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Hawking Energy
The Hawking energy or Hawking mass is one of the possible definitions of mass in general relativity. It is a measure of the bending of ingoing and outgoing rays of light that are orthogonal to a 2-sphere surrounding the region of space whose mass is to be defined. Definition Let (\mathcal^3, g_) be a 3-dimensional sub-manifold of a relativistic spacetime, and let \Sigma \subset \mathcal^3 be a closed 2-surface. Then the Hawking mass m_H(\Sigma) of \Sigma is defined to be :m_H(\Sigma) := \sqrt\left( 1 - \frac\int_\Sigma H^2 da \right), where H is the mean curvature of \Sigma. Properties In the Schwarzschild metric, the Hawking mass of any sphere S_r about the central mass is equal to the value m of the central mass. A result of Geroch implies that Hawking mass satisfies an important monotonicity condition. Namely, if \mathcal^3 has nonnegative scalar curvature, then the Hawking mass of \Sigma is non-decreasing as the surface \Sigma flows outward at a speed equal to the inv ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mass In General Relativity
The concept of mass in general relativity (GR) is more subtle to define than the concept of mass in special relativity. In fact, general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined. The reason for this subtlety is that the energy and momentum in the gravitational field cannot be unambiguously localized. (See Chapter 20 of .) So, rigorous definitions of the mass in general relativity are not local, as in classical mechanics or special relativity, but make reference to the asymptotic nature of the spacetime. A well defined notion of the mass exists for asymptotically flat spacetimes and for asymptotically Anti-de Sitter space. However, these definitions must be used with care in other settings. Defining mass in general relativity: concepts and obstacles In special relativity, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light
Light or visible light is electromagnetic radiation that can be perceived by the human eye. Visible light is usually defined as having wavelengths in the range of 400–700 nanometres (nm), corresponding to frequencies of 750–420 terahertz, between the infrared (with longer wavelengths) and the ultraviolet (with shorter wavelengths). In physics, the term "light" may refer more broadly to electromagnetic radiation of any wavelength, whether visible or not. In this sense, gamma rays, X-rays, microwaves and radio waves are also light. The primary properties of light are intensity, propagation direction, frequency or wavelength spectrum and polarization. Its speed in a vacuum, 299 792 458 metres a second (m/s), is one of the fundamental constants of nature. Like all types of electromagnetic radiation, visible light propagates by massless elementary particles called photons that represents the quanta of electromagnetic field, and can be analyzed as both waves and par ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sphere
A sphere () is a Geometry, geometrical object that is a solid geometry, three-dimensional analogue to a two-dimensional circle. A sphere is the Locus (mathematics), set of points that are all at the same distance from a given point in three-dimensional space.. That given point is the centre (geometry), centre of the sphere, and is the sphere's radius. The earliest known mentions of spheres appear in the work of the Greek mathematics, ancient Greek mathematicians. The sphere is a fundamental object in many fields of mathematics. Spheres and nearly-spherical shapes also appear in nature and industry. Bubble (physics), Bubbles such as soap bubbles take a spherical shape in equilibrium. spherical Earth, The Earth is often approximated as a sphere in geography, and the celestial sphere is an important concept in astronomy. Manufactured items including pressure vessels and most curved mirrors and lenses are based on spheres. Spheres rolling, roll smoothly in any direction, so mos ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mean Curvature
In mathematics, the mean curvature H of a surface S is an ''extrinsic'' measure of curvature that comes from differential geometry and that locally describes the curvature of an embedded surface in some ambient space such as Euclidean space. The concept was used by Sophie Germain in her work on elasticity theory. Jean Baptiste Marie Meusnier used it in 1776, in his studies of minimal surfaces. It is important in the analysis of minimal surfaces, which have mean curvature zero, and in the analysis of physical interfaces between fluids (such as soap films) which, for example, have constant mean curvature in static flows, by the Young-Laplace equation. Definition Let p be a point on the surface S inside the three dimensional Euclidean space . Each plane through p containing the normal line to S cuts S in a (plane) curve. Fixing a choice of unit normal gives a signed curvature to that curve. As the plane is rotated by an angle \theta (always containing the normal line) that curvatur ... [...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 Schwar ... [...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] |
Mass In General Relativity
The concept of mass in general relativity (GR) is more subtle to define than the concept of mass in special relativity. In fact, general relativity does not offer a single definition of the term mass, but offers several different definitions that are applicable under different circumstances. Under some circumstances, the mass of a system in general relativity may not even be defined. The reason for this subtlety is that the energy and momentum in the gravitational field cannot be unambiguously localized. (See Chapter 20 of .) So, rigorous definitions of the mass in general relativity are not local, as in classical mechanics or special relativity, but make reference to the asymptotic nature of the spacetime. A well defined notion of the mass exists for asymptotically flat spacetimes and for asymptotically Anti-de Sitter space. However, these definitions must be used with care in other settings. Defining mass in general relativity: concepts and obstacles In special relativity, the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Yuguang Shi
Shi Yuguang (; born 1969, Yinxian, Zhejiang) is a Chinese mathematician at Peking University. His areas of research are geometric analysis and differential geometry. He was awarded the ICTP Ramanujan Prize in 2010, for "outstanding contributions to the geometry of complete (noncompact) Riemannian manifolds, specifically the positivity of quasi-local mass and rigidity of asymptotically hyperbolic manifolds." He earned his Ph.D. from the Chinese Academy of Sciences in 1996 under the supervision of Ding Weiyue. Technical contributions Shi is well-known for his foundational work with Luen-Fai Tam on compact and smooth Riemannian manifolds-with-boundary whose scalar curvature is nonnegative and whose boundary is mean-convex. In particular, if the manifold has a spin structure, and if each connected component of the boundary can be isometrically embedded as a strictly convex hypersurface in Euclidean space, then the average value of the mean curvature of each boundary component is ... [...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] |
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