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).

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

Contents

1 Structure 1.1 The Minkowski inner product 1.2 Standard basis 1.3 Four-vectors 2 Alternative definition 3 Lorentz transformations 4 Causal structure 5 Locally flat spacetime 6 History 7 Related topics 8 References

Structure

Formally, Minkowski space is a four-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form with signature (+,-,-,-) (used by high energy physicists). The overall sign is a matter of convention and many prefer to use the signature (-,+,+,+) (used by relativists). Minkowski space is often denoted R^1,3 to emphasize the signature, although it is also denoted M ⁴ or simply M.

The Minkowski inner product

The inner product between two vectors v, w in Minkowski space is a map M × M → R, denoted <v, w>, that satisfies four properties. Three of which are that it be

1. bilinear: <au + v, w> = a<u, w> + <v, w>, for all a, u, v, and w
2. symmetric: <v, w> = <w, v> for all v and w, and
3. nondegenerate: if <v, w> = 0 for all w then v = 0,

where a is in R and u, v, w are vectors in M.

Note that this is not an inner product in the usual sense of the word since it is not positive-definite, i.e. the norm-squared of a vector v, defined as ||v||² = <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 The positive-definite condition has been replaced by the weaker condition of nondegeneracy (every positive-definite form is nondegenerate but not vice-versa).

Just as in Euclidean space, two vectors are said to be orthogonal if <v, w> = 0. A vector v is called a unit vector if ||v||² = ±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 1-3 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.

We can then state then fourth condition on the Minkowski inner product:

Standard basis

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

<math>\left(e_0\right)^2 = -(e_1)^2 = -(e_2)^2 = -(e_3)^2 = 1<math>

These conditions can be written compactly in the following form:

<math>\langle e_\mu, e_\nu \rangle = \eta_{\mu\nu}<math>

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

<math>\eta = \begin{pmatrix}1&0&0&0\\0&-1&0&0\\0&0&-1&0\\0&0&0&-1\end{pmatrix}<math>

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 = v^μe_μ. 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

<math>\langle v,w\rangle = \eta_{\mu\nu}v^\mu w^\nu = v^0w^0 - v^1w^1 - v^2w^2 - v^3w^3<math>

and the norm-squared of a vector v is

<math>\|v\|^2 = \eta_{\mu\nu}v^\mu v^\nu = (v^0)^2-(v^1)^2-(v^2)^2-(v^3)^2<math>

Four-vectors

Vectors in Minkowski space are also called four-vectors in order to distinguish them from three-dimensional spatial vectors. In this article, however, we use the two terms interchangeably. We shall refer instead to the spatial component of a four-vector (which, of course, depends on the choice of basis).

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.

Lorentz transformations

See: Lorentz transformations, Lorentz group, Poincaré group

Causal structure

Four-vectors are classified according to the sign of their norm squared. Vectors are said to be timelike or spacelike if their norms squared are positive or negative respectively. Vectors with zero norm are called null or lightlike. This terminology comes from the use of Minkowski space in the theory of relativity. The set of all lightlike vectors constitutes what is called the light cone. Note that all of these notions are independent of one's frame of reference.

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 timelike vectors whose first component is positive, and
2. past 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 null vectors whose first component is positive, and
3. past 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.

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 a 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.

“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

Related topics

• Euclidean space
• spacetime
• speed of light
• world line
• metric tensor
• Lorentzian manifold
• Erlangen program
• split-complex number
• hyperbolic quaternions

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).

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