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Construction Of A Complex Null Tetrad
Calculations in the Newman–Penrose (NP) formalism of general relativity normally begin with the construction of a complex null tetrad \, where \ is a pair of ''real'' null vectors and \ is a pair of ''complex'' null vectors. These tetrad vectors respect the following normalization and metric conditions assuming the spacetime signature (-,+,+,+): *l_a l^a=n_a n^a=m_a m^a=\bar_a \bar^a=0\,; *l_a m^a=l_a \bar^a=n_a m^a=n_a \bar^a=0\,; *l_a n^a=l^a n_a=-1\,,\;\; m_a \bar^a=m^a \bar_a=1\,; *g_=-l_a n_b - n_a l_b +m_a \bar_b +\bar_a m_b\,, \;\; g^=-l^a n^b - n^a l^b +m^a \bar^b +\bar^a m^b\,. Only after the tetrad \ gets constructed can one move forward to compute the directional derivatives, spin coefficients, commutators, Weyl-NP scalars \Psi_i, Ricci-NP scalars \Phi_ and Maxwell-NP scalars \phi_i and other quantities in NP formalism. There are three most commonly used methods to construct a complex null tetrad: # All four tetrad vectors are nonholonomic combinations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Newman–Penrose Formalism
The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients''. Journal of Mathematical Physics, 1963, 4(7): 998. is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a c ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cosmological Horizon
A cosmological horizon is a measure of the distance from which one could possibly retrieve information. This observable constraint is due to various properties of general relativity, the expanding universe, and the physics of Big Bang cosmology. Cosmological horizons set the size and scale of the observable universe. This article explains a number of these horizons. Particle horizon The particle horizon (also called the cosmological horizon, the comoving horizon, or the cosmic light horizon) is the maximum distance from which light from particles could have traveled to the observer in the age of the universe. It represents the boundary between the observable and the unobservable regions of the universe, so its distance at the present epoch defines the size of the observable universe. Due to the expansion of the universe, it is not simply the age of the universe times the speed of light, as in the Hubble horizon, but rather the speed of light multiplied by the conformal time. The e ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gaussian Null Coordinates
Carl Friedrich Gauss (1777–1855) is the eponym of all of the topics listed below. There are over 100 topics all named after this German mathematician and scientist, all in the fields of mathematics, physics, and astronomy. The English eponymous adjective ''Gaussian'' is pronounced . Mathematics Algebra and linear algebra Geometry and differential geometry Number theory Cyclotomic fields * Gaussian period * Gaussian rational *Gauss sum, an exponential sum over Dirichlet characters **Elliptic Gauss sum, an analog of a Gauss sum **Quadratic Gauss sum Analysis, numerical analysis, vector calculus and calculus of variations Complex analysis and convex analysis * Gauss–Lucas theorem * Gauss's continued fraction, an analytic continued fraction derived from the hypergeometric functions * Gauss's criterion – described oEncyclopedia of Mathematics* Gauss's hypergeometric theorem, an identity on hypergeometric series *Gauss plane Statistics *Gauss ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Taylor Expansion
In mathematics, the Taylor series or Taylor expansion of a function is an infinite sum of terms that are expressed in terms of the function's derivatives at a single point. For most common functions, the function and the sum of its Taylor series are equal near this point. Taylor series are named after Brook Taylor, who introduced them in 1715. A Taylor series is also called a Maclaurin series, when 0 is the point where the derivatives are considered, after Colin Maclaurin, who made extensive use of this special case of Taylor series in the mid-18th century. The partial sum formed by the first terms of a Taylor series is a polynomial of degree that is called the th Taylor polynomial of the function. Taylor polynomials are approximations of a function, which become generally better as increases. Taylor's theorem gives quantitative estimates on the error introduced by the use of such approximations. If the Taylor series of a function is convergent, its sum is the limit of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Event Horizon
In astrophysics, an event horizon is a boundary beyond which events cannot affect an observer. Wolfgang Rindler coined the term in the 1950s. In 1784, John Michell proposed that gravity can be strong enough in the vicinity of massive compact objects that even light cannot escape. At that time, the Newtonian theory of gravitation and the so-called corpuscular theory of light were dominant. In these theories, if the escape velocity of the gravitational influence of a massive object exceeds the speed of light, then light originating inside or from it can escape temporarily but will return. In 1958, David Finkelstein used general relativity to introduce a stricter definition of a local black hole event horizon as a boundary beyond which events of any kind cannot affect an outside observer, leading to information and firewall paradoxes, encouraging the re-examination of the concept of local event horizons and the notion of black holes. Several theories were subsequently developed, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Schwarzschild Coordinates
In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of ''nested round spheres''. In such a spacetime, a particularly important kind of coordinate chart is the Schwarzschild chart, a kind of polar spherical coordinate chart on a static and spherically symmetric spacetime, which is ''adapted'' to these nested round spheres. The defining characteristic of Schwarzschild chart is that the radial coordinate possesses a natural geometric interpretation in terms of the surface area and Gaussian curvature of each sphere. However, radial distances and angles are not accurately represented. These charts have many applications in metric theories of gravitation such as general relativity. They are most often used in static spherically symmetric spacetimes. In the case of general relativity, Birkhoff's theorem states that every ''isolated'' spherically symmetric vacuum or electrovacuum solution of the Einstein field equation is static, but this is certa ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isothermal Coordinates
In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form : g = \varphi (dx_1^2 + \cdots + dx_n^2), where \varphi is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.) Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes. In dimensions > 3, a metric is locally conformally fla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Eddington–Finkelstein Coordinates
In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (e.g. a spherically symmetric black hole) which are adapted to radial null geodesics. Null geodesics are the worldlines of photons; radial ones are those that are moving directly towards or away from the central mass. They are named for Arthur Stanley Eddington and David Finkelstein. Although they appear to have inspired the idea, neither ever wrote down these coordinates or the metric in these coordinates. Roger Penrose seems to have been the first to write down the null form but credits it to the above paper by Finkelstein, and, in his Adams Prize essay later that year, to Eddington and Finkelstein. Most influentially, Misner, Thorne and Wheeler, in their book '' Gravitation'', refer to the null coordinates by that name. In these coordinate systems, outward (inward) traveling radial light rays (which each follow a null geodesic) define the surfaces of constan ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isolated Horizons
It was customary to represent black hole horizons via stationary solutions of field equations, i.e., solutions which admit a time-translational Killing vector field everywhere, not just in a small neighborhood of the black hole. While this simple idealization was natural as a starting point, it is overly restrictive. Physically, it should be sufficient to impose boundary conditions at the horizon which ensure only that the black hole itself is isolated. That is, it should suffice to demand only that the intrinsic geometry of the horizon be time independent, whereas the geometry outside may be dynamical and admit gravitational and other radiation. An advantage of isolated horizons over event horizons is that while one needs the entire spacetime history to locate an event horizon, isolated horizons are defined using local spacetime structures only. The laws of black hole mechanics, initially proved for event horizons, are generalized to isolated horizons. An isolated horizon (\Del ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Near-horizon Metric
The near-horizon metric (NHM) refers to the near-horizon limit of the global metric of a black hole. NHMs play an important role in studying the geometry and topology of black holes, but are only well defined for extremal black holes. NHMs are expressed in Gaussian null coordinates, and one important property is that the dependence on the coordinate r is fixed in the near-horizon limit. NHM of extremal Reissner–Nordström black holes The metric of extremal Reissner–Nordström black hole is :ds^2\,=\,-\Big(1-\frac\Big)^2\,dt^2+\Big(1-\frac\Big)^dr^2+r^2\,\big(d\theta^2+\sin^2\theta\,d\phi^2 \big)\,. Taking the near-horizon limit :t\mapsto \frac\,,\quad r\mapsto M+\epsilon\,\tilde\,,\quad \epsilon\to 0\,, and then omitting the tildes, one obtains the near-horizon metric :ds^2=-\frac\,dt^2+\frac\,dr^2+M^2\,\big(d\theta^2+\sin^2\theta\,d\phi^2 \big) NHM of extremal Kerr black holes The metric of extremal Kerr black hole (M=a=J/M) in Boyer–Lindquist coordinates can be ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isothermal Coordinates
In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric. This means that in isothermal coordinates, the Riemannian metric locally has the form : g = \varphi (dx_1^2 + \cdots + dx_n^2), where \varphi is a positive smooth function. (If the Riemannian manifold is oriented, some authors insist that a coordinate system must agree with that orientation to be isothermal.) Isothermal coordinates on surfaces were first introduced by Gauss. Korn and Lichtenstein proved that isothermal coordinates exist around any point on a two dimensional Riemannian manifold. By contrast, most higher-dimensional manifolds do not admit isothermal coordinates anywhere; that is, they are not usually locally conformally flat. In dimension 3, a Riemannian metric is locally conformally flat if and only if its Cotton tensor vanishes. In dimensions > 3, a metric is locally conformally fla ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Stereographic
Stereoscopy (also called stereoscopics, or stereo imaging) is a technique for creating or enhancing the illusion of depth in an image by means of stereopsis for binocular vision. The word ''stereoscopy'' derives . Any stereoscopic image is called a stereogram. Originally, stereogram referred to a pair of stereo images which could be viewed using a stereoscope. Most stereoscopic methods present a pair of two-dimensional images to the viewer. The left image is presented to the left eye and the right image is presented to the right eye. When viewed, the human brain perceives the images as a single 3D view, giving the viewer the perception of 3D depth. However, the 3D effect lacks proper focal depth, which gives rise to the Vergence-Accommodation Conflict. Stereoscopy is distinguished from other types of 3D displays that display an image in three full dimensions, allowing the observer to increase information about the 3-dimensional objects being displayed by head and eye mov ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
