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Frame fields in general relativity

Index Frame fields in general relativity

In general relativity, a frame field (also called a tetrad or vierbein) is a set of four orthonormal vector fields, one timelike and three spacelike, defined on a Lorentzian manifold that is physically interpreted as a model of spacetime. [1]

72 relations: Acceleration, Atlas (topology), Basis (linear algebra), Black hole, Caret, Cartan formalism (physics), Congruence (general relativity), Coordinate singularity, Cotangent bundle, Covariance and contravariance of vectors, Covariant derivative, Differential operator, Dirac equation in curved spacetime, Duality (mathematics), Einstein notation, Electrogravitic tensor, Electrovacuum solution, Event horizon, Exact solutions in general relativity, Fermi–Walker transport, Fiber bundle, Fluid solution, Four-acceleration, Frame bundle, General relativity, Geodesic, Geodesics in general relativity, Georges Lemaître, Gullstrand–Painlevé coordinates, Gyroscope, Hermann Weyl, Hessian matrix, Holonomic basis, Hydrostatic equilibrium, Inner product space, Integral curve, Karl Schwarzschild, Lie algebra, Linearity, Lorentz force, Manifold, Metric tensor, Minkowski space, Moving frame, Multilinear algebra, Order of approximation, Orthonormality, Parallel transport, Parallelizable manifold, Paul Painlevé, ..., Pressure, Principal bundle, Pseudo-Riemannian manifold, Riemannian manifold, Rotation, Schwarzschild metric, Section (fiber bundle), Spacetime, Special relativity, Tangent bundle, Tangent space, Tensor, Tensor product, Test particle, Tetrad formalism, Tidal tensor, Trace (linear algebra), Vacuum solution (general relativity), Vector bundle, Vector field, World line, Yusuke Hagihara. Expand index (22 more) »

Acceleration

In physics, acceleration is the rate of change of velocity of an object with respect to time.

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Atlas (topology)

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

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Basis (linear algebra)

In mathematics, a set of elements (vectors) in a vector space V is called a basis, or a set of, if the vectors are linearly independent and every vector in the vector space is a linear combination of this set.

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Black hole

A black hole is a region of spacetime exhibiting such strong gravitational effects that nothing—not even particles and electromagnetic radiation such as light—can escape from inside it.

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Caret

The caret is an inverted V-shaped grapheme.

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Cartan formalism (physics)

The vierbein or tetrad theory much used in theoretical physics is a special case of the application of Cartan connection in four-dimensional manifolds.

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Congruence (general relativity)

In general relativity, a congruence (more properly, a congruence of curves) is the set of integral curves of a (nowhere vanishing) vector field in a four-dimensional Lorentzian manifold which is interpreted physically as a model of spacetime.

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Coordinate singularity

A coordinate singularity occurs when an apparent singularity or discontinuity occurs in one coordinate frame, which can be removed by choosing a different frame.

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Cotangent bundle

In mathematics, especially differential geometry, the cotangent bundle of a smooth manifold is the vector bundle of all the cotangent spaces at every point in the manifold.

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Covariance and contravariance of vectors

In multilinear algebra and tensor analysis, covariance and contravariance describe how the quantitative description of certain geometric or physical entities changes with a change of basis.

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Covariant derivative

In mathematics, the covariant derivative is a way of specifying a derivative along tangent vectors of a manifold.

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Differential operator

In mathematics, a differential operator is an operator defined as a function of the differentiation operator.

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Dirac equation in curved spacetime

In mathematical physics, the Dirac equation in curved spacetime generalizes the original Dirac equation to curved space.

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Duality (mathematics)

In mathematics, a duality, generally speaking, translates concepts, theorems or mathematical structures into other concepts, theorems or structures, in a one-to-one fashion, often (but not always) by means of an involution operation: if the dual of A is B, then the dual of B is A. Such involutions sometimes have fixed points, so that the dual of A is A itself.

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Einstein notation

In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention that implies summation over a set of indexed terms in a formula, thus achieving notational brevity.

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Electrogravitic tensor

In general relativity, the gravitoelectric tensor or tidal tensor is one of the pieces in the Bel decomposition of the Riemann tensor.

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Electrovacuum solution

In general relativity, an electrovacuum solution (electrovacuum) is an exact solution of the Einstein field equation in which the only nongravitational mass-energy present is the field energy of an electromagnetic field, which must satisfy the (curved-spacetime) source-free Maxwell equations appropriate to the given geometry.

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Event horizon

In general relativity, an event horizon is a region in spacetime beyond which events cannot affect an outside observer.

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Exact solutions in general relativity

In general relativity, an exact solution is a Lorentzian manifold equipped with tensor fields modeling states of ordinary matter, such as a fluid, or classical nongravitational fields such as the electromagnetic field.

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Fermi–Walker transport

Fermi–Walker transport is a process in general relativity used to define a coordinate system or reference frame such that all curvature in the frame is due to the presence of mass/energy density and not to arbitrary spin or rotation of the frame.

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Fiber bundle

In mathematics, and particularly topology, a fiber bundle (or, in British English, fibre bundle) is a space that is locally a product space, but globally may have a different topological structure.

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Fluid solution

In general relativity, a fluid solution is an exact solution of the Einstein field equation in which the gravitational field is produced entirely by the mass, momentum, and stress density of a fluid.

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Four-acceleration

In the theory of relativity, four-acceleration is a four-vector (vector in four-dimensional spacetime) that is analogous to classical acceleration (a three-dimensional vector, see three-acceleration in special relativity).

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Frame bundle

In mathematics, a frame bundle is a principal fiber bundle F(E) associated to any vector bundle E. The fiber of F(E) over a point x is the set of all ordered bases, or frames, for Ex.

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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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Geodesics in general relativity

In general relativity, a geodesic generalizes the notion of a "straight line" to curved spacetime.

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Georges Lemaître

Georges Henri Joseph Édouard Lemaître, RAS Associate (17 July 1894 – 20 June 1966) was a Belgian Catholic Priest, astronomer and professor of physics at the Catholic University of Leuven.

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Gullstrand–Painlevé coordinates

Gullstrand–Painlevé coordinates are a particular set of coordinates for the Schwarzschild metric – a solution to the Einstein field equations which describes a black hole.

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Gyroscope

A gyroscope (from Ancient Greek γῦρος gûros, "circle" and σκοπέω skopéō, "to look") is a device used for measuring or maintaining orientation and angular velocity.

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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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Hessian matrix

In mathematics, the Hessian matrix or Hessian is a square matrix of second-order partial derivatives of a scalar-valued function, or scalar field.

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Holonomic basis

In mathematics and mathematical physics, a coordinate basis or holonomic basis for a differentiable manifold is a set of basis vector fields defined at every point of a region of the manifold as where is the infinitesimal displacement vector between the point and a nearby point whose coordinate separation from is along the coordinate curve (i.e. the curve on the manifold through for which the local coordinate varies and all other coordinates are constant).

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Hydrostatic equilibrium

In fluid mechanics, a fluid is said to be in hydrostatic equilibrium or hydrostatic balance when it is at rest, or when the flow velocity at each point is constant over time.

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Inner product space

In linear algebra, an inner product space is a vector space with an additional structure called an inner product.

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Integral curve

In mathematics, an integral curve is a parametric curve that represents a specific solution to an ordinary differential equation or system of equations.

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Karl Schwarzschild

Karl Schwarzschild (October 9, 1873 – May 11, 1916) was a German physicist and astronomer.

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Lie algebra

In mathematics, a Lie algebra (pronounced "Lee") is a vector space \mathfrak g together with a non-associative, alternating bilinear map \mathfrak g \times \mathfrak g \rightarrow \mathfrak g; (x, y) \mapsto, called the Lie bracket, satisfying the Jacobi identity.

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Linearity

Linearity is the property of a mathematical relationship or function which means that it can be graphically represented as a straight line.

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Lorentz force

In physics (particularly in electromagnetism) the Lorentz force is the combination of electric and magnetic force on a point charge due to electromagnetic fields.

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Manifold

In mathematics, a manifold is a topological space that locally resembles Euclidean space near each point.

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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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Minkowski space

In mathematical physics, Minkowski space (or Minkowski spacetime) is a combining of three-dimensional Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded.

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Moving frame

In mathematics, a moving frame is a flexible generalization of the notion of an ordered basis of a vector space often used to study the extrinsic differential geometry of smooth manifolds embedded in a homogeneous space.

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Multilinear algebra

In mathematics, multilinear algebra extends the methods of linear algebra.

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Order of approximation

In science, engineering, and other quantitative disciplines, orders of approximation refer to formal or informal terms for how precise an approximation is, and to indicate progressively more refined approximations: in increasing order of precision, a zeroth-order approximation, a first-order approximation, a second-order approximation, and so forth.

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Orthonormality

In linear algebra, two vectors in an inner product space are orthonormal if they are orthogonal and unit vectors.

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Parallel transport

In geometry, parallel transport is a way of transporting geometrical data along smooth curves in a manifold.

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Parallelizable manifold

In mathematics, a differentiable manifold M of dimension n is called parallelizable if there exist smooth vector fields on the manifold, such that at any point p of M the tangent vectors provide a basis of the tangent space at p. Equivalently, the tangent bundle is a trivial bundle, so that the associated principal bundle of linear frames has a section on M. A particular choice of such a basis of vector fields on M is called a parallelization (or an absolute parallelism) of M.

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Paul Painlevé

Paul Painlevé (5 December 1863 – 29 October 1933) was a French mathematician and statesman.

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Pressure

Pressure (symbol: p or P) is the force applied perpendicular to the surface of an object per unit area over which that force is distributed.

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Principal bundle

In mathematics, a principal bundle is a mathematical object that formalizes some of the essential features of the Cartesian product of a space with a group.

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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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Riemannian manifold

In differential geometry, a (smooth) Riemannian manifold or (smooth) Riemannian space (M,g) is a real, smooth manifold M equipped with an inner product g_p on the tangent space T_pM at each point p that varies smoothly from point to point in the sense that if X and Y are differentiable vector fields on M, then p \mapsto g_p(X(p),Y(p)) is a smooth function.

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Rotation

A rotation is a circular movement of an object around a center (or point) of rotation.

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Schwarzschild metric

In Einstein's theory of general relativity, the Schwarzschild metric (also known as the Schwarzschild vacuum or Schwarzschild solution) is the solution to the Einstein field equations that describes the gravitational field outside a spherical mass, on the assumption that the electric charge of the mass, angular momentum of the mass, and universal cosmological constant are all zero.

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Section (fiber bundle)

In the mathematical field of topology, a section (or cross section) of a fiber bundle E is a continuous right inverse of the projection function \pi.

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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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Special relativity

In physics, special relativity (SR, also known as the special theory of relativity or STR) is the generally accepted and experimentally well-confirmed physical theory regarding the relationship between space and time.

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Tangent bundle

In differential geometry, the tangent bundle of a differentiable manifold M is a manifold TM which assembles all the tangent vectors in M. As a set, it is given by the disjoint unionThe disjoint union ensures that for any two points x1 and x2 of manifold M the tangent spaces T1 and T2 have no common vector.

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Tangent space

In mathematics, the tangent space of a manifold facilitates the generalization of vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector that gives the displacement of the one point from the other.

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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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Tensor product

In mathematics, the tensor product of two vector spaces and (over the same field) is itself a vector space, together with an operation of bilinear composition, denoted by, from ordered pairs in the Cartesian product into, in a way that generalizes the outer product.

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Test particle

In physical theories, a test particle is an idealized model of an object whose physical properties (usually mass, charge, or size) are assumed to be negligible except for the property being studied, which is considered to be insufficient to alter the behavior of the rest of the system.

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Tetrad formalism

The tetrad formalism is an approach to general relativity that replaces the choice of a coordinate basis by the less restrictive choice of a local basis for the tangent bundle, i.e. a locally defined set of four linearly independent vector fields called a tetrad.

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Tidal tensor

In Newton's theory of gravitation and in various relativistic classical theories of gravitation, such as general relativity, the tidal tensor represents.

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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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Vacuum solution (general relativity)

In general relativity, a vacuum solution is a Lorentzian manifold whose Einstein tensor vanishes identically.

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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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Vector field

In vector calculus and physics, a vector field is an assignment of a vector to each point in a subset of space.

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World line

The world line (or worldline) of an object is the path that object traces in -dimensional spacetime.

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Yusuke Hagihara

was a Japanese astronomer noted for his contributions to celestial mechanics.

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Redirects here:

Co-tetrad, Coframe fields in general relativity, Cotetrad, Cotetrad field, Frame field, Frame fields, Tetrad (general relativity), Tetrads in general relativity.

References

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

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