Curvature of space

This variation of the curvature of space is what really happens in that phenomenon which we call the motion of matter, whether ponderable or ethereal. 

The metric term Eq. (2.8) is important for all cases in which the manifold M has non-zero curvature and is thus nonlinear, e.g. in the cases of Time-Dependent Hartree-Fock (TDHF) and Time-Dependent Multi-Configurational Self-Consistent Field (TDMCSCF) c culations. In such situations the metric tensor varies from point to point and has a nontrivial effect on the time evolution. It plays the role of a time-dependent force (somewhat like the location-dependent gravitational force which arises in general relativity from the curvature of space-time). In the case of flat i.e. linear manifolds, as are found in Time-Dependent Configuration Interaction (TDCI) calculations, the metric is constant and does not have a significant effect on the dynamics. 

The final objective is an equation that relates a geometrical object representing the curvature of space-time to a geometrical object representing the source of the gravitational field. The condition that all affine connections must vanish at a euclidean point, defines a tensor [41]... 

The important result is the obvious symmetry between TM1/ and R u as shown in (42) and (43). Both of these tensors vanish in empty euclidean space and a reciprocal relationship between them is inferred The presence of matter causes space to curl up and curvature of space generates matter. 

Yes. In the foam, adjacent regions of space are continually stealing and giving back energy from one to another. These cause fluctuations in the curvature of space, creating microscopic wormholes. Who knows, someday civilizations might be able to use such wormholes to travel the universe. ... 

I asked Professor Michio Kaku, author of Hyperspace, if the gtavitational curvature of space implies the existence of a foutth dimension. He responded ... 

We do not need a foutth spatial dimension in which to describe the curvature of space. From one point of view, the fourth spatial dimension is fictitious. This is because we can use inttinsic 3-D coordinates in which the only coordinates ate three bent spatial dimensions and one time dimension. Thus, an ant on an ordinaty balloon can only see two dimensions, and says that the third dimension in unnecessary, because the ant cannot travel in the third dimension, which is fictitious from his point of view. 

Einstein [6] illustrated the curvature of space-time by considering two coordinate systems, K and K, with a common origin, one of them stationary and the other rotating (accelerated) about the common Z-axis, in a space free of gravitational fields, shown in Figure 2.4 on the right. A circle around the origin in the X — Y plane of K is also a circle in the X — Y plane of K. Measurement of the circumference S and diameter 2R in the stationary... 

Recall the reciprocity between matter and curvature, implied by the theory of general relativity, to argue that the high-pressure condition at Z/N = 1 corresponds to extreme curvature of space-time caused by massive objects such as quasars, and the like. The argument implies that the Schrodinger solution is valid in empty, flat euclidean space-time, that Z/N = r corresponds to the real world, Z/N = 1 occurs in massive galactic objects where elemental synthesis happens, and Z/N > 1 implies infinite curvature at a space-time singularity. 

Bonds with r < dl < d[ become possible because of nuclear screening (increased bond order), which causes concentration of the bonding pair directly between the nuclei. The exclusion limit is reached at d = t and appears as a geometrical property of space. The distribution of molecular electron density is dictated by the local geometry of space-time. Model functions, such as VSEPR or minimum orbital angular momentum [65], that correctly describe this distribution, do so without dictating the result. The template is provided by the curvature of space-time which appears to be related to the three fundamental constants tt, t and e. 

Anticipating the final conclusion that matter and energy are special configurations of space-time, the investigation starts with the topic of relativity, the only theory that has a direct bearing on the topology of space-time and which demonstrates the equivalence of energy and matter and a reciprocal relationship between matter and the curvature of space. 

Chemical behaviour depends on chemical potential and electromagnetic interaction. Both of these factors depend on the local curvature of space-time, commonly identified with the vacuum. Any chemical or phase transformation is caused by an interaction that changes the symmetry of the gauge field. It is convenient to describe such events in terms of a Lagrangian density which is invariant under gauge transformation and reveals the details of the interaction as a function of the symmetry. The chemically important examples of crystal nucleation and the generation of entropy by time flow will be discussed next. The important conclusion is that in all cases, the gauge field arises from a symmetry of space-time and the nature of chemical matter and interaction reduces to a function of space-time structure. 

In essence, real world-space is not Euclidean and space is generally curved into the time dimension, consistent with the theory of general relativity. The curvature may not be sufficient to become obvious in a local context. However, it is sufficient to break the time-reversal symmetry that seems to characterize the laws of physics. Not only does it cause perpetual time flow with respect to all mass, but actually identifies a fixed direction for this flow. It creates an arrow of time and thereby eliminates an inconsistency in the logic of physics how reversible microscopic laws can underpin an irreversible macroscopic world. General curvature of space breaks the time-reversal symmetry and produces chiral space, manifest in the right-hand... 

Earlier speculations about the effect of the curvature of space on elemental synthesis and the stability of nuclides (2.4.1) are consistent with the interface model. The absolute curvature of the closed double cover of projective space, and the Hubble radius of the universe, together define the golden mean as a universal shape factor [233], characteristic of intergalactic space. This factor regulates the proton neutron ratio of stable nuclides and the detail of elemental periodicity. The self-similarity between material structures at different levels of size, such as elementary particles, atomic nuclei, chemical... 

By assuming that the dynamic variables follow the same rules as for hydrogen in many-electron atoms, there was the expectation that the periodic table of the elements could be reduced to the Schrodinger solution for hydrogen. Apart from a superficial correlation, which is commonly assumed to vindicate this expectation, it has now been shown that the neglect of general-relativistic curvature of space-time prevents such reduction. Once this defect has been rectified the atomic model will be used to investigate commensurability in the self-similar solar system. 

The wave nature of the electron and the physical implications thereof were discussed recently in some detail (Boeyens, 2010). As in the theory of general relativity it is accepted that an empty universe is featureless and flat, but that curvature of space-time causes wavelike distortion of the vacuum. The equivalent of an infinite plane wave in flat space develops interference effects, like wave packets, in curved space, interpreted as units of mass and energy. 

The separation of time-like and space-like events creates the impression of two types of response to increased curvature of space-time. If only space coordinates are curved it results in the inversion of the time coordinate and the conversion of matter into antimatter. This situation will be encountered in the Schwarzschild solution of the gravitational field equations, which serves as a model of a black hole, and assumed here to account for an inversion at Z/A = 1.04. It resembles the limitless time-like accumulation of matter, resulting in a space-time singularity. 

Any astronomically measured frequency shift consists of several components, including the chemical shift, described here. Other contributions include relativistic gravitational redshifting, a distance-dependent redshift caused by the topological curvature of space-time, and a Doppler shift where the source is in relative motion. 

The reciprocal relationship between matter and the curvature of space implies that the occurrence of different periodic functions at Z/N = 1, r and 0.58 arises from a variation of the electronic configuration of atoms with... 

In relativistic terminology the principle is best expressed as the balance between matter density and the curvature of space. The assumption of uniform matter density implies that space is non-Euclidean with constant... 

For a = 0, both A and p must be zero, which implies flat space. The parameter a therefore measures the curvature of space. Although the solution will be shown to define an aesthetically more pleasing cosmology, it has been ignored for many years because it fails to predict a Doppler redshift, does not give a clear definition of the compass of inertia, proposed as rotation axis, and allows closed time loops. 

Although the Godel solution is free of singularities the need to accommodate black holes in the cosmic model requires an interpretation of the Schwarzschild singularity which occurs with infinite curvature of space-time. A new interpretation is rather obvious. Such a high degree of curvature must clearly rupture the interface between adjacent sides of the postulated cosmic double cover. Rather than disappear into a singularity, the matter,... 

The reasonable, but not essential assumption, that the general curvature of space-time be constant, predicts a closed topology in the form of either a hypersphere or a four-dimensional projective plane. Additional evidence is needed to decide between these possibilities. 

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