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Spacetime modeled by four pointwise-orthonormal vector fields
defined on the manifold can be expressed using the frame field and its dual coframe field. Frame fields were introduced into general relativity by Albert Einstein
Frame fields in general relativity
Frame_fields_in_general_relativity
In mathematics, a coframe or coframe field on a smooth manifold M {\displaystyle M} is a system of one-forms or covectors which form a basis of the cotangent
Coframe
Generalization of an ordered basis of a vector space
coordinate expression of the dual coframe, as explained in the next section. A moving frame determines a dual frame or coframe of the cotangent bundle over
Moving_frame
Differential geometry technique
Cartan, the primary geometrical information was expressed in a coframe or collection of coframes on a differentiable manifold. See method of moving frames
Cartan's_equivalence_method
flat. This problem reduces to a question on the coframe bundle of M. Suppose we had such a closed coframe Θ = ( θ 1 , … , θ n ) . {\displaystyle \Theta
Integrability conditions for differential systems
Integrability_conditions_for_differential_systems
Isomorphism between the tangent and cotangent bundles of a manifold
basis), the moving coframe (a moving tangent frame for the cotangent bundle T ∗ M {\displaystyle \mathrm {T} ^{*}M} ; see also coframe) {ei}. Then the pseudo-Riemannian
Musical_isomorphism
Proposed common ancestor to all human languages
PMID 15716951. S2CID 1454595. Nandi, Owi Ivar. 2012. Human Language Evolution, as Coframed by Behavioral and Psychological Universalisms, Bloomington: iUniverse Publishers
Proto-Human_language
Coordinates comprising a distance and an angle
e_{\theta }={\frac {1}{r}}{\frac {\partial }{\partial \theta }},} with dual coframe e r = d r , e θ = r d θ . {\displaystyle e^{r}=dr,\quad e^{\theta }=rd\theta
Polar_coordinate_system
Tool from special relativity
{\displaystyle c=1} and α = 1 {\displaystyle \alpha =1} , it is natural to take the coframe field d σ 0 = x d t , d σ 1 = d x , d σ 2 = d y , d σ 3 = d z {\displaystyle
Rindler_coordinates
Partial differential equation
two-manifold is to use the differential forms method of Élie Cartan. Take the coframe field σ 1 = exp ( p ) d x , σ 2 = exp ( p ) d y {\displaystyle \sigma
Ricci_flow
Coordinate system
coordinates Frame fields in general relativity for more about frame fields and coframe fields. Synge, John Lighton (1960). Relativity: The General Theory. North-Holland
Gaussian_polar_coordinates
Coordinate system in black hole physics
Cartan's exterior calculus method. First, we read off the line element a coframe field, σ 0 = − a ( r ) d t σ 1 = b ( r ) d r σ 2 = r d θ σ 3 = r sin
Schwarzschild_coordinates
Coordinate system used to represent certain spacetimes
Cartan's exterior calculus method. First, we read off the line element a coframe field, σ 0 = − a ( r ) d t {\displaystyle \sigma ^{0}=-a(r)\,dt} σ 1 =
Isotropic_coordinates
Model of hyperbolic geometry
(}1-|\mathbf {x} |^{2}{\Bigr )}{\frac {\partial }{\partial x^{i}}},} with dual coframe of 1-forms θ i = 2 1 − | x | l 2 d x i . {\displaystyle \theta ^{i}={\frac
Poincaré_disk_model
Attempt to extend Yang–Mills theory to gravity
were considered as those of the translation gauge group, and a tetrad (coframe) field was identified with the translation part of an affine connection
Gauge_gravitation_theory
Mathematical concept
Kronecker delta. Then Ei is a Maurer–Cartan frame, and θi is a Maurer–Cartan coframe. Since Ei is left-invariant, applying the Maurer–Cartan form to it simply
Maurer–Cartan_form
Theory of gravity
coefficients) in this global basis. Here ωk is the dual global basis (or coframe) defined by ωi(Xj) = δi j. This is what usually happens in Rn, in any affine
Teleparallelism
metric tensor takes on a particularly convenient form. When allied with coframe fields, frame fields provide a powerful tool for analysing spacetimes and
Mathematics of general relativity
Mathematics_of_general_relativity
Object in differential geometry
the following manner. Let θ i {\displaystyle \theta ^{i}} be a parallel coframe along γ {\displaystyle \gamma } , and let x i {\displaystyle x^{i}} be
Torsion_tensor
Special coordinate system in differential geometry
\tau }(0){\bigr ]}x^{\tau }+O(|x|^{2}).} Similarly we can construct local coframes e ∗ a = e μ ∗ a d x μ {\displaystyle {\mathbf {e}}^{*a}=e_{\mu }^{*a}dx^{\mu
Normal_coordinates
coframes-I: A practical algorithm". Acta Appl. Math. 51 (2): 161–213. doi:10.1023/A:1005878210297. M. Fels and P. J. Olver (1999). "Moving coframes-II:
Moving_frames_method
Exact solution of the Einstein field equations
To prevent misunderstanding, we should emphasize that taking the dual coframe σ 0 = d t + h ( r ) r d φ , σ 1 = 1 f ( r ) d z , σ 2 = 1 f ( r ) d r
Van_Stockum_dust
{\displaystyle M} . On Σ {\displaystyle \Sigma } , define an orthonormal coframe e a {\displaystyle e^{a}} , and let a {\displaystyle a} be the second fundamental
Laguerre_form
Hermann (in French). Paris. Fels, M.; Olver, Peter J. (April 1998). "Moving Coframes: I. A Practical Algorithm". Acta Applicandae Mathematicae. 51 (2): 161–213
Lie_point_symmetry
Coordinates to capture characteristics of rotating frames of reference
Lorentzian manifold by a stationary timelike congruence. If we adopt the coframe θ 1 ^ = d z , θ 2 ^ = d r , θ 3 ^ = r d ϕ 1 − ω 2 r 2 {\displaystyle \theta
Born_coordinates
American mathematician (born 1952)
1137/0147018. ISSN 0036-1399. Fels, Mark; Olver, Peter J. (1998-04-01). "Moving Coframes: I. A Practical Algorithm". Acta Applicandae Mathematicae. 51 (2): 161–213
Peter_J._Olver
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Girl/Female
Arabic, Indian, Malaysian, Muslim, Pakistani
Blue Planet
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Indian, Tamil
Lord Shiva
Girl/Female
Indian
Garland of Lord Vishnu
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Hindu
Friend of fire, Sparkling eyes
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Light of war
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Polish
Polish form of Greek Bartholomaios, BARTÅOMIEJ means "son of Talmai."
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Hindu, Indian, Traditional
Water Clad; A Spring; A Fountain
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Muslim
Pearl
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Tamil
Lord Murugan
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Hindu, Indian, Traditional
Soul of Life
COFRAME
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