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Mathematical notation for tensors and spinors
Abstract index notation (also referred to as slot-naming index notation) is a mathematical notation for tensors and spinors that uses indices to indicate
Abstract_index_notation
Shorthand notation for tensor operations
tensor index notation and the closely related but distinct basis-independent abstract index notation. An index that is summed over is a summation index, in
Einstein_notation
Algebraic object with geometric applications
of basis elements, and requires no symbols for the indices. The abstract index notation is a way to write tensors such that the indices are no longer thought
Tensor
Graphical notation for multilinear algebra calculations
Wikimedia Commons has media related to Penrose graphical notation. Abstract index notation Angular momentum diagrams (quantum mechanics) Braided monoidal
Penrose_graphical_notation
Mathematical notation
Multi-index notation is a mathematical notation that simplifies formulas used in multivariable calculus, partial differential equations and the theory
Multi-index_notation
Tensor index notation for tensor-based calculations
In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields on a differentiable manifold, with
Ricci_calculus
Mathematical Concept
associated names for this idea: Mandel notation, Mandel–Voigt notation and Nye notation are others found. Kelvin notation is a revival by Helbig of old ideas
Voigt_notation
Tensor that describes the 4D geometry of spacetime
{\displaystyle g_{\mu \nu }} themselves as the metric (see, however, abstract index notation). With the quantities d x μ {\displaystyle dx^{\mu }} being regarded
Metric tensor (general relativity)
Metric_tensor_(general_relativity)
Approach to general relativity
basis to reflect important physical aspects of the spacetime. The abstract index notation denotes tensors as if they were represented by their coefficients
Tetrad_formalism
Generalised alphabetical order
order topology on the unit square Lexicographic ordering in tensor abstract index notation Lexicographically minimal string rotation Leximin order Long line
Lexicographic_order
Topological space that locally resembles Euclidean space
of a given manifold is unique. Though useful for definitions, it is an abstract object and not used directly (e.g. in calculations). A manifold can be
Manifold
Note: General relativity articles using tensors will use the abstract index notation. The principle of general covariance was one of the central principles
Mathematics of general relativity
Mathematics_of_general_relativity
Operation in mathematics
2x2; often 3x3 or 4x4 are used, but any size is allowed. In simple index notation, this is written ∑ j = 1 2 a i j × b j k = c i k {\textstyle \sum
Tensor_contraction
Isomorphism between the tangent and cotangent bundles of a manifold
Einstein summation notation: any index may appear at most twice and furthermore a raised index must contract with a lowered index. With these rules we
Musical_isomorphism
Array of numbers
or no columns, called an empty matrix. The specifics of symbolic matrix notation vary widely, with some prevailing trends. Matrices are commonly written
Matrix_(mathematics)
Superenergy tensor of gravitational field flux-energy in a vacuum
differential geometry, the Bel–Robinson tensor is a tensor defined in the abstract index notation by: T a b c d = C a e c f C b e d f + 1 4 ϵ a e h i ϵ b e j k C
Bel–Robinson_tensor
Tensor field in Riemannian geometry
measures the noncommutativity of the second covariant derivative. In abstract index notation, R d c a b Z c = ∇ a ∇ b Z d − ∇ b ∇ a Z d . {\displaystyle
Riemann_curvature_tensor
Tensor in differential geometry
basis v 1 , … , v n {\displaystyle v_{1},\ldots ,v_{n}} . In abstract index notation, R i c a b = R c b c a = R c a c b . {\displaystyle \mathrm {Ric}
Ricci_curvature
1984 graduate textbook by Robert M. Wald
such as causal structure, and quantum effects. The book uses the abstract index notation for tensors. It covers spinors, the variational-principle formulation
General_Relativity_(book)
Algebraic operation on coordinate vectors
specified with respect to an orthonormal basis, is defined, in summation notation, as: a ⋅ b = ∑ i = 1 n a i b i = a 1 b 1 + a 2 b 2 + ⋯ + a n b n {\displaystyle
Dot_product
Covariant derivative of the metric tensor
for X , Y , Z {\displaystyle X,Y,Z} arbitrary vector fields. In abstract index notation, this reads Q a b c = ∇ a g b c {\displaystyle Q_{abc}=\nabla _{a}g_{bc}}
Nonmetricity_tensor
Branch of mathematics
tensors Dyadic tensor Glossary of tensor theory Metric tensor Bra–ket notation Multilinear subspace learning Multivector Geometric algebra Clifford algebra
Multilinear_algebra
Mathematical study of illumination of rooms with mirrored walls
(with Brian W. Aldiss) (1999) Concepts Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Weyl curvature
Illumination_problem
Theorem describing tensor behavior
where C a b c d {\displaystyle C_{abcd}} is the Weyl tensor, and abstract index notation is used. Moreover, in the Petrov classification, C a b c d ( 1
Peeling_theorem
Tensor equal to the negative of any of its transpositions
Antisymmetric permutation object acting on tensors Ricci calculus – Tensor index notation for tensor-based calculations Symmetric tensor – Tensor invariant under
Antisymmetric_tensor
Polynomial in combinatorial mathematics
write the fixed points in the cycle notation for a permutation, but these must be represented in the cycle index. Dixon & Mortimer 1996, pg. 9, Corollary
Cycle_index
Theory of gravitation as curved spacetime
\nu }} is the stress–energy tensor. All tensors are written in abstract index notation. Matching the theory's prediction to observational results for
General_relativity
English mathematician, mathematical physicist (born 1931)
contributions Moore–Penrose inverse Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Illumination
Roger_Penrose
Theory of gravity
manifold M, and xa are coordinates in the fiber Mp. Using the abstract index notation, let a, b, c,… refer to Mp and μ, ν,… refer to the tangent bundle
Teleparallelism
Origin and evolution of the symbols used to write equations and formulas
mathematical notation covers the introduction, development, and cultural diffusion of mathematical symbols and the conflicts between notational methods that
History of mathematical notation
History_of_mathematical_notation
Derivative in differential geometry and vector calculus
_{u,v}^{2}w)^{a}=u^{c}v^{b}\nabla _{c}\nabla _{b}w^{a}} by using abstract index notation. It is also straightforward to verify that ( ∇ u ∇ v w ) a = u
Second_covariant_derivative
1989 book by Roger Penrose
(with Brian W. Aldiss) (1999) Concepts Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Weyl curvature
The_Emperor's_New_Mind
Philosophical argument based on the theory of relativity
(with Brian W. Aldiss) (1999) Concepts Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Weyl curvature
Rietdijk–Putnam_argument
Mathematical object that describes the electromagnetic field in spacetime
}F_{\beta \gamma }+\partial _{\beta }F_{\gamma \alpha }=0} or using the index notation with square brackets[note 1] for the antisymmetric part of the tensor:
Electromagnetic_tensor
Tensor having both covariant and contravariant indices
ones mixed. Notationally, these tensors differ from each other by the covariance/contravariance of their indices. A given contravariant index of a tensor
Mixed_tensor
Method for specifying point positions
elementary mathematics, but may be complex numbers or elements of a more abstract system such as a commutative ring. The use of a coordinate system allows
Coordinate_system
Notation used for Weyl spinors
indices, i.e. "index free notation", an overbar is retained on right-handed spinor, since ambiguity arises between chirality when no index is indicated
Van_der_Waerden_notation
Exterior algebraic map taking tensors from p forms to n-p forms
}(dy\wedge dz)&=dt\wedge dx\,.\end{aligned}}} These are summarized in the index notation as ⋆ ( d x μ ) = η μ λ ε λ ν ρ σ 1 3 ! d x ν ∧ d x ρ ∧ d x σ , ⋆ ( d
Hodge_star_operator
Matrix operation which flips a matrix over its diagonal
another matrix, called the transpose of A and often denoted AT (among other notations). The transpose of a matrix was introduced in 1858 by the British mathematician
Transpose
additional structure on M {\displaystyle M} must be given. Using abstract index notation, the acceleration of a given curve with unit tangent vector ξ a
Acceleration (differential geometry)
Acceleration_(differential_geometry)
Antisymmetric permutation object acting on tensors
lower case epsilon ε or ϵ, or less commonly the Latin lower case e. Index notation allows one to display permutations in a way compatible with tensor analysis:
Levi-Civita_symbol
Scalar measure of the rotational inertia with respect to a fixed axis of rotation
\end{aligned}}} It is common in rigid body mechanics to use notation that explicitly identifies the x {\displaystyle x} , y {\displaystyle y}
Moment_of_inertia
Tensor used in general relativity
a tensor of order 2 defined over pseudo-Riemannian manifolds. In index-free notation it is defined as G = R − 1 2 g R , {\displaystyle {\boldsymbol {G}}={\boldsymbol
Einstein_tensor
Differential form of degree one or section of a cotangent bundle
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
One-form
Straight path on a curved surface or a Riemannian manifold
also great-circle distance). The term has since been generalized to more abstract mathematical spaces; for example, in graph theory, one might consider a
Geodesic
history of the abstract theory see also multilinear algebra. Ricci calculus The earliest foundation of tensor theory – tensor index notation. Order of a
Glossary_of_tensor_theory
Abbreviation in the fields of special and general relativity
four-dimensional spacetime. General four-tensors are usually written in tensor index notation as A ν 1 , ν 2 , . . . , ν m μ 1 , μ 2 , . . . , μ n {\displaystyle
Four-tensor
Set of vectors used to define coordinates
j}y_{j},} for i = 1, ..., n. This formula may be concisely written in matrix notation. Let A be the matrix of the a i , j {\displaystyle a_{i,j}} , and X = [
Basis_(linear_algebra)
Theory of interwoven space and time by Albert Einstein
disconcerting to physicists of the time. Among other things, the presence of an index of refraction term meant that, since n {\displaystyle n} depends on wavelength
Special_relativity
Mathematical function of two variables; outputs 1 if they are equal, 0 otherwise
i = j ] . {\displaystyle \delta _{ij}=[i=j].} Often, a single-argument notation δ i {\displaystyle \delta _{i}} is used, which is equivalent to setting
Kronecker_delta
Vector field on a pseudo-Riemannian manifold that preserves the metric tensor
_{b}K_{c}-\nabla _{b}\nabla _{a}K_{c}=R^{d}{}_{cab}K_{d}} (using abstract index notation) where R a b c d {\displaystyle R^{a}{}_{bcd}} is the Riemann curvature
Killing_vector_field
Branch of mathematics
popularised the tensor calculus of Ricci and Levi-Civita and introduced the notation g {\displaystyle g} for a Riemannian metric, and Γ {\displaystyle \Gamma
Differential_geometry
Specification of a derivative along a tangent vector of a manifold
coordinate-free language and using a local coordinate system and the traditional index notation. The covariant derivative of a tensor field is presented as an extension
Covariant_derivative
Hamiltonian formulation of general relativity
space and time. Most references adopt notation in which four dimensional tensors are written in abstract index notation, and that Greek indices are spacetime
ADM_formalism
Function that is invariant under all permutations of its variables
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Symmetric_function
Vector field in conformal geometry
symmetric part, one can write the conformal Killing equation in abstract index notation as ∇ a X b + ∇ b X a = 2 n g a b ∇ c X c . {\displaystyle \nabla
Conformal Killing vector field
Conformal_Killing_vector_field
Conserved physical quantity; rotational analogue of linear momentum
_{z}\wedge \mathbf {e} _{x}\,,\end{aligned}}} or more compactly in index notation: L i j = x i p j − x j p i . {\displaystyle L_{ij}=x_{i}p_{j}-x_{j}p_{i}\
Angular_momentum
Coordinate-free definition of a tensor
component-free approach to the theory of a tensor views a tensor as an abstract object, expressing some definite type of multilinear concept. Their properties
Tensor_(intrinsic_definition)
Tensor describing energy momentum density in spacetime
superscripted variables (not exponents; see Tensor index notation and Einstein summation notation). The four coordinates of an event of spacetime x are
Stress–energy_tensor
System of moving vectors in differential geometry
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Parallel_transport
Algebra associated to any vector space
given. Then any alternating tensor t ∈ Ar(V) ⊂ Tr(V) can be written in index notation with the Einstein summation convention as t = t i 1 i 2 ⋯ i r e i 1
Exterior_algebra
Property of a mathematical space
configuration spaces such as in Lagrangian or Hamiltonian mechanics; these are abstract spaces, independent of the physical space. In mathematics, the dimension
Dimension
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Symmetrization
Type of derivative in differential geometry
=f{\mathcal {L}}_{X}\omega +df\wedge i_{X}\omega .} In local coordinate notation, for a type ( r , s ) {\displaystyle (r,s)} tensor field T {\displaystyle
Lie_derivative
Effect in special relativity
(with Brian W. Aldiss) (1999) Concepts Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Weyl curvature
Terrell_rotation
is a rank-2 tensor defined over pseudo-Riemannian manifolds. In index-free notation it is defined as G = R − 1 2 g R , {\displaystyle \mathbf {G} =\mathbf
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Assignment of a tensor continuously varying across a region of space
bundle – Construction in differential topology Ricci calculus – Tensor index notation for tensor-based calculations Spinor field – Geometric structurePages
Tensor_field
Vector behavior under coordinate changes
covectors) are said to be contravariant. In Einstein notation (implicit summation over repeated index), contravariant components are denoted with upper indices
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Branch of physics which studies the behavior of materials modeled as continuous media
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Continuum_mechanics
Expression that may be integrated over a region
dependent is zero. A common notation for the wedge product of elementary k {\displaystyle k} -forms is so called multi-index notation: in an n {\displaystyle
Differential_form
Continuous surjection satisfying a local triviality condition
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Fiber_bundle
Array of numbers describing a metric connection
the same notation as tensors with index notation, they do not transform like tensors under a change of coordinates. Contracting the upper index with either
Christoffel_symbols
Operation on differential forms
generalized for any pseudo-Riemannian manifold, and written in coordinate-free notation as follows: grad f ≡ ∇ f = ( d f ) ♯ div F ≡ ∇ ⋅ F = ⋆ d ⋆ ( F ♭ )
Exterior_derivative
Second order tensor in vector algebra
algebra, a dyadic or dyadic tensor is a second-order tensor, written in a notation that fits in with vector algebra. There are numerous ways to multiply two
Dyadics
Spinning motion in theoretical physics
{\mathfrak {se}}(d)} . This article uses Cartesian coordinates and tensor index notation. The Noether current for translations in space is momentum, while the
Spin_tensor
Mapping from p forms to p-1 forms
_{1}a)\,d\xi ^{2}\wedge \dots \wedge d\xi ^{k+1}.\end{aligned}}} Proof by abstract algebra, credited to Shiing-Shen Chern The exterior derivative d {\displaystyle
Interior_product
Type of physical quantity
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Pseudotensor
Data type that represents an ordered collection of elements (values or variables)
use to define such types and declare array variables, and special notation for indexing array elements. For example, in the Pascal programming language
Array_(data_type)
Diagram used to represent quantum field theory calculations
and functions between representations of matrix groups. The diagrammatic notation can thus greatly simplify calculations. Roger Penrose described spin networks
Spin_network
Generalization of the Dirac equation
{\displaystyle \mathbf {g} } , or g a b {\displaystyle g_{ab}} in abstract index notation. We use a set of vierbein or frame fields { e μ } = { e 0 , e 1
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Classification system for symmetry groups in geometry
In geometry, Coxeter notation (also Coxeter symbol) is a system of classifying symmetry groups, describing the angles between fundamental reflections of
Coxeter_notation
Differential form
{\displaystyle \omega } is frequently used to denote the volume form, this notation is not universal; the symbol ω {\displaystyle \omega } often carries many
Volume_form
Quantity in electromagnetism
potential, especially when the Lorenz gauge is used. In particular, in abstract index notation, the set of Maxwell's equations (in the Lorenz gauge) may be written
Magnetic_vector_potential
Model of quantum computing
Abstract index notation Angular momentum diagrams (quantum mechanics) Circuit complexity and BQP Matrix product state uses Penrose graphical notation
Quantum_circuit
Electromagnetism in general relativity
square brackets indicate anti-symmetrization (see Ricci calculus for the notation). The covariant derivative of the electromagnetic field is F α β ; γ =
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Structure defining distance on a manifold
an abstract formulation of "lowering the index" on a vector field. The inverse of Sg is a mapping T*M → TM which, analogously, gives an abstract formulation
Metric_tensor
Formalism of general relativity
black holes. Most references adopt notation in which four dimensional tensors are written in abstract index notation, and that Greek indices are spacetime
BSSN_formalism
Measure of the curvature of a pseudo-Riemannian manifold
v_{3}\right)k\left(v_{1},v_{4}\right)\end{aligned}}} In tensor component notation, this can be written as C i k ℓ m = R i k ℓ m + 1 n − 2 ( R i m g k ℓ −
Weyl_tensor
Affine connection on the tangent bundle of a manifold
derivative and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific
Levi-Civita_connection
Ways of writing certain laws of physics
equations, one for each value of β. Using the antisymmetric tensor notation and comma notation for the partial derivative (see Ricci calculus), the second equation
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Non-tensorial representation of the spin group
form on a complex vector space is equivalent to the standard one, this notation is often used whenever dimℂ(V) = n. If n = 2k is even, then Cℓn(ℂ) is isomorphic
Spinor
Mathematical operation on vector spaces
differentiable, then a */ b is differentiable. However, these kinds of notation are not universally present in array languages. Other array languages may
Tensor_product
Decomposition in multilinear algebra
{\displaystyle M>2} and all I m ≥ 2 {\displaystyle I_{m}\geq 2} . For simplicity in notation, assume without loss of generality that the factors are ordered such that
Tensor_rank_decomposition
British theoretical physicist (born 1929)
(with Brian W. Aldiss) (1999) Concepts Twistor theory Spin network Abstract index notation Black hole bomb Geometry of spacetime Cosmic censorship Weyl curvature
Oliver_Penrose
Tensor invariant under permutations of vectors it acts on
the operator is omitted: T1T2 = T1 ⊙ T2. In some cases an exponential notation is used: v ⊙ k = v ⊙ v ⊙ ⋯ ⊙ v ⏟ k times = v ⊗ v ⊗ ⋯ ⊗ v ⏟ k times =
Symmetric_tensor
Computer programming identifier naming convention
Hungarian notation is an identifier naming convention in computer programming in which the name of a variable or function indicates its intention or kind
Hungarian_notation
Claim that human mathematicians are not describable as formal proof systems
describable as formal proof systems (which theorems can be proved using an abstract object such as a computer), and are therefore running a non-computable
Penrose–Lucas_argument
Study of curves from a differential point of view
phenomena Notation Abstract index notation Einstein notation Index notation Multi-index notation Penrose graphical notation Ricci calculus Tetrad (index notation)
Differentiable_curve
Mathematical function, in linear algebra
Victor (2001) [1994], "Index theory", Encyclopedia of Mathematics, EMS Press: "The main question in index theory is to provide index formulas for classes
Linear_map
Universal construction in multilinear algebra
was actually one and the same thing as ∇ {\displaystyle \nabla } ; and notational sloppiness here would lead to utter chaos. To strengthen this: the tensor
Tensor_algebra
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