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Timelike Congruence
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation. Types of congruences Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called ''timelike'', ''null'', or ''spacelike'' respectively. A congruence is called a ''geodesic congruence'' if it admits a tangent vector field \vec with vanishing covariant derivative, \nabla_ \vec = 0. Relation with vector fields The integral curves of the vector field are a family of ''non-intersecting'' parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization. Many distinct vector fields can give rise to the ''same'' congruenc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time in physics, time, or four-dimensional spacetime. In particular, the ''curvature of spacetime'' is directly related to the energy and momentum of whatever is present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations. Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Wave Equation
The wave equation is a second-order linear partial differential equation for the description of waves or standing wave fields such as mechanical waves (e.g. water waves, sound waves and seismic waves) or electromagnetic waves (including light waves). It arises in fields like acoustics, electromagnetism, and fluid dynamics. This article focuses on waves in classical physics. Quantum physics uses an operator-based wave equation often as a relativistic wave equation. Introduction The wave equation is a hyperbolic partial differential equation describing waves, including traveling and standing waves; the latter can be considered as linear superpositions of waves traveling in opposite directions. This article mostly focuses on the scalar wave equation describing waves in scalars by scalar functions of a time variable (a variable representing time) and one or more spatial variables (variables representing a position in a space under discussion). At the same time, there a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Shear Tensor
Shear may refer to: Textile production *Animal shearing, the collection of wool from various species **Sheep shearing *The removal of nap during wool cloth production *Scissors, a hand-operated cutting equipment Science and technology Engineering *Shear strength (soil), the shear strength of soil under loading *Shear line (locksmithing), where the inner cylinder ends and the outer cylinder begins in a cylinder lock *Shearing (manufacturing), a metalworking process which cuts stock without the formation of chips or the use of burning or melting *Shear (sheet metal), various tools to shear sheet metal * Board shear, in bookbinding, a tool to cut board or paper *Shear pin, in machinery, such as a plough, designed to shear (break) when a certain force is exceeded, to protect other components of the machine. * Shearing interferometer, in optics, a simple and very common means to check the collimation of beams by observing interference *Shearing in computer graphics, more commonly calle ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Expansion Scalar
Expansion may refer to: Arts, entertainment and media * ''L'Expansion'', a French monthly business magazine * ''Expansion'' (album), by American jazz pianist Dave Burrell, released in 2004 * ''Expansions'' (McCoy Tyner album), 1970 * ''Expansions'' (Lonnie Liston Smith album), 1975 * ''Expansión'' (Mexico), a Mexican news portal linked to CNN * Expansion (sculpture) (2004) Bronze sculpture illuminated from within * ''Expansión'' (Spanish newspaper), a Spanish economic daily newspaper published in Spain * Expansion pack in gaming, extra content for games, often simply "expansion" Science, technology, and mathematics * Expansion (geometry), stretching of geometric objects with flat sides * Expansion (model theory), in mathematical logic, a mutual converse of a reduct * Expansion card, in computing, a printed circuit board that can be inserted into an expansion slot * Expansion chamber, on a two-stroke engine, a tuned exhaust system that enhances power output * Expansion joint ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemann Tensor
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to 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 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 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. Early years Riemann was born on 17 September 1826 in Breselenz, a village near Dannenberg in the ... [...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] |
Foliation
In mathematics (differential geometry), a foliation is an equivalence relation on an topological manifold, ''n''-manifold, the equivalence classes being connected, injective function, injectively immersed submanifolds, all of the same dimension ''p'', modeled on the manifold decomposition, decomposition of the real coordinate space R''n'' into the cosets ''x'' + R''p'' of the standardly embedding, embedded subspace topology, subspace R''p''. The equivalence classes are called the leaves of the foliation. If the manifold and/or the submanifolds are required to have a piecewise-linear manifold, piecewise-linear, differentiable manifold, differentiable (of class ''Cr''), or analytic manifold, analytic structure then one defines piecewise-linear, differentiable, or analytic foliations, respectively. In the most important case of differentiable foliation of class ''Cr'' it is usually understood that ''r'' ≥ 1 (otherwise, ''C''0 is a topological foliation). The number ''p'' (the dime ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traceless
In linear algebra, the trace of a square matrix , denoted , is the sum of the elements on its main diagonal, a_ + a_ + \dots + a_. It is only defined for a square matrix (). The trace of a matrix is the sum of its eigenvalues (counted with multiplicities). Also, for any matrices and of the same size. Thus, similar matrices have the same trace. As a consequence, one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the row and column of . The entries of can be real numbers, complex numbers, or more generally elements of a field . The trace is not defined for non-square matrices. Example Let be a matrix, with \mathbf ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Fermi–Walker Transport
Fermi–Walker transport is a process in general relativity used to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not due to arbitrary spin or rotation of the frame. It was discovered by Fermi in 1921 and rediscovered by Walker in 1932. Fermi–Walker differentiation In the theory of Lorentzian manifolds, Fermi–Walker differentiation is a generalization of covariant differentiation. In general relativity, Fermi–Walker derivatives of the spacelike vector fields in a frame field, taken with respect to the timelike unit vector field in the frame field, are used to define non-inertial and non-rotating frames, by stipulating that the Fermi–Walker derivatives should vanish. In the special case of inertial frames, the Fermi–Walker derivatives reduce to covariant derivatives. With a (-+++) sign convention, this is defined for a vector field ''X'' along a curve \gamma(s): :\frac=\frac - \left(X ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality (mathematics), orthogonality is the generalization of the geometric notion of ''perpendicularity''. Although many authors use the two terms ''perpendicular'' and ''orthogonal'' interchangeably, the term ''perpendicular'' is more specifically used for lines and planes that intersect to form a right angle, whereas ''orthogonal'' is used in generalizations, such as ''orthogonal vectors'' or ''orthogonal curves''. ''Orthogonality'' is also used with various meanings that are often weakly related or not related at all with the mathematical meanings. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics Optics In optics, polarization ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Projection (linear Algebra)
In linear algebra and functional analysis, a projection is a linear transformation P from a vector space to itself (an endomorphism) such that P\circ P=P. That is, whenever P is applied twice to any vector, it gives the same result as if it were applied once (i.e. P is idempotent). It leaves its image unchanged. This definition of "projection" formalizes and generalizes the idea of graphical projection. One can also consider the effect of a projection on a geometrical object by examining the effect of the projection on points in the object. Definitions A projection on a vector space V is a linear operator P\colon V \to V such that P^2 = P. When V has an inner product and is complete, i.e. when V is a Hilbert space, the concept of orthogonality can be used. A projection P on a Hilbert space V is called an orthogonal projection if it satisfies \langle P \mathbf x, \mathbf y \rangle = \langle \mathbf x, P \mathbf y \rangle for all \mathbf x, \mathbf y \in V. A projecti ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
FRW Model
The Friedmann equations, also known as the Friedmann–Lemaître (FL) equations, are a set of equations in physical cosmology that govern cosmic expansion in homogeneous and isotropic models of the universe within the context of general relativity. They were first derived by Alexander Friedmann in 1922 from Einstein's field equations of gravitation for the Friedmann–Lemaître–Robertson–Walker metric and a perfect fluid with a given mass density and pressure . (English translation: ). The original Russian manuscript of this paper is preserved in thEhrenfest archive The equations for negative spatial curvature were given by Friedmann in 1924. (English translation: ) The physical models built on the Friedmann equations are called FRW or FLRW models and from the ''Standard Model'' of modern cosmology, although such a description is also associated with the further developed Lambda-CDM model. The FLRW model was developed independently by the named authors in the 1920s and 193 ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
