Minkowski time

The difference between standard time and Minkowski time therefore becomes measurable in the form of alterations produced in the apparent frequency of a freely propagated photon. The relative shift can be calculated as the frequency difference as measured by two observers, one of whom is at rest relative to the other. 

Some common practices further aggravate the situation. The accepted interpretation of special relativity considers all space outside of the Minkowski time cone as non-physical. This prejudice obscures the non-local nature of quantum theory and distorts the common perception of space-time topology. By an equally arbitrary assumption, advanced solutions (in —t) of the three-dimensional wave equation are rejected. This way all perceptions of space-time chirality, the existence of antimatter and non-local correlation are lost. 

In the real world the stress tensor never vanishes and so requires a nonvanishing curvature tensor under all circumstances. Alternatively, the concept of mass is strictly undefined in flat Minkowski space-time. Any mass point in Minkowski space disperses spontaneously, which means that it has a space-like rather than a time-like world line. In perfect analogy a mass point can be viewed as a local distortion of space-time. In euclidean space it can be smoothed away without leaving any trace, but not on a curved manifold. Mass generation therefore resembles distortion of a euclidean cover when spread across a non-euclidean surface. A given degree of curvature then corresponds to creation of a constant quantity of matter, or a constant measure of misfit between cover and surface, that cannot be smoothed away. Associated with the misfit (mass) a strain field appears in the curved surface. 

There is no evidence that Minkowski space is flat on the large scale. The assumption of euclidean Minkowski space could therefore be, and probably is an illusion, like the flat earth. In fact, there is compelling evidence from observed spectroscopic red shifts that space is curved over galactic distances. These red shifts are proportional to distances from the source, precisely as required by a curved space-time[52j. An alternative explanation, in terms of an expanding-universe model that ascribes the red shifts to a Doppler... 

There also exists convincing internal evidence that real Minkowski space must be curved. Euclidean 4-space is commonly represented diagrammati-cally to distinguish between time and space axes as in figure 4. 

Equation (482) is a simple form of the non-Abelian Stokes theorem, a form that is derived by a round trip in Minkowski spacetime [46]. It has been adapted directly for the 0(3) invariant phase factor as in Eq. (547), which gives a simple and accurate description of the Sagnac effect [44], A U(l) invariant electrodynamics has failed to describe the Sagnac effect for nearly 90 years, and kinematic explanations are also unsatisfactory [50], In an 0(3) or SU(2) invariant electrodynamics, the Sagnac effect is simply a round trip in Minkowski space-time and an effect of special relativity and gauge theory, the most successful theory of the late twentieth century. There are open questions in special relativity [108], but no theory has yet evolved to replace it. 

J. P. Vigier, L. de Broglie, D. Bohm, P. Hillion, F. Halbwachs, and T. Takabayasi, Space-time model of relativistic extended particles in Minkowski space II, Phys. Rev. 129, 451 (1963). 

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.

Figure 2. The relation between the Minkowski diagram and the holodiagram designed for the creation and evaluation of holograms. The horizontal intersections of the two cones represent different points of time, and the two sets of circles formed by those intersections are identical to the two sets of circles originally used to produce the ellipses of the holodiagram as later described in Fig. 5.

Figure 6. Spin frame representation of a spin-vector by flagpole normalized pair representation a,b over the Poincare sphere in Minkowski tetrad (l,x,y,z) form (n representation) and for three timeframes or sampling intervals providing overall (t]. r ) a Cartan-Weyl form representation. The sampling intervals reset the clock after every sampling of instantaneous polarization. Thus polarization modulation is represented by the collection of samplings over time. Minkowski form after Penrose and Rindler [28]. This is an SU(2) Gd hx) m C over it representation, not an SO(3) Q(to, 8) in C representation over 2it. This can be seen by noting that an b or bt-z a over it, not 2n, while the polarization modulation in SO(3) repeats at a period of 2it.

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 ... 

Euclidean, 85 Manhattan, 85 Minkowski, 85 distillation time, 27, 29 distributed systems, 188 DNA molecule, 199 DTA curve, 96 dual-slope converter, 170 dynamic range of signals, 157... 

Minkowski therefore considered that the s-diagram represents the universe as it really is. He suggested that the separation of events which exist in the s-diagram into a series of happenings in space and time is due to the one-sided view which any particular observer necessarily gets. This is discussed more fully in Appendix 16. 

Non-locality in terms of special relativity is best explained by the Minkowski space-time diagram, shown in figure 2. A stationary object follows a world-... 

Figure 3.2 Minkowski diagram showing three space dimensions x drawn perpendicular to the direction of time flow t.

The universally observed flow of time is another example of a broken symmetry. A theoretical formulation of this proposition is not known, but in principle it should parallel the theory of superconductivity. A high-symmetry state could be associated with Euclidean Minkowski space that spontaneously transforms into a curved manifold of lower symmetry. In this case the hidden symmetry emerges from a Lagrangian which is invariant under the temporal evolution group... 

The idea of a fourth dimension... was introduced to the modern world by Hermann Minkowski, who pointed out in 1908 that Einstein s Special Theory of Relativity is equivalent to an assertion that the world we live in is not three-dimensional but four-dimensional, the fourth dimension being time. Since "space" implies three-dimensionality, Minkowski referred not to "four-dimensional space" but to the "four-... 

In the flat case of the special theory of relativity the proper time s is given in terms of the space-time displacements by the classical Minkowski expression... 

Notice that the expression equation (26) has the Painleve form a Minkowski ds2 minus the product of a space-time function by the square of a sum of differentials. Furthermore these three sums of differentials given in equations (19), (25) and (26) have the following common point their ds2 can be written... 

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