|
Kaluza–Klein–Einstein Field Equations
In Kaluza–Klein theory, a speculative unification of general relativity and electromagnetism, the five-dimensional Kaluza–Klein–Einstein field equations are created by adding a hypothetical dimension to the four-dimensional Einstein field equations. They use the Kaluza–Klein–Einstein tensor, a generalization of the Einstein tensor, and can be obtained from the Kaluza–Klein–Einstein–Hilbert action, a generalization of the Einstein–Hilbert action. They also feature a phenomenon known as ''Kaluza miracle'', which is that the description of the five-dimensional vacuum perfectly falls apart in a four-dimensional Electrovacuum solution, electrovacuum, Maxwell's equations and an additional Graviscalar, radion field equation for the size of the compactified dimension: : \text\left\{\begin{array}{c} \text{D=4 electrovacuum Einstein field equations} \\ \text{Maxwell's equations} \\ \text{radion field equation} \end{array}\right. The Kaluza–Klein–Einstein field equations ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Kaluza–Klein Theory
In physics, Kaluza–Klein theory (KK theory) is a classical unified field theory of gravitation and electromagnetism built around the idea of a fifth dimension beyond the common 4D of space and time and considered an important precursor to string theory. In their setup, the vacuum has the usual 3 dimensions of space and one dimension of time but with another microscopic extra spatial dimension in the shape of a tiny circle. Gunnar Nordström had an earlier, similar idea. But in that case, a fifth component was added to the electromagnetic vector potential, representing the Newtonian gravitational potential, and writing the Maxwell equations in five dimensions. The five-dimensional (5D) theory developed in three steps. The original hypothesis came from Theodor Kaluza, who sent his results to Albert Einstein in 1919 and published them in 1921. Kaluza presented a purely classical extension of general relativity to 5D, with a metric tensor of 15 components. Ten components are ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tensor Contraction
In multilinear algebra, a tensor contraction is an operation on a tensor that arises from the canonical pairing of a vector space and its dual. In components, it is expressed as a sum of products of scalar components of the tensor(s) caused by applying the summation convention to a pair of dummy indices that are bound to each other in an expression. The contraction of a single mixed tensor occurs when a pair of literal indices (one a subscript, the other a superscript) of the tensor are set equal to each other and summed over. In Einstein notation this summation is built into the notation. The result is another tensor with order reduced by 2. Tensor contraction can be seen as a generalization of the trace. Abstract formulation Let ''V'' be a vector space over a field ''k''. The core of the contraction operation, and the simplest case, is the canonical pairing of ''V'' with its dual vector space ''V''∗. The pairing is the linear map from the tensor product of these two s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Theories Of Gravity
A theory is a systematic and rational form of abstract thinking about a phenomenon, or the conclusions derived from such thinking. It involves contemplative and logical reasoning, often supported by processes such as observation, experimentation, and research. Theories can be scientific, falling within the realm of empirical and testable knowledge, or they may belong to non-scientific disciplines, such as philosophy, art, or sociology. In some cases, theories may exist independently of any formal discipline. In modern science, the term "theory" refers to scientific theories, a well-confirmed type of explanation of nature, made in a way consistent with the scientific method, and fulfilling the criteria required by modern science. Such theories are described in such a way that scientific tests should be able to provide empirical support for it, or empirical contradiction (" falsify") of it. Scientific theories are the most reliable, rigorous, and comprehensive form of scientific ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Gravitational Constant
The gravitational constant is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general relativity, theory of general relativity. It is also known as the universal gravitational constant, the Newtonian constant of gravitation, or the Cavendish gravitational constant, denoted by the capital letter . In Newton's law, it is the proportionality constant connecting the gravitational force between two bodies with the product of their masses and the inverse-square law, inverse square of their distance. In the Einstein field equations, it quantifies the relation between the geometry of spacetime and the energy–momentum tensor (also referred to as the stress–energy tensor). The measured value of the constant is known with some certainty to four significant digits. In SI units, its value is approximately The modern notation of Newton's law involving was introduced i ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
4-manifold
In mathematics, a 4-manifold is a 4-dimensional topological manifold. A smooth 4-manifold is a 4-manifold with a smooth structure. In dimension four, in marked contrast with lower dimensions, topological and smooth manifolds are quite different. There exist some topological 4-manifolds which admit no smooth structure, and even if there exists a smooth structure, it need not be unique (i.e. there are smooth 4-manifolds which are homeomorphic but not diffeomorphic). 4-manifolds are important in physics because in general relativity, spacetime is modeled as a pseudo-Riemannian 4-manifold. Topological 4-manifolds The homotopy type of a simply connected compact 4-manifold only depends on the intersection form on the middle dimensional homology. A famous theorem of implies that the homeomorphism type of the manifold only depends on this intersection form, and on a \Z/2\Z invariant called the Kirby–Siebenmann invariant, and moreover that every combination of unimodular form and ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Circle
A circle is a shape consisting of all point (geometry), points in a plane (mathematics), plane that are at a given distance from a given point, the Centre (geometry), centre. The distance between any point of the circle and the centre is called the radius. The length of a line segment connecting two points on the circle and passing through the centre is called the diameter. A circle bounds a region of the plane called a Disk (mathematics), disc. The circle has been known since before the beginning of recorded history. Natural circles are common, such as the full moon or a slice of round fruit. The circle is the basis for the wheel, which, with related inventions such as gears, makes much of modern machinery possible. In mathematics, the study of the circle has helped inspire the development of geometry, astronomy and calculus. Terminology * Annulus (mathematics), Annulus: a ring-shaped object, the region bounded by two concentric circles. * Circular arc, Arc: any Connected ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Compactification (physics)
In theoretical physics, compactification means changing a theory with respect to one of its Spacetime, space-time dimensions. Instead of having a theory with this dimension being infinite, one changes the theory so that this dimension has a finite length, and may also be Periodic function, periodic. Compactification plays an important part in Thermal quantum field theory, thermal field theory where one compactifies time, in string theory where one compactifies the String theory#Extra dimensions, extra dimensions of the theory, and in two- or one-dimensional Solid-state physics, solid state physics, where one considers a system which is limited in one of the three usual spatial dimensions. At the limit where the size of the compact dimension goes to zero, no fields depend on this extra dimension, and the theory is Dimensional reduction, dimensionally reduced. In string theory In string theory, compactification is a generalization of Kaluza–Klein theory.Dean Rickles (2014). '' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Dimensions Enroulées (cercles)
In physics and mathematics, the dimension of a mathematical space (or object) is informally defined as the minimum number of coordinates needed to specify any point within it. Thus, a line has a dimension of one (1D) because only one coordinate is needed to specify a point on itfor example, the point at 5 on a number line. A surface, such as the boundary of a cylinder or sphere, has a dimension of two (2D) because two coordinates are needed to specify a point on itfor example, both a latitude and longitude are required to locate a point on the surface of a sphere. A two-dimensional Euclidean space is a two-dimensional space on the plane. The inside of a cube, a cylinder or a sphere is three-dimensional (3D) because three coordinates are needed to locate a point within these spaces. In classical mechanics, space and time are different categories and refer to absolute space and time. That conception of the world is a four-dimensional space but not the one that was found nece ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Without Loss Of Generality
''Without loss of generality'' (often abbreviated to WOLOG, WLOG or w.l.o.g.; less commonly stated as ''without any loss of generality'' or ''with no loss of generality'') is a frequently used expression in mathematics. The term is used to indicate the assumption that what follows is chosen arbitrarily, narrowing the premise to a particular case, but does not affect the validity of the proof in general. The other cases are sufficiently similar to the one presented that proving them follows by essentially the same logic. As a result, once a proof is given for the particular case, it is trivial to adapt it to prove the conclusion in all other cases. In many scenarios, the use of "without loss of generality" is made possible by the presence of symmetry. For example, if some property ''P''(''x'',''y'') of real numbers is known to be symmetric in ''x'' and ''y'', namely that ''P''(''x'',''y'') is equivalent to ''P''(''y'',''x''), then in proving that ''P''(''x'',''y'') holds for every ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Prefactor
In mathematics, a coefficient is a multiplicative factor involved in some term of a polynomial, a series, or any other type of expression. It may be a number without units, in which case it is known as a numerical factor. It may also be a constant with units of measurement, in which it is known as a constant multiplier. In general, coefficients may be any expression (including variables such as , and ). When the combination of variables and constants is not necessarily involved in a product, it may be called a ''parameter''. For example, the polynomial 2x^2-x+3 has coefficients 2, −1, and 3, and the powers of the variable x in the polynomial ax^2+bx+c have coefficient parameters a, b, and c. A , also known as constant term or simply constant, is a quantity either implicitly attached to the zeroth power of a variable or not attached to other variables in an expression; for example, the constant coefficients of the expressions above are the number 3 and the parameter ' ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Singular Matrix
A singular matrix is a square matrix that is not invertible, unlike non-singular matrix which is invertible. Equivalently, an n-by-n matrix A is singular if and only if determinant, det(A)=0. In classical linear algebra, a matrix is called ''non-singular'' (or invertible) when it has an inverse; by definition, a matrix that fails this criterion is singular. In more algebraic terms, an n-by-n matrix A is singular exactly when its columns (and rows) are linearly dependent, so that the linear map x\rightarrow Ax is not one-to-one. In this case the kernel ( null space) of A is non-trivial (has dimension ≥1), and the homogeneous system Ax = 0 admits non-zero solutions. These characterizations follow from standard rank-nullity and invertibility theorems: for a square matrix A, det(A) \neq 0 if and only if rank(A)= n, and det(A) = 0 if and only if rank(A)3 then it is a singular matrix. * Numerical noise/ Round off: In numerical computations, a matrix may be nearly singular when its ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Graviphoton
In theoretical physics and quantum physics, a graviphoton or gravivector is a hypothetical particle which emerges as an excitation of the metric tensor (i.e. gravitational field) in spacetime dimensions higher than four, as described in Kaluza–Klein theory. However, its crucial physical properties are analogous to a (massive) photon: it induces a "vector force", sometimes dubbed a "fifth force". The electromagnetic potential A_\mu emerges from an extra component of the metric tensor g_, where the figure 5 labels an additional, fifth dimension. In gravity theories with extended supersymmetry ( extended supergravities), a graviphoton is normally a superpartner of the graviton that behaves like a photon, and is prone to couple with gravitational strength, as was appreciated in the late 1970s. Unlike the graviton, it may provide a ''repulsive'' (as well as an attractive) force, and thus, in some technical sense, a type of anti-gravity. Under special circumstances, in several natur ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
