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Null Hypersurface
In relativity and in pseudo-Riemannian geometry, a null hypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length with respect to the local metric tensor). A light cone is an example. An alternative characterization is that the tangent space at every point of a hypersurface contains a nonzero vector such that the metric applied to such a vector and any vector in the tangent space is zero. Another way of saying this is that the pullback of the metric onto the tangent space is degenerate. For a Lorentzian metric, all the vectors in such a tangent space are space-like except in one direction, in which they are null. Physically, there is exactly one lightlike worldline contained in a null hypersurface through each point that corresponds to the worldline of a particle moving at the speed of light, and no contained worldlines that are time-like. Examples of null hypersurfaces include a light cone, a Killing horizon, and the event horizon of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theory Of Relativity
The theory of relativity usually encompasses two interrelated theories by Albert Einstein: special relativity and general relativity, proposed and published in 1905 and 1915, respectively. Special relativity applies to all physical phenomena in the absence of gravity. General relativity explains the law of gravitation and its relation to the forces of nature. It applies to the cosmological and astrophysical realm, including astronomy. The theory transformed theoretical physics and astronomy during the 20th century, superseding a 200-year-old Classical mechanics, theory of mechanics created primarily by Isaac Newton. It introduced concepts including 4-dimensional spacetime as a unified entity of space and time in physics, time, relativity of simultaneity, kinematics, kinematic and gravity, gravitational time dilation, and length contraction. In the field of physics, relativity improved the science of elementary particles and their fundamental interactions, along with ushering in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pseudo-Riemannian Geometry
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ''n''-dimensional Euclidean space. In a manifold it may only be possible ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hypersurface
In geometry, a hypersurface is a generalization of the concepts of hyperplane, plane curve, and surface. A hypersurface is a manifold or an algebraic variety of dimension , which is embedded in an ambient space of dimension , generally a Euclidean space, an affine space or a projective space. Hypersurfaces share, with surfaces in a three-dimensional space, the property of being defined by a single implicit equation, at least locally (near every point), and sometimes globally. A hypersurface in a (Euclidean, affine, or projective) space of dimension two is a plane curve. In a space of dimension three, it is a surface. For example, the equation :x_1^2+x_2^2+\cdots+x_n^2-1=0 defines an algebraic hypersurface of dimension in the Euclidean space of dimension . This hypersurface is also a smooth manifold, and is called a hypersphere or an -sphere. Smooth hypersurface A hypersurface that is a smooth manifold is called a ''smooth hypersurface''. In , a smooth hypersurface is orienta ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Normal Vector
In geometry, a normal is an object such as a line, ray, or vector that is perpendicular to a given object. For example, the normal line to a plane curve at a given point is the (infinite) line perpendicular to the tangent line to the curve at the point. A normal vector may have length one (a unit vector) or its length may represent the curvature of the object (a ''curvature vector''); its algebraic sign may indicate sides (interior or exterior). In three dimensions, a surface normal, or simply normal, to a surface at point P is a vector perpendicular to the tangent plane of the surface at P. The word "normal" is also used as an adjective: a line ''normal'' to a plane, the ''normal'' component of a force, the normal vector, etc. The concept of normality generalizes to orthogonality (right angles). The concept has been generalized to differentiable manifolds of arbitrary dimension embedded in a Euclidean space. The normal vector space or normal space of a manifold at point P ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Null Vector
In mathematics, given a vector space ''X'' with an associated quadratic form ''q'', written , a null vector or isotropic vector is a non-zero element ''x'' of ''X'' for which . In the theory of real number, real bilinear forms, definite quadratic forms and isotropic quadratic forms are distinct. They are distinguished in that only for the latter does there exist a nonzero null vector. A quadratic space which has a null vector is called a pseudo-Euclidean space. A pseudo-Euclidean vector space may be decomposed (non-uniquely) into orthogonal subspaces ''A'' and ''B'', , where ''q'' is positive-definite on ''A'' and negative-definite on ''B''. The null cone, or isotropic cone, of ''X'' consists of the union of balanced spheres: \bigcup_ \. The null cone is also the union of the isotropic lines through the origin. Examples The Minkowski space#Causal structure, light-like vectors of Minkowski space are null vectors. The four linearly independent biquaternions , , , and are null ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Metric Tensor
In the mathematical field of differential geometry, a metric tensor (or simply metric) is an additional structure on a manifold (such as a surface) that allows defining distances and angles, just as the inner product on a Euclidean space allows defining distances and angles there. More precisely, a metric tensor at a point of is a bilinear form defined on the tangent space at (that is, a bilinear function that maps pairs of tangent vectors to real numbers), and a metric tensor on consists of a metric tensor at each point of that varies smoothly with . A metric tensor is ''positive-definite'' if for every nonzero vector . A manifold equipped with a positive-definite metric tensor is known as a Riemannian manifold. Such a metric tensor can be thought of as specifying ''infinitesimal'' distance on the manifold. On a Riemannian manifold , the length of a smooth curve between two points and can be defined by integration, and the distance between and can be defined as ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Light Cone
In special and general relativity, a light cone (or "null cone") is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime. Details If one imagines the light confined to a two-dimensional plane, the light from the flash spreads out in a circle after the event E occurs, and if we graph the growing circle with the vertical axis of the graph representing time, the result is a cone, known as the future light cone. The past light cone behaves like the future light cone in reverse, a circle which contracts in radius at the speed of light until it converges to a point at the exact position and time of the event E. In reality, there are three space dimensions, so the light would actually form an expanding or contracting sphere in three-dimensional (3D) space rather than a circle in 2D, and the light cone would actually be a four-dimensional version of a ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tangent Space
In mathematics, the tangent space of a manifold generalizes to higher dimensions the notion of '' tangent planes'' to surfaces in three dimensions and ''tangent lines'' to curves in two dimensions. In the context of physics the tangent space to a manifold at a point can be viewed as the space of possible velocities for a particle moving on the manifold. Informal description In differential geometry, one can attach to every point x of a differentiable manifold a ''tangent space''—a real vector space that intuitively contains the possible directions in which one can tangentially pass through x . The elements of the tangent space at x are called the ''tangent vectors'' at x . This is a generalization of the notion of a vector, based at a given initial point, in a Euclidean space. The dimension of the tangent space at every point of a connected manifold is the same as that of the manifold itself. For example, if the given manifold is a 2 -sphere, then one can picture the ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Pullback (differential Geometry)
Suppose that is a smooth map between smooth manifolds ''M'' and ''N''. Then there is an associated linear map from the space of 1-forms on ''N'' (the linear space of sections of the cotangent bundle) to the space of 1-forms on ''M''. This linear map is known as the pullback (by ''φ''), and is frequently denoted by ''φ''∗. More generally, any covariant tensor field – in particular any differential form – on ''N'' may be pulled back to ''M'' using ''φ''. When the map ''φ'' is a diffeomorphism, then the pullback, together with the pushforward, can be used to transform any tensor field from ''N'' to ''M'' or vice versa. In particular, if ''φ'' is a diffeomorphism between open subsets of R''n'' and R''n'', viewed as a change of coordinates (perhaps between different charts on a manifold ''M''), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate dependent) approaches ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Worldline
The world line (or worldline) of an object is the path that an object traces in 4-dimensional spacetime. It is an important concept in modern physics, and particularly theoretical physics. The concept of a "world line" is distinguished from concepts such as an "orbit" or a "trajectory" (e.g., a planet's ''orbit in space'' or the ''trajectory'' of a car on a road) by the ''time'' dimension, and typically encompasses a large area of spacetime wherein perceptually straight paths are recalculated to show their ( relatively) more absolute position states—to reveal the nature of special relativity or gravitational interactions. The idea of world lines originates in physics and was pioneered by Hermann Minkowski. The term is now most often used in relativity theories (i.e., special relativity and general relativity). Usage in physics In physics, a world line of an object (approximated as a point in space, e.g., a particle or observer) is the sequence of spacetime events correspon ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Killing Horizon
In physics, a Killing horizon is a geometrical construct used in general relativity and its generalizations to delineate spacetime boundaries without reference to the dynamic Einstein field equations. Mathematically a Killing horizon is a null hypersurface In relativity and in pseudo-Riemannian geometry, a null hypersurface is a hypersurface whose normal vector at every point is a null vector (has zero length with respect to the local metric tensor). A light cone is an example. An alternative ch ... defined by the vanishing of the norm of a Killing vector field (both are named after Wilhelm Killing). It can also be defined as a null hypersurface generated by a Killing vector, which in turn is null at that surface. After Stephen Hawking, Hawking showed that quantum field theory in curved spacetime (without reference to the Einstein field equations) predicted that a black hole formed by collapse will emit Hawking radiation, thermal radiation, it became clear that there is an une ... [...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, som ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
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