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Belinski–Zakharov Transform
The Belinski–Zakharov (inverse) transform is a nonlinear transformation that generates new exact solutions of the vacuum Einstein's field equation. It was developed by Vladimir Belinski and Vladimir Zakharov in 1978. The Belinski–Zakharov transform is a generalization of the inverse scattering transform. The solutions produced by this transform are called gravitational solitons (gravisolitons). Despite the term 'soliton' being used to describe gravitational solitons, their behavior is very different from other (classical) solitons. In particular, gravitational solitons do not preserve their amplitude and shape in time, and up to June 2012 their general interpretation remains unknown. What is known however, is that most black holes (and particularly the Schwarzschild metric and the Kerr metric) are special cases of gravitational solitons. Introduction The Belinski–Zakharov transform works for spacetime intervals of the form :: ds^2 = f (-d(x^0)^2 + d(x^1)^2) + g_ \, dx^a \, ... [...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 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 the EFE ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir Belinski
Vladimir Alekseevich Belinski (last name is also spelled Belinsky, russian: Владимир Алексеевич Белинский; born 26 March 1941) is a Russian and Italian theoretical physicist involved in research in cosmology and general relativity. He worked at Landau Institute for Theoretical Physics from 1968 to 1989 and got his Habilitation ( Doctor of Sciences) degree at this Institute in 1980. As of 2016, he holds the permanent professor position at International Network of the Centers for Relativistic Astrophysics (ICRANet), Italy. Belinski contributed to the editing of some chapters in Landau and Lifshitz's course of theoretical physics for volume 2. One of his most notable scientific contributions is BKL conjecture (Belinski— Khalatnikov—Lifshitz conjecture) on the behavior of generic solutions of Einstein field equations near a cosmological singularity. He co-invented the Belinski-Zakharov transform in 1978 showing that black holes are a special example of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vladimir E
Vladimir may refer to: Names * Vladimir (name) for the Bulgarian, Croatian, Czech, Macedonian, Romanian, Russian, Serbian, Slovak and Slovenian spellings of a Slavic name * Uladzimir for the Belarusian version of the name * Volodymyr for the Ukrainian version of the name * Włodzimierz (given name) for the Polish version of the name * Valdemar for the Germanic version of the name * Wladimir for an alternative spelling of the name Places * Vladimir, Russia, a city in Russia * Vladimir Oblast, a federal subject of Russia * Vladimir-Suzdal, a medieval principality * Vladimir, Ulcinj, a village in Ulcinj Municipality, Montenegro * Vladimir, Gorj, a commune in Gorj County, Romania * Vladimir, a village in Goiești Commune, Dolj County, Romania * Vladimir (river), a tributary of the Gilort in Gorj County, Romania * Volodymyr (city), a city in Ukraine Religious leaders * Metropolitan Vladimir (other), multiple * Jovan Vladimir (d. 1016), ruler of Doclea and a saint of the S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Scattering Transform
In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations. The method is a non-linear analogue, and in some sense generalization, of the Fourier transform, which itself is applied to solve many linear partial differential equations. The name "inverse scattering method" comes from the key idea of recovering the time evolution of a potential from the time evolution of its scattering data: inverse scattering refers to the problem of recovering a potential from its scattering matrix, as opposed to the direct scattering problem of finding the scattering matrix from the potential. The inverse scattering transform may be applied to many of the so-called exactly solvable models, that is to say completely integrable infinite dimensional systems. Overview The inverse scattering transform was first introduced by for the Korteweg–de Vries equation, and soon extended to the nonlinear Schrödinger equation, the Sine-Gordon e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Solitons
A gravitational soliton is a soliton solution of the Einstein field equation. It can be separated into two kinds, a soliton of the vacuum Einstein field equation generated by the Belinski–Zakharov transform, and a soliton of the Einstein–Maxwell equations 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 Einstein in 1915 in the form ... generated by the Belinski-Zakharov-Alekseev transform. References General relativity {{relativity-stub ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarzschild Metric
In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild solution) is an exact solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero. The solution is a useful approximation for describing slowly rotating astronomical objects such as many stars and planets, including Earth and the Sun. It was found by Karl Schwarzschild in 1916, and around the same time independently by Johannes Droste, who published his more complete and modern-looking discussion four months after Schwarzschild. According to Birkhoff's theorem, the Schwarzschild metric is the most general spherically symmetric vacuum solution of the Einstein field equations. A Schwarzschild black hole or static black hole is a black hole that has neither electric charge nor angular momentum. A S ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kerr Metric
The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially symmetric black hole with a quasispherical event horizon. The Kerr metric is an exact solution of the Einstein field equations of general relativity; these equations are highly non-linear, which makes exact solutions very difficult to find. Overview The Kerr metric is a generalization to a rotating body of the Schwarzschild metric, discovered by Karl Schwarzschild in 1915, which described the geometry of spacetime around an uncharged, spherically symmetric, and non-rotating body. The corresponding solution for a ''charged'', spherical, non-rotating body, the Reissner–Nordström metric, was discovered soon afterwards (1916–1918). However, the exact solution for an uncharged, ''rotating'' black hole, the Kerr metric, remained unsolved until 1963, when it was discovered by Roy Kerr.Melia, Fulvio (2009). "Cracking the Einstein code: relativity and the birth of black hol ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Spacetime Interval
In physics, spacetime is a mathematical model that combines the three dimensions of space and one dimension of time into a single four-dimensional manifold. Spacetime diagrams can be used to visualize relativistic effects, such as why different observers perceive differently where and when events occur. Until the 20th century, it was assumed that the three-dimensional geometry of the universe (its spatial expression in terms of coordinates, distances, and directions) was independent of one-dimensional time. The physicist Albert Einstein helped develop the idea of spacetime as part of his theory of relativity. Prior to his pioneering work, scientists had two separate theories to explain physical phenomena: Isaac Newton's laws of physics described the motion of massive objects, while James Clerk Maxwell's electromagnetic models explained the properties of light. However, in 1905, Einstein based a work on special relativity on two postulates: * The laws of physics are invariant ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein's Summation Convention
In mathematics, especially the usage of linear algebra in Mathematical physics, Einstein notation (also known as the Einstein summation convention or Einstein summation notation) is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving brevity. As part of mathematics it is a notational subset of Ricci calculus; however, it is often used in physics applications that do not distinguish between tangent and cotangent spaces. It was introduced to physics by Albert Einstein in 1916. Introduction Statement of convention According to this convention, when an index variable appears twice in a single term and is not otherwise defined (see Free and bound variables), it implies summation of that term over all the values of the index. So where the indices can range over the set , : y = \sum_^3 c_i x^i = c_1 x^1 + c_2 x^2 + c_3 x^3 is simplified by the convention to: : y = c_i x^i The upper indices are not exponents but are indic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein–Rosen Metric
The Einstein–Rosen metric is an exact solution of Einstein's field equation. It was derived by Albert Einstein Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory ... and Nathan Rosen in 1937. It is the first exact solution of Einstein's equation that described the propagation of a gravitational wave. This metric can be written in a form such that the Belinski–Zakharov transform applies, and thus has the form of a gravitational soliton. In 1972 and 1973, J. R. Rao, A. R. Roy, and R. N. Tiwari published a class of exact solutions involving the Einstein-Rosen metric.\ In 2021 Robert F. Penna found an algebraic derivation of the Einstein-Rosen metric. In the history of science, one might consider as a footnote to the Einstein-Rosen metric that Einstein, for some time, believed th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lax Pair
In mathematics, in the theory of integrable systems, a Lax pair is a pair of time-dependent matrices or operators that satisfy a corresponding differential equation, called the ''Lax equation''. Lax pairs were introduced by Peter Lax to discuss solitons in continuous media. The inverse scattering transform makes use of the Lax equations to solve such systems. Definition A Lax pair is a pair of matrices or operators L(t), P(t) dependent on time and acting on a fixed Hilbert space, and satisfying Lax's equation: :\frac= ,L/math> where ,LPL-LP is the commutator. Often, as in the example below, P depends on L in a prescribed way, so this is a nonlinear equation for L as a function of t. Isospectral property It can then be shown that the eigenvalues and more generally the spectrum of ''L'' are independent of ''t''. The matrices/operators ''L'' are said to be '' isospectral'' as t varies. The core observation is that the matrices L(t) are all similar by virtue of :L(t)=U(t,s) L( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
