Isotropic universe

The simplest method is to assume that all supernovae are identical. This is, of course, not true (see previous paper) but it turns out that the subclass of the Type la supernovae is indeed rather homogeneous. The first to plot a Hubble diagram of Type la Supernovae was Kowal Kowal 1968. There are essentially three quantities that can be derived from such a Hubble diagram in the nearby universe the slope of the expansion line, the scatter around the expansion line and the value of the local Hubble constant from the intercept at zero redshift (e.g. Tammann Leibundgut 1990 Leibundgut. Pinto 1992). The slope gives an indication of the local expansion and for a linear expansion in an isotropic universe it has a fixed value. The scatter around the expansion line provides a measure of the accuracy of the standard candle and the measurement errors. The intercept of the line, finally, together with an estimate of the... 

There are no special points in an isotropic space. An isotropic universe therefore denies the possibility of a unique singular point. Either the big-bang singularity occured at all points of space or not at all. Having agreed that space does not expand, it means that a big bang in an isotropic universe occured everywhere. How can such a universe expand any further and what happened to the infinite mass, which was created instantaneously at every point throughout the universe ... 

The invariant element of separation ds between two neighboring events in a homogeneous and isotropic universe can be expressed by ... 

A recently developed adsorbent version of ORNL s porous carbon fiber-carbon binder eomposite is named carbon fiber composite molecular sieve (CFCMS). The CFCMS monoliths were the product of a collaborative researeh program between ORNL and the University of Kentueky, Center for Applied Energy Researeh (UKCAER) [19-21]. The m.onoliths are manufactured in the manner deseribed in Section 2 from P200 isotropic pitch derived fibers. While development of these materials is in its early stages, a number of potential applieations can be identified. 

For a calculation of d. see R- H. Fowler. Statistical Thermodynamics. Second Edition, Cambridge University Press. 1956. p. 127. In Section 1.5a of Chapter 1 we defined the compressibility and cautioned that this compressibility can be applied rigorously only for gases, liquids, and isotropic solids. For anisotropic solids where the effect of pressure on the volume would not be the same in the three perpendicular directions, more sophisticated relationships are required. Poisson s ratio is the ratio of the strain of the transverse contraction to the strain of the parallel elongation when a rod is stretched by forces applied at the end of the rod in parallel with its axis. 

Fig. 5. The growth of orientational order for the 64 molecules of 5,5-BBCO during a 300 K quench starting from the isotropic liquid. Data from [104], supplied hy C McBride, University of Durham...

Where the Reynolds stress formula (2) and the universal law of the theory of isotropic turbulence apply to the turbulent velocity fluctuations (4), the relationship (20) for the description of the maximum energy dissipation can be derived from the correlation of the particle diameter (see Fig. 9). It includes the geometrical function F and thus provides a detailed description of the stirrer geometry in the investigated range of impeller and reactor geometry 0.225derived from many turbulence measurements, correlation (9). 

Small-step rotational diffusion is the model universally used for characterizing the overall molecular reorientation. If the molecule is of spherical symmetry (or approximately this is generally the case for molecules of important size), a single rotational diffusion coefficient is needed and the molecular tumbling is said isotropic. According to this model, correlation functions obey a diffusion type equation and we can write... 

The Cosmological Principle which states that the Universe is always homogeneous and isotropic and leads to the Robertson-Walker metric... 

The expansion of the Universe is governed by the field equations of General Relativity for isotropic expansion or contraction ... 

Figure 3.13 EPR absorption curves. (A) Isotropic spectrum, gx = gy = gz (B) Axial spectrum, gx = gy gz. (D) Rhombic spectrum, gx / gy / gz. (Reprinted with permission from Figure 4 of Palmer, G., in Que, L., ed. Physical Methods in Bioinorganic Chemistry Spectroscopy and Magnetism. University Science Books, Sausalito, CA, 2000, pp. 121-185. Copyright 2000, University Science Books.)...

One of the principal difficulties faced when employing Lagrangian micromixing models is the determination of tm based on properties of the turbulent flow fields. Researchers have thus attempted to use the universal nature of high-Reynolds-number isotropic turbulence to link tm to the turbulence time scales. For example, in the E-model (Baldyga and Bourne 1989) the engulfment rate essentially controls the rate of micromixing and is defined by... 

Batchelor, G. K. [Symmetrical contraction on isotropic turbulence], p. 74. In The Theory of Homogeneous Turbulence. London Cambridge University Press, 1953. 

Rayleigh, Lord (1889). On the free vibrations of an infinite plate of homogeneous isotropic elastic matter. Proc. Lond. Math. Soc. 20, 225-34. [Reprinted in Scientific papers by Lord Rayleigh, Vol. V. Cambridge University Press (1912) and Dover Publications, New York (1964).] [198]... 

The Roberston-Walker cosmology considers a spacial 3-sphere that evolves through space. It is the simplest of possible models that assumes the universe is isotropic and homogeneous. The metric for this universe is written as... 

The problem of symmetry breaking (SB) is well known and multiply discussed in literature. Briefly, we can formulate it as follows. The Hamiltonian of any system of particles forming the Universe is totally symmetric with respect to rotations and reflections in the isotropic space-time, as well as transmutations of identical and equivalent particles, whereas the real objects of the material world composed by these particles do not possess such symmetry. This is seen already from the examples that we live in a world of particles, not antiparticles, and in condensed matter, we have mostly low-symmetry structures. This circumstance can be expressed by the statement that the world is in a state of broken symmetry. An obvious explanation of the contradiction between the totally symmetric Hamiltonian and the broken symmetry of the real world is that the latter is not a solution of its Schrodinger equation. 

In summary, the thermal history of the early Universe is very simple. It just assumes a global isotropic and uniform Universe. In its simplest version - no structure of any kind on scales larger than individual particles - the contents of the Universe are determined by "standard elmentary physics" i) ag lobal expansion governed by GR, ii) particles interactions governed by the "Standard Model" of Particle Physics, iii) distributions of particles governed by the laws of Statistical Physics. 

In a sense, all these problems had to do with some choice of initial conditions for the Universe. In particular, the Universe had to be supposed initially homogeneous and isotropic. These features had to be put by hand in order the cosmological scenario to work properly, a situation which was of course very uncomfortable. 

Observation suggests that we live in a homogeneous and isotropic patch of the Universe. The hypothesis of homogeneity and isotropy of the Universe translate into the fact that the line element can be written under the form... 

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