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Encyclopedia > Minkowski space

In physics and mathematics, Minkowski space (or Minkowski spacetime) is the mathematical setting in which Einstein's theory of special relativity is most conveniently formulated. In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime. Minkowski space is named for the German mathematician Hermann Minkowski (See History). Since antiquity, people have tried to understand the behavior of matter: why unsupported objects drop to the ground, why different materials have different properties, and so forth. ... Wikibooks Wikiversity has more about this subject: School of Mathematics Wikiquote has a collection of quotations related to: Mathematics Look up Mathematics on Wiktionary, the free dictionary Wikimedia Commons has media related to: Mathematics Bogomolny, Alexander: Interactive Mathematics Miscellany and Puzzles. ... Albert Einstein photographed by Oren J. Turner in 1947. ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... Attempting to understand the nature of space has always been a prime occupation for philosophers and scientists. ... 8:17 am, August 6, 1945, Japanese time. ... This page is about a higher mathematics topic. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... Hermann Minkowski. ...

Note: This article only describes the mathematics of Minkowski space. For physical descriptions see Special relativity.

Contents

A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ...


Structure

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (-,+,+,+). Elements of Minkowski space are called four-vectors. Minkowski space is often denoted R1,3 to emphasize the signature, although it is also denoted M 4 or simply M. In mathematics, the real numbers are intuitively defined as numbers that are in one-to-one correspondence with the points on an infinite line—the number line. ... A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, a bilinear form on a vector space V over a field F is a mapping V × V → F which is linear in both arguments. ... The signature of a metric tensor (or more generally a nondegenerate symmetric bilinear form, thought of as quadratic form) is the number of positive and negative eigenvalues of the metric. ... In relativity, a four-vector is a vector in a four-dimensional real vector space, whose components transform like the space and time coordinates (ct, x, y, z) under spatial rotations and boosts (a change by a constant velocity to another inertial reference frame). ...


The Minkowski inner product

A notion very similar to the inner product, called the Minkowski inner product, can be defined for any two four-vectors of M. Given, V, W in M, the Minkowski inner product is a map eta : M times M rightarrow R, sometimes denoted by <·, ·> that satisfies four properties, three of which are: // Definition Inner Product of two vectors Given twoN-by-1 column vectors v and u, the inner product is defined as the scalar quantity α resulting from where or equivalently indicates the conjugate transpose operator applied to vector v. ...

1.  bilinear: eta (aU + V, W) , = a eta(U, W) + eta(V, W), ( forall a in R and forall U, V, W in M)

2.  symmetric: eta (V, W) , = eta (W, V) (forall V, W in M)

3.  nondegenerate: if eta (V, W) , = 0 forall W in M, then V , = 0,

Note that this is not an inner product in the usual sense, since it is not positive-definite, i.e. the Minkowski norm of a vector V, defined as V^2 , = eta(V, V), need not be positive. The positive-definite condition has been replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa). In mathematics, a definite bilinear form B is one for which B(v,v) has a fixed sign (positive or negative) when it is not 0. ...


Just as in Euclidean space, two vectors are said to be orthogonal if eta (V, W) , = 0. A vector V is called a unit vector if V^2 = pm 1. A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis. In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... In mathematics, orthogonal is synonymous with perpendicular when used as a simple adjective that is not part of any longer phrase with a standard definition. ... In mathematics, a unit vector in a normed vector space is a vector (most commonly a spatial vector) whose length is 1. ... In mathematics, an orthonormal basis of an inner product space V(i. ...


There is a theorem stating that any inner product space satisfying conditions 1-3 above always has an orthonormal basis. Furthermore, the theorem states that the number of positive and negative unit vectors in any such basis is fixed. This pair of numbers is called the signature of the inner product.


Then the fourth condition on η can be stated:

4.  The inner product η has signature (-,+,+,+)

Standard basis

A standard basis for Minkowski space is a set of four mutually orthogonal vectors (e0, e1, e2, e3) such that

-left(e_0right)^2 = (e_1)^2 = (e_2)^2 = (e_3)^2 = 1

These conditions can be written compactly in the following form:

langle e_mu, e_nu rangle = eta_{munu}

where μ and ν run over the values (0, 1, 2, 3) and the matrix η is given by

eta = begin{pmatrix}-1&0&0&00&1&0&00&0&1&00&0&0&1end{pmatrix}

Relative to a standard basis, the components of a vector V are written (V0,V1,V2,V3) and we use the Einstein notation to write V = Vμeμ. The component V0 is called the timelike component of V while the other three components are called the spatial components. For other topics related to Einstein see Einstein (disambig) In mathematics, especially in applications of linear algebra to physics, the Einstein notation or Einstein summation convention is a notational convention useful when dealing with coordinate equations or formulas. ...


In terms of components, the inner product between two vectors V and W is given by

langle V,Wrangle = eta_{munu}V^mu W^nu = -V^0W^0 + V^1W^1 + V^2W^2 + V^3W^3

and the norm-squared of a vector V is

V^2 , = eta_{munu}V^mu V^nu = -(V^0)^2+(V^1)^2+(V^2)^2+(V^3)^2

Alternative definition

The section above defines Minkowski space as a vector space. There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the Poincaré group with the Lorentz group as the stabilizer. See Erlangen program. A vector space (or linear space) is the basic object of study in the branch of mathematics called linear algebra. ... In mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. ... In mathematics, in particular in the theory of Lie groups, algebraic groups and topological groups, a homogeneous space for a group G is a manifold or topological space X on which G acts by symmetry in a transitive way; it is not assumed that the action of G is faithful. ... In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... This article is about the mathematical concept. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ...


Lorentz transformations

See: Lorentz transformations, Lorentz group, Poincaré group The Lorentz transformation (LT), named after its discoverer, the Dutch physicist and mathematician Hendrik Antoon Lorentz (1853-1928), forms the basis for the special theory of relativity, which has been introduced to remove contradictions between the theories of electromagnetism and classical mechanics. ... The Lorentz group is the group of all Lorentz transformations of Minkowski spacetime. ... In physics and mathematics, the Poincaré group is the group of isometries of Minkowski spacetime. ...


Causal structure

Four-vectors are classified according to the sign of their (Minkowski) inner product. For four-vectors, U, V and W, the classification is as follows:

  • V is timelike if and only if: eta_{ab}V^aV^b , = V^aV_a <0
  • U is spacelike if and only if eta_{ab}U^aU^b , = U^aU_a > 0
  • W is null (lightlike) if and only if eta_{ab}W^aW^b , =W^aW_a = 0


This terminology comes from the use of Minkowski space in the theory of relativity. The set of all null vectors at an event of Minkowski space constitutes the light cone of that event. Note that all these notions are independent of the frame of reference. Wikisource has original text related to this article: Relativity: The Special and General Theory Albert Einsteins theory of relativity is a set of two scientific theories in physics: special relativity and general relativity. ... In special relativity, a light cone is the pattern describing the temporal evolution of a flash of light in Minkowski spacetime. ...


A useful result regarding null vectors is that if two null vectors are orthogonal (zero inner product), then they must be proportional.


Once a direction of time is chosen, timelike and null vectors can be further decomposed into various classes. For timelike vectors we have

  1. future directed timelike vectors whose first component is negative, and
  2. past directed timelike vectors whose first component is positive.

Null vectors fall into three class:

  1. the zero vector, whose components in any basis are (0,0,0,0),
  2. future directed null vectors whose first component is negative, and
  3. past directed null vectors whose first component is positive.

Together with spacelike vectors there are 6 classes in all.


An orthonormal basis for Minkowski space necessarily consists of one timelike and three spacelike unit vectors. If one wishes to work with non-orthonormal bases it is possible to have other combinations of vectors. For example, one can easily construct a (non-orthonormal) basis consisting entirely of null vectors, called a null basis.


Locally flat spacetime

Strictly speaking, the use of the Minkowski space to describe physical systems over finite distances applies only in the Newtonian limit of systems without significant gravitation. In the case of significant gravitation, spacetime becomes curved and one must abandon special relativity in favor of the full theory of general relativity. In physics, the Newtonian limit refers to physical systems without significantly intense gravitation, in the sense that Newtons law of universal gravitation may used to obtain values that are correct to a high order. ... Gravitation is the tendency of massive objects to accelerate towards each other. ... A simple introduction to this subject is provided in Special relativity for beginners Special relativity (SR) or the special theory of relativity is the physical theory published in 1905 by Albert Einstein. ... General relativity (GR) is the geometrical theory of gravitation published by Albert Einstein in 1915. ...


Nevertheless, even in such cases, Minkowski space is still a good description in an infinitesimally small region surrounding any point (barring gravitational singularities). More abstractly, we say that in the presence of gravity spacetime is described by a curved 4-dimensional manifold for which the tangent space to any point is a 4-dimensional Minkowski space. Thus, the structure of Minkowski space is still essential in the description of general relativity. This page is about a higher mathematics topic. ... The tangent space of a manifold is a concept which needs to be introduced when generalizing vectors from affine spaces to general manifolds, since in the latter case one cannot simply subtract two points to obtain a vector pointing from one to the other. ...


In the limit of weak gravity, spacetime becomes flat and looks globally, not just locally, like Minkowski space. For this reason Minkowski space is often referred to as flat spacetime.


History

Minkowski space is named for the German mathematician Hermann Minkowski, who around 1907 realized that the theory of special relativity previously worked out by Einstein and Lorentz could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space. Hermann Minkowski. ... 1907 was a common year starting on Tuesday (see link for calendar). ... Albert Einstein photographed by Oren J. Turner in 1947. ... Painting of Hendrik Lorentz by Arnhemensis Hendrik Antoon Lorentz (July 18, 1853, Arnhem – February 4, 1928, Haarlem) was a Dutch physicist and the winner of the 1902 Nobel Prize in Physics for his work on electromagnetic radiation. ...

“The views of space and time which I wish to lay before you have sprung from the soil of experimental physics, and therein lies their strength. They are radical. Henceforth space by itself, and time by itself, are doomed to fade away into mere shadows, and only a kind of union of the two will preserve an independent reality.” – Hermann Minkowski, 1908 1908 is a leap year starting on Wednesday (link will take you to calendar). ...

See also

In mathematics, Euclidean space is a generalization of the 2- and 3-dimensional spaces studied by Euclid. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... Cherenkov effect in a swimming pool nuclear reactor. ... World line of the orbit of the Earth depicted in two spatial dimensions X and Y (the plane of the Earth orbit) and a time dimension, usually put as the vertical axis. ... In mathematics, in Riemannian geometry, the metric tensor is a tensor of rank 2 that is used to measure distance and angle in a space. ... In differential geometry, a pseudo-Riemannian manifold is a smooth manifold equipped with a smooth, symmetric, tensor which is nondegenerate at each point on the manifold. ... In mathematics, hyperbolic n-space, denoted Hn, is the maximally symmetric, simply connected, n-dimensional Riemannian manifold with constant sectional curvature −1. ... An influential research programme and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen. ... In mathematics, the split-complex numbers are an extension of the real numbers defined analogously to the complex numbers. ... A hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association for the Advancement of Science. ...

References

  • Naber, Gregory L., The Geometry of Minkowski Spacetime, Springer-Verlag, New York, 1992. ISBN 0-387-97848-8 (hardcover), ISBN 0-486-43235-1 (Dover paperback edition).

  Results from FactBites:
 
Minkowski space - Wikipedia, the free encyclopedia (965 words)
In this setting the three ordinary dimensions of space are combined with a single dimension of time to form a four-dimensional manifold for representing a spacetime.
There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the Poincaré group with the Lorentz group as the stabilizer.
Minkowski space is named for the German mathematician Hermann Minkowski, who around 1907 realized that the theory of special relativity previously worked out by Einstein and Lorentz could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space.
Hermann Minkowski - Wikipedia, the free encyclopedia (307 words)
Hermann Minkowski (June 22, 1864 - January 12, 1909) was a Jewish German mathematician who developed the geometrical theory of numbers and who used geometrical methods to solve difficult problems in number theory, mathematical physics, and the theory of relativity.
Hermann Minkowski was born in Aleksotas (a suburb of Kaunas, Lithuania), and educated in Germany at the Universities of Berlin and Königsberg, where he achieved his doctorate in 1885.
Minkowski explored the arithmetic of quadratic forms, especially concerning n variables, and his research into that topic led him to consider certain geometric properties in a space of n dimensions.
  More results at FactBites »

 
 

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