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Kaluza–Klein–Riemann Curvature Tensor
In Kaluza–Klein theory, a unification of general relativity and electromagnetism, the five-fimensional Kaluza–Klein–Riemann curvature tensor (or Kaluza–Klein–Riemann–Christoffel curvature tensor) is the generalization of the four-dimensional Riemann curvature tensor (or Riemann–Christoffel curvature tensor). Its contraction with itself is the Kaluza–Klein–Ricci tensor, a generalization of the Ricci tensor. Its contraction with the Kaluza–Klein metric is the Kaluza–Klein–Ricci scalar, a generalization of the Ricci scalar. The Kaluza–Klein–Riemann curvature tensor, Kaluza–Klein–Ricci tensor and scalar are namend after Theodor Kaluza, Oskar Klein, Bernhard Riemann and Gregorio Ricci-Curbastro Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the .... Defini ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaluza–Klein Theory
In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions. The five-dimensional (5D) theory developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Albert Einstein in 1919 and published them in 1921. Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. Ten components are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Oskar Klein
Oskar Benjamin Klein (; 15 September 1894 – 5 February 1977) was a Swedish theoretical physics, theoretical physicist. Oskar Klein is known for his work on Kaluza–Klein theory, which is partially named after him. Biography Klein was born in Danderyd Municipality, Danderyd outside Stockholm, son of the chief rabbi of Stockholm, Gottlieb Klein from Humenné in Kingdom of Hungary, now Slovakia and Antonie (Toni) Levy. He became a student of Svante Arrhenius at the Nobel Institute at a young age and was on the way to Jean-Baptiste Perrin in France when World War I broke out and he was drafted into the military. From 1917, he worked a few years with Niels Bohr in the University of Copenhagen and received his doctoral degree at the University College of Stockholm (now Stockholm University) in 1921. In 1923, he received a professorship at University of Michigan in Ann Arbor, Michigan, Ann Arbor and moved there with his recently wedded wife, Gerda Koch from Denmark. Klein returned ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Physical Cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fundamental questions about its Cosmogony, origin, structure, Chronology of the universe, evolution, and ultimate fate.For an overview, see Cosmology as a science originated with the Copernican principle, which implies that astronomical object, celestial bodies obey identical physical laws to those on Earth, and Newtonian mechanics, which first allowed those physical laws to be understood. Physical cosmology, as it is now understood, began in 1915 with the development of Albert Einstein's general relativity, general theory of relativity, followed by major observational discoveries in the 1920s: first, Edwin Hubble discovered that the universe contains a huge number of external Galaxy, galaxies beyond the Milky Way; then, work by Vesto Sliph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Physics
Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the scale of protons and neutrons, while the study of combinations of protons and neutrons is called nuclear physics. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and bosons (force-carrying particles). There are three Generation (particle physics), generations of fermions, although ordinary matter is made only from the first fermion generation. The first generation consists of Up quark, up and down quarks which form protons and neutrons, and electrons and electron neutrinos. The three fundamental interactions known to be mediated by bosons are electromagnetism, the weak interaction, and the strong interaction. Quark, Quarks cannot exist on their own but form hadrons. Hadrons that ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theories Of Gravity
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of empirical and testable knowledge, or they may belong to non-scientific disciplines, such as philosophy, art, or sociology. In some cases, theories may exist independently of any formal discipline. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction (" falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaluza–Klein–Christoffel Symbol
In Kaluza–Klein theory, a unification of general relativity and electromagnetism, the five-fimensional Kaluza–Klein–Christoffel symbol is the generalization of the four-dimensional Christoffel symbol. They directly appear in the geodesic equations of Kaluza–Klein theory and indirectly through the Kaluza–Klein–Riemann curvature tensor also appear in the Kaluza–Klein–Einstein field equations. The Kaluza–Klein–Christoffel symbols are named after Theodor Kaluza, Oskar Klein and Elwin Bruno Christoffel Elwin Bruno Christoffel (; 10 November 1829 – 15 March 1900) was a German mathematician and physicist. He introduced fundamental concepts of differential geometry, opening the way for the development of tensor calculus, which would later provid .... Definition Let \widetilde_ be the Kaluza–Klein metric. The ''Kaluza–Klein–Christoffel symbols'' are given by:Overduin & Wesson 1997, Equation (4) : \widetilde_^c :=\frac\widetilde^(\partial_a\widetilde_+\pa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gregorio Ricci-Curbastro
Gregorio Ricci-Curbastro (; 12January 1925) was an Italian mathematician. He is most famous as the discoverer of tensor calculus. With his former student Tullio Levi-Civita, he wrote his most famous single publication, a pioneering work on the calculus of tensors, signing it as Gregorio Ricci. This appears to be the only time that Ricci-Curbastro used the shortened form of his name in a publication, and continues to cause confusion. Ricci-Curbastro also published important works in other fields, including a book on higher algebra and infinitesimal analysis, and papers on the theory of real numbers, an area in which he extended the research begun by Richard Dedekind. Early life and education Completing privately his high school studies at only 16 years of age, he enrolled on the course of philosophy-mathematics at Sapienza University of Rome, Rome University (1869). The following year the Vatican State, Papal State fell and so Gregorio was called by his father to the city of hi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bernhard Riemann
Georg Friedrich Bernhard Riemann (; ; 17September 182620July 1866) was a German mathematician who made profound contributions to 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 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 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. Early years Riemann was born on 17 September 1826 in Breselenz, a village near Danne ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theodor Kaluza
Theodor Franz Eduard Kaluza (; 9 November 1885 – 19 January 1954) was a German mathematician and physicist known for the Kaluza–Klein theory, involving field equations in five-dimensional space-time. His idea that fundamental forces can be unified by introducing additional dimensions was reused much later for string theory. Life Kaluza was born to a Roman Catholic family from the town of Ratibor (present-day Racibórz in Poland) in the German Empire's Prussian Province of Silesia. Kaluza himself was born in Wilhelmsthal (a village that was incorporated into Oppeln (presently Opole) in 1899). He spent his youth in Königsberg, where his father, Maximilian "Max" Kaluza, was a professor of the English language. He entered the University of Königsberg to study mathematics and gained his doctorate with a thesis on Tschirnhaus transformations. Kaluza was primarily a mathematician but began studying relativity. In April 1919 Kaluza noticed that when he solved Albert Einstei ... [...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] |
Ricci Scalar
In the mathematical field of Riemannian geometry, the scalar curvature (or the Ricci scalar) is a measure of the curvature of a Riemannian manifold. To each point on a Riemannian manifold, it assigns a single real number determined by the geometry of the metric near that point. It is defined by a complicated explicit formula in terms of partial derivatives of the metric components, although it is also characterized by the volume of infinitesimally small geodesic balls. In the context of the differential geometry of surfaces, the scalar curvature is twice the Gaussian curvature, and completely characterizes the curvature of a surface. In higher dimensions, however, the scalar curvature only represents one particular part of the Riemann curvature tensor. The definition of scalar curvature via partial derivatives is also valid in the more general setting of pseudo-Riemannian manifolds. This is significant in general relativity, where scalar curvature of a Lorentzian metric is one of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaluza–Klein Metric
In Kaluza–Klein theory, a unification of general relativity and electromagnetism, the five-dimensional Kaluza–Klein metric is the generalization of the four-dimensional metric tensor. It additionally includes a scalar field called graviscalar (or radion) and a vector field called graviphoton (or gravivector), which correspond to hypothetical particles. The Kaluza–Klein metric is named after Theodor Kaluza and Oskar Klein. Definition The ''Kaluza–Klein metric'' is given by: : \widetilde_ :=\begin g_+\phi^2A_\mu A_\nu & \phi^2A_\mu \\ \phi^2A_\nu & \phi^2 \end. Its inverse matrix is given by: : \widetilde^ =\begin g^ & -A^\mu \\ -A^\nu & g_A^\mu A^\nu+\phi^ \end. Defining an extended gravivector A_a=(A_\mu,1) shortens the definition to: : \widetilde_ =\operatorname(g_,0) +\phi^2A_aA_b, which also shows that the radion \phi cannot vanish as this would make the metric singular. Properties * A contraction directly shows the passing from four to five dimensions: ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
