Minkowski space

In mathematical physics, Minkowski space or Minkowski spacetime is a combination of Euclidean space and time into a four-dimensional manifold where the spacetime interval between any two events is independent of the inertial frame of reference in which they are recorded. Although initially developed by mathematician Hermann Minkowski for Maxwell's equations of electromagnetism, the mathematical structure of Minkowski spacetime was shown to be an immediate consequence of the postulates of special relativity.^[1]

Minkowski space is closely associated with Einstein's theory of special relativity, and is the most common mathematical structure on which special relativity is formulated. While the individual components in Euclidean space and time will often differ due to length contraction and time dilation, in Minkowski spacetime, all frames of reference will agree on the total distance in spacetime between events.^{[nb 1]} Because it treats time differently than the three spatial dimensions, Minkowski space differs from four-dimensional Euclidean space.^{[nb 2]}

In Euclidean space, the isometry group (the maps preserving the regular inner product) is the Euclidean group. The analogous isometry group for Minkowski space, preserving intervals of spacetime equipped with the associated non-positive definite bilinear form (here called the Minkowski inner product,^{[nb 3]}) is the Poincaré group. The Minkowski inner product is defined as to yield the spacetime interval between two events when given their coordinate difference vector as argument.

History

Hermann Minkowski (1864 – 1909) found that the theory of special relativity, introduced by his former student Albert Einstein, could best be understood in a four-dimensional space, since known as the Minkowski spacetime.

Four-dimensional Euclidean spacetime

In 1905, and later published in 1906, Henri Poincaré showed that by taking time to be an imaginary fourth spacetime coordinate (√−1 c t), a Lorentz transformation can be regarded as a rotation of coordinates in a four-dimensional Euclidean space with three real coordinates representing space, and one imaginary coordinate, representing time, as the fourth dimension. Since the space is then a pseudo-Euclidean space, the rotation is a representation of a hyperbolic rotation, although Poincaré did not give this interpretation, his purpose being only to explain the Lorentz transformation in terms of the familiar Euclidean rotation.^[2]

This idea was elaborated by Hermann Minkowski,^[3] who used it to restate the Maxwell equations in four dimensions, showing directly their invariance under the Lorentz transformation. He further reformulated in four dimensions the then-recent theory of special relativity of Einstein. From this he concluded that time and space should be treated equally, and so arose his concept of events taking place in a unified four-dimensional spacetime continuum.

Minkowski space

In a further development,^[4] he gave an alternative formulation of this idea that used a real time coordinate instead of an imaginary one, representing the four variables (x, y, z, t) of space and time in coordinate form in a four dimensional affine space. Points in this space correspond to events in spacetime. In this space, there is a defined light-cone associated with each point, and events not on the light-cone are classified by their relation to the apex as spacelike or timelike. It is principally this view of spacetime that is current nowadays, although the older view involving imaginary time has also influenced special relativity. Minkowski, aware of the fundamental restatement of the theory which he had made, said

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, 1907^[4]

For further historical information see references Galison (1979), Corry (1997) and Walter (1999).

Mathematical structure

For an overview, Minkowski space is a 4-dimensional real vector space equipped with a nondegenerate, symmetric bilinear form on the tangent space at each point in spacetime, here simply called the Minkowski inner product, with signature either (+,−,−,−) or (−,+,+,+). In practice, one need not be concerned with the tangent spaces. The vector space nature of Minkowski space allows for the canonical identification of vectors in tangent spaces at points (events) with vectors (points, events) in Minkowski space itself.^[5] For some purposes it is desirable to identify tangent vectors at a point p with displacement vectors at p, which is, of course, admissible by essentially the same canonical identification.^[6]

The signature refers to which sign the Minkowski inner product yields when given space and time basis vectors as arguments. In general, mathematicians and general relativists prefer the former while particle physicists tend to use the latter. Arguments for the former (pure space vectors yield positive "norm-squared") include "continuity" from the Euclidean case corresponding to the non-relativistic limit c → ∞. Arguments for the latter (pure space vectors yield negative "norm-squared") include that otherwise ubiquitous minus signs in particle physics go away.

Mathematically associated to this bilinear form is a tensor of type (0,2) at each point in spacetime, called the Minkowski metric. The Minkowski metric, the bilinear form, and the Minkowski inner product are actually all the very same object. In coordinates, this is the 4×4 matrix representing the bilinear form. Keeping this in mind may facilitate reading what follows.

For comparison, in general relativity, a Lorentzian manifold L is likewise equipped with a metric tensor g, which is a nondegenerate symmetric bilinear form on the tangent space T_pL at each point p of L. In coordinates, it may be represented by a 4×4 matrix depending on spacetime position. Minkowski space is thus a comparatively simple special case of a Lorentzian manifold. Its metric tensor, called the Minkowski metric, is in coordinates the same symmetric matrix at every point of M, and its arguments can, per above, be taken as vectors in spacetime itself.

Introducing more terminology (but not more structure), Minkowski space is thus a pseudo-Euclidean space with total dimension n = 4 and signature (3, 1) or (1, 3). Elements of Minkowski space are called events. Minkowski space is often denoted R^3,1 or R^1,3 to emphasize the chosen signature, or just M. It is perhaps the simplest example of a pseudo-Riemannian manifold.

Pseudo-Euclidean metric generalities

The Minkowski metric^{[nb 4]} η is the metric tensor of Minkowski space. It is a Pseudo-Euclidean metric. As such it is a nondegenerate symmetric bilinear form, a type (0,2) tensor. It accepts two arguments u_p, v_p, vectors in T_pM, p ∈ M, the tangent space at p in M. Due to the above-mentioned canonical identification of T_pM with M itself, it accepts arguments u, v with both u and v in M.

As a notational convention, vectors v in M, called 4-vectors, are denoted in sans-serif italics, and not, as is common in the Euclidean setting, with boldface v. The latter is generally reserved for the 3-vector part (to be introduced below) of a 4-vector.

The definition

$u \cdot v =\eta(u, v)$

yields an inner product-like structure on M, previously and also henceforth, called the Minkowski inner product, similar to the Euclidean inner product, but it describes a different geometry. It has the following properties.

• $\eta(au + v, w) = a\eta(u, w) + \eta(v, w), \quad \forall u, v \in M, \forall a \in \mathbb R \qquad \text{(linearity in first slot)}$
• $\eta( u, v) = \eta( v, u) \qquad \text{(symmetry)}$
• $\eta( u, v) = 0 \quad \forall v \in M \Rightarrow u = 0 \qquad \text{(non-degeneracy)}$

The first two conditions imply bilinearity. The defining difference between a pseudo-inner product and an inner product proper is that the former is not required to be positive definite, that is, η(u, u) < 0 is allowed.

Two vectors v and w are said to be orthogonal if η(v, w) = 0.

A vector e is called a unit vector if η(e, e) = ±1. A basis for M consisting of mutually orthogonal unit vectors is called an orthonormal basis.

For a given inertial frame, an orthonormal basis in space, combined by the unit time vector, forms an orthonormal basis in Minkowski space. The number of positive and negative unit vectors in any such basis is a fixed pair of numbers, equal to the signature of the bilinear form associated with the inner product. This is Sylvester's law of inertia.

More terminology (but not more structure): The Minkowski metric is a pseudo-Riemannian metric, more specifically, a Lorentzian metric, even more specifically, the Lorentz metric, reserved for 4-dimensional flat spacetime with the remaining ambiguity only being the signature convention.

Minkowski metric

From the second postulate of special relativity, together with homogeneity of spacetime and isotropy of space, follows that the spacetime interval between two events 1, 2,

$\pm\left[c^2(t_1 - t_2)^2 - (x_1 - x_2)^2 - (y_1 - y_2)^2 - (z_1 - z_2)^2\right],$

is independent of the inertial frame chosen, as is shown here. The factor ± simply means that the choice of signature is left open. The numerical values of η, viewed as a matrix representing the Minkowski inner product, follow from the theory of bilinear forms.

Just as the signature of the metric is differently defined in the literature, this quantity is not consistently named. The interval (as defined here) is sometimes referred to as the interval squared.^[7] Even the square root of the present interval occurs.^[8] When signature and interval are fixed, ambiguity still remains as which coordinate is the time coordinate. It may be the fourth, or it may be the zeroth. This is not an exhaustive list of notational inconsistencies. It is a fact of life that one has to check out the definitions first thing when one consults the relativity literature.

The invariance of the interval under coordinate transformations between inertial frames follows from the invariance of

$\pm\left[c^2t^2 - x^2 - y^2 - z^2\right]$

(with either sign ± preserved), provided the transformations are linear. This quadratic form can be used to define a bilinear form

$u \cdot v = \pm\left[c^2t_1t_2 - x_1x_2 - y_1y_2 - z_1z_2\right].$

via the polarization identity. This bilinear form can in turn be written as

$u \cdot v = u^{\mathrm T}[\eta] v,$

where [η] is a 4×4 matrix associated with η. Possibly confusingly, denote [η] with just η as is common practice. The matrix is read off from the explicit bilinear form as

$\eta = \pm \begin{pmatrix}-1&0&0&0\\0&1&0&0\\0&0&1&0\\0&0&0&1\end{pmatrix},$

and the bilinear form

$u \cdot v =\eta(u, v),$

with which this section started by assuming its existence, is now identified.

For definiteness and shorter presentation, the signature (−,+,+,+) is adopted below. The choice has no (known) physical implications. The symmetry group preserving the bilinear form with one choice of signature is isomorphic (under the map given here) with the symmetry group preserving the other choice of signature. This means that both choices are in accord with the two postulates of relativity.

Standard basis

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

$-\eta(e_0, e_0) = \eta(e_1, e_1) = \eta(e_2, e_2) = \eta(e_3, e_3) = 1 .$

These conditions can be written compactly in the form

$\eta(e_\mu, e_\nu) = \eta_{\mu \nu}.$

Relative to a standard basis, the components of a vector v are written (v⁰, v¹, v², v³) where the Einstein notation is used 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. The spatial components of a 4-vector v may be identified with a 3-vector v = (v₁, v₂, v₃).

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

$\eta(v, w) = \eta_{\mu \nu} v^\mu w^\nu = v^0 w_0 + v^1 w_1 + v^2 w_2 + v^3 w_3 = v^\mu w_\mu = v_\mu w^\mu,$

and

$\eta(v, v) = \eta_{\mu \nu} v^\mu v^\nu = v^0v_0 + v^1 v_1 + v^2 v_2 + v^3 v_3 = v^\mu v_\mu.$

Here lowering of an index with the metric was used. Technically, a non-degenerate bilinear form provides a map between a vector space and its dual, in this context, the map is between the tangent spaces of M and the cotangent spaces of M. At a point in M, the tangent and cotangent spaces are dual. Just as an authentic inner product on a vector space with one argument fixed, by Riesz representation theorem, may be expressed as the action of a linear functional on the vector space, the same holds for the Minkowski inner product of Minkowski space.

Thus if v^μ are the components of a vector in a tangent space, then η_μνv^μ = v_ν are the components of a vector in the cotangent space (a linear functional). Due to the identification of vectors in tangent spaces with vectors in M itself, this is mostly ignored, and vectors with lower indices are referred to as covariant vectors. In this latter interpretation, the covariant vectors are (almost always implicitly) identified with vectors (linear functionals) in the dual of Minkowski space. The ones with upper indices are contravariant vectors. In the same fashion, the inverse of the map from tangent to cotangent spaces, explicitly given by the inverse of η in matrix representation, can be used to define raising of an index. The components of this inverse are denoted η^μν. It happens that η^μν = η_μν. These maps between a vector space and its dual can be denoted η^♭ (eta-flat) and η^♯ (eta-sharp) by the musical analogy.^[9]

The time-proven robustness of the formalism itself, sometimes referred to as index gymnastics, ensures that moving vectors around and changing from contravariant to covariant vectors and vice versa is mathematically sound. Incorrect expressions tend to reveal themselves quickly.

Lorentz transformations and symmetry

Standard configuration of coordinate systems for Lorentz transformations.

The Poincaré group is the group of all transformations preserving the interval. The interval is quite easily seen to be preserved by the translation group in 4 dimensions. The other transformations are those that preserve the interval and leave the origin fixed. Given the bilinear form associated with the Minkowski metric, the appropriate group follows directly from the theory (in particular the definition) of classical groups. In the linked article, one should identify η (in its a matrix representation) with the matrix Φ.

The appropriate group is O(3,1), in this context called the Lorentz group. Its elements are called (homogeneous) Lorentz transformations. For other methods of derivation, with a more physical twist, see derivations of the Lorentz transformations.

Among the simplest Lorentz transformations is a Lorentz boost. For reference, a boost in the x-direction is given by

$\begin{bmatrix} U'_0 \\ U'_1 \\ U'_2 \\ U'_3 \end{bmatrix} = \begin{bmatrix} \gamma&-\beta \gamma&0&0\\ -\beta \gamma&\gamma&0&0\\ 0&0&1&0\\ 0&0&0&1\\ \end{bmatrix} \begin{bmatrix} U_0 \\ U_1 \\ U_2 \\ U_3 \end{bmatrix},$

where

$\gamma = { 1 \over \sqrt{1 - {v^2 \over c^2}} }$

is the Lorentz factor, and

$\beta = { v \over c} \,.$

Other Lorentz transformations are pure rotations, and hence elements of the SO(3) subgroup of O(3,1). A general homogeneous Lorentz transformation is a product of a pure boost and a pure rotation. An inhomogeneous Lorentz transformation is a homogeneous transformation followed by a translation in space and time. Special transformations are those that invert the space coordinates (P) and time coordinate (T) respectively, or both (PT).

All four-vectors in Minkowski space transform, by definition, according to the same formula under Lorentz transformations. Minkowski diagrams illustrate Lorentz transformations.

Causal structure

Subdivision of Minkowski spacetime with respect to an event in four disjoint sets. The light cone, the absolute future, the absolute past, and elsewhere. The terminology is from Sard (1970).

Vectors v = (ct, x, y, z) = (ct, r) are classified according to the sign of c²t² - r². A vector is timelike if c²t² > r², spacelike if c²t² < r², and null or lightlike if c²t² = r². This can be expressed in terms of the sign of η(v,v) as well, but depends on the signature. The classification of any vector will be the same in all frames of reference, because of the invariance of the interval.

The set of all null vectors at an event^{[nb 5]} of Minkowski space constitutes the light cone of that event. Given a timelike vector v, there is a worldline of constant velocity associated with it, represented by a straight line in a Minkowski diagram.

Once a direction of time is chosen,^{[nb 6]} timelike and null vectors can be further decomposed into various classes. For timelike vectors one has

1. future-directed timelike vectors whose first component is positive, (tip of vector located in absolute future in figure) and
2. past-directed timelike vectors whose first component is negative (absolute past).

Null vectors fall into three classes:

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

Spacelike vectors are in elsewhere. The terminology stems from the fact that spacelike separated events are connected by vectors requiring faster-than-light travel, and so cannot possibly influence each other. Together with spacelike and lightlike vectors there are 7 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. Over the reals, if two null vectors are orthogonal (zero Minkowski tensor value), then they must be proportional. However, allowing complex numbers, one can obtain a null tetrad, which is a basis consisting of null vectors, some of which are orthogonal to each other.

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.

Chronological and causality relations

Let x, y ∈ M. We say that

1. x chronologically precedes y if y − x is future-directed timelike. This relation has the transitive property and so can be written x < y.
2. x causally precedes y if y − x is future-directed null or future-directed timelike. It gives a partial ordering of space-time and so can be written x ≤ y.

Reversed triangle inequality

If v and w are both future-directed timelike four-vectors, then in the (+ - - -) sign convention for norm,

$\left\| v+w \right\| \ge \left\| v \right\| + \left\| w \right\| .$

Relationships to other formulations

Different number of dimensions

Strictly speaking, Minkowski space refers to a mathematical formulation in four dimensions. However, the mathematics can easily be extended or simplified to create an analogous "Minkowski space" in any number of dimensions. If n ≥ 2, n-dimensional Minkowski space is a vector space of real dimension n on which there is a constant Lorentz metric of signature (n − 1, 1) or (1, n − 1). These generalizations are used in theories where spacetime is assumed to have more or less than 4 dimensions. String theory and M-theory are two examples where n > 4. In string theory, there appears conformal field theories with 1 + 1 spacetime dimensions.

Flat versus curved space

As a flat spacetime, the three spatial components of Minkowski spacetime always obey the Pythagorean Theorem. Minkowski space is a suitable basis for special relativity, a good description of physical systems over finite distances in systems without significant gravitation. However, in order to take gravity into account, physicists use the theory of general relativity, which is formulated in the mathematics of a non-Euclidean geometry. When this geometry is used as a model of physical space, it is known as curved space.

Even in curved space, Minkowski space is still a good description in an infinitesimal region surrounding any point (barring gravitational singularities).^{[nb 7]} 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.

See also

• Causal structure
• Euclidean space
• Four vector
• Hyperboloid model
• Introduction to mathematics of general relativity
• Lorentzian manifold
• Metric tensor
• Minkowski diagram
• Minkowski plane
• Speed of light
• Super Minkowski space
• World line

Remarks

1. This makes spacetime distance an invariant.
2. Minkowski space can be formulated as an equivalent 4-D Euclidean space if you assume time is always an imaginary number. This is how the spacetime was first formulated, but since Minkowski reworked the structure, time is almost always required to be a real number.
3. Consistent use of the term "Minkowski inner product" is intended for the bilinear form here, since it is in widespread use. It is by no means "standard" in the literature, but no such standard seems to exist.
4. The Minkowski inner product is not an inner product, since it is not positive-definite, i.e. the quadratic form η(v, v) need not be positive for nonzero v. The positive-definite condition has been replaced by the weaker condition of non-degeneracy. The bilinear form is said to be indefinite.
5. Translate the coordinate system so that the event is the new origin.
6. This corresponds to the time coordinate either increasing or decreasing when proper time for any particle increases. An application of T flips this direction.
7. This similarity between flat and curved space at infinitesimally small distance scales is foundational to the definition of a manifold in general.

Notes

1. Landau & Lifshitz 2002, p. 5
2. Poincaré 1905–1906, pp. 129–176 Wikisource translation: On the Dynamics of the Electron
3. Minkowski 1907–1908, pp. 53–111 *Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies.
4. Minkowski 1907–1909, pp. 75–88 Various English translations on Wikisource: Space and Time.
5. Lee 2003, Proposition 3.8. The identification is routinely done in mathematics.
6. Lee 2003, See Lee's discussion on geometric tangent vectors early in chapter 3.
7. Sard 1970, p. 71
8. Landau & Lifshitz 2002, p. 4
9. Lee 2003, The tangent-cotangent isomorphism p. 282.

References

• Corry, L. (1997). "Hermann Minkowski and the postulate of relativity". Arch. Hist. Exact Sci. (Springer-Verlag) 51 (4): 273–314. doi:10.1007/BF00518231. ISSN 0003-9519. (subscription required (help)).
• Catoni, F.; et al. (2008). Mathematics of Minkowski Space. Frontiers in Mathematics. Basel: Birkhäuser Verlag. doi:10.1007/978-3-7643-8614-6. ISBN 978-3-7643-8613-9. ISSN 1660-8046.
• Galison, P. L. (1979). R McCormach; et al., eds. Minkowski's Space-Time: from visual thinking to the absolute world. Historical Studies in the Physical Sciences 10. Johns Hopkins University Press. pp. 85–121. doi:10.2307/27757388. (subscription required (help)).
• Landau, L.D.; Lifshitz, E.M. (2002) [1939]. The Classical Theory of Fields. Course of Theoretical Physics 2 (4th ed.). Butterworth–Heinemann. ISBN 0 7506 2768 9.
• Lee, J. M. (2003). Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics 218. ISBN 0-387-95448-1.
• Minkowski, Hermann (1907–1908), "Die Grundgleichungen für die elektromagnetischen Vorgänge in bewegten Körpern" [The Fundamental Equations for Electromagnetic Processes in Moving Bodies], Nachrichten von der Gesellschaft der Wissenschaften zu Göttingen, Mathematisch-Physikalische Klasse: 53–111 *Wikisource translation: The Fundamental Equations for Electromagnetic Processes in Moving Bodies
• Minkowski, Hermann (1907–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift 10: 75–88 Various English translations on Wikisource: Space and Time
• Naber, G. L. (1992). The Geometry of Minkowski Spacetime. New York: Springer-Verlag. ISBN 0-387-97848-8.
• Penrose, Roger (2005). "18 Minkowskian geometry". Road to Reality : A Complete Guide to the Laws of the Universe. Alfred A. Knopf. ISBN 9780679454434.
• Poincaré, Henri (1905–1906), "Sur la dynamique de l’électron" [On the Dynamics of the Electron], Rendiconti del Circolo matematico di Palermo 21: 129–176, doi:10.1007/BF03013466 Wikisource translation: On the Dynamics of the Electron
• Sard, R. D. (1970). Relativistic Mechanics - Special Relativity and Classical Particle Dynamics. New York: W. A. Benjamin. ISBN 978-0805384918.
• Shaw, R. (1982). "§ 6.6 Minkowski space, § 6.7,8 Canonical forms pp 221–242". Linear Algebra and Group Representations. Academic Press. ISBN 0-12-639201-3.
• Walter, Scott (1999). "Minkowski, Mathematicians, and the Mathematical Theory of Relativity". In Goenner, Hubert (ed.); et al. The Expanding Worlds of General Relativity. Boston: Birkhäuser. pp. 45–86. ISBN 0-8176-4060-6.

External links

Media related to Minkowski diagrams at Wikimedia Commons

• Animation clip on YouTube visualizing Minkowski space in the context of special relativity.
• The Geometry of Special Relativity: The Minkowski Space - Time Light Cone