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Dual Graviton
In theoretical physics, the dual graviton is a hypothetical elementary particle that is a dual of the graviton under electric–magnetic duality, as an S-duality, predicted by some formulations of eleven-dimensional supergravity. The dual graviton was first hypothesized in 1980. It was theoretically modeled in 2000s, which was then predicted in eleven-dimensional mathematics of SO(8) supergravity in the framework of electric–magnetic duality. It again emerged in the ''E''11 generalized geometry in eleven dimensions, and the ''E''7 generalized vielbein-geometry in eleven dimensions. While there is no local coupling between graviton and dual graviton, the field introduced by dual graviton may be coupled to a BF model as non-local gravitational fields in extra dimensions. A ''massive'' dual gravity of Ogievetsky–Polubarinov model can be obtained by coupling the dual graviton field to the curl of its own energy-momentum tensor. The previously mentioned theories of dual ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Elementary Particle
In particle physics, an elementary particle or fundamental particle is a subatomic particle that is not composed of other particles. The Standard Model presently recognizes seventeen distinct particles—twelve fermions and five bosons. As a consequence of flavor and color combinations and antimatter, the fermions and bosons are known to have 48 and 13 variations, respectively. Among the 61 elementary particles embraced by the Standard Model number: electrons and other leptons, quarks, and the fundamental bosons. Subatomic particles such as protons or neutrons, which contain two or more elementary particles, are known as composite particles. Ordinary matter is composed of atoms, themselves once thought to be indivisible elementary particles. The name ''atom'' comes from the Ancient Greek word ''ἄτομος'' ( atomos) which means ''indivisible'' or ''uncuttable''. Despite the theories about atoms that had existed for thousands of years, the factual existence of ato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
De Sitter Space
In mathematical physics, ''n''-dimensional de Sitter space (often denoted dS''n'') is a maximally symmetric Lorentzian manifold with constant positive scalar curvature. It is the Lorentzian analogue of an ''n''-sphere (with its canonical Riemannian metric). The main application of de Sitter space is its use in general relativity, where it serves as one of the simplest mathematical models of the universe consistent with the observed accelerating expansion of the universe. More specifically, de Sitter space is the maximally symmetric vacuum solution of Einstein's field equations in which the cosmological constant \Lambda is positive (corresponding to a positive vacuum energy density and negative pressure). De Sitter space and anti-de Sitter space are named after Willem de Sitter (1872–1934), professor of astronomy at Leiden University and director of the Leiden Observatory. Willem de Sitter and Albert Einstein worked closely together in Le ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric field on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Form
In differential geometry, the curvature form describes curvature of a connection on a principal bundle. The Riemann curvature tensor in Riemannian geometry can be considered as a special case. Definition Let ''G'' be a Lie group with Lie algebra \mathfrak g, and ''P'' → ''B'' be a principal ''G''-bundle. Let ω be an Ehresmann connection on ''P'' (which is a \mathfrak g-valued one-form on ''P''). Then the curvature form is the \mathfrak g-valued 2-form on ''P'' defined by :\Omega=d\omega + omega \wedge \omega= D \omega. (In another convention, 1/2 does not appear.) Here d stands for exterior derivative, cdot \wedge \cdot/math> is defined in the article " Lie algebra-valued form" and ''D'' denotes the exterior covariant derivative. In other terms, :\,\Omega(X, Y)= d\omega(X,Y) + omega(X),\omega(Y)/math> where ''X'', ''Y'' are tangent vectors to ''P''. There is also another expression for Ω: if ''X'', ''Y'' are horizontal vector fields on ''P'', thenProof: \sigma\Omeg ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Young Symmetrizer
In mathematics, a Young symmetrizer is an element of the group algebra of the symmetric group S_n whose natural action on tensor products V^ of a complex vector space V has as image an irreducible representation of the group of invertible linear transformations GL(V). All irreducible representations of GL(V) are thus obtained. It is constructed from the action of S_n on the vector space V^ by permutation of the different factors (or equivalently, from the permutation of the indices of the tensor components). A similar construction works over any field but in characteristic p (in particular over finite fields) the image need not be an irreducible representation. The Young symmetrizers also act on the vector space of functions on Young tableau and the resulting representations are called Specht modules which again construct all complex irreducible representations of the symmetric group while the analogous construction in prime characteristic need not be irreducible. The Young symm ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Massive Gravity
Massive is an adjective related to mass. Massive may refer to: Arts, entertainment, and media * Massive (band), an Australian Hard Rock band * ''Massive'', an album by The Supervillains released in 2008 * Massive Attack, a British musical group, who were temporarily known as Massive in 1991 * Massive (song), "Massive" (song), a song by Drake from the 2022 album ''Honestly, Nevermind'' * Massive (TV series), ''Massive'' (TV series), a situation comedy first aired on BBC3 in September 2008 * Massive! (TV programme), ''Massive!'' (TV programme), a short-lived Saturday morning British television programme (1995–1996) * ''MMO Games Magazine'', formerly ''MASSIVE Magazine'', a short-lived computer magazine * ''Massive: Gay Erotic Manga and the Men Who Make It'', a 2014 manga anthology * The Massive (comics), ''The Massive'' (comics), an ongoing comic series * The Massive, a starship in the U.S. animated television series ''Invader Zim'' * Massive, the villain in ''Loonatics Unleashe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Horndeski's Theory
Horndeski's theory is the most general theory of gravity in four dimensions whose Lagrangian is constructed out of the metric tensor and a scalar field and leads to second order equations of motion. The theory was first proposed by Gregory Horndeski in 1974 and has found numerous applications, particularly in the construction of cosmological models of Inflation and dark energy. Horndeski's theory contains many theories of gravity, including general relativity, Brans–Dicke theory, quintessence, dilaton, chameleon particle and covariant Galileon as special cases. Action Horndeski's theory can be written in terms of an action as S _,\phi= \int\mathrm^x\,\sqrt\left _,\phi.html" ;"title="sum_^\frac\mathcal_[g_,\phi">sum_^\frac\mathcal_[g_,\phi,+\mathcal_[g_,\psi_right] with the Lagrangian (field theory), Lagrangian densities \mathcal_ = G_(\phi,\, X) \mathcal_ = G_(\phi,\,X)\Box\phi \mathcal_ = G_(\phi,\,X)R+G_(\phi,\,X)\left left(\Box\phi\right)^-\phi_\phi^\right/math> \ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hessian Matrix
In mathematics, the Hessian matrix, Hessian or (less commonly) Hesse matrix is a square matrix of second-order partial derivatives of a scalar-valued Function (mathematics), function, or scalar field. It describes the local curvature of a function of many variables. The Hessian matrix was developed in the 19th century by the German mathematician Otto Hesse, Ludwig Otto Hesse and later named after him. Hesse originally used the term "functional determinants". The Hessian is sometimes denoted by H or \nabla\nabla or \nabla^2 or \nabla\otimes\nabla or D^2. Definitions and properties Suppose f : \R^n \to \R is a function taking as input a vector \mathbf \in \R^n and outputting a scalar f(\mathbf) \in \R. If all second-order partial derivatives of f exist, then the Hessian matrix \mathbf of f is a square n \times n matrix, usually defined and arranged as \mathbf H_f= \begin \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \dfrac & \dfrac & \cdots & \dfrac \\[2.2ex] \vdots & \vdot ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Scalar Curvature
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor. The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Curvature
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric bi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Curvature Tensor
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to mathematical analysis, analysis, number theory, and differential geometry. In the field of real analysis, he is mostly known for the first rigorous formulation of the integral, the Riemann integral, and his work on Fourier series. His contributions to complex analysis include most notably the introduction of Riemann surfaces, breaking new ground in a natural, geometric treatment of complex analysis. His On the Number of Primes Less Than a Given Magnitude, 1859 paper on the prime-counting function, containing the original statement of the Riemann hypothesis, is regarded as a foundational paper of analytic number theory. Through his pioneering Riemannian Geometry, contributions to differential geometry, Riemann laid the foundations of the mathematics of general relativity. He is considered by many to be one of the greatest mathematicians of all time. Ear ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Frontiers In Physics
''Frontiers in Physics'' is a peer-reviewed open-access scientific journal covering physics. It was established in 2013 and is published by Frontiers Media. The editor-in-chief is Alex Hansen (Norwegian University of Science and Technology). The scope of the journal covers the entire field of physics, from experimental, to computational and theoretical physics. Abstracting and indexing The journal is abstracted and indexed in Current Contents/Physical, Chemical & Earth Sciences, Science Citation Index Expanded, and Scopus. According to the ''Journal Citation Reports'', the journal has a 2020 impact factor of 3.560. Frontiers Media was included in Jeffrey Beall Jeffrey Beall is an American librarian and library scientist who drew attention to "predatory open access publishing", a term he coined, and created Beall's list, a list of potentially predatory open-access publishers. He is a critic of the o ...'s list of predatory publishers in 2016, though Beall deleted the ent ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
