Space curved

Space Curves Space curves are usually specified as the set of points whose coordinates are given parametrically by a system of equations x =f t), y = g(t), z = h t) in the parameter t. 

Nearly two years ago, studying electrodynamics in curved space-time I found1 that Maxwell s equations impose on space-time a restriction which can be formulated by saying that a certain vector q determined by the curvature field must be the gradient of a scalar function, or... 

When N > 4 there appears to be too many Zn, since N(N — l)/2 > 3N — 6. However, the Zn are not globally redundant. All Zn are needed for a global description of molecular shape, and no subset of ZN — 6 Zn will be adequate everywhere.49 The space of molecular coordinates which defines the shape of a molecule is not a rectilinear or Euclidean space, it is a curved manifold. It is well known in the mathematical literature that you cannot find a single global set of coordinates for such curved spaces. 

Fig. 16.4 Multiple linear model containing phosphorus, calcium and iron contents of the leaves correlated significantly with the polyphenol contents of fruit. Organic (filled symbols) and conventional fruits (open symbols) did not differ significantly in their average polyphenol contents however, trees with a low leaf nutrient status showed a lower polyphenol contents in fruit (R2 = 0.64). Circles = cultivar Glockenapfel , squares = cultivar Idared area between the curves = space where 95% of the modelled values can be expected.

In a curved space the unit vectors of one coordinate system change scale when transformed to another, i.e. 

There is no evidence that Minkowski space is flat on the large scale. The assumption of euclidean Minkowski space could therefore be, and probably is an illusion, like the flat earth. In fact, there is compelling evidence from observed spectroscopic red shifts that space is curved over galactic distances. These red shifts are proportional to distances from the source, precisely as required by a curved space-time[52j. An alternative explanation, in terms of an expanding-universe model that ascribes the red shifts to a Doppler... 

Ahmedov, H. and Duru, I. H. Casimir Energy for a Wedge with Three Surfaces and for a Pyramidal Region, math-ph/0407030, in J. Math. Phys Birrel, N. D. and Davies, P. C.V. Quantum Fields in Curved Spaces. Cambridge University Press, (1982). 

This scenario with curved space is not as zany as it may sound. Georg Bern-hard Riemann (1826—1866), the great nineteenth-century geometer, thought constantly on these issues and profoundly affected the development of modern... 

Whether we will ultimately be able to create furniture from curved space, partake of a multidimensional reality, or directly view all of humanity s alternate histories, becomes less of a issue than being able to fuel the imagination with these endless possibilities. 

M. W. Evans, P. K. Anastasovski, T. E. Bearden, et al., Electromagnetic energy from curved space-time, Optik (in press). 

The metric geometry of equilibrium thermodynamics provides an unusual prototype in the rich spectrum of possibilities of differential geometry. Just as Einstein s general relativistic theory of gravitation enriched the classical Riemann theory of curved spaces, so does its thermodynamic manifestation suggest further extensions of powerful Riemannian concepts. Theorems and tools of the differential geometer may be sharpened or extended by application to the unique Riemannian features of equilibrium chemical and phase thermodynamics. 

Is the gravitational field able, in principle, to produce real, tangible particles in a void, or in the terms of general relativity theory, in curved space-time ... 

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. 

The precise definition of a vacuum in a curved space-time is still subject to some ambiguities. We refer the interested reader to Fulling (1979) Fulling(1989) Birrell Davis (1982) Wald(1994) and to the discussion in Chung, Notari Riotto (2003) and references therein. 

Bernabei, R., et al. 2003. Dark matter search, Riv. Nuovo Cim. 26, 1 Binetruy, P, Girardi, G., Salati, P. 1984. Constraints On A System Of Two Neutral Fermions From Cosmology, Nucl. Phys. B237, 285 Birrell, N. D. Davies, P. C. W., 1982. Quantum Fields in Curved Space (Cambridge Cambridge UniversityPress)... 

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. 

The philosophical appeal of curved space cosmology is that it allows a closed, rather than an infinite, universe. Experimental proof is in the observation of universal background radiation with a Planckian frequency distribution (Figure 2.5). Conjectural implications of closed space-time have been considered before [7, 62, 49]. 

Curved space elements. Membranes, micelles, helices. Higher structures by curvature of lower structures... 

A more general formulation in terms of a non-euclidean manifold (curved space) has several advantages, the most important of which is a geometrical... 

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