# Minkowski Space

(redirected from Minkowski norm)
The following article is from The Great Soviet Encyclopedia (1979). It might be outdated or ideologically biased.

## Minkowski Space

a four-dimensional space, combining the physical three-dimensional space and time; introduced by H. Minkowski in 1907–08. Points in Minkowski space correspond to “events” of the special theory of relativity.

The position of an event in Minkowski space is specified by four coordinates—three space coordinates and one time coordinate. The coordinates that are usually used are x = x, x2 = y, x3 = z, where x, y, and z are rectangular Cartesian coordinates of the event in a given inertial frame of reference, and the coordinate xθ = ct, where t is the time of the event and c is the velocity of light. The imaginary time coordinate x4 = ix0 = ict can be introduced instead of X0.

It follows from the special theory of relativity that space and time are not independent. In passing from one inertial frame of reference to another, the space coordinates and the time are transformed through each other by Lorentz transformations. The introduction of Minkowski space permits the Lorentz transformation to be represented as the transformation of the coordinates x1, x2, x3, x4 of an event in a rotation of the four-dimensional coordinate system in this space.

The chief invariant of Minkowski space is the square of the length of the four-dimensional vector that connects two points—events—and that remains invariant in rotations in Minkowski space and equal in magnitude (but opposite in sign) to the square of the four-dimensional interval (sAB2) of the special theory of relativity:

(x1Ax1B)2 + (x2Ax2B)2 + (x3Ax3B)2 + (x4Ax4B)2 = (xAxB)2 + (yAyB)2 + (zAzB)2 + c2(tAtB)2 = −sAB2

where the subscripts A and B indicate the space coordinates and time of events A and B, respectively. The uniqueness of the geometry of Minkowski space is that this expression contains the squares of the components of a four-dimensional vector along the time and space axes with different signs (such a geometry is said to be pseudo-Euclidean, in contrast to Euclidean geometry in which the square of the distance between two points is determined by the sum of the squares of the components, on the corresponding axes, of the vector that joins the points). As a result, a four-dimensional vector with nonzero components can have zero length. This is the case for the vector that joins two events connected by a light signal:

(xAxB)2 + (yAyB)2 + (zAzB)2 + c2(tAtB)2 = c2(tA - tB)

The geometry of Minkowski space makes it possible to give a lucid interpretation of the kinematic effects of the special theory of relativity (for example, the variation in length and rate of passage of time in passing over from one inertial frame of reference to another). It also serves as the basis for the modern mathematical apparatus of the theory of relativity.

G. A. ZISMAN

The Great Soviet Encyclopedia, 3rd Edition (1970-1979). © 2010 The Gale Group, Inc. All rights reserved.
References in periodicals archive ?
We know, from the case of the spaces A[P.sub.r](R, C), 1 [less than or equal to] r [less than or equal to] 2, that a norm can be defined (the Minkowski norm) for the space [[??].sup.r], 1 [less than or equal to] r [less than or equal to] 2.
A Minkowski norm on a vector space V is a function F : V [right arrow] R such That
A pseudo-manifold geometry is a pseudo-manifold ([M.sup.n], [A.sup.[omega]]) endowed with a Minkowski norm F on T[M.sup.n].
Theorem 4.7.([14]) A pseudo-manifold geometry ([M.sup.n], [[phi].sup.[omega]]) with a Minkowski norm on T[M.sup.n] is a Finsler geometry if and only if all points of ([M.sup.n], [[phi].sup.[omega]]) are euclidean.
Theorem 4.8.([14]) A pseudo-manifold geometry ([M.sup.n.sub.c], [[phi].sup.[omega]]) with a Minkowski norm on T[M.sup.n] is a Kahler geometry if and only if F is a Hermite inner product on [M.sup.n.sub.c] with all points of ([M.sup.n], [[phi].sup.[omega]]) being euclidean.
A combinatorially Fluster geometry is a smoothly combinatorial manifold [??] endowed with a Minkowski norm [??] on [??], denoted by ([??]; [??]).
The above expressions for running times are unchanged when the problem is specified using any Minkowski norm instead of the Euclidean norm.
A Minkowski norm in [R.sup.d] is defined using a convex body C that is symmetric around the origin.
Our algorithm generalizes to the case where distances are measured using any fixed Minkowski norm. The only change is that the value of "m" and "r" changes by some constant factor depending upon the norm.
This is also true when distance is measured using any Minkowski norm (except the constant in the 0([multiplied by]) depends on the norm).
All our algorithms also work, with almost no modification, When distance is measured using any geometric norm (such as [l.sub.p], for p [is greater than or equal to] 1 or other Minkowski norms).

Site: Follow: Share:
Open / Close