World curvature

Here r, 9, 4> are dimensionless co-moving coordinates attached to fundamental observers and R(t) a scale factor with a dimension of length depending only on cosmic time t. k is the curvature constant, which with suitable choice of units takes one of the three values +1 (closed world model with positive curvature), 0 (flat, open model) or —1 (open model with negative curvature). Some consequences of Eq. (4.7) are the relation between redshift and scale factor Eq. (4.2) and the variation of temperature... 

It is proposed to use the terms world space and aether synonymously and to reserve space and vacuum for empty three-dimensional space. Curvature of world space may refer to any of its unknown number of dimensions. It... 

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. 

Gibbs found the solution of the fundamental Equation 9.1 only for the case of moderate surfaces, for which application of the classic capillary laws was not a problem. But, the importance of the world of nanoscale objects was not as pronounced during that period as now. The problem of surface curvature has become very important for the theory of capillary phenomena after Gibbs. R.C. Tolman, F.P. Buff, J.G. Kirkwood, S. Kondo, A.I. Rusanov, RA. Kralchevski, A.W. Neimann, and many other outstanding researchers devoted their work to this field. This problem is directly related to the development of the general theory of condensed state and molecular interactions in the systems of numerous particles. The methods of statistical mechanics, thermodynamics, and other approaches of modem molecular physics were applied [11,22,23],... 

The symmetry between curvature and matter is the most important result of Einstein s gravitational field equations. Both of these tensors vanish in empty euclidean space and the symmetry implies that whereas the presence of matter causes space to curve, curvature of space generates matter. This reciprocity has the important consequence that, because the stress tensor never vanishes in the real world, a non-vanishing curvature tensor must exist everywhere. The simplifying assumption of effective euclidean space-time therefore is a delusion and the simplification it effects is outweighed by the contradiction with reality. Flat space, by definition, is void. 

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. 

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... 

Another consequence of the general curvature of world space is that distant radiation sources are separated from an observer in both space and time [232]. During transit, the photon moves towards an observer which is ahead in time, and therefore appears to lag as if the source was receding. The observed red shift, created by the time difference, will be a function of separation in space and proportional to the time interval, At. The red shift and Hubble s proportionality constant are then defined by... 

However, we are mostly dealing with three-dimensional objects such as bubbles, drops etc. in the real world. In order to describe the curvature of three-dimensional objects, two radii of curvature are needed, and things get a bit more complicated. This is because the curvature can appear different when in different directions. Curvatures may be positive or negative, and here we adopt the convention that a curvature is taken to be positive if the curve turns in the same direction as the surface s chosen normal otherwise it is negative. In Figure 4.5 we see how to obtain these curvatures on a surface shown by the mnpr layer. 

K. Mikula. Solution and application of anisotropic curvature driven evolution of curves (and surfaces). In M. Falcone (ed.) et al., Numerical methods for viscosity solutions and applications. Singapore World Scientific. Ser. Adv. Math. Appl. Sci. 59, 173-196, 2001. 

The recognition of self-similarity simplifies the description of the physical world, all in response of the vacuum to the curvature of space-time, but complicates the perception of three-dimensional beings of their four-dimensional environment, which they are physically unable to visualize. The perceived infinity of space is the illusion created in simply-connected tangent space by the multiply-connected cosmic reality. The large-scale structure of the imiverse is destined to remain unknown for a long time. 

Recent advances in the mechanics of complex buckling [5] come from a recognition that the formation of spherical surfaces of double curvature, which is the converse of drawing a map of the world on a planar surface, demands inplane (membrane) strain as well as bending. It has also been recognised that the simplest problem in this class is the buckling of a circular specimen pushed inwards at three equally spaced points. The material deforms into a dome of... 

Alexandr Alexandrovitch Friedmann (1888-1925), Russian mathematician and physicist, in his articie in Zeit. Phys., 10, 377 (1922), proved on the basis of Einstein s general theory of relativity that the curvature of the Universe must change, which became the basis of cosmological models of the expanding Universe. During World War I, Friedman was a pilot in the Russian army and made bombing raids over my beloved Przemysl. 

A solar concentrating device developed by Wolfgang Scheffler in 1986 produces temperatures between 850 and 1,200 degrees Fahrenheit at a fixed focal point by means of flexible parabolic dishes that track the Sun s diurnal motion and adjust curvature seasonally. By 2008, more than 2,000 large Scheffler cookers had been built, most used for cooking meals. The world s largest system, in Rajasthan, India, can cook up to 35,000 meals daily. 

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