Minkowski Space

(redirected from Minkowski spacetime)

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.


References in periodicals archive ?
That quantization divides the space and time axes of the Minkowski spacetime diagram into equal segments, where the space and time segments are [r.
This is a fundamental misunderstanding of what the Minkowski spacetime interval represents.
We would like to point-out that our study of Frenet formulas for versor fields along a curve in the Minkowski spacetime is more general and, as it can be seen in the previous sections, we had to consider causal characters of the versor fields.
For instance, the Cauchy-Riemann equations, which specify the regularity conditions for a complex-valued function to be analytic (expressible as a power series), generalize to the Lanczos equations in Minkowski spacetime, and then generalize further to the Nijenhuis tensor equations for holomorphic functions in n-dimensional space.
Here, we come to the same conclusion, but it is argued that the lack of gravitation can be entirely explained in Minkowski spacetime with assumption of the Newtonian approximation for gravity.
It is plainly evident that metric (1) changes its signature from (+, -, -, -) to (-, +, -, -) when 0 < r < 2m, despite the fact that metric (1) is supposed to be a generalisation of Minkowski spacetime, described by (using c = 1),
The Minkowski spacetime diagrams from which the LT and its inverse have sometimes been derived for two frames in relative motion along their collinear x'--and x-axes, are shown as Figs.
in which R is an arbitrary function of r within the limit that metrics must be asymptotically Minkowski spacetime, i.
We report the discovery of an exact mapping from Galilean time and space coordinates to Minkowski spacetime coordinates, showing that Lorentz covariance and the spacetime construct are consistent with the existence of a dynamical 3-space, and "absolute motion".
Misconceptions as to the relationship between Minkowski spacetime and Special Relativity are also discussed, along with their relationships to the pseudo-Riemannian metric manifold of Einstein's gravitational field, and their fundamental geometric structures pertaining to spherical symmetry.
In order to describe the physics at infinity we will recur to Penrose's ideas [12] of conformal compactifications of Minkowski spacetime by attaching the light-cones at conformal infinity.
d] space (the hyperboloid) degenerates into a flat Minkowski spacetime and the coordinates [q.