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Geroch Energy
The Geroch energy or Geroch mass is one of the possible definitions of mass in general relativity. It can be derived from the Hawking energy, itself 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, by leaving out certain (positive) terms related to the sphere's external and internal curvature. The equation for the mass of a 2-sphere is based on the equation for the ingoing and outgoing ray of light corresponding to an electron on its surface, expressed as: γ = 2 γ. The inverse of this equation represents the value of the Hawking energy , and the result is the mass of an entangled pair of photons: the two photons can not be entangled if only they are located at the same location on the surface. But in this case, both photons must be propagating simultaneously. So the two photons, or their antiparticles, must be travelling through a field that is 3-dimensional, and the field ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mass
Mass is an intrinsic property of a body. It was traditionally believed to be related to the quantity of matter in a physical body, until the discovery of the atom and particle physics. It was found that different atoms and different elementary particles, theoretically with the same amount of matter, have nonetheless different masses. Mass in modern physics has multiple definitions which are conceptually distinct, but physically equivalent. Mass can be experimentally defined as a measure of the body's inertia, meaning the resistance to acceleration (change of velocity) when a net force is applied. The object's mass also determines the strength of its gravitational attraction to other bodies. The SI base unit of mass is the kilogram (kg). In physics, mass is not the same as weight, even though mass is often determined by measuring the object's weight using a spring scale, rather than balance scale comparing it directly with known masses. An object on the Moon would weigh le ... [...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] |
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] |
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] |
Curvature
In mathematics, curvature is any of several strongly related concepts in geometry. Intuitively, the curvature is the amount by which a curve deviates from being a straight line, or a surface deviates from being a plane. For curves, the canonical example is that of a circle, which has a curvature equal to the reciprocal of its radius. Smaller circles bend more sharply, and hence have higher curvature. The curvature ''at a point'' of a differentiable curve is the curvature of its osculating circle, that is the circle that best approximates the curve near this point. The curvature of a straight line is zero. In contrast to the tangent, which is a vector quantity, the curvature at a point is typically a scalar quantity, that is, it is expressed by a single real number. For surfaces (and, more generally for higher-dimensional manifolds), that are embedded in a Euclidean space, the concept of curvature is more complex, as it depends on the choice of a direction on the surface or man ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Journal Of Mathematical Physics
The ''Journal of Mathematical Physics'' is a peer-reviewed journal published monthly by the American Institute of Physics devoted to the publication of papers in mathematical physics. The journal was first published bimonthly beginning in January 1960; it became a monthly publication in 1963. The current editor is Jan Philip Solovej from University of Copenhagen The University of Copenhagen ( da, Københavns Universitet, KU) is a prestigious public university, public research university in Copenhagen, Copenhagen, Denmark. Founded in 1479, the University of Copenhagen is the second-oldest university in .... Its 2018 Impact Factor is 1.355 Abstracting and indexing This journal is indexed by the following services: 2013. References External links [...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] |
Robert Geroch
Robert Geroch (born 1 June 1942 in Akron, Ohio) is an American theoretical physicist and professor at the University of Chicago. He has worked prominently on general relativity and mathematical physics and has promoted the use of category theory in mathematics and physics. He was the Ph.D. supervisor for Abhay Ashtekar, Basilis Xanthopoulos and Gary Horowitz. He also proved an important theorem in spin geometry. Education Geroch obtained his Ph.D. degree from Princeton University in 1967 under the supervision of John Archibald Wheeler, with a thesis on ''Singularities in the spacetime of general relativity: their definition, existence, and local characterization''. Writings Chapters *Geroch R.P. (1977) "Asymptotic Structure of Space-Time", p.1--105 in: Esposito F.P., Witten L. (eds) Asymptotic Structure of Space-Time. Springer, Boston, MA. https://doi.org/10.1007/978-1-4684-2343-3_1 *Horowitz, G.T and Geroch, R.P. (1979) "Global structure of spacetimes", p.212--293. In Hawki ... [...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] |
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