Space Minkowski

The Lorentz transformation is an orthogonal transformation in the four dimensions of Minkowski space. The condition of constant c is equivalent to the requirement that the magnitude of the 4-vector s be held invariant under the transformation. In matrix notation... 

Equation (29) is therefore equivalent to rotation in the X3X4 plane of Minkowski space through an imaginary angle (f>, such that... 

The relativistic invariance of the electromagnetic field is conveniently expressed in tensor notation. Factorized in Minkowski space the Maxwell equa-... 

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. 

To treat the general situation, compatible with cartesian geometries, we will consider the (l+N)-dimensional Minkowski space. Then, taking... 

The price which must be paid in order to make the action local is that the spatial dimension must be augmented by one. Hence, the integral must be performed over a five-dimensional manifold whose boundary (M4) is ordinary Minkowski space. In [27, 32, 36] the constant C has been shown to be the... 

The conformal group 0(1,3), if the Yang-Mills equations are defined in Minkowski space... 

To the best of our knowledge, the first paper devoted to symmetry reduction of the 57/(2) Yang-Mills equations in Minkowski space has been published by Fushchych and Shtelen [27] (see also Ref. 21). They use two conformally invariant ansatzes in order to perform reduction of Eqs. (1) to systems of ordinary differential equations. Integrating the latter yields several exact solutions of Yang-Mills equations (1). 

The principal aim of the present chapter is twofold. First, we will review the already known ideas, methods, and results centered around the solution techniques that are based on the symmetry reduction method for the Yang-Mills equations (1) in Minkowski space. Second, we will describe the general reduction routine, developed by us in the 1990s, which enables the unified treatment of both the classical and nonclassical symmetry reduction approaches for an arbitrary relativistically invariant system of partial differential equations. As a byproduct, this approach yields exhaustive solution of the problem of... 

In this section we apply the technique described above in order to perform in-depth analysis of the problems of symmetry reduction and construction of exact invariant solutions of the SU(2) Yang-Mills equations in the (l+3)-dimensional Minkowski space of independent variables. Since the general method to be used relies heavily on symmetry properties of the equations under study, we will briefly review the group-theoretic properties of the SU(2) Yang-Mills equations. 

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). 

But the Minkowski spacetime R4 has trivial cohomology. This means that the Maxwell equation implies that. is a closed 2-form, so it is also an exact form and we can write. = d d, where ( is another potential 1-form in the Minkowski space. Now the dynamical equation becomes another Bianchi identity. This simple idea is a consequence of the electromagnetic duality, which is an exact symmetry in vacuum. In tensor components, with sJ = A dx and ((i = C(1dxt we have b iV = c, /tv — and b iV = SMCV - SvC or, in vector components... 

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 ... 

Although in Minkowski space, this corresponds to energy and momentum conservation, this is no longer the case in an expanding universe. 

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

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-... 

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... 

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