Space-time geometry

As an example of affine transformation, consider a square transformed under shear and under strain, as shown in figure 9. Lines that originally 

A more general view is provided by absolute geometry that makes no 

Projective geometry was created by Renaissance artists. To make a correct perspective drawing the scene of interest is projected into a plane that intersects all lines between points on the object and the eye of the artist. Affine 

As it views a scene the eye does not respond directly to the objects in the scene, but to light rays that travel along straight lines from points in the scene to the eye. It follows that radial lines will look like points and radial planes look like lines. As a result radial dimensions are lost. A further consequence is that non-radial lines acquire an extra point at infinity. As shown in figure 11 the non-radial line L is observed through a set of radial lines in the plane OL. Only one radial line in OL does not connect a point on L to the eye. That one exception is the radial line parallel to L. This line Pqo appears to the eye as a point at infinity on L. The two parallel lines that meet at infinity are considered to intersect at an angle of zero. 

The same reasoning shows that non-radial planes acquire an extra line at infinity. It follows that projective space may be regarded as affine space plus a plane at infinity. One of the most elegant properties of projective geometry is the principle of duality which asserts that, in a projective plane every definition or theorem remains valid on the consistent interchange of the words line and point. 

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Note that the operator, c(r) > 0, depends on the coordinate r of the particle m, with the origin at the centre of mass of Af. It is important to distinguish between the coordinates r (and t) of a flat Euclidean space and the scales defining the curved space-time geometry defined by the (operator)-secular problem Eq. (14). We get directly... 

If the superposition of point-source solutions is determined by a Green function , then by analogy with Wheeler-Feynman absorber theory the total interaction results from a symmetrical combination of retarded and advanced Green functions. On dehning these Green functions to be compatible with space-time geometry (Riemann tensor) the interaction is shown to be consistent with Einstein s field equations. 

Equations (6.4) describe the balance between two unknowns - the fundamental tensor and the distribution of matter in a system of interest. Although the distribution function is not conditioned by the theory of relativity in any way, it assumes critical importance in deciding the appropriate space-time geometry in cosmological applications. This way the metric tensor is defined, not on the basis of relativistic considerations, but on Newtonian principles. Cosmological models arrived at in this way we consider non-relativistic, unless the metric tensor has the correct relativistic signature. To explain the reasoning we consider a few elementary models. 

The universal space-time geometry itself is not necessarily directly observed no apparent departures from a Euclidean model have been found by classical measurements. It might be argued that curved space-time coordinates are split into flat space and time coordinates. There is no direct observational basis for asserting that the Cosmos is Minkowskian at large distances and times. 

Figure 5.18 Comparison of space-time yields of direct fluorination of toluene for the falling film micro reactor (FFMR), micro bubble column (MBC) and laboratory bubble column (LBC) referred to the reaction volume (a) and referred to an idealized reactor geometry (b) [38],...

The above-mentioned space-time yields were referred solely to the reaction volume, i.e. the micro channel volume. When defining this quantity via an idealized reactor geometry, taking into accoimt the construction material as well, natarally the difference in space-time yield of the micro reactors from the laboratory bubble column becomes smaller. Still, the performance of the micro reactors is more than one order of magnitude better [38], The space-time yields for the micro reactors defined in this way ranged from about 200 to 1100 mol monofluorinated product... 

The product selectivity is strongly affected by the flow rate, reactor geometry (i.e., internal diameter and "heated zone) and weight of catalyst. On this account, the space time yield to HCHO - or HCHO productivity -( HCHO Scat h ) appears to be the more definite parameter to evaluate the reactivity of the partial oxi tion catalysts. 

M. Requardt, Spectral analysis and operator theory on (infinite) graphs..JPA (in press) math-ph/0001026 M. Requardt (Quantum) space-time as a statistical geometry of lumps in random networks, Class. Quant. Grav. 17, 2029 (2000) gr-qc/9912059. 

M. Requardt and Sisir Roy, (Quantum)Space-Time as a Statistical Geometry of Fuzzy Lumps and the Connection with Random Metric Spaces - to be published in Classical Quantum Gravity (2001). 

Incidentally we find that the positive operator x(r) >0 depends formally on the coordinate r of the particle m, with origin at the center of mass of M. Since the dimensions or scales x and r are subject to the description of the conjugate problem, we will on balance recover a geometry of curved space-time scales reminiscent of the classical theories, see more below. 

Geometry and Dynamics Geometry of 4-dimensional space-time... 

The conscious final decision to take the risk, with the current sequence, should be read as a personal conviction that the beauty of chemistry can never be fully appreciated unless viewed against the background in which all matter originates - space-time, or the vacuum. Not only matter, but all modes of interaction are shaped by the geometry of space, which at the moment remains a matter of conjecture. However, the theory of general relativity points the way by firmly demonstrating that the known material world can only exist in curved space-time. The theory of special relativity affirms that space-time has a minimum of four dimensions. Again, spaces of more dimensions are conjectural at present. 

Equation (2.11) with variable metric tensor describes the invariance in the gravitational case which is characterized by curved space-time. The summation extends over all values of y, and u, so that the sum consists of 4 x 4 terms, of which 12 are equal in pairs, hence 10 independent functions. The motion of a free material point in this field will take the form of curvilinear non-uniform motion. If the matrix of the metric tensor can be diagonalized it is independent of position and the corresponding geometry is said to be flat, which is the special case of SR. 

All solutions of Einstein s equations are conditioned by the need of some ad hoc assumption about the geometry of space-time. The only indisputably valid assumption is that space-time is of absolute non-euclidean geometry. It is interesting to note that chiral space-time, probably demanded by the existence of antimatter and other chiral forms of matter, rules out the possibility of affine geometry, the standard assumption of modern TGR [7]. 

The conclusion that it is the geometry of the space-time manifold that generates all known fields that operate in the physical world seems to be unassailable, without further proof. One circumstance that spontaneously produces a constant potential in space is an interface. It has been speculated... 

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. 

Covalent interaction in diatomic molecules depends on the golden mean t, the interatomic distance d and the radius ratio x r /r2 of the constituent atoms, as summarized in Figure 5.6. The golden mean is a universal constant that matches the geometry and topology of space-time, the radius ratio is a known function of atomic number and dl relates to the optimal wave-mechanical distribution of valence-electron density in the diatomic system. 

Interatomic distances are determined by steric factors, of which the most important is the exclusion principle that depends directly on the geometry of space-time, observed as the golden ratio. Bond order depends on the ratio between the number of valence electrons and the number of first neighbours, or ligands, and affects interatomic distances by the screening of internuclear repulsion. 

Fig. 48. (a) Three-dimensional geometry used in the calculations, (b) and (c) Calculated space-time plots of the double layer potential in a gray scale representation where all model parameters were identical but the circumference of the CE, rce (b) rce = 11.7 cm. (c) rcE = 5.0 cm. (Reproduced from A. Birzu, B. J. Green, N. I. Jaeger, J. L. Hudson, J. Electroanal. Chem. 504 (2001) 126, (2001), with permission from Elsevier Science.)... 

Note that interpretations of the time-reversal experiments are only valid in strictly euclidean space-time. This condition is rarely emphasized by authors who state that all laws of physics are time-reversible, except for the law of entropy. Fact is that entropy is the only macroscopic state function which is routinely observed to be irreversible. One common explanation is to hint that entropy is an emergent property of macro systems and hence undefined for microsystems. Even so, the mystery of the microscopic origin of entropy remains. A plausible explanation may be provided if the assumed euclidean geometry of space-time is recognized as an approximate symmetry as demanded by general relativity. 

The broken symmetries of chapter 1, assumed responsible for shaping the physical world, refer to the symmetry of the vacuum and thus finally to the geometry of space-time. It is not immediately obvious that chemical theories could also be reduced to the same cause. While physics produced quantum theory and general relativity, the fundamental contribution from chemistry was the periodic table of the elements. Although the structure of individual atomic nuclei may be considered shaped by local space-time symmetry, the functional relationship between different nuclides needs further study. 

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