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Conformastatic Spacetimes
Conformastatic spacetimes refer to a special class of static solutions to Einstein's equation in general relativity. Introduction The line element for the conformastatic class of solutions in Weyl's canonical coordinates readsJohn Lighton Synge. ''Relativity: The General Theory'', Chapter VIII. Amsterdam: North-Holland Publishing Company (Interscience), 1960.Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt . ''Exact Solutions of Einstein's Field Equations'' (2nd Edition), Chapter 18. Cambridge: Cambridge University Press, 2003.Guillermo A Gonzalez, Antonio C Gutierrez-Pineres, Paolo A Ospina. ''Finite axisymmetric charged dust disks in conformastatic spacetimes''. Physical Review D 78 (2008): 064058arXiv:0806.4285[gr-qc]/ref>F D Lora-Clavijo, P A Ospina-Henao, J F Pedraza. ''Charged annular disks and Reissner–Nordström type black holes from extremal dust''. Physical Review D 82 (2010): 084005arXiv:1009.1005[gr-qc]/ref>Ivan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein's Equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of t ... [...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] |
Line Element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by ''ds''. Line elements are used in physics, especially in theories of gravitation (most notably general relativity) where spacetime is modelled as a curved Pseudo-Riemannian manifold with an appropriate metric tensor.Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, General formulation Definition of the line element and arc length The coordinate-independent definition of the square of the line element ''ds'' in an ''n''-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement d\mathbfTensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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] |
Einstein's Field Equation
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Albert Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Laplace Operator
In mathematics, the Laplace operator or Laplacian is a differential operator given by the divergence of the gradient of a Scalar field, scalar function on Euclidean space. It is usually denoted by the symbols \nabla\cdot\nabla, \nabla^2 (where \nabla is the Del, nabla operator), or \Delta. In a Cartesian coordinate system, the Laplacian is given by the sum of second partial derivatives of the function with respect to each independent variable. In other coordinate systems, such as cylindrical coordinates, cylindrical and spherical coordinates, the Laplacian also has a useful form. Informally, the Laplacian of a function at a point measures by how much the average value of over small spheres or balls centered at deviates from . The Laplace operator is named after the French mathematician Pierre-Simon de Laplace (1749–1827), who first applied the operator to the study of celestial mechanics: the Laplacian of the gravitational potential due to a given mass density distributio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gradient Operator
Del, or nabla, is an operator used in mathematics (particularly in vector calculus) as a vector differential operator, usually represented by the nabla symbol ∇. When applied to a function defined on a one-dimensional domain, it denotes the standard derivative of the function as defined in calculus. When applied to a ''field'' (a function defined on a multi-dimensional domain), it may denote any one of three operations depending on the way it is applied: the gradient or (locally) steepest slope of a scalar field (or sometimes of a vector field, as in the Navier–Stokes equations); the divergence of a vector field; or the curl (rotation) of a vector field. Del is a very convenient mathematical notation for those three operations (gradient, divergence, and curl) that makes many equations easier to write and remember. The del symbol (or nabla) can be formally defined as a vector operator whose components are the corresponding partial derivative operators. As a vector operato ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Metrics
In general relativity, the Weyl metrics (named after the German-American mathematician Hermann Weyl) are a class of ''static'' and ''axisymmetric'' solutions to Einstein's field equation. Three members in the renowned Kerr–Newman family solutions, namely the Schwarzschild, nonextremal Reissner–Nordström and extremal Reissner–Nordström metrics, can be identified as Weyl-type metrics. Standard Weyl metrics The Weyl class of solutions has the generic formJeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 10.Hans Stephani, Dietrich Kramer, Malcolm MacCallum, Cornelius Hoenselaers, Eduard Herlt. ''Exact Solutions of Einstein's Field Equations''. Cambridge: Cambridge University Press, 2003. Chapter 20. where \psi(\rho,z) and \gamma(\rho,z) are two metric potentials dependent on ''Weyl's canonical coordinates'' \. The coordinate system \ serves best for symmetries of Weyl's space ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Reissner–Nordström Metric
In physics and astronomy, the Reissner–Nordström metric is a Static spacetime, static solution to the Einstein–Maxwell equations, Einstein–Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass ''M''. The analogous solution for a charged, rotating body is given by the Kerr–Newman metric. The metric was discovered between 1916 and 1921 by Hans Reissner, Hermann Weyl, Gunnar Nordström and George Barker Jeffery independently. Metric In spherical coordinates , the Reissner–Nordström metric (i.e. the line element) is : ds^2 = c^2\, d\tau^2 = \left( 1 - \frac + \frac \right) c^2\, dt^2 -\left( 1 - \frac + \frac \right)^ \, dr^2 - ~ r^2 \, d\theta^2 - ~ r^2\sin^2\theta \, d\varphi^2 , where * c is the speed of light * \tau is the proper time * t is the time coordinate (measured by a stationary clock at infinity). * r is the radial coordinate * (\theta, \varphi) are the spherical angles * ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
