# Minkowski space

## Information about 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.

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. Therefore the symmetry group of a Euclidean space is the Euclidean group and for a Minkowski space it is the PoincarÃ© group.

## 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 other words, Minkowski space is a pseudo-Euclidean space with n = 4 and nk = 1 (in a broader definition any n>1 is allowed). Elements of Minkowski space are called events or four-vectors. 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.

### 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. Let M be a 4-dimensional real vector space. The Minkowski inner product is a map η: M × MR (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.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 The positive-definite condition has been replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa).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. 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. 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.

A vector v is called a unit vector if v2 = ±1. A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis.

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 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

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.

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. There is an alternative definition of Minkowski space as an affine space which views Minkowski space as a homogeneous space of the PoincarÃ© group with the Lorentz group as the stabilizer. See Erlangen program.

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.e., in the above terminology, n "pluses" and one "minus".

## Lorentz transformations

See: Lorentz transformations, Lorentz group, PoincarÃ© group

## Causal structure

Vectors are classified according to the sign of their (Minkowski) norm. 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. 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.

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.

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, yM. We say that
1. x chronologically precedes y if yx is future directed timelike.
2. x causally precedes y if yx is future directed null

## Reversed triangle inequality

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

where

## 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.

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.

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.
“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 fact, as a mathematical structure, Minkowski space can be taken as hyperbolic quaternions, minus the multiplicative product, and retaining only the bilinear form
η(p,q) = −(pq* + (pq*)*)/2
which is generated by the hyperbolic quaternion product pq*.

## References

Physics is the science of matter[1] and its motion[2][3], as well as space and time[4][5] —the science that deals with concepts such as force, energy, mass, and charge.
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Mathematics (colloquially, maths or math) is the body of knowledge centered on such concepts as quantity, structure, space, and change, and also the academic discipline that studies them. Benjamin Peirce called it "the science that draws necessary conclusions".
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special theory of relativity was proposed in 1905 by Albert Einstein in his article "On the Electrodynamics of Moving Bodies". Some three centuries earlier, Galileo's principle of relativity had stated that all uniform motion was relative, and that there was no absolute and
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The term SPACE (capitalized) can refer to:
• , a Canadian science-fiction channel
• The Society for Promotion of Alternative Computing and Employment
• DSPACE, a term in computational complexity theory

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time.

One view is that time is part of the fundamental structure of the universe, a dimension in which events occur in sequence, and time itself is something that can be measured.
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manifold is an abstract mathematical space in which every point has a neighborhood which resembles Euclidean space, but in which the global structure may be more complicated. In discussing manifolds, the idea of dimension is important.
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spacetime is any mathematical model that combines space and time into a single construct called the space-time continuum. Spacetime is usually interpreted with space being three-dimensional and time playing the role of the fourth dimension.
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Anthem
"Das Lied der Deutschen" (third stanza)
also called "Einigkeit und Recht und Freiheit"
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mathematician is a person whose primary area of study and research is the field of mathematics.

## Problems in mathematics

Some people incorrectly believe that mathematics has been fully understood, but the publication of new discoveries in mathematics continues at an immense
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Hermann Minkowski

Born May 22 1864
Aleksotas, Kaunas, Lithuania
Died January 12 1909 (aged 46)
GÃ¶ttingen, Germany
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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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.
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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.
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rotation (symmetry) group of the figure.]]

The symmetry group of an object (image, signal, etc., e.g. in 1D, 2D or 3D) is the group of all isometries under which it is invariant with composition as the operation.
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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In mathematics, the Euclidean group E(n), sometimes called ISO(n) or similar, is the symmetry group of n-dimensional Euclidean space. Its elements, the isometries associated with the Euclidean metric, are called Euclidean moves.
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In physics and mathematics, the PoincarÃ© group, named after Henri PoincarÃ©, is the group of isometries of Minkowski spacetime. It is a 10-dimensional noncompact Lie group. The abelian group of translations is a normal subgroup while the Lorentz group is a subgroup, the stabilizer
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In mathematics, the real numbers may be described informally as numbers that can be given by an infinite decimal representation, such as 2.4871773339…. The real numbers include both rational numbers, such as 42 and −23/129, and irrational numbers, such as π and
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In mathematics, a vector space (or linear space) is a collection of objects (called vectors) that, informally speaking, may be scaled and added. More formally, a vector space is a set on which two operations, called (vector) addition and (scalar) multiplication, are
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In mathematics, a bilinear form on a vector space V is a bilinear mapping V Ã— V → F, where F is the field of scalars.
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This article or section may be confusing or unclear for some readers.
Please [improve the article] or discuss this issue on the talk page. This article has been tagged since December 2006.
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A pseudo-Euclidean space is a finite-dimensional real vector space together with a non-degenerate indefinite quadratic form. Such a quadratic form can, after a change of coordinates, be written as

where is the dimension of the space, and
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In relativity, a four-vector is a vector in a four-dimensional real vector space, called Minkowski space. The usage of the four-vector name tacitly assumes that its components refer to a standard basis.
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In differential geometry, a pseudo-Riemannian manifold (also called a semi-Riemannian manifold) is a generalization of a Riemannian manifold. It is one of many things named after Bernhard Riemann.
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inner product space is a vector space of arbitrary (possibly infinite) dimension with additional structure, which, among other things, enables generalization of concepts from two or three-dimensional Euclidean geometry.
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In mathematics, a definite bilinear form is a bilinear form B such that

B(x, x)

has a fixed sign (positive or negative) when x is not 0.
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Euclidean space. Most of this article is devoted to developing the modern language necessary for the conceptual leap to higher dimensions.

An essential property of a Euclidean space is its flatness. Other spaces exist in geometry that are not Euclidean.
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In mathematics, orthogonal, as a simple adjective, not part of a longer phrase, is a generalization of perpendicular. It means at right angles, from the Greek ὀρθός orthos
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