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Magnetogravitic Tensor
In semi-Riemannian geometry, the Bel decomposition, taken with respect to a specific timelike congruence, is a way of breaking up the Riemann tensor of a pseudo-Riemannian manifold into lower order tensors with properties similar to the electric field and magnetic field. Such a decomposition was partially described by Alphonse Matte in 1953 and by Lluis Bel in 1958. This decomposition is particularly important in general relativity. This is the case of four-dimensional Lorentzian manifolds, for which there are only three pieces with simple properties and individual physical interpretations. Decomposition of the Riemann tensor In four dimensions the Bel decomposition of the Riemann tensor, with respect to a timelike unit vector field \vec, not necessarily geodesic or hypersurface orthogonal, consists of three pieces: # the ''electrogravitic tensor'' E[\vec]_ = R_ \, X^m \, X^n #* Also known as the tidal tensor. It can be physically interpreted as giving the tidal stresses on smal ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Semi-Riemannian Geometry
In differential geometry, a pseudo-Riemannian manifold, also called a semi-Riemannian manifold, is a differentiable manifold with a metric tensor that is everywhere non-degenerate bilinear form, nondegenerate. This is a generalization of a Riemannian manifold in which the requirement of Positive-definite bilinear form, positive-definiteness is relaxed. Every tangent space of a pseudo-Riemannian manifold is a pseudo-Euclidean vector space. A special case used in general relativity is a four-dimensional Lorentzian manifold for modeling spacetime, where tangent vectors can be classified as Causal structure, timelike, null, and spacelike. Introduction Manifolds In differential geometry, a differentiable manifold is a space which is locally similar to a Euclidean space. In an ''n''-dimensional Euclidean space any point can be specified by ''n'' real numbers. These are called the coordinates of the point. An ''n''-dimensional differentiable manifold is a generalisation of ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Vacuum Solution
In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically. According to the Einstein field equation, this means that the stress–energy tensor also vanishes identically, so that no matter or non-gravitational fields are present. These are distinct from the electrovacuum solutions, which take into account the electromagnetic field in addition to the gravitational field. Vacuum solutions are also distinct from the lambdavacuum solutions, where the only term in the stress–energy tensor is the cosmological constant term (and thus, the lambdavacuums can be taken as cosmological models). More generally, a vacuum region in a Lorentzian manifold is a region in which the Einstein tensor vanishes. Vacuum solutions are a special case of the more general exact solutions in general relativity. Equivalent conditions It is a mathematical fact that the Einstein tensor vanishes if and only if the Ricci tensor vanishes. This follows from th ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Curvature Invariant
In Riemannian geometry and pseudo-Riemannian geometry, curvature invariants are scalar quantities constructed from tensors that represent curvature. These tensors are usually the Riemann tensor, the Weyl tensor, the Ricci tensor and tensors formed from these by the operations of taking dual contractions and covariant differentiations. Types of curvature invariants The invariants most often considered are ''polynomial invariants''. These are polynomials constructed from contractions such as traces. Second degree examples are called ''quadratic invariants'', and so forth. Invariants constructed using covariant derivatives up to order n are called n-th order ''differential invariants''. The Riemann tensor is a multilinear operator of fourth rank acting on tangent vectors. However, it can also be considered a linear operator acting on bivectors, and as such it has a characteristic polynomial, whose coefficients and roots (eigenvalues) are polynomial scalar invariants. Physi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Tidal Tensor
{{disambiguation ...
Tidal is the adjectival form of tide. Tidal may also refer to: * ''Tidal'' (album), a 1996 album by Fiona Apple * Tidal (king), a king involved in the Battle of the Vale of Siddim * TidalCycles, a live coding environment for music * Tidal (service), a music streaming service * Tidal, Manitoba, Canada ** Tidal station, Tidal, Manitoba See also * Tidal flow (traffic), the flow of traffic thought of as an analogy with the flow of tides * Tidal force, a secondary effect of the force of gravity and is responsible for the tides * Tide (other) A tide is the rise and fall of a sea level caused by the Moon's gravity and other factors. Tide may also refer to: Media * ''The Tide'' (Nigeria), a newspaper * ''Tide'' (TV series), 2019 Irish/Welsh/Scottish documentary series * WTKN, a radio s ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Ricci Decomposition
In the mathematical fields of Riemannian and pseudo-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a Riemannian or pseudo-Riemannian manifold into pieces with special algebraic properties. This decomposition is of fundamental importance in Riemannian and pseudo-Riemannian geometry. Definition of the decomposition Let (''M'',''g'') be a Riemannian or pseudo-Riemannian ''n''-manifold. Consider its Riemann curvature, as a (0,4)-tensor field. This article will follow the sign convention :R_=g_\Big(\partial_i\Gamma_^p-\partial_j\Gamma_^p+\Gamma_^p\Gamma_^q-\Gamma_^p\Gamma_^q\Big); written multilinearly, this is the convention :\operatorname(W,X,Y,Z)=g\Big(\nabla_W\nabla_XY-\nabla_X\nabla_WY-\nabla_Y,Z\Big). With this convention, the Ricci tensor is a (0,2)-tensor field defined by ''Rjk''=''gilRijkl'' and the scalar curvature is defined by ''R''=''gjkRjk.'' Define the traceless Ricci tensor :Z_=R_-\fracRg_, and then define three (0, ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Bel–Robinson Tensor
In general relativity and differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by: :T_=C_C_ ^ _ ^ + \frac\epsilon_^ \epsilon_^_ C_ C_^_^ Alternatively, :T_ = C_C_ ^ _ ^ - \frac g_ C_ C^_^ where C_ is the Weyl tensor. It was introduced by Lluís Bel in 1959. The Bel–Robinson tensor is constructed from the Weyl tensor in a manner analogous to the way the electromagnetic stress–energy tensor is built from the electromagnetic tensor. Like the electromagnetic stress–energy tensor, the Bel–Robinson tensor is totally symmetric and traceless: :\begin T_ &= T_ \\ T^_ &= 0 \end In general relativity, there is no unique definition of the local energy of the gravitational field. The Bel–Robinson tensor is a possible definition for local energy, since it can be shown that whenever the Ricci tensor In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, is a geometric object which is determi ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Traceless
In linear algebra, the trace of a square matrix , denoted , is defined to be the sum of elements on the main diagonal (from the upper left to the lower right) of . The trace is only defined for a square matrix (). It can be proved that the trace of a matrix is the sum of its (complex) eigenvalues (counted with multiplicities). It can also be proved that for any two matrices and . This implies that similar matrices have the same trace. As a consequence one can define the trace of a linear operator mapping a finite-dimensional vector space into itself, since all matrices describing such an operator with respect to a basis are similar. The trace is related to the derivative of the determinant (see Jacobi's formula). Definition The trace of an square matrix is defined as \operatorname(\mathbf) = \sum_^n a_ = a_ + a_ + \dots + a_ where denotes the entry on the th row and th column of . The entries of can be real numbers or (more generally) complex numbers. The trace is not def ... [...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] |
Test Particle
In physical theories, a test particle, or test charge, is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system. The concept of a test particle often simplifies problems, and can provide a good approximation for physical phenomena. In addition to its uses in the simplification of the dynamics of a system in particular limits, it is also used as a diagnostic in computer simulations of physical processes. Classical gravity The easiest case for the application of a test particle arises in Newtonian gravity. The general expression for the gravitational force between any two point masses m_1 and m_2 is: :F = -G \frac, where \mathbf_1 and \mathbf_2 represent the position of each particle in space. In the general solution for this equation, both masses rotate around their center of mass R, in this ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |
Timelike Congruence
In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime. Often this manifold will be taken to be an exact or approximate solution to the Einstein field equation. Types of congruences Congruences generated by nowhere vanishing timelike, null, or spacelike vector fields are called ''timelike'', ''null'', or ''spacelike'' respectively. A congruence is called a '' geodesic congruence'' if it admits a tangent vector field \vec with vanishing covariant derivative, \nabla_ \vec = 0. Relation with vector fields The integral curves of the vector field are a family of ''non-intersecting'' parameterized curves which fill up the spacetime. The congruence consists of the curves themselves, without reference to a particular parameterization. Many distinct vector fields can give rise to the ''same'' cong ... [...More Info...] [...Related Items...] OR: [Wikipedia] [Google] [Baidu] |