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Tensors In General Relativity
In mathematics, a tensor is an algebraic object that describes a multilinear relationship between sets of algebraic objects associated with a vector space. Tensors may map between different objects such as vectors, scalars, and even other tensors. There are many types of tensors, including scalars and vectors (which are the simplest tensors), dual vectors, multilinear maps between vector spaces, and even some operations such as the dot product. Tensors are defined independent of any basis, although they are often referred to by their components in a basis related to a particular coordinate system; those components form an array, which can be thought of as a high-dimensional matrix. Tensors have become important in physics because they provide a concise mathematical framework for formulating and solving physics problems in areas such as mechanics ( stress, elasticity, quantum mechanics, fluid mechanics, moment of inertia, ...), electrodynamics (electromagnetic tensor, Ma ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Components Stress Tensor
Component may refer to: In engineering, science, and technology Generic systems *System components, an entity with discrete structure, such as an assembly or software module, within a system considered at a particular level of analysis *Lumped element model, a model of spatially distributed systems Electrical *Component video, a type of analog video information that is transmitted or stored as two or more separate signals *Electronic component, a constituent of an electronic circuit *Symmetrical components, in electrical engineering, analysis of unbalanced three-phase power systems Mathematics *Color model, a way of describing how colors can be represented, typically as multiple values or color components *Component (group theory), a quasi-simple subnormal sub-group *Connected component (graph theory), a maximal connected subgraph *Connected component (topology), a maximal connected subspace of a topological space *Vector component, result of the decomposition of a vector into va ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Elasticity (physics)
In physics and materials science, elasticity is the ability of a body to resist a distorting influence and to return to its original size and shape when that influence or force is removed. Solid objects will deform when adequate loads are applied to them; if the material is elastic, the object will return to its initial shape and size after removal. This is in contrast to ''plasticity'', in which the object fails to do so and instead remains in its deformed state. The physical reasons for elastic behavior can be quite different for different materials. In metals, the Crystal structure, atomic lattice changes size and shape when forces are applied (energy is added to the system). When forces are removed, the lattice goes back to the original lower energy state. For rubber elasticity, rubbers and other polymers, elasticity is caused by the stretching of polymer chains when forces are applied. Hooke's law states that the force required to deform elastic objects should be Prop ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Tullio Levi-Civita
Tullio Levi-Civita, (; ; 29 March 1873 – 29 December 1941) was an Italian mathematician, most famous for his work on absolute differential calculus ( tensor calculus) and its applications to the theory of relativity, but who also made significant contributions in other areas. He was a pupil of Gregorio Ricci-Curbastro, the inventor of tensor calculus. His work included foundational papers in both pure and applied mathematics, celestial mechanics (notably on the three-body problem), analytic mechanics (the Levi-Civita separability conditions in the Hamilton–Jacobi equation) and hydrodynamics. Biography Born into an Italian Jewish family in Padua, Levi-Civita was the son of Giacomo Levi-Civita, a lawyer and former senator. He graduated in 1892 from the University of Padua Faculty of Mathematics. In 1894 he earned a teaching diploma after which he was appointed to the Faculty of Science teacher's college in Pavia. In 1898 he was appointed to the Padua Chair of Rational Me ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Tensor Field
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space or manifold) or of the physical space. Tensor fields are used in differential geometry, algebraic geometry, general relativity, in the analysis of stress and strain in material object, and in numerous applications in the physical sciences. As a tensor is a generalization of a scalar (a pure number representing a value, for example speed) and a vector (a magnitude and a direction, like velocity), a tensor field is a generalization of a ''scalar field'' and a ''vector field'' that assigns, respectively, a scalar or vector to each point of space. If a tensor is defined on a vector fields set over a module , we call a tensor field on . A tensor field, in common usage, is often referred to in the shorter form "tensor". For example, the ''Riemann curvature tensor'' refers a tensor ''field'', as it associates a tensor to each ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Stress–energy Tensor
The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor physical quantity that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics. It is an attribute of matter, radiation, and non-gravitational force fields. This density and flux of energy and momentum are the sources of the gravitational field in the Einstein field equations of general relativity, just as mass density is the source of such a field in Newtonian gravity. Definition The stress–energy tensor involves the use of superscripted variables ( exponents; see ''Tensor index notation'' and '' Einstein summation notation''). If Cartesian coordinates in SI units are used, then the components of the position four-vector are given by: . In traditional Cartesian coordinates these are instead customarily written , where is coordinate time, and , , and are coordinate distances. Th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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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] |
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Magnetic Susceptibility
In electromagnetism, the magnetic susceptibility (; denoted , chi) is a measure of how much a material will become magnetized in an applied magnetic field. It is the ratio of magnetization (magnetic moment per unit volume) to the applied magnetic field intensity . This allows a simple classification, into two categories, of most materials' responses to an applied magnetic field: an alignment with the magnetic field, , called paramagnetism, or an alignment against the field, , called diamagnetism. Magnetic susceptibility indicates whether a material is attracted into or repelled out of a magnetic field. Paramagnetic materials align with the applied field and are attracted to regions of greater magnetic field. Diamagnetic materials are anti-aligned and are pushed away, toward regions of lower magnetic fields. On top of the applied field, the magnetization of the material adds its own magnetic field, causing the field lines to concentrate in paramagnetism, or be excluded in diamagn ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Permittivity
In electromagnetism, the absolute permittivity, often simply called permittivity and denoted by the Greek letter (epsilon), is a measure of the electric polarizability of a dielectric material. A material with high permittivity polarizes more in response to an applied electric field than a material with low permittivity, thereby storing more energy in the material. In electrostatics, the permittivity plays an important role in determining the capacitance of a capacitor. In the simplest case, the electric displacement field resulting from an applied electric field E is \mathbf = \varepsilon\ \mathbf ~. More generally, the permittivity is a thermodynamic State function, function of state. It can depend on the Dispersion (optics), frequency, Nonlinear optics, magnitude, and Anisotropy, direction of the applied field. The International System of Units, SI unit for permittivity is farad per meter (F/m). The permittivity is often represented by the relative permittivity which is ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Maxwell Stress Tensor
The Maxwell stress tensor (named after James Clerk Maxwell) is a symmetric second-order tensor in three dimensions that is used in classical electromagnetism to represent the interaction between electromagnetic forces and mechanical momentum. In simple situations, such as a point charge moving freely in a homogeneous magnetic field, it is easy to calculate the forces on the charge from the Lorentz force law. When the situation becomes more complicated, this ordinary procedure can become impractically difficult, with equations spanning multiple lines. It is therefore convenient to collect many of these terms in the Maxwell stress tensor, and to use tensor arithmetic to find the answer to the problem at hand. In the relativistic formulation of electromagnetism, the nine components of the Maxwell stress tensor appear, negated, as components of the electromagnetic stress–energy tensor, which is the electromagnetic component of the total stress–energy tensor. The latter describe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Electromagnetic Tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was developed by Arnold Sommerfeld after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written concisely, and allows for the quantization of the electromagnetic field by the Lagrangian formulation described below. Definition The electromagnetic tensor, conventionally labelled ''F'', is defined as the exterior derivative of the electromagnetic four-potential, ''A'', a differential 1-form: :F \ \stackrel\ \mathrmA. Therefore, ''F'' is a differential 2-form— an antisymmetric rank-2 tensor field—on Minkowski space. In component form, :F_ = \partial_\mu A_\nu - \partial_\nu A_\mu. where \partial is the four-gradient and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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Classical Electromagnetism
Classical electromagnetism or classical electrodynamics is a branch of physics focused on the study of interactions between electric charges and electrical current, currents using an extension of the classical Newtonian model. It is, therefore, a classical field theory. The theory provides a description of electromagnetic phenomena whenever the relevant length scales and field strengths are large enough that quantum mechanical effects are negligible. For small distances and low field strengths, such interactions are better described by quantum electrodynamics which is a quantum field theory. History The physical phenomena that electromagnetism describes have been studied as separate fields since antiquity. For example, there were many advances in the field of History of optics, optics centuries before light was understood to be an electromagnetic wave. However, the theory of electromagnetism, as it is currently understood, grew out of Michael Faraday's experiments suggesting t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |