Minkowski space-time

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. 

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. 

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 ... 

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-... 

Special relativistic notation Minkowski space-time. Lorentz transformations... 

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 field equations of general relativity are rarely used without simplifying assumptions. The most common application treats of a mass, sufficiently distant from other masses, so as to move uniformly in a straight line. All applications of special relativity are of this type, in order to stay in Minkowski space-time. A body that moves inertially (or at rest) is thus assumed to have four-dimensionally straight world lines from which they deviate only under acceleration or rotation. The well-known Minkowski diagram of special relativity is a graphical representation of this assumption and therefore refers to a highly idealized situation. 

Figure 4.2 Two-dimensional representation of Minkowski space-time...

The Minkowski space-time of special relativity differs from conventional Euclidean space only in the number of dimensions and gives the correct description of all forms of uniform relative motion. However, it fails when applied to accelerated motion, of which circular motion at constant orbital speed is the simplest example. Relativistic contraction only occurs in the direction of motion, but not in the perpendicular radial direction towards the centre of the orbit. The simple Euclidean formula that relates the circumference of the circle to its radius therefore no longer holds. The inevitable conclusion is that relativistic acceleration implies non-Euclidean geometry. 

The Vanishing of Apparent Forces The Galilean Transformation The Michelson-Morley Experiment The Galilean Transformation Crashes The Lorentz Transformation New Law of Adding Velocities The Minkowski Space-Time Continuum How do we Gel E =... 

Lorentz transformation (p. 113) Michelson-Morley experiment (p. 1111 Minkowski space-time (p. 177) negative energy continuum (p. 125) positive energy continuum (p. 125) positron (p. 126)... 

D Alembertian Symbol (sometimes printed D ). An operator that is the analogue of the Laplace operator in four-dimensional Minkowski space-time, i.e. 

Minkowski space-time Universe is a reality, just like the quadrivec-tors in Minkowski s terms, expressed at a conference in Cologne on September 21 (1908). ""The views on space and time that I want to develop for you were horn on the experimental-physical ground. In this lays their strength. Their tendency is to give radical demonstrations. From now on, the space for itself should completely disappear in shadow, considering only the existence of an association of the two. ""... 

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-... 

Minkowski space-time (p. 104) time dilation (p. 105) relativistic mass (p. 107)... 

The unexpected appearance of complex operators is also associated with nonzero commutators and reflects the essential two-dimensional representation in MP Minkowski space-time. In four-dimensional space-time, all commutators are non-zero, as appropriate for wave motion of both quantum and relativity theories. An important consequence is that local observation has no validity on global extrapolation, as evidenced by the appearance of cosmical red shifts in the curved manifold and the illusion of an expanding universe. 

When contemplating the formulation of four-dimensional theories the first measure would be the use of Minkowski space-time, which is tangent to the underlying curved manifold and adequate, to first approximation, for the analysis of macroscopic local phenomena. At the sub-atomic or galactic level the effects of curvature cannot be ignored. 

With a wave model in mind as a chemical theory it is helpful to first examine wave motion in fewer dimensions. In all cases periodic motion is associated with harmonic functions, best known of which are defined by Laplace s equation in three dimensions. It occurs embedded in Schrodinger s equation of wave mechanics, where it generates the complex surface-harmonic operators which produce the orbital angular momentum eigenvectors of the hydrogen electron. If the harmonic solutions of the four-dimensional analogue of Laplace s equation are to be valid in the Minkowski space-time of special relativity, they need to be Lorentz invariant. This means that they should not be separable in the normal sense of Sturm-Liouville problems. In standard wave mechanics this is exactly the way in which space and time variables are separated to produce a three-dimensional wave equation. 

The instanton theory was invented in the field theory by introducing imaginary time to Minkowski space-time. The instanton is the classical object in the Euclidean space-time that gives a finite action. The instanton is also called pseudo-particle. Here the theory is explained by following Coleman [2]. See also References [17,39,43,46]. 

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