Euclidean universe

It is therefore a Euclidean universe which lies at the meeting point between data from remote supernova studies and observations of the cosmic background radiation. Such a universe contains just enough matter and energy to keep the geometry Euclidean. In fact the Euclidean cosmology fits our Universe like a glove. 

On weighing the impartial evidence it becomes obvious that standard cosmology underestimates the age of the world by the same margin that it overestimates the size of the universe. Both factors relate to the Doppler interpretation of redshifts, which should be re-examined, and to geometry and the topology of space-time. The obvious alternative to an infinite Euclidean universe is a world closed in space-time. As originally pointed out by Weyl (1922)(p.278) ... 

Interpreted, as it is, within the standard model, Higgs theory has little meaning in the real world, failing, as it does to relate the broken symmetry of the field to the chirality of space, time and matter. Only vindication of the conjecture is expected to be the heralded observation of the field bosons at stupendous temperatures in monstrous particle accelerators of the future. However, the mathematical model, without cosmological baggage, identifies important structural characteristics of any material universe. The most obvious stipulation is to confirm that inertial matter cannot survive in high-symmetry euclidean space. 

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... 

The angular diameter of the observed patches is about 1°. Their (real) linear diameter can be estimated as ct, where c is the speed of light and t = 300 000 years is the time when the Universe became transparent. An angular diameter of 1° measured 14 billion years later means that the light rays remained parallel over the whole path and hence that the Universe is globally Euclidean. 

An often-overlooked issue is the inherent non-orthogonality of coordinate systems used to portray data points. Almost universally a Euclidean coordinate system is used. This assumes that the original variables are orthogonal, that is, are uncorrelated, when it is well known that this is generally not the case. Typically, principal component analysis (PCA) is performed to generate a putative orthogonal coordinate system each of whose axes correspond to directions of maximum variance in the transformed space. This, however, is not quite cor-... 

Many of you are probably asking what is outside the universe The answer is unclear. I must reiterate that this question supposes that the ultimate physical reality must be a Euclidean space of some dimension. That is, it presumes that if space is a hypersphere, then the hypersphere must sit in a four-dimensional Euclidean space, allowing us to view it from the outside. As the authors of the Scientific American article point out, nature need not adhere to this notion. It would be perfectly acceptable for the universe to be a hypersphere and not be embedded in any higher-dimensional space. We have difficulty visualizing this because we are used to viewing shapes from the outside. But there need not be an outside. ... 

Scientific theories themselves can be distinguished as deductive or inductive in nature, according to the underlying character of their premises. In a deductive theory, the fundamental premises are axioms or postulates that are neither questionable nor explainable within the theory itself. Outstanding examples of deductive theories include Euclidean geometry (based on Euclid s five axioms) and quantum mechanics (based on Schrodinger s prescription for converting classical trajectory equations into wave equations). An inductive theory, on the other hand, is based on universal laws of experience that express what has always been found to be true in the past, and may therefore be reasonably expected to hold in the future. Thermodynamics is the pre-eminent example of an inductive theory. 

Cosmic structure based on a vacuum interface has been proposed before [49, 7] as a device to rationalize quantum events. To avoid partitioning the universe into regions of opposite chirality the two sides of the interface are joined together with an involution. The one-dimensional analogue is a Mobius strip. Matter on opposite sides of the interface has mutually inverted chirality - matter and anti-matter - but transplantation along the double cover gradually interconverts the two chiral forms. The amounts of matter and anti-matter in such a universe are equal, as required by symmetry, but only one form is observed to predominate in any local environment. Because of the curvature, which is required to close the universe, space itself is chiral, as observed in the structure of the electromagnetic field. This property does not appear in a euclidean Robertson-Walker sub-space. 

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... 

It was at the University of Kazan, in the Russian province of Kazakhstan, that Nicolai Ivanovitch Lobachevsky made his contributions in Non-Euclidean geometry. In his early days at the university, he did try to find a proof of the parallel postulate, but later changed direction. As early as 1826, he made use of the hypothesis of the acute angle already developed by Saccheri and... 

Although, in general, a, H, and f are all time-dependent, eq. 2.11 reveals that if ever Q < 1, then it will always be < 1 and in this case the universe is open (k < 0). Similarly, if ever Q > 1, then it will always be > 1 and in this case the universe is closed (re > 0). For the special case of Q = ]. where the density is equal to the critical density Pent = 3H2/8itG, Q is always unity and the universe is flat (Euclidean 3-space sections re = 0). 

The physical interpretation is that matter, which seems to disappear into a black hole, has an escape route through a non-Euclidean throat that connects one part of the universe to a related region of opposite chirality, which implies conversion of matter into anti-matter. 

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