Minkowski coordinates

As a consequence of translational invariance both quantities are functions of the difference of the Minkowski coordinates only, so that their four-dimensional Fourier transform can be written as... 

Distance measures were already discussed in Section 2.4. The most widely used distance measure for cluster analysis is the Euclidean distance. The Manhattan distance would be less dominated by far outlying objects since it is based on absolute rather than squared differences. The Minkowski distance is a generalization of both measures, and it allows adjusting the power of the distances along the coordinates. All these distance measures are not scale invariant. This means that variables with higher scale will have more influence to the distance measure than variables with smaller scale. If this effect is not wanted, the variables need to be scaled to equal variance. 

Figure 1. An ultrashort light pulse is emitted at A, which is the apex of a Minkowski lightcone. In our coordinate system x-y represent two axes of our ordinary world, while the third axis ct represents time. The widening of the cone upward represents the radius of the sphere of light as it increases with time. An ultrashort observation is later made at B, the apex of an inverted cone. The only way for light to be transmitted from A to B is by scattering objects placed where the two cones intersect. If the observer s velocity (v) is high, this intersection will be an ellipse that is inclined in relation to our stationary world.

We can add the time as the fourth coordinate, to build the equivalent of the Minkowski space-time element. We then get the Robertson-Walker line element after the change of variables f> —> r ... 

Figure 2.4 (Left) Accelerated motion in Minkowski space. (Right) Two coordinate systems in relative rotational motion.

An event in Minkowski space-time is defined, relative to a coordinate frame 5, by a 4-vector jc = (/t = 0,1,2,3) where Jt = ctis the time coordinate... 

The metric tensor g i, in four-dimensional Minkowski space reads in Cartesian coordinates as... 

The only way in which to transform Minkowski space into a closed manifold is by adding a point at infinity to each coordinate axis, to produce a multiply-connected non-orientable hypersurface, known as a projective plane. General relativity should therefore ideally be formulated as a field theory in projective, and not in affine, space. 

Theories like those of Lemaitre or Friedmann, which predict an expanding universe, are all based on forcing an affine metric, such as the Robertson-Walker metric, on the projective geometry of space-time. This has the effect of splitting local Minkowski space into separate space and time coordinates, without the natural complex relationship that ties space and time together. 

The 2 X 2 submatrix of 3 3 and X4 resembles the rotation matrix, equation (3.3) for rotation about one axis in a three-dimensional coordinate system. Correspondingly the transformation (4.3) can be said to be a rotation in the X3X4 plane of four-dimensional Minkowski space, through an imaginary angle , such that... 

As a test particle crosses the coordinate singularity at r = 2m the radial coordinate becomes time-like, which means that it resembles a stationary particle in Minkowski space. The consequence of this is that the residence time within the inversion zone (or bridge) is extended almost indefinitely. 

The velocity of light in a radial Robertson-Walker direction can be calculated by putting ds = 0, as in Minkowski space, together with angular coordinates d9 and d(p equal to zero, such that... 

As p TV, at the antipodal point in the projective space, the redshift approaches totality as 2 —> oo. In the stationary Minkowski frame the antipode is infinitely distant at an infinite time coordinate. The conditions leading to the derivation of the redshift formula are not met for radiation with a propagation interval close to a half-circuit in space, and such photons will appear entirely delocalized and severely redshifted, constituting the isotropic microwave background. Segal ascribes the Planddan distribution to the conservation of energy, which is tantamount to the fact that any closed space must eventually impose a Planckian spectrum on stray radiation. 

Another way of looking at four-dimensional projective space is by adding a point at infinity on each coordinate axis of Minkowski space. Any displacement implies a change in all coordinates. The only difference between stationary antipodal points is an inversion of local chirality, which includes the direction of time flow. There is no possibility of communication between such points which move apart on their respective world lines. 

By relating the rest mass to the internal motion, quantum theory brings an insight into the bearing of such relativistic concepts as Lorentz-invariant, Minkowski s proper interval Xq. As the property moC is the residual momentum when the linear partp is subtracted from the total entity m c (Eq. 2.7b), the property xo is the residual interval when the space coordinate is subtracted from the time coordinate c f (Eq. 2.5b). 

This equation and Eq. (3.28) enabled Minkowski to interpret the Lorentz transformation [Eq. (3.28)] as a rotation of the event (x, ct) in the Minkowski space about the origin of the coordinate system (since any rotation preserves the distance from the rotation axis). 

In the Minkowski space, the distance of any event from the origin (and both coordinate systems fly apart ... 

Hermann Minkowski introduced the seminal concept of the four-dimensional space-time continuum (jc, y, z, ct). In our one-dimensional space, the elements of the Minkowski space-time continuum are events, i.e. vectors (x,ct), something happens at space coordinate x at time t, when the event is observed from coordinate system O. When the same event is observed in two coordinate tys-... 

The Lorentz transformations have been identified as those coordinate transformations of the four-dimensional space-time (Minkowski space) that leave the four-dimensional (squared) distance s 2 between any two events Ei and Ez invariant, S12 = emphasize that both events have to be described... 

As this rotation mathematically interchanges time and space coordinates, it means that they are symmetry related and no longer separable in the usual way. It is therefore more appropriate to deal with four-dimensional space-time, rather than the traditional three-dimensional space and absolute time. To visualize Minkowski space, it is useful first to review some properties of the complex plane. 

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