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Physics concept
In physics, a covariant transformation is a rule that specifies how certain entities, such as vectors or tensors, change under a change of basis. The transformation
Covariant_transformation
Specification of a derivative along a tangent vector of a manifold
derivative transforms covariantly under a general coordinate transformation, that is, linearly via the Jacobian matrix of the transformation. This article presents
Covariant_derivative
Vector behavior under coordinate changes
and a covariant vector is a list of numbers that transforms in the same way. Contravariant vectors are often just called vectors and covariant vectors
Covariance and contravariance of vectors
Covariance_and_contravariance_of_vectors
Symmetries in a gravitational theory
general covariant transformations are symmetries of gravitation theory on a world manifold X {\displaystyle X} . They are gauge transformations whose parameter
General covariant transformations
General_covariant_transformations
Algebraic object with geometric applications
some combination of covariant and contravariant transformations, with one transformation law for each index. If the transformation matrix of an index is
Tensor
Differential form of degree one or section of a cotangent bundle
one-form has a covariant transformation law on passing from one coordinate system to another. Thus a one-form is an order 1 covariant tensor field. The
One-form
Family of linear transformations
In physics, the Lorentz transformations are a six-parameter family of linear transformations from a coordinate frame in spacetime to another frame that
Lorentz_transformation
Derivative used in gauge theories
such gauge transformations, because they depend on the local frame. However, when gauge transformations act on fields and the gauge covariant derivative
Gauge_covariant_derivative
Principle stating that physical laws are the same in all coordinate systems
contravariance Covariant derivative Fictitious force Galilean invariance Gauge covariant derivative General covariant transformations Harmonic coordinate
General_covariance
Mathematical function, in linear algebra
two vector spaces are the same, a linear map is also called a linear transformation or linear endomorphism. A linear map is a homomorphism of vector spaces
Linear_map
vector is called covariant or contravariant depending on how the transformation of the vector's components is related to the transformation of coordinates
Introduction to the mathematics of general relativity
Introduction_to_the_mathematics_of_general_relativity
Tensor index notation for tensor-based calculations
summarizes how the manipulation of covariant and contravariant indices fit in with invariance under a passive transformation between bases, with the components
Ricci_calculus
Mapping between categories
→ C o v a r i a n t {\displaystyle \mathrm {Covariant} \circ \mathrm {Covariant} \to \mathrm {Covariant} } C o n t r a v a r i a n t ∘ C o n t r a v
Functor
Concept in relativistic physics
In particular, a Lorentz covariant scalar (e.g., the space-time interval) remains the same under Lorentz transformations and is said to be a Lorentz
Lorentz_covariance
Ways of writing certain laws of physics
The covariant formulation of classical electromagnetism refers to ways of writing the laws of classical electromagnetism (in particular, Maxwell's equations
Covariant formulation of classical electromagnetism
Covariant_formulation_of_classical_electromagnetism
Attempt to extend Yang–Mills theory to gravity
internal symmetries encountered a problem of treating general covariant transformations and establishing the gauge status of a pseudo-Riemannian metric
Gauge_gravitation_theory
Physical theory with fields invariant under the action of local "gauge" Lie groups
\mathbf {F} } transforms covariantly. Not all gauge transformations can be generated by infinitesimal gauge transformations in general. An example is
Gauge_theory
Central object of study in category theory
G^{\text{op}}} are thus "turned around". Forming the opposite group becomes a (covariant) functor from Grp {\displaystyle {\textbf {Grp}}} to Grp {\displaystyle
Natural_transformation
Concept in physics and mathematics
In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion
Galilean_transformation
Embedding of categories into functor categories
m ( A , − ) {\displaystyle h_{A}(-)\equiv \mathrm {Hom} (A,-)} . The (covariant) hom-functor h A {\displaystyle h_{A}} sends X ∈ C {\displaystyle X\in
Yoneda_lemma
Type of observable in a physical system
these coordinate transformations. Thus, a physicist might say that these equations are covariant. Despite this usage of "covariant", it is more accurate
Invariant_(physics)
Group of gauge symmetries in Yang–Mills theory
on a principal frame bundle whose gauge symmetries are general covariant transformations which are not elements of a gauge group. In the physical literature
Gauge_group_(mathematics)
Structure defining distance on a manifold
matrix whose entries transform covariantly under changes to the coordinate system. Thus a metric tensor is a covariant symmetric tensor. From the coordinate-independent
Metric_tensor
Relationship between relativity and pre-quantum electromagnetism
convenient notation for the laws of electromagnetism, namely the "manifestly covariant" tensor form. Maxwell's equations, when they were first stated in their
Classical electromagnetism and special relativity
Classical_electromagnetism_and_special_relativity
General theory of mathematical structures
a contravariant functor acts as a covariant functor from the opposite category Cop to D. A natural transformation is a relation between two functors
Category_theory
Procedure of coping with redundant degrees of freedom in physical field theories
however, not Lorentz covariant. If a Lorentz transformation to a new inertial frame is carried out, a further gauge transformation has to be made to retain
Gauge_fixing
Topics referred to by the same term
linear transformations. Covariance and contravariance of vectors, properties of how vector coordinates change under a change of basis Covariant transformation
Covariance_(disambiguation)
Field-equations in general relativity
partial derivatives, denoted by a comma, is fully covariant, despite the individual terms not being covariant. Brown, Harvey (2005). Physical Relativity. Oxford
Einstein_field_equations
Coordinate system whose directions vary in space
components, Si j the mixed right-covariant components, Si j the mixed left-covariant components, and Sij the covariant components of the second-order tensor
Curvilinear_coordinates
Method for specifying point positions
relationship between different systems is described by coordinate transformations, which give formulas for the coordinates in one system in terms of
Coordinate_system
Relativistic quantum mechanical wave equation
Dirac spinor under Lorentz transformations. In natural units where ℏ = c = 1 {\displaystyle \hbar =c=1} , the Lorentz covariant formulation of the Dirac
Dirac_equation
Defines a notion of parallel transport on a bundle
transport must be linear. A linear connection is equivalently specified by a covariant derivative, an operator that differentiates sections of the bundle along
Connection_(vector_bundle)
Mathematical operation
{\displaystyle M} ), then the pullback and pushforward describe the transformation properties of covariant and contravariant tensors used in more traditional (coordinate
Pullback (differential geometry)
Pullback_(differential_geometry)
Principle in relativity
quantities must transform covariantly, that is, under a certain representation of the group of coordinate transformations between admissible frames of
Principle_of_covariance
Array of numbers describing a metric connection
^{i}}_{jk}={\Gamma ^{i}}_{kj}.} The index-less transformation properties of a tensor are given by pullbacks for covariant indices, and pushforwards for contravariant
Christoffel_symbols
Representation of a tensor in Euclidean space
components from one such basis to another is done through an orthogonal transformation. The most familiar coordinate systems are the two-dimensional and three-dimensional
Cartesian_tensor
pluriharmonic functions on CR manifolds of real dimension three. Hirachi's covariant transformation formula for P 4 {\displaystyle P_{4}} on three dimensional CR manifolds
Paneitz_operator
Functor mapping hom objects to an underlying category
Note that fixing the first argument of Hom naturally gives rise to a covariant functor and fixing the second argument naturally gives a contravariant
Hom_functor
Identities in general relativity
the scalar curvature, and ∇ ρ {\displaystyle \nabla _{\rho }} indicates covariant differentiation. These identities are named after Luigi Bianchi, although
Contracted_Bianchi_identities
Tensor field in Riemannian geometry
invariant of Riemannian metrics that measures the failure of the second covariant derivatives to commute. A Riemannian manifold has zero curvature if and
Riemann_curvature_tensor
Shorthand notation for tensor operations
with covariant and contravariant vectors, where the position of an index indicates the type of vector, the first case usually applies; a covariant vector
Einstein_notation
Local rescaling of a metric tensor
}\varphi +kB_{\mu }\varphi } is covariant and has conformal weight k − 1 {\displaystyle k-1} . For the transformation g a b = f ( ϕ ( x ) ) g ¯ a b {\displaystyle
Weyl_transformation
Mathematical description of spacetime used in relativity
vectors with lower indices are referred to as covariant vectors. In this latter interpretation, the covariant vectors are (almost always implicitly) identified
Minkowski_spacetime
Formalism in string theory
the construction of covariant string field theories (preserving manifest Lorentz invariance) was the construction of a covariant kinetic term. This kinetic
String_field_theory
Notion in geometry
the covariant derivative. The linear transformation w ↦ R ( u , v ) w {\displaystyle w\mapsto R(u,v)w} is also called the curvature transformation or
Curvature of Riemannian manifolds
Curvature_of_Riemannian_manifolds
Vector in relativity
are given as contravariant vectors, there are also the corresponding covariant vectors xμ, pμ and Aμ(x). These transform according to the rule X ′ =
Four-vector
Study of vector bundles, principal bundles, and fibre bundles
Since the exterior covariant derivative in degree 0 is the same as the regular covariant derivative, the connection or covariant derivative itself is
Gauge_theory_(mathematics)
Matrix operation which flips a matrix over its diagonal
Tensor operator Tensor bundle Two-point tensor Operations Covariant derivative Exterior covariant derivative Exterior derivative Exterior product Hodge star
Transpose
Quantum field giving rise to gluons
position r and time t. Matrix exponentiation is used in the transformation. The gauge covariant derivative transforms similarly. The functions θn here are
Gluon_field
Generalization of tensor fields
\sigma }\right)} , the determinant of the metric tensor expressed with covariant indices. With this choice, classical densities, like charge density, will
Tensor_density
Gives the total power radiated by an accelerating, nonrelativistic point charge
accelerators. The radiated power is actually a Lorentz scalar, given in covariant form as P = − 2 3 q 2 m 2 c 3 d p μ d τ d p μ d τ . {\displaystyle P=-{\frac
Larmor_formula
Systematic procedure of turning a classical theory into a quantum one
perform a canonical quantization without having to resort to the non covariant approach of foliating spacetime and choosing a Hamiltonian. This method
Quantization_(physics)
Functor between abelian categories
family of natural transformations that, for each short exact sequence, commute with the morphisms δ. For example, in the case of two covariant cohomological
Delta-functor
Mathematical technique used in data compression and analysis
Lindeberg, T. (January 23, 2023). "A time-causal and time-recursive scale-covariant scale-space representation of temporal signals and past time". Biological
Wavelet_transform
Shielding an object from view using materials made to redirect light
render an object seemingly invisible. Metamaterial cloaking, based on transformation optics, describes the process of shielding something from view by controlling
Metamaterial_cloaking
Affine connection on the tangent bundle of a manifold
theorems). The covariant derivative is defined given any affine connection. In the theory of Riemannian and pseudo-Riemannian manifolds, the "covariant derivative"
Levi-Civita_connection
Four-vector analogue of the gradient operation
{\mathbf {a} }}\cdot {\vec {\mathbf {b} }}} The 4-gradient covariant components compactly written in four-vector and Ricci calculus notation
Four-gradient
Generalization of the Dirac equation
checked that under the transformation ψ ↦ ρ ( Λ ) ψ , {\displaystyle \psi \mapsto \rho (\Lambda )\psi ,} if we define the covariant derivative D μ ψ = ∂
Dirac equation in curved spacetime
Dirac_equation_in_curved_spacetime
Theory of supergravity in four dimensions
supersymmetry transformations, this Lagrangian also is Lorentz invariant, gauge invariant, and Kähler transformation invariant, with covariant derivatives
4D_N_=_1_supergravity
that the laws of physics should transform from one frame to another covariantly, that is, according to a representation of the covariance group. Special
Covariance_group
Transformations induced by a mathematical group
of performing the transformations of the group of transformations. The reason for distinguishing the group from the transformations is that, generally
Group_action
Tensor describing energy momentum density in spacetime
stress–energy tensor. However, it is often convenient to work with the covariant form, T μ ν = T α β g α μ g β ν , {\displaystyle T_{\mu \nu }=T^{\alpha
Stress–energy_tensor
Development of linear transformations forming the Lorentz group
are covariant with respect to these transformations, irrespective of the choice of λ. These variants of conformal or Lie sphere transformations were
History of Lorentz transformations
History_of_Lorentz_transformations
Axiomatization of quantum field theory
axioms have position-dependent operators called quantum fields, which form covariant representations of the Poincaré group. Since quantum field theory suffers
Wightman_axioms
Mathematical object that describes the electromagnetic field in spacetime
c} . This is the exterior derivative of its 1-form antiderivative, the covariant form of the four-potential, is A = ( ϕ / c ) d t − A x d x − A y d y −
Electromagnetic_tensor
Topics referred to by the same term
after) each iteration A data type in method overriding that is neither covariant nor contravariant Class invariant, an invariant used to constrain objects
Invariant
Assignment of a tensor continuously varying across a region of space
transformations: x k ↦ A j k x j + a k {\displaystyle x^{k}\mapsto A_{j}^{k}x^{j}+a^{k}} (with n-dimensional indices, summation implied). A covariant
Tensor_field
Theory of interwoven space and time by Albert Einstein
special relativity, although they are easier to manipulate in a manifestly covariant form, that is, in the language of tensor calculus. Special relativity
Special_relativity
Fundamental construction of differential calculus
a unique preferred torsion-free covariant derivative, known as the Levi-Civita connection. See also gauge covariant derivative for a treatment oriented
Generalizations of the derivative
Generalizations_of_the_derivative
Equations describing classical electromagnetism
value problem, analytical mechanics, or for use in quantum mechanics. The covariant formulation (on spacetime rather than space and time separately) makes
Maxwell's_equations
Philosophical argument against general covariance
distinguish it from coordinate transformations (passive diffeomorphisms). Einstein failed to find non-generally covariant field equations only to return
Hole_argument
Breakdown of general covariance at the quantum level
\langle \;\;\;\rangle } . Let us label the Lorentz, Einstein and Weyl transformations respectively by their parameters α , ξ , σ {\displaystyle \alpha ,\
Gravitational_anomaly
Operation in mathematics
+T_{n}^{n}} . A general contraction is denoted by labeling one covariant index and one contravariant index with the same letter, summation over
Tensor_contraction
Dimension of the column space of a matrix
of type (1,1), having one row index and one column index, also called covariant order 1 and contravariant order 1; see Tensor (intrinsic definition) for
Rank_(linear_algebra)
Gauge symmetries of metric-affine gravitation theory are general covariant transformations. It is essential that, given a pseudo-Riemannian metric g {\displaystyle
Metric-affine gravitation theory
Metric-affine_gravitation_theory
Electromagnetism in general relativity
under arbitrary curvilinear coordinate transformations. Thus, if one replaced the partial derivatives with covariant derivatives, the extra terms thereby
Maxwell's equations in curved spacetime
Maxwell's_equations_in_curved_spacetime
Vector operator in vector calculus
Y_{q-1}){\Big )};} that is, we take the trace over the first two covariant indices of the covariant derivative. The ♯ {\displaystyle \sharp } symbol refers to
Divergence
Type of derivative in differential geometry
respect to connections, the exterior derivative of totally antisymmetric covariant tensors, i.e. differential forms. The main difference between the Lie
Lie_derivative
Force acting on charged particles in electric and magnetic fields
flux in the fields to the force exerted on a charge distribution. (See Covariant formulation of classical electromagnetism for more details.) The power
Lorentz_force
defined as elements of the tangent space and covectors (sometimes termed covariant vectors, but more commonly dual vectors or one-forms) are elements of
Mathematics of general relativity
Mathematics_of_general_relativity
Application of topology
manifold are natural bundles characterized by general covariant transformations. These transformations are gauge symmetries of gravitation theory on a world
World_manifold
some group of linear transformations. See also covariant, contravariant, concomitant. inversion An inversion is a transformation of order 2 exchanging
Glossary of classical algebraic geometry
Glossary_of_classical_algebraic_geometry
Thought experiment in physics
frames. Norton, John D. (2004), "Einstein's Investigations of Galilean Covariant Electrodynamics prior to 1905", Archive for History of Exact Sciences
Moving magnet and conductor problem
Moving_magnet_and_conductor_problem
Type of physical quantity
{\displaystyle B^{l_{p}}{}_{j_{p}}} is the transition matrix for the covariant indices, and ( − 1 ) A = s i g n ( det ( A i q k q ) ) = ± 1 . {\displaystyle
Pseudotensor
4-dimensional analogue of force used in theories of relativity
four-force are related to the elements of the four-momentum through a covariant derivative with respect to proper time. F λ := D P λ d τ = d P λ d τ +
Four-force
Physical quantity that changes sign with improper rotation
rigid transformations such as rotations or translations, but which does not transform like a vector under certain discontinuous rigid transformations such
Pseudovector
Differential geometry construct on fiber bundles
classical covariant derivatives, covariance is an a posteriori feature of the derivative. In their construction one specifies the transformation law of the
Ehresmann_connection
Branch of optics which studies how EM radiation can be manipulated with metamaterials
Transformation optics is a branch of optics which applies metamaterials to produce spatial variations, derived from coordinate transformations, which can
Transformation_optics
Function in mathematics
along a curve. An affine connection is typically given in the form of a covariant derivative, which gives a means for taking directional derivatives of
Connection_(mathematics)
Vector describing a wave; often its propagation direction
inverse wavelength λ. When written out explicitly its contravariant and covariant forms are: K μ = ( ω c , k x , k y , k z ) K μ = ( ω c , − k x , − k y
Wave_vector
Set of coordinates where the coordinate hypersurfaces all meet at right angles
common since it is more complicated. The basis vectors shown above are covariant basis vectors (because they "co-vary" with vectors). In the case of orthogonal
Orthogonal_coordinates
Tensor formulation of non-relativistic physics
The Galilei-covariant tensor formulation is a method for treating non-relativistic physics using the extended Galilei group as the representation group
Galilei-covariant tensor formulation
Galilei-covariant_tensor_formulation
Principle in physics
extended Lagrangian is covariant with respect to a new extended group of local transformations. Gauge theory Gauge covariant derivative Gauge fixing
Gauge_principle
Mathematical description of fermions
energy), while the vector combination carries momentum and current, being covariant under the action of the Lorentz group. The angular momentum is carried
Dirac_spinor
Mathematics of smooth surfaces
earlier notion of covariant derivative, because it is the monodromy of the ordinary differential equation on the curve defined by the covariant derivative with
Differential geometry of surfaces
Differential_geometry_of_surfaces
Millennium Prize Problem
axioms have position dependent operators called quantum fields which form covariant representations of the Poincaré group. The group of space-time translations
Yang–Mills existence and mass gap
Yang–Mills_existence_and_mass_gap
Elliptic differential operators in geometry mathematics
the Laplace–Beltrami operator. It is defined as the trace of the second covariant derivative: Δ T = tr ∇ 2 T , {\displaystyle \Delta T={\text{tr}}\;\nabla
Laplace operators in differential geometry
Laplace_operators_in_differential_geometry
Construct allowing differentiation of tangent vector fields of manifolds
derivative of one vector field along another transformed covariantly under coordinate transformations — these correction terms subsequently came to be known
Affine_connection
form of field equations is covariant and it is equivalent to the Euler-Lagrange equations when the Legendre transformation to the variables pia and H
De_Donder–Weyl_theory
System of moving vectors in differential geometry
a manifold. If the manifold is equipped with an affine connection (a covariant derivative or connection on the tangent bundle), then this connection
Parallel_transport
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