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Pp-waves
In general relativity, the pp-wave spacetimes, or pp-waves for short, are an important family of exact solutions of Einstein's field equation. The term ''pp'' stands for ''plane-fronted waves with parallel propagation'', and was introduced in 1962 by Jürgen Ehlers and Wolfgang Kundt. Overview The pp-waves solutions model radiation moving at the speed of light. This radiation may consist of: * electromagnetic radiation, * gravitational radiation, * massless radiation associated with Weyl fermions, * ''massless'' radiation associated with some hypothetical distinct type relativistic classical field, or any combination of these, so long as the radiation is all moving in the ''same'' direction. A special type of pp-wave spacetime, the plane wave spacetimes, provide the most general analogue in general relativity of the plane waves familiar to students of electromagnetism. In particular, in general relativity, we must take into account the gravitational effects of the energy d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Aichelburg–Sexl Ultraboost
In general relativity, the Aichelburg–Sexl ultraboost is an exact solution which models the spacetime of an observer moving towards or away from a spherically symmetric gravitating object at nearly the speed of light. It was introduced by Peter C. Aichelburg and Roman U. Sexl in 1971. The original motivation behind the ultraboost was to consider the gravitational field of massless point particles within general relativity. It can be considered an approximation to the gravity well of a photon or other lightspeed particle, although it does not take into account quantum uncertainty in particle position or momentum. The metric tensor can be written, in terms of Brinkmann coordinates, as : ds^2 = -8m \, \delta(u) \, \log r \, du^2 + 2 \, du \, dv + dr^2 + r^2 \, d\theta^2, : -\infty < u,v < \infty, \, 0 < r < \infty, \, -\pi < \theta < \pi The ultraboost can be obtained as the limit of a metric, which is also an exact solution, at least if one admits impulsive curvature ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Jürgen Ehlers
Jürgen Ehlers (; 29 December 1929 – 20 May 2008) was a German physicist who contributed to the understanding of Albert Einstein's theory of general relativity. From graduate and postgraduate work in Pascual Jordan's relativity research group at Hamburg University, he held various posts as a lecturer and, later, as a professor before joining the Max Planck Institute for Astrophysics in Munich as a director. In 1995, he became the founding director of the newly created Max Planck Institute for Gravitational Physics in Potsdam, Germany. Ehlers' research focused on the foundations of general relativity as well as on the theory's applications to astrophysics. He formulated a suitable classification of exact solutions to Einstein's field equations and proved the Ehlers–Geren–Sachs theorem that justifies the application of simple, general-relativistic model universes to modern cosmology. He created a spacetime-oriented description of gravitational lensing and clarified the rel ... [...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 gravitat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate Vector
In linear algebra, a coordinate vector is a representation of a vector as an ordered list of numbers (a tuple) that describes the vector in terms of a particular ordered basis. An easy example may be a position such as (5, 2, 1) in a 3-dimensional Cartesian coordinate system with the basis as the axes of this system. Coordinates are always specified relative to an ordered basis. Bases and their associated coordinate representations let one realize vector spaces and linear transformations concretely as column vectors, row vectors, and matrices; hence, they are useful in calculations. The idea of a coordinate vector can also be used for infinite-dimensional vector spaces, as addressed below. Definition Let ''V'' be a vector space of dimension ''n'' over a field ''F'' and let : B = \ be an ordered basis for ''V''. Then for every v \in V there is a unique linear combination of the basis vectors that equals '' v '': : v = \alpha _1 b_1 + \alpha _2 b_2 + \cdots + \alpha _n b_n . ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Covariant Derivative
In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold. Alternatively, the covariant derivative is a way of introducing and working with a connection on a manifold by means of a differential operator, to be contrasted with the approach given by a principal connection on the frame bundle – see affine connection. In the special case of a manifold isometrically embedded into a higher-dimensional Euclidean space, the covariant derivative can be viewed as the orthogonal projection of the Euclidean directional derivative onto the manifold's tangent space. In this case the Euclidean derivative is broken into two parts, the extrinsic normal component (dependent on the embedding) and the intrinsic covariant derivative component. The name is motivated by the importance of changes of coordinate in physics: the covariant derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobia ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real number, real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space which has a null vector is called a pseudo-Euclidean space. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces ''A'' and ''B'', , where ''q'' is positive-definite on ''A'' and negative-definite on ''B''. The null cone, or isotropic cone, of ''X'' consists of the union of balanced spheres: \bigcup_ \. The null cone is also the union of the isotropic lines through the origin. Examples The Minkowski space#Causal structure, light-like vectors of Minkowski space are null vectors. The four linearly independent biquaternions , , , and are null ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Coordinate-free
A coordinate-free, or component-free, treatment of a scientific theory or mathematical topic develops its concepts on any form of manifold without reference to any particular coordinate system. Benefits Coordinate-free treatments generally allow for simpler systems of equations and inherently constrain certain types of inconsistency, allowing greater mathematical elegance at the cost of some abstraction from the detailed formulae needed to evaluate these equations within a particular system of coordinates. In addition to elegance, coordinate-free treatments are crucial in certain applications for proving that a given definition is well formulated. For example, for a vector space V with basis v_1, ..., v_n, it may be tempting to construct the dual space V^* as the formal span of the symbols v_1^*, ..., v_n^* with bracket \langle \sum \alpha_i v_i , \sum \beta_i v_i^* \rangle := \sum \alpha_i \beta_i, but it is not immediately clear that this construction is independent of the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all Order of derivation, orders in its Domain of a function, domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Brinkmann Coordinates
Brinkmann coordinates are a particular coordinate system for a spacetime belonging to the family of pp-wave metrics. They are named for Hans Brinkmann. In terms of these coordinates, the metric tensor can be written as :ds^2 = H(u,x,y) du^2 + 2 du dv + dx^2 + dy^2 where \partial_, the coordinate vector field dual to the covector field dv, is a null vector field. Indeed, geometrically speaking, it is a null geodesic congruence with vanishing optical scalars. Physically speaking, it serves as the wave vector defining the direction of propagation for the pp-wave. The coordinate vector field \partial_ can be spacelike, null, or timelike at a given event in the spacetime, depending upon the sign of H(u,x,y) at that event. The coordinate vector fields \partial_, \partial_ are both spacelike vector fields. Each surface u=u_, v=v_ can be thought of as a wavefront. In discussions of exact solutions to the Einstein field equation, many authors fail to specify the intended ran ... [...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 tensor 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] |
Lorentzian Manifold
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional Euclidean space. In a manifold it may only be possible to defi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Nathan Rosen
Nathan Rosen (Hebrew: נתן רוזן; March 22, 1909 – December 18, 1995) was an American-Israeli physicist noted for his study on the structure of the hydrogen atom and his work with Albert Einstein and Boris Podolsky on entangled wave functions and the EPR paradox. The Einstein–Rosen bridge, later named the wormhole, was a theory of Nathan Rosen. Background Nathan Rosen was born into a Jewish family in Brooklyn, New York. He attended MIT during the Great Depression, where he received a bachelor's degree in electromechanical engineering and later a master's and a doctorate in physics. As a student he published several papers of note, one being "The Neutron," which attempted to explain the structure of the atomic nucleus a year before their discovery by James Chadwick. He also developed an interest in wave functions, and later, gravitation, when he worked as a fellow at the University of Michigan and Princeton University. State of science At the beginning of the 20th century ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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