View source for Minkowski space

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A four-dimensional [[Pseudo-Euclidean space|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|Lorentz transformation]]).

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

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



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

====References====
<table><TR><TD valign="top">[a1]</TD> <TD valign="top">  M. Dillard-Bleik,   "Analysis, manifolds and physics" , North-Holland  (1977)  (Translated from French)</TD></TR><TR><TD valign="top">[a2]</TD> <TD valign="top">  S. Weinberg,   "Gravitation and cosmology" , Wiley  (1972)  pp. Chapt. 3</TD></TR><TR><TD valign="top">[a3]</TD> <TD valign="top">  A.L. Besse,   "Einstein manifolds" , Springer  (1987)</TD></TR><TR><TD valign="top">[a4]</TD> <TD valign="top">  S.W. Hawking,   G.F.R. Ellis,   "The large scale structure of space-time" , Cambridge Univ. Press  (1973)</TD></TR><TR><TD valign="top">[a5]</TD> <TD valign="top">  C.W. Misner,   K.S. Thorne,   J.A. Wheeler,   "Gravitation" , Freeman  (1973)</TD></TR><TR><TD valign="top">[a6]</TD> <TD valign="top">  B. O'Neill,   "Semi-Riemannian geometry (with applications to relativity)" , Acad. Press  (1983)</TD></TR></table>