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Spinning motion in theoretical physics
theoretical physics, the spin tensor is a quantity used to describe the rotational motion of particles in spacetime. The spin tensor has application in general
Spin_tensor
Mathematical model for describing material deformation under stress
deformation tensors. In 1839, George Green introduced a deformation tensor known as the right Cauchy–Green deformation tensor or Green's deformation tensor (the
Finite_strain_theory
Tensor describing energy momentum density in spacetime
stress-energy tensor The stress–energy tensor, sometimes called the stress–energy–momentum tensor or the energy–momentum tensor, is a tensor field quantity
Stress–energy_tensor
Classical theory of gravitation
be related to the spin tensor. This canonical energy–momentum tensor is related to the more familiar symmetric energy–momentum tensor by the Belinfante–Rosenfeld
Einstein–Cartan_theory
Non-tensorial representation of the spin group
decompositions are possible on the tensor product of one spin representation with another. These decompositions express the tensor product in terms of the alternating
Spinor
Algebraic object with geometric applications
(electromagnetic tensor, Maxwell tensor, permittivity, magnetic susceptibility, etc.), and general relativity (stress–energy tensor, curvature tensor, etc.). In
Tensor
Antisymmetric permutation object acting on tensors
independent of any metric tensor and coordinate system. Also, the specific term "symbol" emphasizes that it is not a tensor because of how it transforms
Levi-Civita_symbol
of tensor theory. For expositions of tensor theory from different points of view, see: Tensor Tensor (intrinsic definition) Application of tensor theory
Glossary_of_tensor_theory
Object in differential geometry
differential geometry, the torsion tensor is a tensor that is associated to any affine connection. The torsion tensor is a bilinear map of two input vectors
Torsion_tensor
Mathematical object that describes the electromagnetic field in spacetime
electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a tensor that describes
Electromagnetic_tensor
Tensor index notation for tensor-based calculations
notation and manipulation for tensors and tensor fields on a differentiable manifold, with or without a metric tensor or connection. It is also the modern
Ricci_calculus
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
inertia tensor of a body calculated at its center of mass, and R {\displaystyle \mathbf {R} } be the displacement vector of the body. The inertia tensor of
Moment_of_inertia
Graphical notation for multilinear algebra calculations
essentially the composition of functions. In the language of tensor algebra, a particular tensor is associated with a particular shape with many lines projecting
Penrose_graphical_notation
Tensor field in Riemannian geometry
mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the
Riemann_curvature_tensor
Generalization of tensor fields
differential geometry, a tensor density or relative tensor is a generalization of the tensor field concept. A tensor density transforms as a tensor field when passing
Tensor_density
Tensor operator generalizes the notion of operators which are scalars and vectors
graphics, a tensor operator generalizes the notion of operators which are scalars and vectors. A special class of these are spherical tensor operators which
Tensor_operator
Structure defining distance on a manifold
metric field on M consists of a metric tensor at each point p of M that varies smoothly with p. A metric tensor g is positive-definite if g ( v , v ) >
Metric_tensor
Conserved physical quantity; rotational analogue of linear momentum
as an anti-symmetric second order tensor, with components ωij. The relation between the two anti-symmetric tensors is given by the moment of inertia which
Angular_momentum
Branch of mathematics
Pseudovector Spinor Tensor Tensor algebra, Free algebra Tensor contraction Symmetric algebra, Symmetric power Symmetric tensor Mixed tensor Pandey, Divyanshu;
Multilinear_algebra
Shorthand notation for tensor operations
the multiplication. Given a tensor, one can raise an index or lower an index by contracting the tensor with the metric tensor, g μ ν {\displaystyle g_{\mu
Einstein_notation
Abbreviation in the fields of special and general relativity
relativity, a four-tensor is an abbreviation for a tensor in a four-dimensional spacetime. General four-tensors are usually written in tensor index notation
Four-tensor
Geometric algebra approach to gravity
Einstein field equations are derivable from a variational principle. A spin tensor can also be supported in a manner similar to Einstein–Cartan–Sciama–Kibble
Gauge_theory_gravity
Mathematical wave functions
Tensor networks or tensor network states are a class of variational wave functions used in the study of many-body quantum systems and fluids. Tensor networks
Tensor_network
Assignment of a tensor continuously varying across a region of space
In mathematics and physics, a tensor field is a function assigning a tensor to each point of a region of a mathematical space (typically a Euclidean space
Tensor_field
Tensor in differential geometry
converge. Formally, it is a symmetric rank-two tensor obtained by taking a trace of the Riemann curvature tensor of a Riemannian or pseudo-Riemannian metric
Ricci_curvature
Tensor equal to the negative of any of its transpositions
tensor is antisymmetric with respect to its first three indices. If a tensor changes sign under exchange of each pair of its indices, then the tensor
Antisymmetric_tensor
Intrinsic quantum property of particles
of SU(2) Spin angular momentum of light Spin engineering Spin-flip Spin isomers of hydrogen Spin–orbit interaction Spin tensor Spintronics Spin wave Yrast
Spin_(physics)
Tensor that describes the 4D geometry of spacetime
manifold M {\displaystyle M} and the metric tensor is given as a covariant, second-degree, symmetric tensor on M {\displaystyle M} , conventionally denoted
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Hypothetical elementary particle that mediates gravity
must be a spin-2 boson because the source of gravitation is the stress–energy tensor, a second-order tensor (compared with electromagnetism's spin-1 photon
Graviton
Algebraic operation on coordinate vectors
(single-) dot product between a tensor of order n {\displaystyle n} and a tensor of order m {\displaystyle m} is a tensor of order n + m − 2 {\displaystyle
Dot_product
Mathematical operation on vector spaces
two vectors is sometimes called an elementary tensor or a decomposable tensor. The elementary tensors span V ⊗ W {\displaystyle V\otimes W} in the sense
Tensor_product
Tensor having both covariant and contravariant indices
In tensor analysis, a mixed tensor is a tensor which is neither strictly covariant nor strictly contravariant; at least one of the indices of a mixed
Mixed_tensor
Concept in mathematics
In mathematics, the tensor bundle of a manifold is the direct sum of all tensor products of the tangent bundle and the cotangent bundle of that manifold
Tensor_bundle
Tensor invariant under permutations of vectors it acts on
In mathematics, a symmetric tensor is an unmixed tensor that is invariant under a permutation of its vector arguments: T ( v 1 , v 2 , … , v r ) = T (
Symmetric_tensor
Universal construction in multilinear algebra
the tensor algebra of a vector space V, denoted T(V) or T•(V), is the algebra of tensors on V (of any order) with multiplication being the tensor product
Tensor_algebra
Exterior algebraic map taking tensors from p forms to n-p forms
space L ( V , V ) {\displaystyle L(V,V)} is naturally isomorphic to the tensor product V ∗ ⊗ V ≅ V ⊗ V {\displaystyle V^{*}\!\!\otimes V\cong V\otimes
Hodge_star_operator
Topics referred to by the same term
{\displaystyle {\mathfrak {spin}}(n)} Spin tensor, a tensor quantity for describing spinning motion in special relativity and general relativity Spin (aerodynamics)
Spin
Tensor used in general relativity
differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature
Einstein_tensor
Operation in mathematics
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. This example
Tensor_contraction
Dirac equation for self-interacting fermions
equations, the torsion tensor is a homogeneous, linear function of the spin tensor. The minimal coupling between torsion and Dirac spinors thus generates an
Nonlinear_Dirac_equation
Ways of writing certain laws of physics
t^{2}}-\nabla ^{2}.} The signs in the following tensor analysis depend on the convention used for the metric tensor. The convention used here is (+ − − −), corresponding
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Measure of the curvature of a pseudo-Riemannian manifold
Riemann curvature tensor, the Weyl tensor expresses the tidal force that a body feels when moving along a geodesic. The Weyl tensor differs from the Riemann
Weyl_tensor
Geometric structure
a spin structure on an n {\displaystyle n} -dimensional orientable Riemannian manifold ( M , g ) , {\displaystyle (M,g),\,} one defines the spinor bundle
Spinor_bundle
Coordinate-free definition of a tensor
mathematics, the modern component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear
Tensor_(intrinsic_definition)
Operation that pairs a left and a right R-module into an abelian group
universal property of the tensor product of vector spaces extends to more general situations in abstract algebra. The tensor product of an algebra and
Tensor_product_of_modules
Specification of a derivative along a tangent vector of a manifold
fields) and to arbitrary tensor fields, in a unique way that ensures compatibility with the tensor product and trace operations (tensor contraction). Given
Covariant_derivative
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
thought of as a tensor, and is written δ j i {\displaystyle \delta _{j}^{i}} . Sometimes the Kronecker delta is called the substitution tensor. When juxtaposition
Kronecker_delta
Topological space that locally resembles Euclidean space
curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein
Manifold
Decomposition in multilinear algebra
multilinear algebra, the tensor rank decomposition or rank-R decomposition is the decomposition of a tensor as a sum of R rank-1 tensors, where R is minimal
Tensor_rank_decomposition
Method for specifying point positions
curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein
Coordinate_system
Differential form of degree one or section of a cotangent bundle
one coordinate system to another. Thus a one-form is an order 1 covariant tensor field. The most basic non-trivial differential one-form is the "change in
One-form
Isomorphism between the tangent and cotangent bundles of a manifold
index of an ( r , s ) {\displaystyle (r,s)} tensor gives a ( r − 1 , s + 1 ) {\displaystyle (r-1,s+1)} tensor, while raising an index gives a ( r + 1 ,
Musical_isomorphism
Vector behavior under coordinate changes
consequently a vector is called a contravariant tensor. A vector, which is an example of a contravariant tensor, has components that transform inversely to
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
field. Tensors also have extensive applications in physics: Electromagnetic tensor (or Faraday's tensor) in electromagnetism Finite deformation tensors for
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Second-rank tensor in quantum chromodynamics
In theoretical particle physics, the gluon field strength tensor is a second-order tensor field characterizing the gluon interaction between quarks. The
Gluon_field_strength_tensor
Type of derivative in differential geometry
differentiable manifold. Functions, tensor fields and forms can be differentiated with respect to a vector field. If T is a tensor field and X is a vector field
Lie_derivative
Array of numbers
multiplication can be defined with entries objects of a category equipped with a "tensor product" similar to multiplication in a ring, having coproducts similar
Matrix_(mathematics)
Property of a mathematical space
curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein
Dimension
Continuous surjection satisfying a local triviality condition
curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein
Fiber_bundle
Representation of mechanical stress at every point within a deformed 3D object
Cauchy stress tensor (symbol σ {\displaystyle {\boldsymbol {\sigma }}} , named after Augustin-Louis Cauchy), also called true stress tensor or simply stress
Cauchy_stress_tensor
Covariant derivative of the metric tensor
In mathematics, the nonmetricity tensor in differential geometry is the covariant derivative of the metric tensor. It can be interpreted as the failure
Nonmetricity_tensor
Theory of gravitation as curved spacetime
stress–energy tensor, which includes both energy and momentum densities as well as stress: pressure and shear. Using the equivalence principle, this tensor is readily
General_relativity
Branch of mathematics
where N J {\displaystyle N_{J}} is a tensor of type (2, 1) related to J {\displaystyle J} , called the Nijenhuis tensor (or sometimes the torsion). An almost
Differential_geometry
Construct in differenital geometry
the field strength tensor, a classical one using R as the curvature tensor, and the classical notation for the Riemann curvature tensor, most of which can
Metric_connection
Matrix operation which flips a matrix over its diagonal
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Transpose
Angular momentum in special and general relativity
momentum tensor is expressed in terms of the stress–energy tensor of the rotating object. In special relativity alone, in the rest frame of a spinning object
Relativistic_angular_momentum
Straight path on a curved surface or a Riemannian manifold
and real trees. In a Riemannian manifold M {\displaystyle M} with metric tensor g {\displaystyle g} , the length L {\displaystyle L} of a continuously differentiable
Geodesic
Set of vectors used to define coordinates
of redirect targets Spherical basis – Basis used to express spherical tensors Brown, William A. (1991). Matrices and vector spaces. New York: M. Dekker
Basis_(linear_algebra)
Second order tensor in vector algebra
mathematics, specifically multilinear algebra, a dyadic or dyadic tensor is a second-order tensor, written in a notation that fits in with vector algebra. There
Dyadics
Theory of interwoven space and time by Albert Einstein
coordinates are divided by c or factors of c±2 are included in the metric tensor. These numerous conventions can be superseded by using natural units where
Special_relativity
Array of numbers describing a metric connection
corresponding gravitational potential being the metric tensor. When the coordinate system and the metric tensor share some symmetry, many of the Γijk are zero
Christoffel_symbols
Electromagnetism in general relativity
inverse of the metric tensor g α β {\displaystyle g_{\alpha \beta }} , and g {\displaystyle g} is the determinant of the metric tensor. Notice that A α {\displaystyle
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Branch of physics which studies the behavior of materials modeled as continuous media
stress tensor, and ρ 0 {\displaystyle \rho _{0}} is the mass density in the reference configuration. The first Piola-Kirchhoff stress tensor is related
Continuum_mechanics
Operation on differential forms
curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein
Exterior_derivative
Quantum state of a system
Diradical Angular momentum Pauli matrices Spin multiplicity Spin quantum number Spin-1/2 Spin tensor Spinor Borden, Weston Thatcher; Hoffmann, Roald;
Triplet_state
Mathematical notation
curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein
Multi-index_notation
Mathematical notation for tensors and spinors
between tensor factors of type V {\displaystyle V} and those of type V ∗ {\displaystyle V^{*}} . A general homogeneous tensor is an element of a tensor product
Abstract_index_notation
Type of physical quantity
spacetime Tensor – Algebraic object with geometric applications Tensor density – Generalization of tensor fields Tensor field – Assignment of a tensor continuously
Pseudotensor
Affine connection on the tangent bundle of a manifold
components of a contravariant vector. This discovery was the real beginning of tensor analysis. In 1906, L. E. J. Brouwer was the first mathematician to consider
Levi-Civita_connection
System of moving vectors in differential geometry
curvature tensor Torsion tensor Weyl tensor Physics Moment of inertia Angular momentum tensor Spin tensor Cauchy stress tensor stress–energy tensor Einstein
Parallel_transport
Mapping from p forms to p-1 forms
generalized dot productPages displaying short descriptions of redirect targets Tensor contraction – Operation in mathematics Tu, Sec 20.5. There is another formula
Interior_product
Expression that may be integrated over a region
covariant tensor field of rank k {\displaystyle k} . The differential forms on M {\displaystyle M} are in one-to-one correspondence with such tensor fields
Differential_form
Object in differential geometry
The contorsion tensor (or contortion tensor) in differential geometry is the difference between a connection with and without torsion in it. It commonly
Contorsion_tensor
Function that is invariant under all permutations of its variables
functions, tensors that act as functions of several vectors can be symmetric, and in fact the space of symmetric k {\displaystyle k} -tensors on a vector
Symmetric_function
Topic in semi-Riemannian geometry
electrogravitic tensor E [ X → ] a b = R a m b n X m X n {\displaystyle E[{\vec {X}}]_{ab}=R_{ambn}\,X^{m}\,X^{n}} Also known as the tidal tensor. It can be
Bel_decomposition
Mathematical function, in linear algebra
linear maps are said to be 1-co- 1-contra-variant objects, or type (1, 1) tensors. A linear transformation between topological vector spaces, for example
Linear_map
Algebra associated to any vector space
alternating tensor subspace. On the other hand, the image A ( T ( V ) ) {\displaystyle {\mathcal {A}}(\mathrm {T} (V))} is always the alternating tensor graded
Exterior_algebra
that is 0 whenever arguments are linearly dependent Antisymmetric tensor – Tensor equal to the negative of any of its transpositions Hazewinkel (1990)
Symmetrization
Concept in physics
In continuum mechanics, the strain-rate tensor or rate-of-strain tensor is a physical quantity that describes the rate of change of the strain (i.e.,
Strain-rate_tensor
Mathematical Concept
notation is as follows: Write down the second order tensor in matrix form (in the example, the stress tensor) Strike out the diagonal Continue on the third
Voigt_notation
Differential form
absolute value of the determinant of the matrix representation of the metric tensor on the manifold. The volume form is denoted variously by ω = v o l n = ε
Volume_form
energy–momentum tensor and the Petrov classification of the Weyl tensor. There are various methods of classifying these tensors, some of which use tensor invariants
Mathematics of general relativity
Mathematics_of_general_relativity
Math/physics concept
arbitrary E-valued forms, thus regarding it as a differential operator on the tensor product of E with the full exterior algebra of differential forms. Given
Connection_form
Vector in relativity
definition, see tensor. All four-vectors transform in the same way, and this can be generalized to four-dimensional relativistic tensors; see special relativity
Four-vector
Construct allowing differentiation of tangent vector fields of manifolds
known as tensor calculus) by Gregorio Ricci-Curbastro and his student Tullio Levi-Civita between 1880 and the turn of the 20th century. Tensor calculus
Affine_connection
Mathematics of smooth surfaces
due to Élie Cartan. In the language of tensor calculus, making use of natural metrics and connections on tensor bundles, the Gauss equation can be written
Differential geometry of surfaces
Differential_geometry_of_surfaces
Study of curves from a differential point of view
notation Tensor definitions Tensor (intrinsic definition) Tensor field Tensor density Tensors in curvilinear coordinates Mixed tensor Antisymmetric tensor Symmetric
Differentiable_curve
Notation used for Weyl spinors
Van der Waerden notation refers to the usage of two-component spinors (Weyl spinors) in four spacetime dimensions. This is standard in twistor theory
Van_der_Waerden_notation
Basis used to express spherical tensors
a basis in a 3-dimensional space is a valid definition for a spherical tensor, it only covers the case for when the rank k {\displaystyle k} is 1. For
Spherical_basis
Element of an exterior algebra
tangent vector space of a smooth manifold; that is, it is an antisymmetric tensor obtained by taking linear combinations of the exterior product of k tangent
Multivector
SPIN TENSOR
SPIN TENSOR
Boy/Male
Egyptian
Light skin.
Biblical
a bush, enmity
Boy/Male
British, Danish, English, Norwegian
Skin; Parchment
Female/Male/Unisex
Korean
Korean name SHIN means "faith, trust." Compare with another form of Shin.
Surname or Lastname
English
English : from Middle English spink ‘chaffinch’ (probably of imitative origin), hence a nickname bestowed on account of some fancied resemblance to the bird.
Girl/Female
Muslim
Glowing skin
Male
Babylonian
, I trust in Sin!
Male
Japanese
(1-晋, 2-信, 3-紳, 4-心, 5-慎, 6-新, 7-進, 8-真) Japanese name SHIN means 1) "advancing," 2) "belief," 3) "gentleman," 4) "heart," 5) "humble," 6) "new," 7) "progressive," and 8) "true." Compare with another form of Shin.
Girl/Female
Australian, Indian, Punjabi, Sikh
Quite and Gentle
Boy/Male
Indian
Life Span
Girl/Female
Biblical
Rare, precious.
Girl/Female
Native American
Spins.
Girl/Female
Australian, Biblical, Kurdish
Bush
Male
French
Old French name, possibly derived from the word pepin/pipin, PÉPIN means "seed of a fruit."
Boy/Male
Indian, Sanskrit
Skin of a Goat; Tiger Skin
Surname or Lastname
English and Irish
English and Irish : (of Norman origin): habitational name from Épaignes in Eure, recorded in the Latin form Hispania in the 12th century. It seems to have been so called because it was established by colonists from Spain during the Roman Empire.English and Irish : habitational name from Espinay in Ille-et-Vilaine, Brittany, so called from a collective of Old French espine ‘thorn bush’.English and Irish : ethnic name for a Spaniard or, in the case of the Irish name, for someone returning from Spain (from Gaelic Spainneach ‘Spanish’); many Irish took refuge in Spain during the 17th century wars.
Boy/Male
Australian, Spanish
Innocent
Biblical
rare; precious
Girl/Female
Christian, Hindu, Indian
Dark Skin
Girl/Female
Indian
Glowing skin
SPIN TENSOR
SPIN TENSOR
Girl/Female
Australian, French, Greek, Italian, Latin
Devoted to God; Similar to Elissa; From the Blessed Isles
Surname or Lastname
Irish
Irish : reduced Anglicized form of Gaelic Ó Seighin ‘descendant of Seighin’, possibly a derivative of séaghainn ‘accomplished’.English : nickname from Middle English schene ‘beautiful’ (Old English scīene).Americanized form of Ashkenazic Jewish or German Schein.
Boy/Male
Australian, Irish
Black
Biblical
brother of goodness
Surname or Lastname
English and Irish
English and Irish : variant spelling of Cullen.
Boy/Male
Arabic
Lucky
Boy/Male
Polynesian
Familiar.
Boy/Male
Gaelic
Son of the man from the ford by the oak trees.
Boy/Male
Russian Slavic
Eagle.
Girl/Female
Arabic, Muslim, Parsi
Rose-coloured
SPIN TENSOR
SPIN TENSOR
SPIN TENSOR
SPIN TENSOR
SPIN TENSOR
v. t.
To cause to turn round rapidly; to whirl; to twirl; as, to spin a top.
v. i.
To move swifty; as, to spin along the road in a carriage, on a bicycle, etc.
a.
Like a spine in shape; slender.
v. t.
To draw out tediously; to form by a slow process, or by degrees; to extend to a great length; -- with out; as, to spin out large volumes on a subject.
n.
To thrust a spit through; to fix upon a spit; hence, to thrust through or impale; as, to spit a loin of veal.
n.
A sin offering; a sacrifice for sin.
imp. & p. p.
of Spit
v. t.
To draw out, and twist into threads, either by the hand or machinery; as, to spin wool, cotton, or flax; to spin goat's hair; to produce by drawing out and twisting a fibrous material.
v. t.
To cover with skin, or as with skin; hence, to cover superficially.
v. i.
To attend to a spit; to use a spit.
v. i.
To practice spinning; to work at drawing and twisting threads; to make yarn or thread from fiber; as, the woman knows how to spin; a machine or jenny spins with great exactness.
imp. & p. p.
of Spin
n.
The act of spinning; as, the spin of a top; a spin a bicycle.
imp.
of Spin
v. t.
To strip off the skin or hide of; to flay; to peel; as, to skin an animal.
v. t.
To measure by the span of the hand with the fingers extended, or with the fingers encompassing the object; as, to span a space or distance; to span a cylinder.
a.
Full of spines; thorny; as, a spiny tree.
v. t.
To protract; to spend by delays; as, to spin out the day in idleness.