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 after 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 Poincaré group.

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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 (+,−,−,−) but in general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter.) 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). 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 × 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.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 The positive-definite condition has been replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa).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 η(v,v) = ±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 \eta can be stated:

4. signature The bilinear form η has signature (-,+,+,+) or (+,-,-,-)

Which signature is used is a matter of convention. Both are fairly common.

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. 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 Poincaré 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, Poincaré group

Causal structure

Main article: 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.

Given a timelike vector v, there is a worldline of constant velocity associated with it. 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. In the plane spanned by v and such a w in the hyperplane, the relation of w to v is hyperbolic-orthogonal.

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, 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 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 realm 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
\eta(p,q) = -\frac{pq^\ast + (pq^\ast)^\ast}{2}
which is generated by the hyperbolic quaternion product pq^\ast.

See also

References

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