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Weyl tensor

Index Weyl tensor

In differential geometry, the Weyl curvature tensor, named after Hermann Weyl, is a measure of the curvature of spacetime or, more generally, a pseudo-Riemannian manifold. [1]

45 relations: Antisymmetric tensor, Atlas (topology), Christoffel symbols, Conformal map, Conformally flat manifold, Cotton tensor, Curvature, Curvature of Riemannian manifolds, Differential geometry, Einstein field equations, Einstein manifold, General relativity, Geodesic, Gravitational wave, Hermann Weyl, Integrability conditions for differential systems, Irreducible representation, Isothermal coordinates, Kulkarni–Nomizu product, Lanczos tensor, Method of characteristics, Metric tensor, Necessity and sufficiency, Nordström's theory of gravitation, Orthogonal group, Orthogonality, Peeling theorem, Petrov classification, Plebanski tensor, Pseudo-Riemannian manifold, Ricci curvature, Ricci decomposition, Ricci-flat manifold, Riemann curvature tensor, Scalar curvature, Schouten tensor, Spacetime, Springer Science+Business Media, Tensor, The Large Scale Structure of Space-Time, Tidal force, Trace (linear algebra), Vector bundle, Weyl curvature hypothesis, Weyl scalar.

Antisymmetric tensor

In mathematics and theoretical physics, a tensor is antisymmetric on (or with respect to) an index subset if it alternates sign (+/−) when any two indices of the subset are interchanged.

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Atlas (topology)

In mathematics, particularly topology, one describes a manifold using an atlas.

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Christoffel symbols

In mathematics and physics, the Christoffel symbols are an array of numbers describing a metric connection.

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Conformal map

In mathematics, a conformal map is a function that preserves angles locally.

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Conformally flat manifold

A (pseudo-)Riemannian manifold is conformally flat if each point has a neighborhood that can be mapped to flat space by a conformal transformation.

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Cotton tensor

In differential geometry, the Cotton tensor on a (pseudo)-Riemannian manifold of dimension n is a third-order tensor concomitant of the metric, like the Weyl tensor.

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Curvature

In mathematics, curvature is any of a number of loosely related concepts in different areas of geometry.

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Curvature of Riemannian manifolds

In mathematics, specifically differential geometry, the infinitesimal geometry of Riemannian manifolds with dimension greater than 2 is too complicated to be described by a single number at a given point.

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Differential geometry

Differential geometry is a mathematical discipline that uses the techniques of differential calculus, integral calculus, linear algebra and multilinear algebra to study problems in geometry.

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Einstein field equations

The Einstein field equations (EFE; also known as Einstein's equations) comprise the set of 10 equations in Albert Einstein's general theory of relativity that describe the fundamental interaction of gravitation as a result of spacetime being curved by mass and energy.

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Einstein manifold

In differential geometry and mathematical physics, an Einstein manifold is a Riemannian or pseudo-Riemannian differentiable manifold whose Ricci tensor is proportional to the metric.

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General relativity

General relativity (GR, also known as the general theory of relativity or GTR) is the geometric theory of gravitation published by Albert Einstein in 1915 and the current description of gravitation in modern physics.

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Geodesic

In differential geometry, a geodesic is a generalization of the notion of a "straight line" to "curved spaces".

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Gravitational wave

Gravitational waves are the disturbance in the fabric ("curvature") of spacetime generated by accelerated masses and propagate as waves outward from their source at the speed of light.

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Hermann Weyl

Hermann Klaus Hugo Weyl, (9 November 1885 – 8 December 1955) was a German mathematician, theoretical physicist and philosopher.

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Integrability conditions for differential systems

In mathematics, certain systems of partial differential equations are usefully formulated, from the point of view of their underlying geometric and algebraic structure, in terms of a system of differential forms.

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Irreducible representation

In mathematics, specifically in the representation theory of groups and algebras, an irreducible representation (\rho, V) or irrep of an algebraic structure A is a nonzero representation that has no proper subrepresentation (\rho|_W,W), W \subset V closed under the action of \. Every finite-dimensional unitary representation on a Hermitian vector space V is the direct sum of irreducible representations.

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Isothermal coordinates

In mathematics, specifically in differential geometry, isothermal coordinates on a Riemannian manifold are local coordinates where the metric is conformal to the Euclidean metric.

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Kulkarni–Nomizu product

In the mathematical field of differential geometry, the Kulkarni–Nomizu product (named for Ravindra Shripad Kulkarni and Katsumi Nomizu) is defined for two (0,2)-tensors and gives as a result a (0,4)-tensor.

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Lanczos tensor

The Lanczos tensor or Lanczos potential is a rank 3 tensor in general relativity that generates the Weyl tensor.

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Method of characteristics

In mathematics, the method of characteristics is a technique for solving partial differential equations.

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Metric tensor

In the mathematical field of differential geometry, a metric tensor is a type of function which takes as input a pair of tangent vectors and at a point of a surface (or higher dimensional differentiable manifold) and produces a real number scalar in a way that generalizes many of the familiar properties of the dot product of vectors in Euclidean space.

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Necessity and sufficiency

In logic, necessity and sufficiency are terms used to describe an implicational relationship between statements.

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Nordström's theory of gravitation

In theoretical physics, Nordström's theory of gravitation was a predecessor of general relativity.

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Orthogonal group

In mathematics, the orthogonal group in dimension, denoted, is the group of distance-preserving transformations of a Euclidean space of dimension that preserve a fixed point, where the group operation is given by composing transformations.

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Orthogonality

In mathematics, orthogonality is the generalization of the notion of perpendicularity to the linear algebra of bilinear forms.

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Peeling theorem

In general relativity, the peeling theorem describes the asymptotic behavior of the Weyl tensor as one goes to.

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Petrov classification

In differential geometry and theoretical physics, the Petrov classification (also known as Petrov–Pirani–Penrose classification) describes the possible algebraic symmetries of the Weyl tensor at each event in a Lorentzian manifold.

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Plebanski tensor

The Plebanski tensor is a order 4 tensor in general relativity constructed from the trace-free Ricci tensor.

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Pseudo-Riemannian manifold

In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold in which the metric tensor need not be positive-definite, but need only be a non-degenerate bilinear form, which is a weaker condition.

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Ricci curvature

In differential geometry, the Ricci curvature tensor, named after Gregorio Ricci-Curbastro, represents the amount by which the volume of a small wedge of a geodesic ball in a curved Riemannian manifold deviates from that of the standard ball in Euclidean space.

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Ricci decomposition

In semi-Riemannian geometry, the Ricci decomposition is a way of breaking up the Riemann curvature tensor of a pseudo-Riemannian manifold into pieces with useful individual algebraic properties.

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Ricci-flat manifold

In mathematics, Ricci-flat manifolds are Riemannian manifolds whose Ricci curvature vanishes.

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Riemann curvature tensor

In the mathematical field of differential geometry, the Riemann curvature tensor or Riemann–Christoffel tensor (after Bernhard Riemann and Elwin Bruno Christoffel) is the most common method used to express the curvature of Riemannian manifolds.

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Scalar curvature

In Riemannian geometry, the scalar curvature (or the Ricci scalar) is the simplest curvature invariant of a Riemannian manifold.

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Schouten tensor

In Riemannian geometry, the Schouten tensor is a second-order tensor introduced by Jan Arnoldus Schouten.

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Spacetime

In physics, spacetime is any mathematical model that fuses the three dimensions of space and the one dimension of time into a single four-dimensional continuum.

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Springer Science+Business Media

Springer Science+Business Media or Springer, part of Springer Nature since 2015, is a global publishing company that publishes books, e-books and peer-reviewed journals in science, humanities, technical and medical (STM) publishing.

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Tensor

In mathematics, tensors are geometric objects that describe linear relations between geometric vectors, scalars, and other tensors.

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The Large Scale Structure of Space-Time

The Large Scale Structure of Space-Time is 1973 book by Stephen Hawking and George Ellis on the theoretical physics of spacetime.

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Tidal force

The tidal force is an apparent force that stretches a body towards the center of mass of another body due to a gradient (difference in strength) in gravitational field from the other body; it is responsible for the diverse phenomena, including tides, tidal locking, breaking apart of celestial bodies and formation of ring systems within Roche limit, and in extreme cases, spaghettification of objects.

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Trace (linear algebra)

In linear algebra, the trace of an n-by-n square matrix A is defined to be the sum of the elements on the main diagonal (the diagonal from the upper left to the lower right) of A, i.e., where aii denotes the entry on the ith row and ith column of A. The trace of a matrix is the sum of the (complex) eigenvalues, and it is invariant with respect to a change of basis.

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Vector bundle

In mathematics, a vector bundle is a topological construction that makes precise the idea of a family of vector spaces parameterized by another space X (for example X could be a topological space, a manifold, or an algebraic variety): to every point x of the space X we associate (or "attach") a vector space V(x) in such a way that these vector spaces fit together to form another space of the same kind as X (e.g. a topological space, manifold, or algebraic variety), which is then called a vector bundle over X.

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Weyl curvature hypothesis

The Weyl curvature hypothesis, which arises in the application of Albert Einstein's general theory of relativity to physical cosmology, was introduced by the British mathematician and theoretical physicist Sir Roger Penrose in an article in 1979 in an attempt to provide explanations for two of the most fundamental issues in physics.

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Weyl scalar

In the Newman–Penrose (NP) formalism of general relativity, Weyl scalars refer to a set of five complex scalars \ which encode the ten independent components of the Weyl tensors of a four-dimensional spacetime.

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Redirects here:

Conformal tensor, Weyl curvature, Weyl curvature tensor.

References

[1] https://en.wikipedia.org/wiki/Weyl_tensor

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