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Globally Hyperbolic
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is * t^2 - r^2 = T^2 (t and r being the usual variables of time and radius) which is one of the usual equations representing an hyperbola. But this expression is only true relative to the ordinary origin; this article then outline bases for generalizing the concept to any pair of points in spacetime. This is relevant to Albert Einstein's theory of general relativity, and potentially to other metric gravitational theories. Definitions There are several equivalent definitions of global hyperbolicity. Let ''M'' be a smooth connected Lorentzian manifold without boundary. We make the following preliminary definitions: * ''M'' is ''non-totally vicious'' if there is at least one point such that no closed timelike curve passes through it. * '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Geometry And Physics
The ''Journal of Geometry and Physics'' is a scientific journal in mathematical physics. Its scope is to stimulate the interaction between geometry and physics by publishing primary research and review articles which are of common interest to practitioners in both fields. The journal is published by Elsevier since 1984. The Journal covers the following areas of research: ''Methods of:'' * Algebraic and Differential Topology * Algebraic Geometry * Real and Complex Differential Geometry * Riemannian and Finsler Manifolds * Symplectic Geometry * Global Analysis, Analysis on Manifolds * Geometric Theory of Differential Equations * Geometric Control Theory * Lie Groups and Lie Algebras * Supermanifolds and Supergroups * Discrete Geometry * Spinors and Twistors ''Applications to:'' * Strings and Superstrings * Noncommutative Topology and Geometry * Quantum Groups * Geometric Methods in Statistics and Probability * Geometry Approaches to Thermodynamics * Classical and Quantum Dynamica ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The University Of Chicago Press
The University of Chicago Press is the largest and one of the oldest university presses in the United States. It is operated by the University of Chicago and publishes a wide variety of academic titles, including ''The Chicago Manual of Style'', numerous academic journals, and advanced monographs in the academic fields. One of its quasi-independent projects is the BiblioVault, a digital repository for scholarly books. The Press building is located just south of the Midway Plaisance on the University of Chicago campus. History The University of Chicago Press was founded in 1890, making it one of the oldest continuously operating university presses in the United States. Its first published book was Robert F. Harper's ''Assyrian and Babylonian Letters Belonging to the Kouyunjik Collections of the British Museum''. The book sold five copies during its first two years, but by 1900 the University of Chicago Press had published 127 books and pamphlets and 11 scholarly journals, includ ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cambridge University Press
Cambridge University Press is the university press of the University of Cambridge. Granted letters patent by Henry VIII of England, King Henry VIII in 1534, it is the oldest university press A university press is an academic publishing house specializing in monographs and scholarly journals. Most are nonprofit organizations and an integral component of a large research university. They publish work that has been reviewed by schola ... in the world. It is also the King's Printer. Cambridge University Press is a department of the University of Cambridge and is both an academic and educational publisher. It became part of Cambridge University Press & Assessment, following a merger with Cambridge Assessment in 2021. With a global sales presence, publishing hubs, and offices in more than 40 Country, countries, it publishes over 50,000 titles by authors from over 100 countries. Its publishing includes more than 380 academic journals, monographs, reference works, school and uni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light Cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime. Details If one imagines the light confined to a two-dimensional plane, the light from the flash spreads out in a circle after the event E occurs, and if we graph the growing circle with the vertical axis of the graph representing time, the result is a cone, known as the future light cone. The past light cone behaves like the future light cone in reverse, a circle which contracts in radius at the speed of light until it converges to a point at the exact position and time of the event E. In reality, there are three space dimensions, so the light would actually form an expanding or contracting sphere in three-dimensional (3D) space rather than a circle in 2D, and the light cone would actually be a four-dimensional version of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal Structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events. The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature. Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships. Tangent vectors If \,(M,g) is a Lorentzian manifold (for metric g on manifold M) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector X is: * timelike if \,g(X,X) 0 Here we use the ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causality Conditions
In the study of Lorentzian manifold spacetimes there exists a hierarchy of causality conditions which are important in proving mathematical theorems about the global structure of such manifolds. These conditions were collected during the late 1970s.E. Minguzzi and M. Sanchez, ''The causal hierarchy of spacetimes'' in H. Baum and D. Alekseevsky (eds.), vol. Recent developments in pseudo-Riemannian geometry, ESI Lect. Math. Phys., (Eur. Math. Soc. Publ. House, Zurich, 2008), pp. 299–358, , arXiv:gr-qc/0609119 The weaker the causality condition on a spacetime, the more ''unphysical'' the spacetime is. Spacetimes with closed timelike curves, for example, present severe interpretational difficulties. See the grandfather paradox. It is reasonable to believe that any physical spacetime will satisfy the strongest causality condition: global hyperbolicity. For such spacetimes the equations in general relativity can be posed as an initial value problem on a Cauchy surface. The hierar ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Initial Value Formulation (general Relativity)
The initial value formulation of general relativity is a reformulation of Albert Einstein's theory of general relativity that describes a universe evolving over time. Each solution of the Einstein field equations encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify evolution equations for physical systems; if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For Einstein's equations, there appear to be subtle differences compared with other fields: they are self-interacting (that is, non-linear even in the absence of other fields); they are diffeomorphism invariant, so to obtain a unique solution, a fixed background metric and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Communications In Mathematical Physics
''Communications in Mathematical Physics'' is a peer-reviewed academic journal published by Springer. The journal publishes papers in all fields of mathematical physics, but focuses particularly in analysis related to condensed matter physics, statistical mechanics and quantum field theory, and in operator algebras, quantum information and relativity. History Rudolf Haag conceived this journal with Res Jost, and Haag became the Founding Chief Editor. The first issue of ''Communications in Mathematical Physics'' appeared in 1965. Haag guided the journal for the next eight years. Then Klaus Hepp succeeded him for three years, followed by James Glimm, for another three years. Arthur Jaffe began as chief editor in 1979 and served for 21 years. Michael Aizenman became the fifth chief editor in the year 2000 and served in this role until 2012. The current editor-in-chief is Horng-Tzer Yau. Archives Articles from 1965 to 1997 are available in electronic form free of charge, via Pro ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Helga Baum
Helga Baum (née Dlubek, born 1954) is a German mathematician. She is professor for differential geometry and global analysis in the Institute for Mathematics of the Humboldt University of Berlin Humboldt-Universität zu Berlin (german: Humboldt-Universität zu Berlin, abbreviated HU Berlin) is a German public research university in the central borough of Mitte in Berlin. It was established by Frederick William III on the initiative o .... Education Baum earned a doctorate (Dr. sc. nat.) in mathematics in 1980 at the Humboldt University of Berlin. Her dissertation, ''Spin-Strukturen und Dirac-Operatoren über Pseudoriemannschen Mannigfaltigkeiten'', was supervised by . Books Baum of the author or coauthor of books including: *''Conformal differential geometry: Q-curvature and conformal holonomy'', with Andreas Juhl, Birkhäuser, 2010 *''Eichfeldtheorie: Eine Einführung in die Differentialgeometrie auf Faserbündeln'' 'Gauge theory: An introduction into differential geom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Causal Structure
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events. The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature. Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships. Tangent vectors If \,(M,g) is a Lorentzian manifold (for metric g on manifold M) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector X is: * timelike if \,g(X,X) 0 Here we use the ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Classical And Quantum Gravity
''Classical and Quantum Gravity'' is a peer-reviewed journal that covers all aspects of gravitational physics and the theory of spacetime. Its scope includes: *Classical general relativity *Applications of relativity *Experimental gravitation *Cosmology and the early universe *Quantum gravity *Supergravity, superstrings and supersymmetry *Mathematical physics relevant to gravitation The editor-in-chief is Gabriela González at Louisiana State University. The 2018 impact factor is 3.487 according to Journal Citation Reports. As of October 2015, the journal publishes letters in addition to regular articles. There was a companion website to the main journal, CQG+ which highlighted high quality papers published in the journal to raise the visibility of those papers. It also featured film reviews related to gravity such as '' Interstellar'' and '' The Theory of Everything ''. ''Classical and Quantum Gravity'' also supports the field of gravitational physics through sponsorship ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
