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Penrose Inequality
In mathematical general relativity, the Penrose inequality, first conjectured by Sir Roger Penrose, estimates the mass of a spacetime in terms of the total area of its black holes and is a generalization of the positive mass theorem. The Riemannian Penrose inequality is an important special case. Specifically, if (''M'', ''g'') is an asymptotically flat Riemannian 3-manifold with nonnegative scalar curvature and ADM mass ''m'', and ''A'' is the area of the outermost minimal surface (possibly with multiple connected components), then the Riemannian Penrose inequality asserts : m \geq \sqrt. This is purely a geometrical fact, and it corresponds to the case of a complete three-dimensional, space-like, totally geodesic submanifold of a (3 + 1)-dimensional spacetime. Such a submanifold is often called a time-symmetric initial data set for a spacetime. The condition of (''M'', ''g'') having nonnegative scalar curvature is equivalent to the spacetime obeying the ... [...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] |
Gerhard Huisken
Gerhard Huisken (born 20 May 1958) is a German mathematician whose research concerns differential geometry and partial differential equations. He is known for foundational contributions to the theory of the mean curvature flow, including Huisken's monotonicity formula, which is named after him. With Tom Ilmanen, he proved a version of the Riemannian Penrose inequality, which is a special case of the more general Penrose conjecture in general relativity. Education and career After finishing high school in 1977, Huisken took up studies in mathematics at Heidelberg University. In 1982, one year after his diploma graduation, he completed his PhD at the same university under the direction of Claus Gerhardt. The topic of his dissertation were non-linear partial differential equations (''Reguläre Kapillarflächen in negativen Gravitationsfeldern''). From 1983 to 1984, Huisken was a researcher at the Centre for Mathematical Analysis at the Australian National University (ANU) in C ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Inequalities
Geometry (; ) is, with arithmetic, one of the oldest branches of mathematics. It is concerned with properties of space such as the distance, shape, size, and relative position of figures. A mathematician who works in the field of geometry is called a ''geometer''. Until the 19th century, geometry was almost exclusively devoted to Euclidean geometry, which includes the notions of point, line, plane, distance, angle, surface, and curve, as fundamental concepts. During the 19th century several discoveries enlarged dramatically the scope of geometry. One of the oldest such discoveries is Carl Friedrich Gauss' ("remarkable theorem") that asserts roughly that the Gaussian curvature of a surface is independent from any specific embedding in a Euclidean space. This implies that surfaces can be studied ''intrinsically'', that is, as stand-alone spaces, and has been expanded into the theory of manifolds and Riemannian geometry. Later in the 19th century, it appeared that geometries wi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Geometry
Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a ''Riemannian metric'', i.e. with an inner product on the tangent space at each point that varies smoothly from point to point. This gives, in particular, local notions of angle, length of curves, surface area and volume. From those, some other global quantities can be derived by integrating local contributions. Riemannian geometry originated with the vision of Bernhard Riemann expressed in his inaugural lecture "''Ueber die Hypothesen, welche der Geometrie zu Grunde liegen''" ("On the Hypotheses on which Geometry is Based.") It is a very broad and abstract generalization of the differential geometry of surfaces in R3. Development of Riemannian geometry resulted in synthesis of diverse results concerning the geometry of surfaces and the behavior of geodesics on them, with techniques that can be applied to the study of differentiable manifolds of higher dim ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
The Big Bang Theory
''The Big Bang Theory'' is an American television sitcom created by Chuck Lorre and Bill Prady, both of whom served as executive producers on the series, along with Steven Molaro, all of whom also served as head writers. It premiered on CBS on September 24, 2007, and concluded on May 16, 2019, having broadcast 279 episodes over 12 seasons. The show originally centered on five characters living in Pasadena, California: Leonard Hofstadter (Johnny Galecki) and Sheldon Cooper (Jim Parsons), both physicists at Caltech, who share an apartment; Penny (The Big Bang Theory), Penny (Kaley Cuoco), a waitress and aspiring actress who lives across the hall; and Leonard and Sheldon's similarly geeky and socially awkward friends and coworkers, aerospace engineer Howard Wolowitz (Simon Helberg) and astrophysicist Raj Koothrappali (Kunal Nayyar). Over time, supporting characters were promoted to starring roles, including neuroscientist Amy Farrah Fowler (Mayim Bialik), microbiologist Bernadet ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Apparent Horizon
In general relativity, an apparent horizon is a surface that is the boundary between light rays that are directed outwards and moving outwards and those directed outward but moving inward. Apparent horizons are not invariant properties of spacetime, and in particular, they are distinct from event horizons. Within an apparent horizon, light does not move outward; this is in contrast with the event horizon. In a dynamical spacetime, there can be outgoing light rays exterior to an apparent horizon (but still interior to the event horizon). An apparent horizon is a ''local'' notion of the boundary of a black hole, whereas an event horizon is a global notion. The notion of a horizon in general relativity is subtle and depends on fine distinctions. Definition The notion of an "apparent horizon" begins with the notion of a trapped null surface. A (compact, orientable, spacelike) surface always has two independent forward-in-time pointing, lightlike, normal directions. For example, a (sp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarzschild Spacetime
Schwarzschild () is a German surnameIt is likely to be misspelled and/or mispronounced by native English speakers, particularly involving failure to grasp that * German ''sch'' (at the beginning of ''each'' of the two syllables) is pronounced as English ''sh'', * German ''w'' is pronounced as English ''v'', and * German ''z'' is pronounced as English ''ts''. Some English speakers omit the ''s'' that begins the second syllable and pronounce ''child'' and ''childe'' (instead of approximating either English ''shield'' or German ). meaning "black sign" or "black shield". Those bearing the name include: * Karl Schwarzschild (1873–1916), physicist and astronomer * Steven Schwarzschild (1924–1989), philosopher and rabbi * Henry Schwarzschild (1926–1996), civil rights activist * Martin Schwarzschild (1912–1997), astronomer * Shimon Schwarzschild (1925–), environmental activist * Luise Hercus (née Schwarzschild) (1926–), linguist * Leopold_Schwarzschil ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmic Censorship Hypothesis
The weak and the strong cosmic censorship hypotheses are two mathematical conjectures about the structure of gravitational singularities arising in general relativity. Singularities that arise in the solutions of Einstein's equations are typically hidden within event horizons, and therefore cannot be observed from the rest of spacetime. Singularities that are not so hidden are called ''naked''. The weak cosmic censorship hypothesis was conceived by Roger Penrose in 1969 and posits that no naked singularities exist in the universe. Basics Since the physical behavior of singularities is unknown, if singularities can be observed from the rest of spacetime, causality may break down, and physics may lose its predictive power. The issue cannot be avoided, since according to the Penrose–Hawking singularity theorems, singularities are inevitable in physically reasonable situations. Still, in the absence of naked singularities, the universe, as described by the general theory of relativi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hawking Area Theorem
{{DEFAULTSORT:Hawking ...
Hawking may refer to: People * Stephen Hawking (1942–2018), English theoretical physicist and cosmologist *Hawking (surname), a family name (including a list of other persons with the name) Film * ''Hawking'' (2004 film), about Stephen Hawking * ''Hawking'' (2013 film), about Stephen Hawking Animals *Hawking (birds), in birds, catching flying insects *Hawking (falconry), the sport of hunting with hawks Outer space *7672 Hawking, a minor planet *Hawking radiation, thermal radiation emitted outside a black hole Music *Hawking (band), a Canadian alternative rock band Trade * Street hawking, vending merchandise on the street See also * Hawker (other) * Hawk (other) The hawk is a predatory bird. Hawk or The Hawk may also refer to: People * Hawk (nickname), a list of people * Hawk (surname), a list of people * Road Warrior Hawk, or Hawk, the ring name of Michael Hegstrand (1957–2003), American professio ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Geometric Flow
In the mathematical field of differential geometry, a geometric flow, also called a geometric evolution equation, is a type of partial differential equation for a geometric object such as a Riemannian metric or an embedding. It is not a term with a formal meaning, but is typically understood to refer to parabolic partial differential equations. Certain geometric flows arise as the gradient flow associated to a functional on a manifold which has a geometric interpretation, usually associated with some extrinsic or intrinsic curvature. Such flows are fundamentally related to the calculus of variations, and include mean curvature flow and Yamabe flow. Examples Extrinsic Extrinsic geometric flows are flows on embedded submanifolds, or more generally immersed submanifolds. In general they change both the Riemannian metric and the immersion. * Mean curvature flow, as in soap films; critical points are minimal surfaces * Curve-shortening flow, the one-dimensional case of the mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hubert Bray
Hubert Lewis Bray is a mathematician and differential geometry, differential geometer. He is known for having proved the Riemannian Penrose inequality. He works as professor of mathematics and physics at Duke University. Early life and education Hubert is the brother of Clark Bray and grandson of Hubert Evelyn Bray, also American mathematicians. He earned his B.A. and B.S. degrees in Mathematics and Physics, respectively, in 1992 from Rice University and obtained his Ph.D. in 1997 from Stanford University, under the mentorship of Richard Melvin Schoen. Career He was an invited speaker at the 2002 International Congress of Mathematicians in Beijing (in the section of differential geometry). He is one of the inaugural fellows of the American Mathematical Society. Hubert was appointed Professor of Mathematics in 2004, an additionally Professor of Physics in 2019. In 2019, he was appointed Director of Undergraduate Studies of Duke's Mathematics Department. Personal life Hubert ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Inverse Mean Curvature Flow
In the mathematical fields of differential geometry and geometric analysis, inverse mean curvature flow (IMCF) is a geometric flow of submanifolds of a Riemannian or pseudo-Riemannian manifold. It has been used to prove a certain case of the Riemannian Penrose inequality, which is of interest in general relativity. Formally, given a pseudo-Riemannian manifold and a smooth manifold , an inverse mean curvature flow consists of an open interval and a smooth map from into such that :\frac=\frac, where is the mean curvature vector of the immersion . If is Riemannian, if is closed with , and if a given smooth immersion of into has mean curvature which is nowhere zero, then there exists a unique inverse mean curvature flow whose "initial data" is . Gerhardt's convergence theorem A simple example of inverse mean curvature flow is given by a family of concentric round hyperspheres in Euclidean space. If the dimension of such a sphere is and its radius is , then its mean ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
