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Kaluza–Klein Metric
In Kaluza–Klein theory, a unification of general relativity and electromagnetism, the five-dimensional Kaluza–Klein metric is the generalization of the four-dimensional metric tensor. It additionally includes a scalar field called graviscalar (or radion) and a vector field called graviphoton (or gravivector), which correspond to hypothetical particles. The Kaluza–Klein metric is named after Theodor Kaluza and Oskar Klein. Definition The ''Kaluza–Klein metric'' is given by: : \widetilde_ :=\begin g_+\phi^2A_\mu A_\nu & \phi^2A_\mu \\ \phi^2A_\nu & \phi^2 \end. Its inverse matrix is given by: : \widetilde^ =\begin g^ & -A^\mu \\ -A^\nu & g_A^\mu A^\nu+\phi^ \end. Defining an extended gravivector A_a=(A_\mu,1) shortens the definition to: : \widetilde_ =\operatorname(g_,0) +\phi^2A_aA_b, which also shows that the radion \phi cannot vanish as this would make the metric singular. Properties * A contraction directly shows the passing from four to five dimensions: ... [...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] |
Physical Cosmology
Physical cosmology is a branch of cosmology concerned with the study of cosmological models. A cosmological model, or simply cosmology, provides a description of the largest-scale structures and dynamics of the universe and allows study of fundamental questions about its Cosmogony, origin, structure, Chronology of the universe, evolution, and ultimate fate.For an overview, see Cosmology as a science originated with the Copernican principle, which implies that astronomical object, celestial bodies obey identical physical laws to those on Earth, and Newtonian mechanics, which first allowed those physical laws to be understood. Physical cosmology, as it is now understood, began in 1915 with the development of Albert Einstein's general relativity, general theory of relativity, followed by major observational discoveries in the 1920s: first, Edwin Hubble discovered that the universe contains a huge number of external Galaxy, galaxies beyond the Milky Way; then, work by Vesto Sliph ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Particle Physics
Particle physics or high-energy physics is the study of Elementary particle, fundamental particles and fundamental interaction, forces that constitute matter and radiation. The field also studies combinations of elementary particles up to the scale of protons and neutrons, while the study of combinations of protons and neutrons is called nuclear physics. The fundamental particles in the universe are classified in the Standard Model as fermions (matter particles) and bosons (force-carrying particles). There are three Generation (particle physics), generations of fermions, although ordinary matter is made only from the first fermion generation. The first generation consists of Up quark, up and down quarks which form protons and neutrons, and electrons and electron neutrinos. The three fundamental interactions known to be mediated by bosons are electromagnetism, the weak interaction, and the strong interaction. Quark, Quarks cannot exist on their own but form hadrons. Hadrons that ... [...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] |
Nuclear Physics B
Nuclear may refer to: Physics Relating to the nucleus of the atom: *Nuclear engineering *Nuclear physics *Nuclear power *Nuclear reactor *Nuclear weapon *Nuclear medicine *Radiation therapy *Nuclear warfare Mathematics *Nuclear space *Nuclear operator * Nuclear congruence *Nuclear C*-algebra Biology Relating to the nucleus of the cell: * Nuclear DNA Society *Nuclear family, a family consisting of a pair of adults and their children Music * "Nuclear" (band), chilean thrash metal band * "Nuclear" (Ryan Adams song), 2002 *"Nuclear", a song by Mike Oldfield from his ''Man on the Rocks'' album * ''Nu.Clear'' (EP) by South Korean girl group CLC Films * ''Nuclear'' (film), a 2022 documentary by Oliver Stone. See also *Nucleus (other) *Nucleolus *Nucleation *Nucleic acid Nucleic acids are large biomolecules that are crucial in all cells and viruses. They are composed of nucleotides, which are the monomer components: a pentose, 5-carbon sugar, a phosphate group and a ni ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Matrix Determinant Lemma
In mathematics, in particular linear algebra, the matrix determinant lemma computes the determinant of the sum of an invertible matrix A and the dyadic product, uvT, of a column vector u and a row vector vT. Statement Suppose A is an invertible square matrix and u, v are column vectors. Then the matrix determinant lemma states that :\det(\mathbf + \mathbf^\textsf) = (1 + \mathbf^\textsf\mathbf^\mathbf)\,\det(\mathbf)\,. Here, uvT is the outer product of two vectors u and v. The theorem can also be stated in terms of the adjugate matrix of A: :\det(\mathbf + \mathbf^\textsf) = \det(\mathbf) + \mathbf^\textsf\mathrm(\mathbf)\mathbf\,, in which case it applies whether or not the matrix A is invertible. Proof First the proof of the special case A = I follows from the equality: : \begin \mathbf & 0 \\ \mathbf^\textsf & 1 \end \begin \mathbf + \mathbf^\textsf & \mathbf \\ 0 & 1 \end \begin \mathbf & 0 \\ -\mathbf^\textsf & 1 \end = \begin \mathbf & \mathbf \\ 0 & 1 + \m ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Special Relativity
In physics, the special theory of relativity, or special relativity for short, is a scientific theory of the relationship between Spacetime, space and time. In Albert Einstein's 1905 paper, Annus Mirabilis papers#Special relativity, "On the Electrodynamics of Moving Bodies", the theory is presented as being based on just Postulates of special relativity, two postulates: # The laws of physics are Invariant (physics), invariant (identical) in all Inertial frame of reference, inertial frames of reference (that is, Frame of reference, frames of reference with no acceleration). This is known as the principle of relativity. # The speed of light in vacuum is the same for all observers, regardless of the motion of light source or observer. This is known as the principle of light constancy, or the principle of light speed invariance. The first postulate was first formulated by Galileo Galilei (see ''Galilean invariance''). Background Special relativity builds upon important physics ide ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Lorentz Factor
The Lorentz factor or Lorentz term (also known as the gamma factor) is a dimensionless quantity expressing how much the measurements of time, length, and other physical properties change for an object while it moves. The expression appears in several equations in special relativity, and it arises in derivations of the Lorentz transformations. The name originates from its earlier appearance in Lorentz ether theory, Lorentzian electrodynamics – named after the Netherlands, Dutch physicist Hendrik Lorentz. It is generally denoted (the Greek lowercase letter gamma). Sometimes (especially in discussion of superluminal motion) the factor is written as (Greek uppercase-gamma) rather than . Definition The Lorentz factor is defined as \gamma = \frac = \frac = \frac , where: * is the relative velocity between inertial reference frames, * is the speed of light in vacuum, * is the ratio of to , * is coordinate time, * is the proper time for an observer (measuring time intervals in ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Line Element
In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space. The length of the line element, which may be thought of as a differential arc length, is a function of the metric tensor and is denoted by ''ds''. Line elements are used in physics, especially in theories of gravitation (most notably general relativity) where spacetime is modelled as a curved Pseudo-Riemannian manifold with an appropriate metric tensor.Gravitation, J.A. Wheeler, C. Misner, K.S. Thorne, W.H. Freeman & Co, 1973, General formulation Definition of the line element and arc length The coordinate-independent definition of the square of the line element ''ds'' in an ''n''-dimensional Riemannian or Pseudo Riemannian manifold (in physics usually a Lorentzian manifold) is the "square of the length" of an infinitesimal displacement d\mathbfTensor Calculus, D.C. Kay, Schaum’s Outlines, McGraw Hill ( ... [...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] |
General Relativity
General relativity, also known as the general theory of relativity, and as Einstein's theory of gravity, is the differential geometry, geometric theory of gravitation published by Albert Einstein in 1915 and is the current description of gravitation in modern physics. General theory of relativity, relativity generalizes special relativity and refines Newton's law of universal gravitation, providing a unified description of gravity as a geometric property of space and time in physics, time, or four-dimensional spacetime. In particular, the ''curvature of spacetime'' is directly related to the energy and momentum of whatever is present, including matter and radiation. The relation is specified by the Einstein field equations, a system of second-order partial differential equations. Newton's law of universal gravitation, which describes gravity in classical mechanics, can be seen as a prediction of general relativity for the almost flat spacetime geometry around stationary mass ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
