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Minkowskian
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (relativity), events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Albert Einstein, Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime betwee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
General Relativity
General relativity, also known as the general theory of relativity and Einstein's theory of gravity, is the geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General 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 or four-dimensional spacetime. In particular, the ' 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 second order partial differential equations. Newton's law of universal gravitation, which describes classical gravity, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass distributions. Some predictions of general relativity, however, are beyond Newton's law of universal gra ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-dimensional
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled ''x'', ''y'', and ''z''). The idea of adding a fourth dimension began with Jean le Rond d'Alembert's "Dimensions" being published in 1754, was followed by Joseph-Louis Lagrange in the mid-1700s, and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. In 1880, Charles Howard Hinton popularized these insights in an essay titled "What is the Fourth Dimension?", which explained the concept of a " four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. The simplest form ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inertial Frame Of Reference
In classical physics and special relativity, an inertial frame of reference (also called inertial reference frame, inertial frame, inertial space, or Galilean reference frame) is a frame of reference that is not undergoing any acceleration. It is a frame in which an isolated physical object — an object with zero net force acting on it — is perceived to move with a constant velocity (it might be a zero velocity) or, equivalently, it is a frame of reference in which Newton's first law of motion holds. All inertial frames are in a state of constant, rectilinear motion with respect to one another; in other words, an accelerometer moving with any of them would detect zero acceleration. It has been observed that celestial objects which are far away from other objects and which are in uniform motion with respect to the cosmic microwave background radiation maintain such uniform motion. Measurements in one inertial frame can be converted to measurements in another by a si ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Four-dimensional Space
A four-dimensional space (4D) is a mathematical extension of the concept of three-dimensional or 3D space. Three-dimensional space is the simplest possible abstraction of the observation that one only needs three numbers, called ''dimensions'', to describe the sizes or locations of objects in the everyday world. For example, the volume of a rectangular box is found by measuring and multiplying its length, width, and height (often labeled ''x'', ''y'', and ''z''). The idea of adding a fourth dimension began with Jean le Rond d'Alembert's "Dimensions" being published in 1754, was followed by Joseph-Louis Lagrange in the mid-1700s, and culminated in a precise formalization of the concept in 1854 by Bernhard Riemann. In 1880, Charles Howard Hinton popularized these insights in an essay titled " What is the Fourth Dimension?", which explained the concept of a " four-dimensional cube" with a step-by-step generalization of the properties of lines, squares, and cubes. The simplest fo ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Time Dilation
In physics and relativity, time dilation is the difference in the elapsed time as measured by two clocks. It is either due to a relative velocity between them ( special relativistic "kinetic" time dilation) or to a difference in gravitational potential between their locations ( general relativistic gravitational time dilation). When unspecified, "time dilation" usually refers to the effect due to velocity. After compensating for varying signal delays due to the changing distance between an observer and a moving clock (i.e. Doppler effect), the observer will measure the moving clock as ticking slower than a clock that is at rest in the observer's own reference frame. In addition, a clock that is close to a massive body (and which therefore is at lower gravitational potential) will record less elapsed time than a clock situated further from the said massive body (and which is at a higher gravitational potential). These predictions of the theory of relativity have been repeated ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Degenerate Bilinear Form
In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space ''V'' is a bilinear form such that the map from ''V'' to ''V''∗ (the dual space of ''V'' ) given by is not an isomorphism. An equivalent definition when ''V'' is finite-dimensional is that it has a non-trivial kernel: there exist some non-zero ''x'' in ''V'' such that :f(x,y)=0\, for all \,y \in V. Nondegenerate forms A nondegenerate or nonsingular form is a bilinear form that is not degenerate, meaning that v \mapsto (x \mapsto f(x,v)) is an isomorphism, or equivalently in finite dimensions, if and only if :f(x,y)=0 for all y \in V implies that x = 0. The most important examples of nondegenerate forms are inner products and symplectic forms. Symmetric nondegenerate forms are important generalizations of inner products, in that often all that is required is that the map V \to V^* be an isomorphism, not positivity. For example, a manifold with an inner product structure on ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galilean Group
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group (assumed throughout below). Without the translations in space and time the group is the homogeneous Galilean group. The Galilean group is the group of motions of Galilean relativity acting on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view. In special relativity the homogenous and inhomogenous Galilean transformations are, respectively, replaced by the Lorentz transformations and Poincaré transformations; conversely, the group contraction in the classical limit of Poincaré transformations yields Galilean transformations. The equations below are only physically valid in a Newtonian frame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Galilean Boost
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics. These transformations together with spatial rotations and translations in space and time form the inhomogeneous Galilean group (assumed throughout below). Without the translations in space and time the group is the homogeneous Galilean group. The Galilean group is the group of motions of Galilean relativity acting on the four dimensions of space and time, forming the Galilean geometry. This is the passive transformation point of view. In special relativity the homogenous and inhomogenous Galilean transformations are, respectively, replaced by the Lorentz transformations and Poincaré transformations; conversely, the group contraction in the classical limit of Poincaré transformations yields Galilean transformations. The equations below are only physically valid in a Newtonian frame ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Translation (geometry)
In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure, shape or space by the same distance in a given direction. A translation can also be interpreted as the addition of a constant vector to every point, or as shifting the origin of the coordinate system. In a Euclidean space, any translation is an isometry. As a function If \mathbf is a fixed vector, known as the ''translation vector'', and \mathbf is the initial position of some object, then the translation function T_ will work as T_(\mathbf)=\mathbf+\mathbf. If T is a translation, then the image of a subset A under the function T is the translate of A by T . The translate of A by T_ is often written A+\mathbf . Horizontal and vertical translations In geometry, a vertical translation (also known as vertical shift) is a translation of a geometric object in a direction parallel to the vertical axis of the Cartesian coordinate system. Often, vertical translation ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reflection (mathematics)
In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection. The image of a figure by a reflection is its mirror image in the axis or plane of reflection. For example the mirror image of the small Latin letter p for a reflection with respect to a vertical axis would look like q. Its image by reflection in a horizontal axis would look like b. A reflection is an involution: when applied twice in succession, every point returns to its original location, and every geometrical object is restored to its original state. The term ''reflection'' is sometimes used for a larger class of mappings from a Euclidean space to itself, namely the non-identity isometries that are involutions. Such isometries have a set of fixed points (the "mirror") that is an affine subspace, but is possibly smaller than a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Rotation Matrix
In linear algebra, a rotation matrix is a transformation matrix that is used to perform a rotation in Euclidean space. For example, using the convention below, the matrix :R = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end rotates points in the plane counterclockwise through an angle with respect to the positive axis about the origin of a two-dimensional Cartesian coordinate system. To perform the rotation on a plane point with standard coordinates , it should be written as a column vector, and multiplied by the matrix : : R\mathbf = \begin \cos \theta & -\sin \theta \\ \sin \theta & \cos \theta \end \begin x \\ y \end = \begin x\cos\theta-y\sin\theta \\ x\sin\theta+y\cos\theta \end. If and are the endpoint coordinates of a vector, where is cosine and is sine, then the above equations become the trigonometric summation angle formulae. Indeed, a rotation matrix can be seen as the trigonometric summation angle formulae in matrix form. On ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Euclidean Group
In mathematics, a Euclidean group is the group of (Euclidean) isometries of a Euclidean space \mathbb^n; that is, the transformations of that space that preserve the Euclidean distance between any two points (also called Euclidean transformations). The group depends only on the dimension ''n'' of the space, and is commonly denoted E(''n'') or ISO(''n''). The Euclidean group E(''n'') comprises all translations, rotations, and reflections of \mathbb^n; and arbitrary finite combinations of them. The Euclidean group can be seen as the symmetry group of the space itself, and contains the group of symmetries of any figure (subset) of that space. A Euclidean isometry can be ''direct'' or ''indirect'', depending on whether it preserves the handedness of figures. The direct Euclidean isometries form a subgroup, the special Euclidean group, often denoted SE(''n''), whose elements are called rigid motions or Euclidean motions. They comprise arbitrary combinations of translations and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
