Minkowski space

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A four-dimensional pseudo-Euclidean space of signature $(1,3)$, suggested by H. Minkowski (1908) as a geometric model of space-time in the special theory of relativity (see ). Corresponding to each event there is a point of Minkowski space, three coordinates of which represent its coordinates in the three-dimensional space; the fourth coordinate is $ct$, where $c$ is the velocity of light and $t$ is the time of the event. The space-time relationship between two events is characterized by the so-called square interval:

$$s^2=c^2(\Delta t)^2-(\Delta x)^2-(\Delta y)^2-(\Delta z)^2.$$

The interval in Minkowski space plays a role similar to that of distance in Euclidean geometry. A vector with positive square interval is called a time-like vector, one with negative square interval, a space-like vector, one with square interval zero, a null or isotropic vector. A curve with a time-like tangent vector at each point is called a time-like curve. Space-like and isotropic curves are similarly defined. An event at a given moment of time is called a world point; a set of world points describing the development of some process or phenomenon through time is called a world line. If a vector joining neighbouring world points is time-like, then there is a frame of reference in which the events project to one and the same point of three-dimensional space. The time separating the events in this frame of reference is equal to $\Delta t=\tau=s/c$, where $\tau$ is the so-called proper time. There is no frame of reference in which these events can be simultaneous (that is, have the same time coordinate $t$). If the vector joining the world points of two events is space-like, then there is a frame of reference in which these two events occur simultaneously; they are not connected by a causal relation; the modulus of the square interval defines the spatial distance between the points (events) in this frame of reference. A tangent vector to a world line is a time-like vector. The tangent vector to a light ray is an isotropic vector.

The motions of Minkowski space, that is, the interval-preserving transformations, are the Lorentz transformations (cf. Lorentz transformation).

A generalization of Minkowski space is the pseudo-Riemannian space used in the construction of the theory of gravitation.


[1a] H. Minkowski, "Raum und Zeit" Phys. Z. Sowjetunion , 10 (1909) pp. 104- ((Reprint in: Lorentz–Einstein–Minkowski, Teubner, 1922))
[1b] H. Minkowski, "Das Relativitätsprinzip" Jahresber. Deutsch. Math. Verein , 24 (1915) pp. 372-
[2] L.D. Landau, E.M. Lifshitz, "The classical theory of fields" , Addison-Wesley (1962) (Translated from Russian)
[3] V.A. [V.A. Fok] Fock, "The theory of space, time and gravitation" , Macmillan (1954) (Translated from Russian)
[4] P.K. [P.K. Rashevskii] Rashewski, "Riemannsche Geometrie und Tensoranalyse" , Deutsch. Verlag Wissenschaft. (1959) (Translated from Russian)
[5] J.L. Synge, "Relativity: the general theory" , North-Holland (1960)


See in particular [a3], pp. 10-11 and Chapt. 3, and [a4][a6] for material on the pseudo-Riemannian spaces used in gravitational theories.


[a1] M. Dillard-Bleik, "Analysis, manifolds and physics" , North-Holland (1977) (Translated from French)
[a2] S. Weinberg, "Gravitation and cosmology" , Wiley (1972) pp. Chapt. 3
[a3] A.L. Besse, "Einstein manifolds" , Springer (1987)
[a4] S.W. Hawking, G.F.R. Ellis, "The large scale structure of space-time" , Cambridge Univ. Press (1973)
[a5] C.W. Misner, K.S. Thorne, J.A. Wheeler, "Gravitation" , Freeman (1973)
[a6] B. O'Neill, "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press (1983)
How to Cite This Entry:
Minkowski space. Encyclopedia of Mathematics. URL:
This article was adapted from an original article by D.D. Sokolov (originator), which appeared in Encyclopedia of Mathematics - ISBN 1402006098. See original article