# Minkowski Space

(redirected from Lightlike)
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 It might be outdated or ideologically biased.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

References in periodicals archive ?
where: p,u [member of] [R.sup.4] For an arbitrary curve a I [subset] R [right arrow] [E.sup.4.sub.1] [alpha] [member of] [E.sup.4.sub.1] is called a lightlike, timelike or spacelike curve if velocity vector of the curve satisfies [mathematical expression not reproducible] for each t [member of] I, respectively.
Because of H' [??] [E.sup.4.sub.2], we can define the timelike, spacelike, and lightlike quaternions for q = ([q.sub.0], [q.sub.1], [q.sub.2], [q.sub.3]) as follows:
However, likewise, adding and subtracting a gauge fixing of (1/2[xi])[(n x A).sup.2], with n being a lightlike vector, [n.sup.2] = 0, one gets (in the limit [xi] [right arrow] 0, [9])
Define [N.sub.[delta]](t) as the lightlike field on [delta] given by (D[T.sub.[delta]]/dt)(t) = [N.sub.[delta]](t) and denote by [B.sub.[delta]](t) the only lightlike vector such that ([N.sub.[delta]], [B.sub.[delta]]) = 1 and ([T.sub.[delta]], [B.sub.[delta]]) = 0.
Recent developments are described in Einstein metrics on Lie groups, Einstein-Weyl structures on 3-dimensional manifolds, quaternionic CR manifolds, geodesic surfaces in symmetric spaces, lightlike manifolds, and complex statistical manifolds.
For a lightlike radial trajectory leading towards the event horizon, the Schwarzschild metric reduces to [(dr/dt).sup.2] = [(2m/r - 1).sup.2].
Kang, "On lightlike hypersurfaces of a GRW space-time," Bulletin of the Korean Mathematical Society, vol.
We say that the vector v [member of] [E.sup.2n+1.sub.v] is spacelike, lightlike or timelike if (v,v) > 0 or v = 0, (v,v) = 0 and v [not equal to] 0, and (v,v) < 0, respectively, [8].
"What a waste, what a waste," an unknown beggarwoman sighed, clutching her saddened children against her bosom asEduardo's coffin passed her by, his face iridescent with lightlike an angel.
Since g is an indefinite metric, recall that a vector v [member of] [E.sup.3.sub.1] can have one of three Lorentzian causal characters: it can be spacelike if g(v, v) > 0 or v =0, timelike if g(v, v) < 0, and null (lightlike) if g(v, v) = 0 and v [not equal to] 0.
Recall that a nonzero vector [??] [member of] [E.sup.3.sub.1] is spacelike if [<[??], [??]>.sub.L] > 0, timelike if [<[??], [??]>.sub.L] < 0, and null (lightlike) if [<[??], [??]>.sub.L] = 0.
A vector x of [R.sup.3.sub.1] is said to be spacelike if <x, x> > 0 or x = 0, timelike if <x, x> < 0, and lightlike or null if <x, x> = 0 and x [not equal to] 0.

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