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Minkowski space

Index 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. [1]

146 relations: Abuse of notation, Academic Press, Affine space, Albert Einstein, Alfred A. Knopf, Annals of Mathematics, Basis (linear algebra), Bilinear form, Birkhäuser, Born coordinates, Cambridge University Press, Cartesian coordinate system, Classical group, Conformal field theory, Cornelius Lanczos, Cotangent space, Course of Theoretical Physics, Covariance and contravariance of vectors, Curved space, De Sitter space, Definite quadratic form, Degenerate bilinear form, Derivations of the Lorentz transformations, Differential form, Differential geometry, Directional derivative, Dot product, Dual basis, Dual space, Einstein field equations, Einstein notation, Electromagnetic tensor, Euclidean distance, Euclidean group, Euclidean space, Event (relativity), Exterior derivative, Faster-than-light, Four-dimensional space, Four-momentum, Four-vector, Galilean invariance, Galilean transformation, General relativity, Gravitation (book), Gravity, Henri Poincaré, Hermann Minkowski, Hyperbolic geometry, Hyperbolic orthogonality, ..., Hyperbolic space, Hyperboloid, Hyperboloid model, Hyperplane, Imaginary time, Imaginary unit, Inclusion map, Induced metric, Inertial frame of reference, Inner product space, Introduction to the mathematics of general relativity, Invariant (physics), Isometry group, Jacobian matrix and determinant, Johns Hopkins University Press, Kernel (linear algebra), Length contraction, Light cone, Light-cone coordinates, Line element, Linear form, Local reference frame, Lorentz factor, Lorentz group, Lorentz transformation, M-theory, Manifold, Mathematical physics, Matter wave, Maxwell's equations, McGraw-Hill Education, Metric signature, Metric space, Metric tensor, Metric tensor (general relativity), Minkowski diagram, Minkowski plane, Nash embedding theorem, Non-Euclidean geometry, One-form, Orthogonal group, Orthogonality, Orthonormal basis, Orthonormality, Oxford University Press, Parallel transport, Partially ordered set, Poincaré disk model, Poincaré group, Poincaré half-plane model, Polarization identity, Postulates of special relativity, Proper time, Pseudo-Euclidean space, Pseudo-Riemannian manifold, Pullback (differential geometry), Pushforward (differential), Pythagorean theorem, Quadratic form, Raising and lowering indices, Real number, Reflection (mathematics), Relativistic mechanics, Relativity of simultaneity, Riemannian geometry, Riemannian manifold, Riesz representation theorem, Rotation group SO(3), Rotation matrix, Sign convention, Spacetime, Special relativity, Speed of light, Sphere, Springer Science+Business Media, Stereographic projection, Steven Weinberg, String theory, Submanifold, Sylvester's law of inertia, Symmetric bilinear form, T-symmetry, Tangent space, Tensor, Tensor product, Three-dimensional space, Time, Time dilation, Transitive relation, Translation (geometry), Transpose of a linear map, Unit vector, Vector field, Vector space, World line, 3-sphere. Expand index (96 more) »

Abuse of notation

In mathematics, abuse of notation occurs when an author uses a mathematical notation in a way that is not formally correct but that seems likely to simplify the exposition or suggest the correct intuition (while being unlikely to introduce errors or cause confusion).

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Academic Press

Academic Press is an academic book publisher.

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Affine space

In mathematics, an affine space is a geometric structure that generalizes the properties of Euclidean spaces in such a way that these are independent of the concepts of distance and measure of angles, keeping only the properties related to parallelism and ratio of lengths for parallel line segments.

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Albert Einstein

Albert Einstein (14 March 1879 – 18 April 1955) was a German-born theoretical physicist who developed the theory of relativity, one of the two pillars of modern physics (alongside quantum mechanics).

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Alfred A. Knopf

Alfred A. Knopf, Inc. is a New York publishing house that was founded by Alfred A. Knopf Sr. and Blanche Knopf in 1915.

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Annals of Mathematics

The Annals of Mathematics is a bimonthly mathematical journal published by Princeton University and the Institute for Advanced Study.

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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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Bilinear form

In mathematics, more specifically in abstract algebra and linear algebra, a bilinear form on a vector space V is a bilinear map, where K is the field of scalars.

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Birkhäuser

Birkhäuser is a former Swiss publisher founded in 1879 by Emil Birkhäuser.

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Born coordinates

In relativistic physics, the Born coordinate chart is a coordinate chart for (part of) Minkowski spacetime, the flat spacetime of special relativity.

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Cambridge University Press

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

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Cartesian coordinate system

A Cartesian coordinate system is a coordinate system that specifies each point uniquely in a plane by a pair of numerical coordinates, which are the signed distances to the point from two fixed perpendicular directed lines, measured in the same unit of length.

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Classical group

In mathematics, the classical groups are defined as the special linear groups over the reals, the complex numbers and the quaternions together with special automorphism groups of symmetric or skew-symmetric bilinear forms and Hermitian or skew-Hermitian sesquilinear forms defined on real, complex and quaternionic finite-dimensional vector spaces.

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Conformal field theory

A conformal field theory (CFT) is a quantum field theory that is invariant under conformal transformations.

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Cornelius Lanczos

Cornelius (Cornel) Lanczos (Lánczos Kornél,, born as Kornél Lőwy, until 1906: Löwy (Lőwy) Kornél) was a Jewish Hungarian mathematician and physicist, who was born on February 2, 1893, and died on June 25, 1974.

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Cotangent space

In differential geometry, one can attach to every point x of a smooth (or differentiable) manifold a vector space called the cotangent space at x. Typically, the cotangent space is defined as the dual space of the tangent space at x, although there are more direct definitions (see below).

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Course of Theoretical Physics

The Course of Theoretical Physics is a ten-volume series of books covering theoretical physics that was initiated by Lev Landau and written in collaboration with his student Evgeny Lifshitz starting in the late 1930s.

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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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Curved space

Curved space often refers to a spatial geometry which is not "flat" where a flat space is described by Euclidean 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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Definite quadratic form

In mathematics, a definite quadratic form is a quadratic form over some real vector space that has the same sign (always positive or always negative) for every nonzero vector of.

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Degenerate bilinear form

In mathematics, specifically linear algebra, a degenerate bilinear form on a vector space V is a bilinear form such that the map from V to V∗ (the dual space of V) given by is not an isomorphism.

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Derivations of the Lorentz transformations

There are many ways to derive the Lorentz transformations utilizing a variety of physical principles, ranging from Maxwell's equations to Einstein's postulates of special relativity, and mathematical tools, spanning from elementary algebra and hyperbolic functions, to linear algebra and group theory.

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Differential form

In the mathematical fields of differential geometry and tensor calculus, differential forms are an approach to multivariable calculus that is independent of coordinates.

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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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Directional derivative

In mathematics, the directional derivative of a multivariate differentiable function along a given vector v at a given point x intuitively represents the instantaneous rate of change of the function, moving through x with a velocity specified by v. It therefore generalizes the notion of a partial derivative, in which the rate of change is taken along one of the curvilinear coordinate curves, all other coordinates being constant.

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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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Dual basis

In linear algebra, given a vector space V with a basis B of vectors indexed by an index set I (the cardinality of I is the dimensionality of V), its dual set is a set B∗ of vectors in the dual space V∗ with the same index set I such that B and B∗ form a biorthogonal system.

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Dual space

In mathematics, any vector space V has a corresponding dual vector space (or just dual space for short) consisting of all linear functionals on V, together with the vector space structure of pointwise addition and scalar multiplication by constants.

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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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Electromagnetic tensor

In electromagnetism, the electromagnetic tensor or electromagnetic field tensor (sometimes called the field strength tensor, Faraday tensor or Maxwell bivector) is a mathematical object that describes the electromagnetic field in spacetime.

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Euclidean distance

In mathematics, the Euclidean distance or Euclidean metric is the "ordinary" straight-line distance between two points in Euclidean space.

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Euclidean group

In mathematics, the Euclidean group E(n), also known as ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space.

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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 (relativity)

In physics, and in particular relativity, an event is the instantaneous physical situation or occurrence associated with a point in spacetime (that is, a specific place and time).

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Exterior derivative

On a differentiable manifold, the exterior derivative extends the concept of the differential of a function to differential forms of higher degree.

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Faster-than-light

Faster-than-light (also superluminal or FTL) communication and travel are the conjectural propagation of information or matter faster than the speed of light.

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Four-dimensional space

A four-dimensional space or 4D space is a mathematical extension of the concept of three-dimensional or 3D space.

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Four-momentum

In special relativity, four-momentum is the generalization of the classical three-dimensional momentum to four-dimensional spacetime.

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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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Galilean invariance

Galilean invariance or Galilean relativity states that the laws of motion are the same in all inertial frames.

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Galilean transformation

In physics, a Galilean transformation is used to transform between the coordinates of two reference frames which differ only by constant relative motion within the constructs of Newtonian physics.

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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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Gravitation (book)

Gravitation is a physics book on Einstein's theory of gravity, written by Charles W. Misner, Kip S. Thorne, and John Archibald Wheeler and originally published by W. H. Freeman and Company in 1973.

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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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Henri Poincaré

Jules Henri Poincaré (29 April 1854 – 17 July 1912) was a French mathematician, theoretical physicist, engineer, and philosopher of science.

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Hermann Minkowski

Hermann Minkowski (22 June 1864 – 12 January 1909) was a German mathematician and professor at Königsberg, Zürich and Göttingen.

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Hyperbolic geometry

In mathematics, hyperbolic geometry (also called Bolyai–Lobachevskian geometry or Lobachevskian geometry) is a non-Euclidean geometry.

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Hyperbolic orthogonality

In plane geometry, two lines are hyperbolic orthogonal when they are reflections of each other over the asymptote of a given hyperbola.

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Hyperbolic space

In mathematics, hyperbolic space is a homogeneous space that has a constant negative curvature, where in this case the curvature is the sectional curvature.

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Hyperboloid

In geometry, a hyperboloid of revolution, sometimes called circular hyperboloid, is a surface that may be generated by rotating a hyperbola around one of its principal axes.

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Hyperboloid model

In geometry, the hyperboloid model, also known as the Minkowski model or the Lorentz model (after Hermann Minkowski and Hendrik Lorentz), is a model of n-dimensional hyperbolic geometry in which points are represented by the points on the forward sheet S+ of a two-sheeted hyperboloid in (n+1)-dimensional Minkowski space and m-planes are represented by the intersections of the (m+1)-planes in Minkowski space with S+.

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Hyperplane

In geometry, a hyperplane is a subspace whose dimension is one less than that of its ambient space.

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Imaginary time

Imaginary time is a mathematical representation of time which appears in some approaches to special relativity and quantum mechanics.

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Imaginary unit

The imaginary unit or unit imaginary number is a solution to the quadratic equation.

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Inclusion map

In mathematics, if A is a subset of B, then the inclusion map (also inclusion function, insertion, or canonical injection) is the function \iota that sends each element, x, of A to x, treated as an element of B: A "hooked arrow" is sometimes used in place of the function arrow above to denote an inclusion map; thus: \iota: A\hookrightarrow B. (On the other hand, this notation is sometimes reserved for embeddings.) This and other analogous injective functions from substructures are sometimes called natural injections.

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Induced metric

In mathematics and theoretical physics, the induced metric is the metric tensor defined on a submanifold which is calculated from the metric tensor on a larger manifold into which the submanifold is embedded.

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Inertial frame of reference

An inertial frame of reference in classical physics and special relativity is a frame of reference in which a body with zero net force acting upon it is not accelerating; that is, such a body is at rest or it is moving at a constant speed in a straight line.

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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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Introduction to the mathematics of general relativity

The mathematics of general relativity is complex.

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Invariant (physics)

In mathematics and theoretical physics, an invariant is a property of a system which remains unchanged under some transformation.

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Isometry group

In mathematics, the isometry group of a metric space is the set of all bijective isometries (i.e. bijective, distance-preserving maps) from the metric space onto itself, with the function composition as group operation.

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Jacobian matrix and determinant

In vector calculus, the Jacobian matrix is the matrix of all first-order partial derivatives of a vector-valued function.

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Johns Hopkins University Press

The Johns Hopkins University Press (also referred to as JHU Press or JHUP) is the publishing division of Johns Hopkins University.

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Kernel (linear algebra)

In mathematics, and more specifically in linear algebra and functional analysis, the kernel (also known as null space or nullspace) of a linear map between two vector spaces V and W, is the set of all elements v of V for which, where 0 denotes the zero vector in W. That is, in set-builder notation,.

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Length contraction

Length contraction is the phenomenon that a moving object's length is measured to be shorter than its proper length, which is the length as measured in the object's own rest frame.

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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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Light-cone coordinates

In special relativity, light-cone coordinates is a special coordinate system where two of the coordinates, x+ and x− are null coordinates and all the other coordinates are spatial.

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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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Linear form

In linear algebra, a linear functional or linear form (also called a one-form or covector) is a linear map from a vector space to its field of scalars.

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Local reference frame

In theoretical physics, a local reference frame (local frame) refers to a coordinate system or frame of reference that is only expected to function over a small region or a restricted region of space or spacetime.

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Lorentz factor

The Lorentz factor or Lorentz term is the factor by which time, length, and relativistic mass change for an object while that object is moving.

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Lorentz group

In physics and mathematics, the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical and quantum setting for all (nongravitational) physical phenomena.

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Lorentz transformation

In physics, the Lorentz transformations (or transformation) are coordinate transformations between two coordinate frames that move at constant velocity relative to each other.

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M-theory

M-theory is a theory in physics that unifies all consistent versions of superstring theory.

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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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Mathematical physics

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

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Matter wave

Matter waves are a central part of the theory of quantum mechanics, being an example of wave–particle duality.

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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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McGraw-Hill Education

McGraw-Hill Education (MHE) is a learning science company and one of the "big three" educational publishers that provides customized educational content, software, and services for pre-K through postgraduate education.

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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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Metric space

In mathematics, a metric space is a set for which distances between all members of the set are defined.

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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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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 diagram

The Minkowski diagram, also known as a spacetime diagram, was developed in 1908 by Hermann Minkowski and provides an illustration of the properties of space and time in the special theory of relativity.

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Minkowski plane

In mathematics, a Minkowski plane (named after Hermann Minkowski) is one of the Benz planes: Möbius plane, Laguerre plane and Minkowski plane.

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Nash embedding theorem

The Nash embedding theorems (or imbedding theorems), named after John Forbes Nash, state that every Riemannian manifold can be isometrically embedded into some Euclidean space.

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Non-Euclidean geometry

In mathematics, non-Euclidean geometry consists of two geometries based on axioms closely related to those specifying Euclidean geometry.

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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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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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Orthonormal basis

In mathematics, particularly linear algebra, an orthonormal basis for an inner product space V with finite dimension is a basis for V whose vectors are orthonormal, that is, they are all unit vectors and orthogonal to each other.

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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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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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Parallel transport

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

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Partially ordered set

In mathematics, especially order theory, a partially ordered set (also poset) formalizes and generalizes the intuitive concept of an ordering, sequencing, or arrangement of the elements of a set.

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Poincaré disk model

In geometry, the Poincaré disk model, also called the conformal disk model, is a model of 2-dimensional hyperbolic geometry in which the points of the geometry are inside the unit disk, and the straight lines consist of all segments of circles contained within that disk that are orthogonal to the boundary of the disk, plus all diameters of the disk.

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Poincaré group

The Poincaré group, named after Henri Poincaré (1906), was first defined by Minkowski (1908) as the group of Minkowski spacetime isometries.

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Poincaré half-plane model

In non-Euclidean geometry, the Poincaré half-plane model is the upper half-plane, denoted below as H \, together with a metric, the Poincaré metric, that makes it a model of two-dimensional hyperbolic geometry.

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Polarization identity

In mathematics, the polarization identity is any one of a family of formulas that express the inner product of two vectors in terms of the norm of a normed vector space.

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Postulates of special relativity

1.

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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-Euclidean space

In mathematics and theoretical physics, a pseudo-Euclidean space is a finite-dimensional ''n''-space together with a non-degenerate quadratic form.

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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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Pullback (differential geometry)

Suppose that φ:M→ N is a smooth map between smooth manifolds M and N; then there is an associated linear map from the space of 1-forms on N (the linear space of sections of the cotangent bundle) to the space of 1-forms on M. This linear map is known as the pullback (by φ), and is frequently denoted by φ*.

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Pushforward (differential)

Suppose that is a smooth map between smooth manifolds; then the differential of φ at a point x is, in some sense, the best linear approximation of φ near x. It can be viewed as a generalization of the total derivative of ordinary calculus.

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Pythagorean theorem

In mathematics, the Pythagorean theorem, also known as Pythagoras' theorem, is a fundamental relation in Euclidean geometry among the three sides of a right triangle.

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Quadratic form

In mathematics, a quadratic form is a homogeneous polynomial of degree two in a number of variables.

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Raising and lowering indices

In mathematics and mathematical physics, raising and lowering indices are operations on tensors which change their type.

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Real number

In mathematics, a real number is a value of a continuous quantity that can represent a distance along a line.

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Reflection (mathematics)

In mathematics, a reflection (also spelled reflexion) is a mapping from a Euclidean space to itself that is an isometry with a hyperplane as a set of fixed points; this set is called the axis (in dimension 2) or plane (in dimension 3) of reflection.

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Relativistic mechanics

In physics, relativistic mechanics refers to mechanics compatible with special relativity (SR) and general relativity (GR).

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Relativity of simultaneity

In physics, the relativity of simultaneity is the concept that distant simultaneity – whether two spatially separated events occur at the same time – is not absolute, but depends on the observer's reference frame.

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Riemannian geometry

Riemannian geometry is the branch of differential geometry that studies Riemannian manifolds, smooth manifolds with a Riemannian metric, i.e. with an inner product on the tangent space at each point that varies smoothly from point to point.

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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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Riesz representation theorem

There are several well-known theorems in functional analysis known as the Riesz representation theorem.

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Rotation group SO(3)

In mechanics and geometry, the 3D rotation group, often denoted SO(3), is the group of all rotations about the origin of three-dimensional Euclidean space R3 under the operation of composition.

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Rotation matrix

In linear algebra, a rotation matrix is a matrix that is used to perform a rotation in Euclidean space.

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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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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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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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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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Stereographic projection

In geometry, the stereographic projection is a particular mapping (function) that projects a sphere onto a plane.

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Steven Weinberg

Steven Weinberg (born May 3, 1933) is an American theoretical physicist and Nobel laureate in Physics for his contributions with Abdus Salam and Sheldon Glashow to the unification of the weak force and electromagnetic interaction between elementary particles.

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String theory

In physics, string theory is a theoretical framework in which the point-like particles of particle physics are replaced by one-dimensional objects called strings.

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Submanifold

In mathematics, a submanifold of a manifold M is a subset S which itself has the structure of a manifold, and for which the inclusion map S → M satisfies certain properties.

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Sylvester's law of inertia

Sylvester's law of inertia is a theorem in matrix algebra about certain properties of the coefficient matrix of a real quadratic form that remain invariant under a change of basis.

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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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T-symmetry

T-symmetry or time reversal symmetry is the theoretical symmetry of physical laws under the transformation of time reversal: T-symmetry can be shown to be equivalent to the conservation of entropy, by Noether's Theorem.

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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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Three-dimensional space

Three-dimensional space (also: 3-space or, rarely, tri-dimensional space) is a geometric setting in which three values (called parameters) are required to determine the position of an element (i.e., point).

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Time

Time is the indefinite continued progress of existence and events that occur in apparently irreversible succession from the past through the present to the future.

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Time dilation

According to the theory of relativity, time dilation is a difference in the elapsed time measured by two observers, either due to a velocity difference relative to each other, or by being differently situated relative to a gravitational field.

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Transitive relation

In mathematics, a binary relation over a set is transitive if whenever an element is related to an element and is related to an element then is also related to.

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Translation (geometry)

In Euclidean geometry, a translation is a geometric transformation that moves every point of a figure or a space by the same distance in a given direction.

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Transpose of a linear map

In linear algebra, the transpose of a linear map between two vector spaces, defined over the same field, is an induced map between the dual spaces of the two vector spaces.

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Unit vector

In mathematics, a unit vector in a normed vector space is a vector (often a spatial vector) of length 1.

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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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Vector space

A vector space (also called a linear space) is a collection of objects called vectors, which may be added together and multiplied ("scaled") by numbers, called scalars.

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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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3-sphere

In mathematics, a 3-sphere, or glome, is a higher-dimensional analogue of a sphere.

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4-dimensional spacetime, Flat space, Flat spacetime, Four-dimensional spacetime, Geometry of special relativity, Hyperspace theory, Lightlike, Lightlike separation, Lm distance, Locally flat spacetime, Minkowski Metric, Minkowski Space, Minkowski geometry, Minkowski inner product, Minkowski metric, Minkowski metric tensor, Minkowski metrics, Minkowski signature, Minkowski space-time, Minkowski spacetime, Minkowski tensor, Minkowskian, Minkowskian geometry, Null vector (Minkowski space), One timelike dimension, Spacelike vector, Timelike vector.

References

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

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