that are invariant under some symmetry transformation groups, when they are invariant under a transformation identically performed at every point in the spacetime in which the physical processes occur, they are said to have a global symmetry. Local symmetry, the cornerstone of gauge theories, is a stricter constraint, in fact, a global symmetry is just a local symmetry whose groups parameters are fixed in spacetime. Gauge theories are important as the field theories explaining the dynamics of elementary particles. Quantum electrodynamics is a gauge theory with the symmetry group U and has one gauge field. The Standard Model is a gauge theory with the symmetry group U×SU×SU and has a total of twelve gauge bosons. Gauge theories are important in explaining gravitation in the theory of general relativity. Its case is unusual in that the gauge field is a tensor. Theories of quantum gravity, beginning with gauge gravitation theory, also postulate the existence of a gauge boson known as the graviton, both gauge invariance and diffeomorphism invariance reflect a redundancy in the description of the system. An alternative theory of gravitation, gauge theory gravity, replaces the principle of covariance with a true gauge principle with new gauge fields. Historically, these ideas were first stated in the context of classical electromagnetism, however, the modern importance of gauge symmetries appeared first in the relativistic quantum mechanics of electrons – quantum electrodynamics, elaborated on below

7.
Solvable group
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In mathematics, more specifically in the field of group theory, a solvable group or soluble group is a group that can be constructed from abelian groups using extensions. Equivalently, a group is a group whose derived series terminates in the trivial subgroup. Historically, the word solvable arose from Galois theory and the proof of the unsolvability of quintic equation. Specifically, an equation is solvable by radicals if and only if the corresponding Galois group is solvable. Or equivalently, if its derived series, the normal series G ▹ G ▹ G ▹ ⋯. These two definitions are equivalent, since for every group H and every normal subgroup N of H, the least n such that G = is called the derived length of the solvable group G. For finite groups, an equivalent definition is that a group is a group with a composition series all of whose factors are cyclic groups of prime order. This is equivalent because a group has finite composition length. The Jordan–Hölder theorem guarantees that if one composition series has this property, for the Galois group of a polynomial, these cyclic groups correspond to nth roots over some field. All abelian groups are trivially solvable – a subnormal series being given by just the group itself, but non-abelian groups may or may not be solvable. More generally, all nilpotent groups are solvable, in particular, finite p-groups are solvable, as all finite p-groups are nilpotent. A small example of a solvable, non-nilpotent group is the symmetric group S3, in fact, as the smallest simple non-abelian group is A5, it follows that every group with order less than 60 is solvable. The group S5 is not solvable — it has a series, giving factor groups isomorphic to A5 and C2. Generalizing this argument, coupled with the fact that An is a normal, maximal, non-abelian simple subgroup of Sn for n >4, we see that Sn is not solvable for n >4. This is a key step in the proof that for every n >4 there are polynomials of n which are not solvable by radicals. This property is used in complexity theory in the proof of Barringtons theorem. The celebrated Feit–Thompson theorem states that every group of odd order is solvable. In particular this implies that if a group is simple