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Causal Past
In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold. Introduction In modern physics (especially general relativity) spacetime is represented by a Lorentzian manifold. The causal relations between points in the manifold are interpreted as describing which events in spacetime can influence which other events. The causal structure of an arbitrary (possibly curved) Lorentzian manifold is made more complicated by the presence of curvature. Discussions of the causal structure for such manifolds must be phrased in terms of smooth curves joining pairs of points. Conditions on the tangent vectors of the curves then define the causal relationships. Tangent vectors If \,(M,g) is a Lorentzian manifold (for metric g on manifold M) then the nonzero tangent vectors at each point in the manifold can be classified into three disjoint types. A tangent vector X is: * timelike if \,g(X,X) 0 Here we use the ( ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Mathematical Physics
Mathematical physics refers to the development of mathematics, mathematical methods for application to problems in physics. The ''Journal of Mathematical Physics'' defines the field as "the application of mathematics to problems in physics and the development of mathematical methods suitable for such applications and for the formulation of physical theories". An alternative definition would also include those mathematics that are inspired by physics (also known as physical mathematics). Scope There are several distinct branches of mathematical physics, and these roughly correspond to particular historical periods. Classical mechanics The rigorous, abstract and advanced reformulation of Newtonian mechanics adopting the Lagrangian mechanics and the Hamiltonian mechanics even in the presence of constraints. Both formulations are embodied in analytical mechanics and lead to understanding the deep interplay of the notions of symmetry (physics), symmetry and conservation law, con ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Minkowski Metric
In mathematical physics, Minkowski space (or Minkowski spacetime) () is a combination of Three-dimensional space, three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two Event (relativity), events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be implied by the postulates of special relativity. Minkowski space is closely associated with Albert Einstein, Einstein's theories of special relativity and general relativity and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time may differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime betwee ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Closed Timelike Curve
In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime, that is "closed", returning to its starting point. This possibility was first discovered by Willem Jacob van Stockum in 1937 and later confirmed by Kurt Gödel in 1949,Stephen Hawking, '' My Brief History'', chapter 11 who discovered a solution to the equations of general relativity (GR) allowing CTCs known as the Gödel metric; and since then other GR solutions containing CTCs have been found, such as the Tipler cylinder and traversable wormholes. If CTCs exist, their existence would seem to imply at least the theoretical possibility of time travel backwards in time, raising the spectre of the grandfather paradox, although the Novikov self-consistency principle seems to show that such paradoxes could be avoided. Some physicists speculate that the CTCs which appear in certain GR solutions might be ruled out by a future theory of quantum gravity whic ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
World Line
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] |
Monotonic Function
In mathematics, a monotonic function (or monotone function) is a function between ordered sets that preserves or reverses the given order. This concept first arose in calculus, and was later generalized to the more abstract setting of order theory. In calculus and analysis In calculus, a function f defined on a subset of the real numbers with real values is called ''monotonic'' if and only if it is either entirely non-increasing, or entirely non-decreasing. That is, as per Fig. 1, a function that increases monotonically does not exclusively have to increase, it simply must not decrease. A function is called ''monotonically increasing'' (also ''increasing'' or ''non-decreasing'') if for all x and y such that x \leq y one has f\!\left(x\right) \leq f\!\left(y\right), so f preserves the order (see Figure 1). Likewise, a function is called ''monotonically decreasing'' (also ''decreasing'' or ''non-increasing'') if, whenever x \leq y, then f\!\left(x\right) \geq f\!\left(y\ri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Diffeomorphism
In mathematics, a diffeomorphism is an isomorphism of smooth manifolds. It is an invertible function that maps one differentiable manifold to another such that both the function and its inverse are differentiable. Definition Given two manifolds M and N, a differentiable map f \colon M \rightarrow N is called a diffeomorphism if it is a bijection and its inverse f^ \colon N \rightarrow M is differentiable as well. If these functions are r times continuously differentiable, f is called a C^r-diffeomorphism. Two manifolds M and N are diffeomorphic (usually denoted M \simeq N) if there is a diffeomorphism f from M to N. They are C^r-diffeomorphic if there is an r times continuously differentiable bijective map between them whose inverse is also r times continuously differentiable. Diffeomorphisms of subsets of manifolds Given a subset X of a manifold M and a subset Y of a manifold N, a function f:X\to Y is said to be smooth if for all p in X there is a neighbor ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Homeomorphism
In the mathematical field of topology, a homeomorphism, topological isomorphism, or bicontinuous function is a bijective and continuous function between topological spaces that has a continuous inverse function. Homeomorphisms are the isomorphisms in the category of topological spaces—that is, they are the mappings that preserve all the topological properties of a given space. Two spaces with a homeomorphism between them are called homeomorphic, and from a topological viewpoint they are the same. The word ''homeomorphism'' comes from the Greek words '' ὅμοιος'' (''homoios'') = similar or same and '' μορφή'' (''morphē'') = shape or form, introduced to mathematics by Henri Poincaré in 1895. Very roughly speaking, a topological space is a geometric object, and the homeomorphism is a continuous stretching and bending of the object into a new shape. Thus, a square and a circle are homeomorphic to each other, but a sphere and a torus are not. However, this desc ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Smooth Function
In mathematical analysis, the smoothness of a function (mathematics), function is a property measured by the number of Continuous function, continuous Derivative (mathematics), derivatives it has over some domain, called ''differentiability class''. At the very minimum, a function could be considered smooth if it is differentiable everywhere (hence continuous). At the other end, it might also possess derivatives of all Order of derivation, orders in its Domain of a function, domain, in which case it is said to be infinitely differentiable and referred to as a C-infinity function (or C^ function). Differentiability classes Differentiability class is a classification of functions according to the properties of their derivatives. It is a measure of the highest order of derivative that exists and is continuous for a function. Consider an open set U on the real line and a function f defined on U with real values. Let ''k'' be a non-negative integer. The function f is said to be of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Arrow Of Time
The arrow of time, also called time's arrow, is the concept positing the "one-way direction" or " asymmetry" of time. It was developed in 1927 by the British astrophysicist Arthur Eddington, and is an unsolved general physics question. This direction, according to Eddington, could be determined by studying the organization of atoms, molecules, and bodies, and might be drawn upon a four-dimensional relativistic map of the world ("a solid block of paper"). Physical processes at the microscopic level are believed to be either entirely or mostly time-symmetric: if the direction of time were to reverse, the theoretical statements that describe them would remain true. Yet at the macroscopic level it often appears that this is not the case: there is an obvious direction (or ''flow'') of time. Overview The symmetry of time (T-symmetry) can be understood simply as the following: if time were perfectly symmetrical, a video of real events would seem realistic whether played forward ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Class
In mathematics, when the elements of some set S have a notion of equivalence (formalized as an equivalence relation), then one may naturally split the set S into equivalence classes. These equivalence classes are constructed so that elements a and b belong to the same equivalence class if, and only if, they are equivalent. Formally, given a set S and an equivalence relation \,\sim\, on S, the of an element a in S, denoted by is the set \ of elements which are equivalent to a. It may be proven, from the defining properties of equivalence relations, that the equivalence classes form a partition of S. This partition—the set of equivalence classes—is sometimes called the quotient set or the quotient space of S by \,\sim\,, and is denoted by S / \sim. When the set S has some structure (such as a group operation or a topology) and the equivalence relation \,\sim\, is compatible with this structure, the quotient set often inherits a similar structure from its parent set. Examp ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Equivalence Relation
In mathematics, an equivalence relation is a binary relation that is reflexive, symmetric and transitive. The equipollence relation between line segments in geometry is a common example of an equivalence relation. Each equivalence relation provides a partition of the underlying set into disjoint equivalence classes. Two elements of the given set are equivalent to each other if and only if they belong to the same equivalence class. Notation Various notations are used in the literature to denote that two elements a and b of a set are equivalent with respect to an equivalence relation R; the most common are "a \sim b" and "", which are used when R is implicit, and variations of "a \sim_R b", "", or "" to specify R explicitly. Non-equivalence may be written "" or "a \not\equiv b". Definition A binary relation \,\sim\, on a set X is said to be an equivalence relation, if and only if it is reflexive, symmetric and transitive. That is, for all a, b, and c in X: * a \sim a ( ref ... [...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] |
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