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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. Physics (Greek Physis - φύσις in everyday terms is the Science of Matter and its motion. Mathematics is the body of Knowledge and Academic discipline that studies such concepts as Quantity, Structure, Space and Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial 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. Space is the extent within which Matter is physically extended and objects and Events have positions relative to one another For other uses see Time (disambiguation Time is a component of a measuring system used to sequence events to compare the durations of A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Minkowski space is named after the German mathematician Hermann Minkowski. Germany, officially the Federal Republic of Germany ( ˈbʊndəsʁepuˌbliːk ˈdɔʏtʃlant is a Country in Central Europe. A mathematician is a person whose primary area of study and research is the field of Mathematics. Hermann Minkowski ( June 22 1864 – January 12 1909) was a Russian born German Mathematician, of Jewish

In theoretical physics, Minkowski space is often compared to Euclidean space. While a Euclidean space has only spacelike dimensions, a Minkowski space has also one timelike dimension. SpaceTime is a patent-pending three dimensional graphical user interface that allows end users to search their content such as Google Google Images Yahoo! YouTube eBay Amazon and RSS Therefore the symmetry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the Poincaré group. The Symmetry group of an object ( Image, signal, etc eg in 1D 2D or 3D is the group of all Isometries under which it is In Mathematics, the Euclidean group E ( n) sometimes called ISO( n) or similar is the Symmetry group of n -dimensional In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime

Contents

Structure

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (−,+,+,+) (Some may also prefer the alternative signature (+,−,−,−)). In Mathematics, the real numbers may be described informally in several different ways In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V  ×  V  →  F, where The signature of a Metric tensor (or more generally a nondegenerate Symmetric bilinear form, thought of as Quadratic form) is the number of positive In other words, Minkowski space is a pseudo-Euclidean space with n = 4 and n−k = 1 (in a broader definition any n>1 is allowed). A pseudo-Euclidean space is a finite- Dimensional real Vector space together with a non-degenerate indefinite Quadratic form Elements of Minkowski space are called events or four-vectors. In relativity, a four-vector is a vector in a four-dimensional real Vector space, called Minkowski space. Minkowski space is often denoted R1,3 to emphasize the signature, although it is also denoted M4 or simply M. It is perhaps the simplest example of a pseudo-Riemannian manifold. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold.

The Minkowski inner product

This inner product is similar to the usual, Euclidean, inner product, but is used to describe a different geometry; the geometry is usually associated with relativity. In Mathematics, an inner product space is a Vector space with the additional Structure of inner product. Let M be a 4-dimensional real vector space. The Minkowski inner product is a map η: M × M → R (i. e. given any two vectors v, w in M we define η(v,w) as a real number) which satisfies properties (1), (2), (3) listed here, as well as property (4) given below:

1. bilinear η(au + v, w) = aη(u, w) + η(v, w)

for all a ∈ R and u, v, w in M.

2 symmetric η(v,w) = η(w,v)

for all v,w in M.

3. nondegenerate if η(v,w) = 0 for all w ∈ M then v = 0.

Note that this is not an inner product in the usual sense, since it is not positive-definite, i. In Mathematics, a definite bilinear form is a Bilinear form B such that B ( x, x) has a fixed e. the Minkowski norm of a vector v, defined as v2 = η(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). The inner product is said to be indefinite.

Just as in Euclidean space, two vectors v and w are said to be orthogonal if η(v, w) = 0. In Mathematics, two Vectors are orthogonal if they are Perpendicular, i But there is a paradigm shift in Minkowski space to include hyperbolic-orthogonal events in case v and w span a plane where η takes negative values. Paradigm shift, sometimes known as extraordinary science or revolutionary science, is the term first used by Thomas Kuhn in his influential In Mathematics, two points in the Cartesian plane are hyperbolically orthogonal if the Slopes of their rays from the origin are Reciprocal to This shift to a new paradigm is clarified by comparing the Euclidean structure of the ordinary complex number plane to the structure of the plane of split-complex numbers. The word paradigm ( Greek:παράδειγμα (paradigmacomposite from para- and the verb δείχνυμι "to show" as a whole -roughly- meaning "example" Complex plane In Mathematics, the complex numbers are an extension of the Real numbers obtained by adjoining an Imaginary unit, denoted In Linear algebra, a split-complex number is of the form z = x + y j where j2 = +1, and x and y are Real

A vector v is called a unit vector if η(v,v) = ±1. In Mathematics, a unit vector in a Normed vector space is a vector (often a spatial vector) whose length is 1 (the unit length A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis. 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 to 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. signature The bilinear form 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

−(e0)2 = (e1)2 = (e2)2 = (e3)2 = 1

These conditions can be written compactly in the following form:

〈 eμ , eν 〉 = ημν

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&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{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. In Mathematics, especially in applications of Linear algebra to Physics, the Einstein notation or Einstein summation convention is a notational The component v0 is called the timelike component of v while the other three components are called the spatial components.

In terms of components, the inner product between two vectors v and w is given by

〈 v,w 〉 = ημνvμ wν = −v0w0 + v1w1 + v2w2 + v3w3

and the norm-squared of a vector v is

v2 = ημν vμvν = −(v0)2 + (v1)2 + (v2)2 + (v3)2

Alternative definition

The section above defines Minkowski space as a vector space. In Mathematics, a vector space (or linear space) is a collection of objects (called vectors) that informally speaking may be scaled and added 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. In Mathematics, an affine space is an abstract structure that generalises the affine-geometric properties of Euclidean space. In Mathematics, particularly in the theories of Lie groups Algebraic groups and Topological groups a homogeneous space for a group In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Algebra and Geometry, a group action is a way of describing symmetries of objects using groups. See Erlangen program. An influential research program and manifesto was published in 1872 by Felix Klein, under the title Vergleichende Betrachtungen über neuere geometrische Forschungen

Note also that the term "Minkowski space" is also used for analogues in any dimension: n+1 dimensional Minkowski space is a vector space or affine space of real dimension n+1 on which there is an inner product or pseudo-Riemannian metric of signature (n,1), i. In Differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. e. , in the above terminology, n "pluses" and one "minus".

Lorentz transformations

See: Lorentz transformations, Lorentz group, Poincaré group

Causal structure

Main article: Causal structure

Vectors are classified according to the sign of their (Minkowski) norm. In Physics, the Lorentz transformation converts between two different observers' measurements of space and time where one observer is in constant motion with respect to In Physics (and mathematics the Lorentz group is the group of all Lorentz transformations of Minkowski spacetime, the classical setting In Physics and Mathematics, the Poincaré group, named after Henri Poincaré, is the group of isometries of Minkowski spacetime The causal structure of a Lorentzian manifold describes the causal relationships between points in the manifold A vector v is:

Timelike if η(v,v) < 0
Spacelike if η(v,v) > 0
Null (or lightlike) if η(v,v) = 0

This terminology comes from the use of Minkowski space in the theory of relativity. This page is about the scientific concept of relativity for philosophical or sociological theories about relativity see Relativism. The set of all null vectors at an event of Minkowski space constitutes the light cone of that event. In Special relativity, a light cone (or null cone) is the pattern describing the temporal evolution of a flash of Light in Minkowski spacetime Note that all these notions are independent of the frame of reference.

Given a timelike vector v, there is a worldline of constant velocity associated with it. In physics the world line of an object is the unique path of that object as it travels through 4- Dimensional Spacetime. The set {w : η(w,v) = 0 } corresponds to the simultaneous hyperplane at the origin of this worldline. Minkowski space exhibits relativity of simultaneity since this hyperplane depends on v. The relativity of simultaneity is the concept that simultaneity is not absolute but dependent on the observer A hyperplane is a concept in Geometry. It is a higher-dimensional generalization of the concepts of a line in Euclidean plane geometry and a In the plane spanned by v and such a w in the hyperplane, the relation of w to v is hyperbolic-orthogonal. In Mathematics, two points in the Cartesian plane are hyperbolically orthogonal if the Slopes of their rays from the origin are Reciprocal to

Vector fields are called timelike, spacelike or null if the associated vectors are timelike, spacelike or null at each point where the field is defined. In Mathematics a vector field is a construction in Vector calculus which associates a vector to every point in a (locally Euclidean space.

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

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 positive, and
  3. past directed null vectors whose first component is negative.

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.

Causality relations

Let x, y ∈ M. We say that

  1. x chronologically precedes y if y − x is future directed timelike.
  2. x causally precedes y if y − x is future directed null

Reversed triangle inequality

If v and w are two equally directed timelike four-vectors then

|v+w| \ge |v|+|w|

where

|v|:=\sqrt{-\eta_{\mu \nu}v^\mu v^\nu}

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 Physics, the Newtonian limit refers to Physical systems without significantly intense Gravitation, in the sense that Newton's Law Gravitation is a natural Phenomenon by which objects with Mass attract one another In the case of significant gravitation, spacetime becomes curved and one must abandon special relativity in favor of the full theory of general relativity. Special relativity (SR (also known as the special theory of relativity or STR) is the Physical theory of Measurement in Inertial General relativity or the general theory of relativity is the geometric theory of Gravitation published by Albert Einstein in 1916

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. A manifold is a mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be In Mathematics, the tangent space of a Manifold is a concept which facilitates the generalization of vectors from Affine spaces to general manifolds since Thus, the structure of Minkowski space is still essential in the description of general relativity.

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 developed by Einstein) could be elegantly described using a four-dimensional spacetime, which combines the dimension of time with the three dimensions of space. Hermann Minkowski ( June 22 1864 – January 12 1909) was a Russian born German Mathematician, of Jewish Albert Einstein ( German: ˈalbɐt ˈaɪ̯nʃtaɪ̯n; English: ˈælbɝt ˈaɪnstaɪn (14 March 1879 – 18 April 1955 was a German -born theoretical

“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

The way had been prepared for Minkowski's space by the development of hyperbolic quaternions in the 1890s. In Mathematics, a hyperbolic quaternion is a mathematical concept first suggested by Alexander MacFarlane in 1891 in a speech to the American Association In fact, as a mathematical structure, Minkowski space can be taken as hyperbolic quaternions, minus the multiplicative product, and retaining only the bilinear form
\eta(p,q) = -\frac{pq^\ast + (pq^\ast)^\ast}{2}
which is generated by the hyperbolic quaternion product pq^\ast. In Mathematics, a bilinear form on a Vector space V is a Bilinear mapping V  ×  V  →  F, where

See also

References

External links

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