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Einstein field equations

Einstein field equations
The Einstein field equations (EFE) or Einstein - Hilbert equations are a set of 10 equations in Albert Einstein's general theory of relativity which describe the fundamental interaction of gravitation as a result of spacetime being curved by matter and energy.[1] First published by Einstein in 1915[2] as a tensor equation, the EFE equate local spacetime curvature (expressed by the Einstein tensor) with the local energy and momentum within that spacetime (expressed by the stress–energy tensor).[3] As well as obeying local energy-momentum conservation, the EFE reduce to Newton's law of gravitation where the gravitational field is weak and velocities are much less than the speed of light.[4] Exact solutions for the EFE can only be found under simplifying assumptions such as symmetry. Special classes of exact solutions are most often studied as they model many gravitational phenomena, such as rotating black holes and the expanding universe. Mathematical form[edit] where the scalar curvature, Related:  wikipediaPhysics

Miguel Alcubierre Miguel Alcubierre Moya (born 1964, Mexico City) is a Mexican theoretical physicist.[1] He obtained a degree in physics, and a Master of Science in theoretical physics at the School of Science of Universidad Nacional Autónoma de México (UNAM). Biography[edit] At the end of 1990, Alcubierre moved to Wales to attend graduate school at the University of Wales, Cardiff, receiving his doctorate through study of numerical general relativity.[1][2][3] After 1996 he worked at the Max Planck Institute for Gravitational Physics in Potsdam, Germany, developing new numerical techniques used in the description of black holes. On 11 June 2012, Miguel Alcubierre was appointed Director of the Nuclear Sciences Institute at the National Autonomous University of Mexico (UNAM). May 1994 paper[edit] Media appearances[edit]

Mathematics of general relativity The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity. The main tools used in this geometrical theory of gravitation are tensor fields defined on a Lorentzian manifold representing spacetime. This article is a general description of the mathematics of general relativity. Note: General relativity articles using tensors will use the abstract index notation. Why tensors? The principle of general covariance states that the laws of physics should take the same mathematical form in all reference frames and was one of the central principles in the development of general relativity. Spacetime as a manifold[edit] Most modern approaches to mathematical general relativity begin with the concept of a manifold. The rationale for choosing a manifold as the fundamental mathematical structure is to reflect desirable physical properties. Local versus global structure[edit] At at .

Deriving the Schwarzschild solution The Schwarzschild solution is one of the simplest and most useful solutions of the Einstein field equations (see general relativity). It describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object. It is worthwhile deriving this metric in some detail; the following is a reasonably rigorous derivation that is not always seen in the textbooks. Assumptions and notation[edit] Working in a coordinate chart with coordinates labelled 1 to 4 respectively, we begin with the metric in its most general form (10 independent components, each of which is a smooth function of 4 variables). (1) A spherically symmetric spacetime is one in which all metric components are unchanged under any rotation-reversal or (2) A static spacetime is one in which all metric components are independent of the time coordinate (so that ) and the geometry of the spacetime is unchanged under a time-reversal (3) A vacuum solution is one that satisfies the equation . (after contracting and putting and with

Spacetime In non-relativistic classical mechanics, the use of Euclidean space instead of spacetime is appropriate, as time is treated as universal and constant, being independent of the state of motion of an observer.[disambiguation needed] In relativistic contexts, time cannot be separated from the three dimensions of space, because the observed rate at which time passes for an object depends on the object's velocity relative to the observer and also on the strength of gravitational fields, which can slow the passage of time for an object as seen by an observer outside the field. Until the beginning of the 20th century, time was believed to be independent of motion, progressing at a fixed rate in all reference frames; however, later experiments revealed that time slows at higher speeds of the reference frame relative to another reference frame. Such slowing, called time dilation, is explained in special relativity theory. Spacetime in literature[edit] Mathematical concept[edit] is that

Lightnin' Hopkins Sam John Hopkins (March 15, 1912 – January 30, 1982[1]), better known as Lightnin’ Hopkins, was an American country blues singer, songwriter, guitarist and occasional pianist, from Houston, Texas. Rolling Stone magazine included Hopkins at number 71 on their list of the 100 greatest guitarists of all time.[2] Musicologist Robert "Mack" McCormick opined that Hopkins "is the embodiment of the jazz-and-poetry spirit, representing its ancient form in the single creator whose words and music are one act".[3] Life[edit] In 1959, Hopkins was contacted by Mack McCormick, who hoped to bring him to the attention of the broader musical audience, which was caught up in the folk revival.[1] McCormack presented Hopkins to integrated audiences first in Houston and then in California. Houston's poet-in-residence for 35 years, Hopkins recorded more albums than any other bluesman.[3] Hopkins died of esophageal cancer in Houston on January 30, 1982, at the age of 69. References in popular culture[edit]

Friedmann equations and pressure . The equations for negative spatial curvature were given by Friedmann in 1924.[2] Assumptions[edit] The Friedmann equations start with the simplifying assumption that the universe is spatially homogeneous and isotropic, i.e. the Cosmological Principle; empirically, this is justified on scales larger than ~100 Mpc. where is a three-dimensional metric that must be one of (a) flat space, (b) a sphere of constant positive curvature or (c) a hyperbolic space with constant negative curvature. Einstein's equations now relate the evolution of this scale factor to the pressure and energy of the matter in the universe. The equations[edit] There are two independent Friedmann equations for modeling a homogeneous, isotropic universe. which is derived from the 00 component of Einstein's field equations. is the spatial curvature in any time-slice of the universe; it is equal to one-sixth of the spatial Ricci curvature scalar R since in the Friedmann model. which eliminates to give: Here . . Set

Einstein tensor In differential geometry, the Einstein tensor (also trace-reversed Ricci tensor), named after Albert Einstein, is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, the Einstein tensor occurs in the Einstein field equations for gravitation describing spacetime curvature in a manner consistent with energy considerations. Definition[edit] The Einstein tensor is a rank 2 tensor defined over pseudo-Riemannian manifolds. where is the Ricci tensor, is the metric tensor and is the scalar curvature. The Einstein tensor is symmetric and, like the stress–energy tensor, divergenceless Explicit form[edit] The Ricci tensor depends only on the metric tensor, so the Einstein tensor can be defined directly with just the metric tensor. is the Kronecker tensor and the Christoffel symbol is defined as Before cancellations, this formula results in individual terms. where square brackets conventionally denote antisymmetrization over bracketed indices, i.e. Trace[edit] . See also[edit]

Web application A web application or web app is any software that runs in a web browser. It is created in a browser-supported programming language (such as the combination of JavaScript, HTML and CSS) and relies on a web browser to render the application.[1][2][3] History[edit] In earlier computing models, e.g. in client-server, the load for the application was shared between code on the server and code installed on each client locally. In contrast, web applications use web documents written in a standard format such as HTML and JavaScript, which are supported by a variety of web browsers. In 1995 Netscape introduced a client-side scripting language called JavaScript allowing programmers to add some dynamic elements to the user interface that ran on the client side. In 2005, the term Ajax was coined, and applications like Gmail started to make their client sides more and more interactive. Interface[edit] Structure[edit] There are some who view a web application as a two-tier architecture. Benefits[edit]

Non-orientable wormhole In topology, this sort of connection is referred to as an Alice handle. Theory[edit] "Normal" wormhole connection[edit] Matt Visser has described a way of visualising wormhole geometry: take a "normal" region of space"surgically remove" spherical volumes from two regions ("spacetime surgery")associate the two spherical bleeding edges, so that a line attempting to enter one "missing" spherical volume encounters one bounding surface and then continues outward from the other. For a "conventional" wormhole, the network of points will be seen at the second surface to be inverted, as if one surface was the mirror image of the other—countries will appear back-to-front, as will any text written on the map. "Reversed" wormhole connection[edit] The alternative way of connecting the surfaces makes the "connection map" appear the same at both mouths. This configuration reverses the "handedness" or "chirality" of any objects passing through. Consequences[edit] Alice universe[edit] Notes[edit]