|
Initial Value Formulation (general Relativity)
The initial value formulation of general relativity is a reformulation of Albert Einstein's theory of general relativity that describes a universe evolving over time. Each solution of the Einstein field equations encompasses the whole history of a universe – it is not just some snapshot of how things are, but a whole spacetime: a statement encompassing the state of matter and geometry everywhere and at every moment in that particular universe. By this token, Einstein's theory appears to be different from most other physical theories, which specify evolution equations for physical systems; if the system is in a given state at some given moment, the laws of physics allow you to extrapolate its past or future. For Einstein's equations, there appear to be subtle differences compared with other fields: they are self-interacting (that is, non-linear even in the absence of other fields); they are diffeomorphism invariant, so to obtain a unique solution, a fixed background metric and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Albert Einstein
Albert Einstein ( ; ; 14 March 1879 – 18 April 1955) was a German-born theoretical physicist, widely acknowledged to be one of the greatest and most influential physicists of all time. Einstein is best known for developing the theory of relativity, but he also made important contributions to the development of the theory of quantum mechanics. Relativity and quantum mechanics are the two pillars of modern physics. His mass–energy equivalence formula , which arises from relativity theory, has been dubbed "the world's most famous equation". His work is also known for its influence on the philosophy of science. He received the 1921 Nobel Prize in Physics "for his services to theoretical physics, and especially for his discovery of the law of the photoelectric effect", a pivotal step in the development of quantum theory. His intellectual achievements and originality resulted in "Einstein" becoming synonymous with "genius". In 1905, a year sometimes described as his ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Interior Metric
Interior may refer to: Arts and media * ''Interior'' (Degas) (also known as ''The Rape''), painting by Edgar Degas * ''Interior'' (play), 1895 play by Belgian playwright Maurice Maeterlinck * ''The Interior'' (novel), by Lisa See * Interior design, the trade of designing an architectural interior Places * Interior, South Dakota * Interior, Washington * Interior Township, Michigan * British Columbia Interior, commonly known as "The Interior" Government agencies * Interior ministry, sometimes called the ministry of home affairs * United States Department of the Interior Other uses * Interior (topology), mathematical concept that includes, for example, the inside of a shape * Interior FC, a football team in Gambia See also * * * List of geographic interiors * Interiors (other) * Inter (other) * Inside (other) Inside may refer to: * Insider, a member of any group of people of limited number and generally restricted access Film * ''In ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Collapse
Gravitational collapse is the contraction of an astronomical object due to the influence of its own gravity, which tends to draw matter inward toward the center of gravity. Gravitational collapse is a fundamental mechanism for structure formation in the universe. Over time an initial, relatively smooth distribution of matter will collapse to form pockets of higher density, typically creating a hierarchy of condensed structures such as clusters of galaxies, stellar groups, stars and planets. A star is born through the gradual gravitational collapse of a cloud of interstellar matter. The compression caused by the collapse raises the temperature until thermonuclear fusion occurs at the center of the star, at which point the collapse gradually comes to a halt as the outward thermal pressure balances the gravitational forces. The star then exists in a state of dynamic equilibrium. Once all its energy sources are exhausted, a star will again collapse until it reaches a new equilibrium ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Black Hole
A black hole is a region of spacetime where gravitation, gravity is so strong that nothing, including light or other Electromagnetic radiation, electromagnetic waves, has enough energy to escape it. The theory of general relativity predicts that a sufficiently compact mass can deform spacetime to form a black hole. The boundary (topology), boundary of no escape is called the event horizon. Although it has a great effect on the fate and circumstances of an object crossing it, it has no locally detectable features according to general relativity. In many ways, a black hole acts like an ideal black body, as it reflects no light. Moreover, quantum field theory in curved spacetime predicts that event horizons emit Hawking radiation, with thermal radiation, the same spectrum as a black body of a temperature inversely proportional to its mass. This temperature is of the order of billionths of a kelvin for stellar black holes, making it essentially impossible to observe directly. Obje ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Numerical Relativity
Numerical relativity is one of the branches of general relativity that uses numerical methods and algorithms to solve and analyze problems. To this end, supercomputers are often employed to study black holes, gravitational waves, neutron stars and many other phenomena governed by Einstein's theory of general relativity. A currently active field of research in numerical relativity is the simulation of relativistic binaries and their associated gravitational waves. Overview A primary goal of numerical relativity is to study spacetimes whose exact form is not known. The spacetimes so found computationally can either be fully dynamical, stationary or static and may contain matter fields or vacuum. In the case of stationary and static solutions, numerical methods may also be used to study the stability of the equilibrium spacetimes. In the case of dynamical spacetimes, the problem may be divided into the initial value problem and the evolution, each requiring different methods. Numeri ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Isometry
In mathematics, an isometry (or congruence, or congruent transformation) is a distance-preserving transformation between metric spaces, usually assumed to be bijective. The word isometry is derived from the Ancient Greek: ἴσος ''isos'' meaning "equal", and μέτρον ''metron'' meaning "measure". Introduction Given a metric space (loosely, a set and a scheme for assigning distances between elements of the set), an isometry is a transformation which maps elements to the same or another metric space such that the distance between the image elements in the new metric space is equal to the distance between the elements in the original metric space. In a two-dimensional or three-dimensional Euclidean space, two geometric figures are congruent if they are related by an isometry; the isometry that relates them is either a rigid motion (translation or rotation), or a composition of a rigid motion and a reflection. Isometries are often used in constructions where one space i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Cauchy Surface
In the mathematical field of Lorentzian geometry, a Cauchy surface is a certain kind of submanifold of a Lorentzian manifold. In the application of Lorentzian geometry to the physics of general relativity, a Cauchy surface is usually interpreted as defining an "instant of time"; in the mathematics of general relativity, Cauchy surfaces are important in the formulation of the Einstein equations as an evolutionary problem. They are named for French mathematician Augustin-Louis Cauchy (1789-1857) due to their relevance for the Cauchy problem of general relativity. Informal introduction Although it is usually phrased in terms of general relativity, the formal notion of a Cauchy surface can be understood in familiar terms. Suppose that humans can travel at a maximum speed of 20 miles per hour. This places constraints, for any given person, upon where they can reach by a certain time. For instance, it is impossible for a person who is in Mexico at 3 o'clock to arrive in Libya by 4 o'cl ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Globally Hyperbolic Manifold
In mathematical physics, global hyperbolicity is a certain condition on the causal structure of a spacetime manifold (that is, a Lorentzian manifold). It's called hyperbolic because the fundamental condition that generates the Lorentzian manifold is * t^2 - r^2 = T^2 (t and r being the usual variables of time and radius) which is one of the usual equations representing an hyperbola. But this expression is only true relative to the ordinary origin; this article then outline bases for generalizing the concept to any pair of points in spacetime. This is relevant to Albert Einstein's theory of general relativity, and potentially to other metric gravitational theories. Definitions There are several equivalent definitions of global hyperbolicity. Let ''M'' be a smooth connected Lorentzian manifold without boundary. We make the following preliminary definitions: * ''M'' is ''non-totally vicious'' if there is at least one point such that no closed timelike curve passes through it. * '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Uniqueness Theorem
In mathematics, a uniqueness theorem, also called a unicity theorem, is a theorem asserting the uniqueness of an object satisfying certain conditions, or the equivalence of all objects satisfying the said conditions. Examples of uniqueness theorems include: * Alexandrov's uniqueness theorem of three-dimensional polyhedra * Black hole uniqueness theorem * Cauchy–Kowalevski theorem is the main local existence and uniqueness theorem for analytic partial differential equations associated with Cauchy initial value problems. * Cauchy–Kowalevski–Kashiwara theorem is a wide generalization of the Cauchy–Kowalevski theorem for systems of linear partial differential equations with analytic coefficients. * Division theorem, the uniqueness of quotient and remainder under Euclidean division. * Fundamental theorem of arithmetic, the uniqueness of prime factorization. * Holmgren's uniqueness theorem for linear partial differential equations with real analytic coefficients. * Picard–Lind ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Existence Theorem
In mathematics, an existence theorem is a theorem which asserts the existence of a certain object. It might be a statement which begins with the phrase " there exist(s)", or it might be a universal statement whose last quantifier is existential (e.g., "for all , , ... there exist(s) ..."). In the formal terms of symbolic logic, an existence theorem is a theorem with a prenex normal form involving the existential quantifier, even though in practice, such theorems are usually stated in standard mathematical language. For example, the statement that the sine function is continuous everywhere, or any theorem written in big O notation, can be considered as theorems which are existential by nature—since the quantification can be found in the definitions of the concepts used. A controversy that goes back to the early twentieth century concerns the issue of purely theoretic existence theorems, that is, theorems which depend on non-constructive foundational material such as the axiom ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Constraint (mathematics)
In mathematics, a constraint is a condition of an optimization problem that the solution must satisfy. There are several types of constraints—primarily equality constraints, inequality constraints, and integer constraints. The set of candidate solutions that satisfy all constraints is called the feasible set. Example The following is a simple optimization problem: :\min f(\mathbf x) = x_1^2+x_2^4 subject to :x_1 \ge 1 and :x_2 = 1, where \mathbf x denotes the vector (''x''1, ''x''2). In this example, the first line defines the function to be minimized (called the objective function, loss function, or cost function). The second and third lines define two constraints, the first of which is an inequality constraint and the second of which is an equality constraint. These two constraints are hard constraints, meaning that it is required that they be satisfied; they define the feasible set of candidate solutions. Without the constraints, the solution would be (0,0), whe ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Hamiltonian Mechanics
Hamiltonian mechanics emerged in 1833 as a reformulation of Lagrangian mechanics. Introduced by Sir William Rowan Hamilton, Hamiltonian mechanics replaces (generalized) velocities \dot q^i used in Lagrangian mechanics with (generalized) ''momenta''. Both theories provide interpretations of classical mechanics and describe the same physical phenomena. Hamiltonian mechanics has a close relationship with geometry (notably, symplectic geometry and Poisson structures) and serves as a link between classical and quantum mechanics. Overview Phase space coordinates (p,q) and Hamiltonian H Let (M, \mathcal L) be a mechanical system with the configuration space M and the smooth Lagrangian \mathcal L. Select a standard coordinate system (\boldsymbol,\boldsymbol) on M. The quantities \textstyle p_i(\boldsymbol,\boldsymbol,t) ~\stackrel~ / are called ''momenta''. (Also ''generalized momenta'', ''conjugate momenta'', and ''canonical momenta''). For a time instant t, the Legendre transformat ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
.jpg "Click for more on -> Albert Einstein")
