Minkowski space

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For spacetime graphics, see Minkowski diagram. For Minkowski space associated to a number field, see Minkowski space (number field). For geometry of the Minkowski plane, see Minkowski plane .


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 spacial dimensions, Minkowski space differs from four-dimensional Euclidean space.[nb 2]

The isometry group, preserving Euclidean distances of a Euclidean space equipped with the regular inner product is the Euclidean group. The analogous isometry group for Minkowski apace, 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.


Hermann Minkowski (1864 – 1909) was a German mathematician. He 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[edit]

In 1905, with the publication 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]


Minkowski space[edit]



— Hermann Minkowski, 1907[4]


Mathematical structure[edit]

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.



Pseudo-Euclidean metric generalities[edit]

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 up, vp, vectors in TpM, pM, the tangent space at p in M. Due to the above mentioned canonical identification of TpM 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 Eucliedean 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)


  • \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)}






Minkowski metric[edit]

From the two postulates of special relativity 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. 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.


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


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


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



and the bilinear form

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


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[edit]


-\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}.



\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,


\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.


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[edit]

Standard configuration of coordinate systems for Lorentz transformations.




U_0 \\ U_1 \\ U_2 \\ U_3



is the Lorentz factor, and

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



Causal structure[edit]

Main article: Causal structure

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


Null vectors fall into three classes:





Chronological and causality relations[edit]

Let x, yM. We say that


Reversed triangle inequality[edit]


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

Relationships to other formulations[edit]

Different number of dimensions[edit]

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[edit]



See also[edit]


  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.
  5. ^ Translate the coordinate system so that the event is the new origin.


  1. ^ Landau & Lifshitz 2002, p. 5
  4. ^ a b 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.
  7. ^ Sard 1970, p. 71
  8. ^ Landau & Lifshitz 2002, p. 4
  9. ^ Lee 2003, The tangent-cotangent isomorphism p. 282.


  • Lee, J. M. (2003). Introduction to Smooth manifolds. Springer Graduate Texts in Mathematics 218. ISBN 0-387-95448-1. 
  • Minkowski, Hermann (1907–1909), "Raum und Zeit" [Space and Time], Physikalische Zeitschrift 10: 75–88  Various English translations on Wikisource: Space and Time
  • 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. 

External links[edit]

Media related to Minkowski diagrams at Wikimedia Commons