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Newman–Penrose Formalism
The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.Ezra T Newman, Roger Penrose. ''Errata: An Approach to Gravitational Radiation by a Method of Spin Coefficients''. Journal of Mathematical Physics, 1963, 4(7): 998. is a set of notation developed by Ezra T. Newman and Roger Penrose for general relativity (GR). Their notation is an effort to treat general relativity in terms of spinor notation, which introduces complex forms of the usual variables used in GR. The NP formalism is itself a special case of the tetrad formalism, where the tensors of the theory are projected onto a complete vector basis at each point in spacetime. Usually this vector basis is chosen to reflect some symmetry of the spacetime, leading to simplified expressions for physical observables. In the case of the NP formalism, the vector basis chosen is a null tetrad: a set of four null vectors—two real, and a com ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ezra T
Ezra (; he, עֶזְרָא, '; fl. 480–440 BCE), also called Ezra the Scribe (, ') and Ezra the Priest in the Book of Ezra, was a Jewish scribe (''sofer'') and priest (''kohen''). In Greco-Latin Ezra is called Esdras ( grc-gre, Ἔσδρας). According to the Hebrew Bible he was a descendant of Sraya, the last High Priest to serve in the First Temple, and a close relative of Joshua, the first High Priest of the Second Temple. He returned from Babylonian exile and reintroduced the Torah in Jerusalem. According to 1 Esdras, a Greek translation of the Book of Ezra still in use in Eastern Orthodoxy, he was also a High Priest. Rabbinic tradition holds that he was an ordinary member of the priesthood. Several traditions have developed over his place of burial. One tradition says that he is buried in al-Uzayr near Basra (Iraq), while another tradition alleges that he is buried in Tadif near Aleppo, in northern Syria. His name may be an abbreviation of ', " Yah helps". In t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Orthogonal Tetrad
In mathematics, orthogonality is the generalization of the geometric notion of ''perpendicularity''. By extension, orthogonality is also used to refer to the separation of specific features of a system. The term also has specialized meanings in other fields including art and chemistry. Etymology The word comes from the Ancient Greek ('), meaning "upright", and ('), meaning "angle". The Ancient Greek (') and Classical Latin ' originally denoted a rectangle. Later, they came to mean a right triangle. In the 12th century, the post-classical Latin word ''orthogonalis'' came to mean a right angle or something related to a right angle. Mathematics Physics * In optics, polarization states are said to be orthogonal when they propagate independently of each other, as in vertical and horizontal linear polarization or right- and left-handed circular polarization. * In special relativity, a time axis determined by a rapidity of motion is hyperbolic-orthogonal to a space axis of s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Field Equations
In the general theory of relativity, the Einstein field equations (EFE; also known as Einstein's equations) relate the geometry of spacetime to the distribution of matter within it. The equations were published by Einstein in 1915 in the form of a tensor equation which related the local ' (expressed by the Einstein tensor) with the local energy, momentum and stress within that spacetime (expressed by the stress–energy tensor). Analogously to the way that electromagnetic fields are related to the distribution of charges and currents via Maxwell's equations, the EFE relate the spacetime geometry to the distribution of mass–energy, momentum and stress, that is, they determine the metric tensor of spacetime for a given arrangement of stress–energy–momentum in the spacetime. The relationship between the metric tensor and the Einstein tensor allows the EFE to be written as a set of nonlinear partial differential equations when used in this way. The solutions of the EFE are t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Sign Convention
In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary. "Arbitrary" here means that the same physical system can be correctly described using different choices for the signs, as long as one set of definitions is used consistently. The choices made may differ between authors. Disagreement about sign conventions is a frequent source of confusion, frustration, misunderstandings, and even outright errors in scientific work. In general, a sign convention is a special case of a choice of coordinate system for the case of one dimension. Sometimes, the term "sign convention" is used more broadly to include factors of '' i'' and 2 π, rather than just choices of sign. Relativity Metric signature In relativity, the metric signature can be either or . (Note that throughout this article we are displaying the signs of the eigenvalues of the metric in the order that presents t ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Maxwell Equations
Maxwell's equations, or Maxwell–Heaviside equations, are a set of coupled partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits. The equations provide a mathematical model for electric, optical, and radio technologies, such as power generation, electric motors, wireless communication, lenses, radar etc. They describe how electric and magnetic fields are generated by charges, currents, and changes of the fields.''Electric'' and ''magnetic'' fields, according to the theory of relativity, are the components of a single electromagnetic field. The equations are named after the physicist and mathematician James Clerk Maxwell, who, in 1861 and 1862, published an early form of the equations that included the Lorentz force law. Maxwell first used the equations to propose that light is an electromagnetic phenomenon. The modern form of the equations in their most common formulati ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electromagnetic Tensor
In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime. The field tensor was first used after the four-dimensional tensor formulation of special relativity was introduced by Hermann Minkowski. The tensor allows related physical laws to be written very concisely, and allows for the quantization of the electromagnetic field by Lagrangian formulation described below. Definition The electromagnetic tensor, conventionally labelled ''F'', is defined as the exterior derivative of the electromagnetic four-potential, ''A'', a differential 1-form: :F \ \stackrel\ \mathrmA. Therefore, ''F'' is a differential 2-form—that is, an antisymmetric rank-2 tensor field—on Minkowski space. In component form, :F_ = \partial_\mu A_\nu - \partial_\nu A_\mu. where \partial is the four-gradient and A is the four ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Riemannian Connection On A Surface
In mathematics, the Riemannian connection on a surface or Riemannian 2-manifold refers to several intrinsic geometric structures discovered by Tullio Levi-Civita, Élie Cartan and Hermann Weyl in the early part of the twentieth century: parallel transport, covariant derivative and connection form. These concepts were put in their current form with principal bundles only in the 1950s. The classical nineteenth century approach to the differential geometry of surfaces, due in large part to Carl Friedrich Gauss, has been reworked in this modern framework, which provides the natural setting for the classical theory of the moving frame as well as the Riemannian geometry of higher-dimensional Riemannian manifolds. This account is intended as an introduction to the theory of connections. Historical overview After the classical work of Gauss on the differential geometry of surfaces and the subsequent emergence of the concept of Riemannian manifold initiated by Bernhard Riemann in the mi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Electrovacuum Solution
In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass–energy present is the field energy of an electromagnetic field, which must satisfy the (curved-spacetime) ''source-free'' Maxwell equations appropriate to the given geometry. For this reason, electrovacuums are sometimes called (source-free) ''Einstein–Maxwell solutions''. Definition In general relativity, the geometric setting for physical phenomena is a Lorentzian manifold, which is interpreted as a curved spacetime, and which is specified by defining a metric tensor g_ (or by defining a frame field). The Riemann curvature tensor R_ of this manifold and associated quantities such as the Einstein tensor G^, are well-defined. In general relativity, they can be interpreted as geometric manifestations (curvature and forces) of the gravitational field. We also need to specify an electromagnetic field by defining an electr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Einstein Tensor
In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold. In general relativity, it occurs in the Einstein field equations for gravitation that describe spacetime curvature in a manner that is consistent with conservation of energy and momentum. Definition The Einstein tensor \mathbf is a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as \mathbf=\mathbf-\frac\mathbfR, where \mathbf is the Ricci tensor, \mathbf is the metric tensor and R is the scalar curvature, which is computed as the trace of the Ricci Tensor R_ by R = g^R_ = R_\mu^\mu. In component form, the previous equation reads as G_ = R_ - g_R . The Einstein tensor is symmetric G_ = G_ and, like the on shell stress–energy tensor, and has zero divergence: \nabla_\mu G^ = 0\,. Explicit form The Ricci tensor depends only on the metric ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Trace-free Ricci Tensor
In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determined by a choice of Riemannian or pseudo-Riemannian metric on a manifold. It can be considered, broadly, as a measure of the degree to which the geometry of a given metric tensor differs locally from that of ordinary Euclidean space or pseudo-Euclidean space. The Ricci tensor can be characterized by measurement of how a shape is deformed as one moves along geodesics in the space. In general relativity, which involves the pseudo-Riemannian setting, this is reflected by the presence of the Ricci tensor in the Raychaudhuri equation. Partly for this reason, the Einstein field equations propose that spacetime can be described by a pseudo-Riemannian metric, with a strikingly simple relationship between the Ricci tensor and the matter content of the universe. Like the metric tensor, the Ricci tensor assigns to each tangent space of the manifold a symmetric b ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Weyl Scalar
In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars \ which encode the ten independent components of the Weyl tensor of a four-dimensional spacetime. Definitions Given a complex null tetrad \ and with the convention \, the Weyl-NP scalars are defined byJeremy Bransom Griffiths, Jiri Podolsky. ''Exact Space-Times in Einstein's General Relativity''. Cambridge: Cambridge University Press, 2009. Chapter 2.Valeri P Frolov, Igor D Novikov. ''Black Hole Physics: Basic Concepts and New Developments''. Berlin: Springer, 1998. Appendix E.Abhay Ashtekar, Stephen Fairhurst, Badri Krishnan. ''Isolated horizons: Hamiltonian evolution and the first law''. Physical Review D, 2000, 62(10): 104025. Appendix Bgr-qc/0005083/ref> :\Psi_0 := C_ l^\alpha m^\beta l^\gamma m^\delta\ , :\Psi_1 := C_ l^\alpha n^\beta l^\gamma m^\delta\ , :\Psi_2 := C_ l^\alpha m^\beta \bar^\gamma n^\delta\ , :\Psi_3 := C_ l^\alpha n^\beta \bar^\gamma n^\d ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Commutator
In mathematics, the commutator gives an indication of the extent to which a certain binary operation fails to be commutative. There are different definitions used in group theory and ring theory. Group theory The commutator of two elements, and , of a group , is the element : . This element is equal to the group's identity if and only if and commute (from the definition , being equal to the identity if and only if ). The set of all commutators of a group is not in general closed under the group operation, but the subgroup of ''G'' generated by all commutators is closed and is called the ''derived group'' or the ''commutator subgroup'' of ''G''. Commutators are used to define nilpotent and solvable groups and the largest abelian quotient group. The definition of the commutator above is used throughout this article, but many other group theorists define the commutator as :. Identities (group theory) Commutator identities are an important tool in group theory. The expr ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
