Space-time curved manifold

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. 

It has already been shown that for constant a this invariance (symmetry) implies conservation of the charge of a free particle. In general relativity, which is based on a curved manifold rather than flat space with a globally fixed coordinate system, each point has its own coordinate system and hence its own gauge factor. This means that the gauge factor a is no longer a constant, but a function of space-time, i.e. 

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... 

An elegant but simple model of a five-dimensional universe has been proposed by Thierrin [224]. It is of particular interest as a convincing demonstration of how a curved four-dimensional manifold can be embedded in a Euclidean five-dimensional space-time in which the perceived anomalies such as coordinate contraction simply disappear. The novel proposal is that the constant speed of light that defines special relativity has a counterpart for all types of particle/wave entities, such that the constant speed for each type, in an appropriate inertial system, are given by the relationship... 

The idea of a closed space-time manifold with an involution has been mooted on the basis of nuclear synthesis (figure 2.6), number theory (figure 2.8), historical argument (4.4), absorber theory (figure 4.8) and chirality (5.9.3). All of these schemes can now be combined into a single construct based on curved Thierrin space-time. 

An age-old argument about the heat-death of the universe is also settled by the interface model. It relates to the problem that the second law of thermodynamics is time-irreversible, but based on time-reversible laws of physics. It has been argued (Boeyens, 2005) that, because the world lines in neighbouring tangent spaces of the curved manifold are not parallel, a static distribution of mass points must be inherently unstable. As systems with non-parallel world lines interact a chaotic situation such as the motion in an ideal gas occurs, which means that time flow generates entropy. 

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. 

We contend that the shape of large molecules in empty space is affected by the topology of the four-dimensional space-time manifold. Guided by the principle of cosmic self-similarity, it is reasonable to assume that, like many spiral galaxies, extended molecules tend to curve like the surface of a golden spiral. It lies in an... 

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. 

Structure and can be used to construct, via the TDVP, approximations of the TDSE as a classical dynamical system. If one used the manifold constituted by whole Hilbert space (not an AGP manifold) then the classical Hamilton equations plus an equation for a time-dependent phase would be equivalent to the TDSE, and their solutions would provide solutions to the Schrbdinger equation. The classical equations restricted to the AGP manifolds produce paths of AGP states that approximate the exact paths. One can consider the classical linear response of approximate stationary states found on these manifolds to external time-dependent perturbations either wifliin the curved phase or in the tangent space at these points [37]. This allows a variety of linear response approximations schemes to be developed that generdize and clarify propagator and RPAs. 

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