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Exact solutions in general relativity

Index 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. [1]

89 relations: Alcubierre drive, Bäcklund transform, Belinski–Zakharov transform, Binary pulsar, Cambridge University Press, Casimir effect, Classical field theory, Closed timelike curve, Conformal group, Constraint counting, Cosmological constant, Covariant derivative, Curvature form, Dark energy, Demetrios Christodoulou, Deriving the Schwarzschild solution, Differentiable manifold, Edward Witten, Eigenvalues and eigenvectors, Einstein field equations, Einstein tensor, Electromagnetic field, Electrovacuum solution, Emmy Noether, Energy condition, Ernst equation, Fluid, Fluid solution, Frame fields in general relativity, Friedmann–Lemaître–Robertson–Walker metric, Gödel metric, General relativity, Geometrized unit system, Gravitational constant, Gravitational wave, Group action, Heat equation, Integrable system, Inverse scattering transform, Korteweg–de Vries equation, Lagrangian (field theory), Lambdavacuum solution, Linear map, Local spacetime structure, Mathematical physics, Maxwell's equations, Meson, Metric tensor (general relativity), Minkowski space, Newman–Penrose formalism, ..., Nonlinear Schrödinger equation, Nonlinear system, Null dust solution, Oxford University Press, Partial differential equation, Perfect fluid, Perturbation theory, Petrov classification, Point reflection, Positive energy theorem, Post-Newtonian expansion, Pseudo-Riemannian manifold, Quintessence (physics), Ricci curvature, Ricci decomposition, Richard Schoen, Riemann curvature tensor, Roger Penrose, Rotation around a fixed axis, Scalar field, Scalar field solution, Scholarpedia, Segre classification, Sergiu Klainerman, Shing-Tung Yau, Solid mechanics, Soliton, Solutions of the Einstein field equations, Sophus Lie, Spacetime topology, Speed of light, Static spherically symmetric perfect fluid, Stationary spacetime, Stress–energy tensor, Tensor, Time translation symmetry, Vacuum solution (general relativity), Weyl tensor, Wormhole. Expand index (39 more) »

Alcubierre drive

The Alcubierre drive or Alcubierre warp drive (or Alcubierre metric, referring to metric tensor) is a speculative idea based on a solution of Einstein's field equations in general relativity as proposed by Mexican theoretical physicist Miguel Alcubierre, by which a spacecraft could achieve apparent faster-than-light travel if a configurable energy-density field lower than that of vacuum (that is, negative mass) could be created.

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Bäcklund transform

In mathematics, Bäcklund transforms or Bäcklund transformations (named after the Swedish mathematician Albert Victor Bäcklund) relate partial differential equations and their solutions.

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Belinski–Zakharov transform

The Belinski–Zakharov (inverse) transform is a nonlinear transformation that generates new exact solutions of the vacuum Einstein's field equation.

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Binary pulsar

A binary pulsar is a pulsar with a binary companion, often a white dwarf or neutron star.

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Cambridge University Press

Cambridge University Press (CUP) is the publishing business of the University of Cambridge.

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Casimir effect

In quantum field theory, the Casimir effect and the Casimir–Polder force are physical forces arising from a quantized field.

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Classical field theory

A classical field theory is a physical theory that predicts how one or more physical fields interact with matter through field equations.

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Closed timelike curve

In mathematical physics, a closed timelike curve (CTC) is a world line in a Lorentzian manifold, of a material particle in spacetime that is "closed", returning to its starting point.

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Conformal group

In mathematics, the conformal group of a space is the group of transformations from the space to itself that preserve angles.

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Constraint counting

In mathematics, constraint counting is counting the number of constraints in order to compare it with the number of variables, parameters, etc.

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Cosmological constant

In cosmology, the cosmological constant (usually denoted by the Greek capital letter lambda: Λ) is the value of the energy density of the vacuum of space.

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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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Curvature form

In differential geometry, the curvature form describes the curvature of a connection on a principal bundle.

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Dark energy

In physical cosmology and astronomy, dark energy is an unknown form of energy which is hypothesized to permeate all of space, tending to accelerate the expansion of the universe.

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Demetrios Christodoulou

Demetrios Christodoulou (Δημήτριος Χριστοδούλου; born October 19, 1951) is a Greek mathematician and physicist, who first became well known for his proof, together with Sergiu Klainerman, of the nonlinear stability of the Minkowski spacetime of special relativity in the framework of general relativity.

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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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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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Edward Witten

Edward Witten (born August 26, 1951) is an American theoretical physicist and professor of mathematical physics at the Institute for Advanced Study in Princeton, New Jersey.

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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 tensor

In differential geometry, the Einstein tensor (named after Albert Einstein; also known as the trace-reversed Ricci tensor) is used to express the curvature of a pseudo-Riemannian manifold.

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Electromagnetic field

An electromagnetic field (also EMF or EM field) is a physical field produced by electrically charged objects.

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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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Emmy Noether

Amalie Emmy NoetherEmmy is the Rufname, the second of two official given names, intended for daily use.

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Energy condition

In relativistic classical field theories of gravitation, particularly general relativity, an energy condition is one of various alternative conditions which can be applied to the matter content of the theory, when it is either not possible or desirable to specify this content explicitly.

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Ernst equation

In mathematics, the Ernst equation is a non-linear partial differential equation, named after the American physicist.

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Fluid

In physics, a fluid is a substance that continually deforms (flows) under an applied shear stress.

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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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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.

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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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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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Geometrized unit system

A geometrized unit system or geometric unit system is a system of natural units in which the base physical units are chosen so that the speed of light in vacuum, c, and the gravitational constant, G, are set equal to unity.

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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 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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Group action

In mathematics, an action of a group is a formal way of interpreting the manner in which the elements of the group correspond to transformations of some space in a way that preserves the structure of that space.

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Heat equation

The heat equation is a parabolic partial differential equation that describes the distribution of heat (or variation in temperature) in a given region over time.

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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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Inverse scattering transform

In mathematics, the inverse scattering transform is a method for solving some non-linear partial differential equations.

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Korteweg–de Vries equation

In mathematics, the Korteweg–de Vries equation (KdV equation for short) is a mathematical model of waves on shallow water surfaces.

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Lagrangian (field theory)

Lagrangian field theory is a formalism in classical field theory.

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Lambdavacuum solution

In general relativity, a lambdavacuum solution is an exact solution to the Einstein field equation in which the only term in the stress–energy tensor is a cosmological constant term.

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Linear map

In mathematics, a linear map (also called a linear mapping, linear transformation or, in some contexts, linear function) is a mapping between two modules (including vector spaces) that preserves (in the sense defined below) the operations of addition and scalar multiplication.

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Local spacetime structure

Local spacetime structure refers to the structure of spacetime on a local level, i.e. only considering those points in an open region of a point.

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Mathematical physics

Mathematical physics refers to the development of mathematical methods for application to problems in physics.

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Maxwell's equations

Maxwell's equations are a set of partial differential equations that, together with the Lorentz force law, form the foundation of classical electromagnetism, classical optics, and electric circuits.

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Meson

In particle physics, mesons are hadronic subatomic particles composed of one quark and one antiquark, bound together by strong interactions.

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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.

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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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Newman–Penrose formalism

The Newman–Penrose (NP) formalism The original paper by Newman and Penrose, which introduces the formalism, and uses it to derive example results.

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Nonlinear Schrödinger equation

In theoretical physics, the (one-dimensional) nonlinear Schrödinger equation (NLSE) is a nonlinear variation of the Schrödinger equation.

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Nonlinear system

In mathematics and science, a nonlinear system is a system in which the change of the output is not proportional to the change of the input.

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Null dust solution

In mathematical physics, a null dust solution (sometimes called a null fluid) is a Lorentzian manifold in which the Einstein tensor is null.

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Oxford University Press

Oxford University Press (OUP) is the largest university press in the world, and the second oldest after Cambridge University Press.

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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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Perfect fluid

In physics, a perfect fluid is a fluid that can be completely characterized by its rest frame mass density \rho_m; and isotropic pressure p. Real fluids are "sticky" and contain (and conduct) heat.

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Perturbation theory

Perturbation theory comprises mathematical methods for finding an approximate solution to a problem, by starting from the exact solution of a related, simpler problem.

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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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Point reflection

In geometry, a point reflection or inversion in a point (or inversion through a point, or central inversion) is a type of isometry of Euclidean space.

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Positive energy theorem

In general relativity, the positive energy theorem (more commonly known as the positive mass conjecture) states that, assuming the dominant energy condition, the mass of an asymptotically flat spacetime is non-negative; furthermore, the mass is zero only for Minkowski spacetime.

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Post-Newtonian expansion

Post-Newtonian expansions in general relativity are used for finding an approximate solution of the Einstein field equations for the metric 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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Quintessence (physics)

In physics, quintessence is a hypothetical form of dark energy, more precisely a scalar field, postulated as an explanation of the observation of an accelerating rate of expansion of the universe.

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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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Richard Schoen

Richard Melvin Schoen (born October 23, 1950) is an American mathematician known for his work in differential geometry.

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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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Roger Penrose

Sir Roger Penrose (born 8 August 1931) is an English mathematical physicist, mathematician and philosopher of science.

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Rotation around a fixed axis

Rotation around a fixed axis or about a fixed axis of revolution or motion with respect to a fixed axis of rotation is a special case of rotational motion.

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Scalar field

In mathematics and physics, a scalar field associates a scalar value to every point in a space – possibly physical space.

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Scalar field solution

In general relativity, a scalar field solution is an exact solution of the Einstein field equation in which the gravitational field is due entirely to the field energy and momentum of a scalar field.

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Scholarpedia

Scholarpedia is an English-language online wiki-based encyclopedia with features commonly associated with open-access online academic journals, which aims to have quality content.

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Segre classification

The Segre classification is an algebraic classification of rank two symmetric tensors.

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Sergiu Klainerman

Sergiu Klainerman (born May 13, 1950) is a mathematician known for his contributions to the study of hyperbolic differential equations and general relativity.

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Shing-Tung Yau

Shing-Tung Yau (born April 4, 1949) is a chinese and naturalized American mathematician.

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Solid mechanics

Solid mechanics is the branch of continuum mechanics that studies the behavior of solid materials, especially their motion and deformation under the action of forces, temperature changes, phase changes, and other external or internal agents.

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Soliton

In mathematics and physics, a soliton is a self-reinforcing solitary wave packet that maintains its shape while it propagates at a constant velocity.

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Solutions of the Einstein field equations

Solutions of the Einstein field equations are spacetimes that result from solving the Einstein field equations (EFE) of general relativity.

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Sophus Lie

Marius Sophus Lie (17 December 1842 – 18 February 1899) was a Norwegian mathematician.

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Spacetime topology

Spacetime topology is the topological structure of spacetime, a topic studied primarily in general relativity.

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Speed of light

The speed of light in vacuum, commonly denoted, is a universal physical constant important in many areas of physics.

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Static spherically symmetric perfect fluid

In metric theories of gravitation, particularly general relativity, a static spherically symmetric perfect fluid solution (a term which is often abbreviated as ssspf) is a spacetime equipped with suitable tensor fields which models a static round ball of a fluid with isotropic pressure.

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Stationary spacetime

In general relativity, specifically in the Einstein field equations, a spacetime is said to be stationary if it admits a Killing vector that is asymptotically timelike.

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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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Tensor

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

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Time translation symmetry

Time translation symmetry or temporal translation symmetry (TTS) is a mathematical transformation in physics that moves the times of events through a common interval.

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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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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.

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Wormhole

A wormhole is a concept that represents a solution of the Einstein field equations: a non-trivial resolution of the Ehrenfest paradox structure linking separate points in spacetime.

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Exact solution (general relativity), Exact solutions of Einstein's field equations.

References

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

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