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Metric tensor (general relativity)

Index Metric tensor (general relativity)

In general relativity, the metric tensor (in this context often abbreviated to simply the metric) is the fundamental object of study. [1]

82 relations: Abstract index notation, Alternatives to general relativity, Anti-de Sitter space, Atlas (topology), Causal structure, Christoffel symbols, Connection (mathematics), Covariance and contravariance of vectors, Curvature, De Sitter space, Degeneracy (mathematics), Deriving the Schwarzschild solution, Determinant, Differentiable manifold, Dot product, Eddington–Finkelstein coordinates, Eigenvalues and eigenvectors, Einstein field equations, Einstein notation, Energy, Euclidean space, Event horizon, Four-vector, Friedmann–Lemaître–Robertson–Walker metric, Fundamental theorem of Riemannian geometry, Gödel metric, General relativity, Gradient, Gravitational constant, Gravitational potential, Gravitational singularity, Gravity, Gullstrand–Painlevé coordinates, Integrable system, Introduction to the mathematics of general relativity, Invertible matrix, Isotropic coordinates, Kerr metric, Kerr–Newman metric, Kruskal–Szekeres coordinates, Lemaître coordinates, Lemaître–Tolman metric, Levi-Civita connection, Light cone, Line element, Local coordinates, Mass, Mathematics of general relativity, Matter, Metric signature, ..., Minkowski space, One-form, Partial differential equation, Peres metric, Positive definiteness, Proper length, Proper time, Pseudo-Riemannian manifold, Region (mathematics), Reissner–Nordström metric, Ricci calculus, Ricci curvature, Riemann curvature tensor, Rindler coordinates, Scalar curvature, Schwarzschild metric, Sign convention, Spacetime, Special relativity, Sphere, Spherical coordinate system, Stress–energy tensor, Symmetric bilinear form, Symmetric matrix, Symmetric tensor, Tangent space, Tensor, Tensor field, Tensor product, Torsion tensor, Volume form, Weyl−Lewis−Papapetrou coordinates. Expand index (32 more) »

Abstract index notation

Abstract index notation is a mathematical notation for tensors and spinors that uses indices to indicate their types, rather than their components in a particular basis.

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Alternatives to general relativity

Alternatives to general relativity are physical theories that attempt to describe the phenomenon of gravitation in competition to Einstein's theory of general relativity.

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Anti-de Sitter space

In mathematics and physics, n-dimensional anti-de Sitter space (AdSn) is a maximally symmetric Lorentzian manifold with constant negative scalar curvature.

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Atlas (topology)

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

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Causal structure

In mathematical physics, the causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold.

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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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Connection (mathematics)

In geometry, the notion of a connection makes precise the idea of transporting data along a curve or family of curves in a parallel and consistent manner.

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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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Curvature

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

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De Sitter space

In mathematics and physics, a de Sitter space is the analog in Minkowski space, or spacetime, of a sphere in ordinary Euclidean space.

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Degeneracy (mathematics)

In mathematics, a degenerate case is a limiting case in which an element of a class of objects is qualitatively different from the rest of the class and hence belongs to another, usually simpler, class.

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Deriving the Schwarzschild solution

The Schwarzschild solution describes spacetime in the vicinity of a non-rotating massive spherically-symmetric object.

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Determinant

In linear algebra, the determinant is a value that can be computed from the elements of a square matrix.

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Differentiable manifold

In mathematics, a differentiable manifold (also differential manifold) is a type of manifold that is locally similar enough to a linear space to allow one to do calculus.

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Dot product

In mathematics, the dot product or scalar productThe term scalar product is often also used more generally to mean a symmetric bilinear form, for example for a pseudo-Euclidean space.

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Eddington–Finkelstein coordinates

In general relativity, Eddington–Finkelstein coordinates are a pair of coordinate systems for a Schwarzschild geometry (i.e. a spherically symmetric black hole) which are adapted to radial null geodesics.

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Eigenvalues and eigenvectors

In linear algebra, an eigenvector or characteristic vector of a linear transformation is a non-zero vector that changes by only a scalar factor when that linear transformation is applied to it.

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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 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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Energy

In physics, energy is the quantitative property that must be transferred to an object in order to perform work on, or to heat, the object.

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Euclidean space

In geometry, Euclidean space encompasses the two-dimensional Euclidean plane, the three-dimensional space of Euclidean geometry, and certain other spaces.

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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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Four-vector

In special relativity, a four-vector (also known as a 4-vector) is an object with four components, which transform in a specific way under Lorentz transformation.

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Friedmann–Lemaître–Robertson–Walker metric

The Friedmann–Lemaître–Robertson–Walker (FLRW) metric is an exact solution of Einstein's field equations of general relativity; it describes a homogeneous, isotropic, expanding or contracting universe that is path connected, but not necessarily simply connected.

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Fundamental theorem of Riemannian geometry

In Riemannian geometry, the fundamental theorem of Riemannian geometry states that on any Riemannian manifold (or pseudo-Riemannian manifold) there is a unique torsion-free metric connection, called the Levi-Civita connection of the given metric.

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Gödel metric

The Gödel metric is an exact solution of the Einstein field equations in which the stress–energy tensor contains two terms, the first representing the matter density of a homogeneous distribution of swirling dust particles (dust solution), and the second associated with a nonzero cosmological constant (see lambdavacuum solution).

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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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Gradient

In mathematics, the gradient is a multi-variable generalization of the derivative.

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Gravitational constant

The gravitational constant (also known as the "universal gravitational constant", the "Newtonian constant of gravitation", or the "Cavendish gravitational constant"), denoted by the letter, is an empirical physical constant involved in the calculation of gravitational effects in Sir Isaac Newton's law of universal gravitation and in Albert Einstein's general theory of relativity.

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Gravitational potential

In classical mechanics, the gravitational potential at a location is equal to the work (energy transferred) per unit mass that would be needed to move the object from a fixed reference location to the location of the object.

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Gravitational singularity

A gravitational singularity or spacetime singularity is a location in spacetime where the gravitational field of a celestial body becomes infinite in a way that does not depend on the coordinate system.

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Gravity

Gravity, or gravitation, is a natural phenomenon by which all things with mass or energy—including planets, stars, galaxies, and even light—are brought toward (or gravitate toward) one another.

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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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Integrable system

In the context of differential equations to integrate an equation means to solve it from initial conditions.

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Introduction to the mathematics of general relativity

The mathematics of general relativity is complex.

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Invertible matrix

In linear algebra, an n-by-n square matrix A is called invertible (also nonsingular or nondegenerate) if there exists an n-by-n square matrix B such that where In denotes the n-by-n identity matrix and the multiplication used is ordinary matrix multiplication.

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Isotropic coordinates

In the theory of Lorentzian manifolds, spherically symmetric spacetimes admit a family of nested round spheres.

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Kerr metric

The Kerr metric or Kerr geometry describes the geometry of empty spacetime around a rotating uncharged axially-symmetric black hole with a spherical event horizon.

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Kerr–Newman metric

The Kerr–Newman metric is a solution of the Einstein–Maxwell equations in general relativity that describes the spacetime geometry in the region surrounding a charged, rotating mass.

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Kruskal–Szekeres coordinates

In general relativity Kruskal–Szekeres coordinates, named after Martin Kruskal and George Szekeres, are a coordinate system for the Schwarzschild geometry for a black hole.

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Lemaître coordinates

Lemaître coordinates are a particular set of coordinates for the Schwarzschild metric – a spherically symmetric solution to the Einstein field equations in a vacuum – obtained by Monsignor Georges Lemaître in 1932.

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Lemaître–Tolman metric

In mathematical physics, the Lemaître–Tolman metric is the spherically symmetric dust solution of Einstein's field equations.

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Levi-Civita connection

In Riemannian geometry, the Levi-Civita connection is a specific connection on the tangent bundle of a manifold.

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Light cone

In special and general relativity, a light cone is the path that a flash of light, emanating from a single event (localized to a single point in space and a single moment in time) and traveling in all directions, would take through spacetime.

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Line element

In geometry, the line element or length element can be informally thought of as a line segment associated with an infinitesimal displacement vector in a metric space.

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Local coordinates

Local coordinates are measurement indices into a local coordinate system or a local coordinate space.

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Mass

Mass is both a property of a physical body and a measure of its resistance to acceleration (a change in its state of motion) when a net force is applied.

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Mathematics of general relativity

The mathematics of general relativity refers to various mathematical structures and techniques that are used in studying and formulating Albert Einstein's theory of general relativity.

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Matter

In the classical physics observed in everyday life, matter is any substance that has mass and takes up space by having volume.

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Metric signature

The signature of a metric tensor g (or equivalently, a real quadratic form thought of as a real symmetric bilinear form on a finite-dimensional vector space) is the number (counted with multiplicity) of positive and zero eigenvalues of the real symmetric matrix of the metric tensor with respect to a basis.

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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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One-form

In linear algebra, a one-form on a vector space is the same as a linear functional on the space.

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Partial differential equation

In mathematics, a partial differential equation (PDE) is a differential equation that contains unknown multivariable functions and their partial derivatives.

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Peres metric

In mathematical physics, the Peres metric is defined by the proper time ^.

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Positive definiteness

In mathematics, positive definiteness is a property of any object to which a bilinear form or a sesquilinear form may be naturally associated, which is positive definite.

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Proper length

Proper length or rest length refers to the length of an object in the object's rest frame.

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Proper time

In relativity, proper time along a timelike world line is defined as the time as measured by a clock following that line.

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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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Region (mathematics)

In mathematical analysis, the word region usually refers to a subset of \R^n or \C^n that is open (in the standard Euclidean topology), connected and non-empty.

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Reissner–Nordström metric

In physics and astronomy, the Reissner–Nordström metric is a static solution to the Einstein-Maxwell field equations, which corresponds to the gravitational field of a charged, non-rotating, spherically symmetric body of mass M. The metric was discovered by Hans Reissner, Hermann Weyl, Gunnar Nordström and G. B. Jeffery.

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Ricci calculus

In mathematics, Ricci calculus constitutes the rules of index notation and manipulation for tensors and tensor fields.

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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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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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Rindler coordinates

In relativistic physics, the coordinates of a hyperbolically accelerated reference frame constitute an important and useful coordinate chart representing part of flat Minkowski spacetime.

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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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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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Sign convention

In physics, a sign convention is a choice of the physical significance of signs (plus or minus) for a set of quantities, in a case where the choice of sign is arbitrary.

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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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Sphere

A sphere (from Greek σφαῖρα — sphaira, "globe, ball") is a perfectly round geometrical object in three-dimensional space that is the surface of a completely round ball (viz., analogous to the circular objects in two dimensions, where a "circle" circumscribes its "disk").

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Spherical coordinate system

In mathematics, a spherical coordinate system is a coordinate system for three-dimensional space where the position of a point is specified by three numbers: the radial distance of that point from a fixed origin, its polar angle measured from a fixed zenith direction, and the azimuth angle of its orthogonal projection on a reference plane that passes through the origin and is orthogonal to the zenith, measured from a fixed reference direction on that plane.

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Stress–energy tensor

The stress–energy tensor (sometimes stress–energy–momentum tensor or energy–momentum tensor) is a tensor quantity in physics that describes the density and flux of energy and momentum in spacetime, generalizing the stress tensor of Newtonian physics.

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Symmetric bilinear form

A symmetric bilinear form on a vector space is a bilinear map from two copies of the vector space to the field of scalars such that the order of the two vectors does not affect the value of the map.

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Symmetric matrix

In linear algebra, a symmetric matrix is a square matrix that is equal to its transpose.

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Symmetric tensor

In mathematics, a symmetric tensor is a tensor that is invariant under a permutation of its vector arguments: for every permutation σ of the symbols Alternatively, a symmetric tensor of order r represented in coordinates as a quantity with r indices satisfies The space of symmetric tensors of order r on a finite-dimensional vector space is naturally isomorphic to the dual of the space of homogeneous polynomials of degree r on V. Over fields of characteristic zero, the graded vector space of all symmetric tensors can be naturally identified with the symmetric algebra on V. A related concept is that of the antisymmetric tensor or alternating form.

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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 field

In mathematics and physics, a tensor field assigns a tensor to each point of a mathematical space (typically a Euclidean space or manifold).

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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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Torsion tensor

In differential geometry, the notion of torsion is a manner of characterizing a twist or screw of a moving frame around a curve.

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Volume form

In mathematics, a volume form on a differentiable manifold is a top-dimensional form (i.e., a differential form of top degree).

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Weyl−Lewis−Papapetrou coordinates

In general relativity, the Weyl−Lewis−Papapetrou coordinates are a set of coordinates, used in the solutions to the vacuum region surrounding an axisymmetric distribution of mass–energy.

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Metric (general relativity), Metric theory of gravitation, Spacetime metric.

References

[1] https://en.wikipedia.org/wiki/Metric_tensor_(general_relativity)

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