Einstein’s formulation

On the right we have introduced Einstein s formulation for the relation between energy density u and momentum density p of electromagnetic radiation, namely pc = u. A classical derivation of the above expression was furnished in the preceding section. 

Thus the symmetric second-rank metric tensor g v of Einstein s formulation of general relativity corresponds to the symmetric sum from the quaternion theory, (—+ qvq ). [The factor (—5) is chosen in anticipation of the normalization of the quaternion variables.] Thus we see that ds is a factorization of the standard Riemannian squared differential metric ds2 = g dx dxv. 

In 1934 Kuhn (56] sought to combine the statistics of flexible polymer chains with Einstein s formulation. He showed that from random flight statistics (57] that the mean square end-to-end distance and essentially the diameter of the equivalent sphere (radius of gyration 5 ) of the polymer chain (57] was... 

Historical Background.—Relativistic quantum mechanics had its beginning in 1900 with Planck s formulation of the law of black body radiation. Perhaps its inception should be attributed more accurately to Einstein (1905) who ascribed to electromagnetic radiation a corpuscular character the photons. He endowed the photons with an energy and momentum hv and hv/c, respectively, if the frequency of the radiation is v. These assignments of energy and momentum for these zero rest mass particles were consistent with the postulates of relativity. It is to be noted that zero rest mass particles can only be understood within the framework of relativistic dynamics. 

To appreciate the predictive properties of Kieffer s model, it is sufficient to compare calculated and experimental entropy values for several phases of geochemical interest in table 3.1, which also lists entropy values obtained through apphcation of Debye s and Einstein s models. One advantage of Kieffer s model with respect to the two preceding formulations is its wider T range of applicability (Debye s model is appropriate to low frequencies and hence to low T, whereas Einstein s model is appropriate to high frequencies and hence to high T). 

Many similar formulations have also been advanced (Cussler, 1997). One is by Sutherland (1905), predating Einstein s work, who used the slip condition, so that the total drag is 4nr au instead of 6nr au. The result is a diffusivity that is 1.5 times the Einstein diffusivity... 

Oddly enough. Diamond and Scheibel found Einstein to have fewer glial cells than expected in Area 39 of Einstein s left brain hemisphere. Could Einstein s low number of glial cells account for his slower (or unusual) speech development We know that this area impacts speech because lesions in this area lead to dyslexia. As we discussed in Chapter 2, Einstein once mentioned that written and spoken words were not important when he formulated his theories. Imagery and emotion actually... 

This success of density functional theory allows the whole question of bonding and structure to be formulated within an effective one-electron framework that is so beloved by chemists in their molecular orbital description of molecules and by physicists in their band theory description of solids. In this book I have tried to follow Einstein s dictum by simplifying the one-electron problem to the barest... 

The depolarization of light by dense systems of spherical atoms or molecules has been known as an experimental fact for a long time. It is, however, discordant with Smoluchowski s and Einstein s celebrated theories of light scattering which were formulated in the early years of this century. These theories consider the effects of fluctuation of density and other thermodynamic variables [371, 144]. 

The mathematical treatment of the Rutherford-Bohr atom was especially productive in Denmark and Germany. It led directly to quantum mechanics, which treated electrons as particles. Electrons, however, like light, were part of electromagnetic radiation, and radiation was generally understood to be a wave phenomenon. In 1924, the French physicist Prince Louis de Broglie (1892-1987), influenced by Einstein s work on the photoelectric effect, showed that electrons had both wave and particle aspects. Wave mechanics, an alternative approach to quantum physics, was soon developed, based on the wave equation formulated in 1926 by the Austrian-born Erwin Schrodinger (1887-1961). Quantum mechanics and wave mechanics turned out to be complementary and both were fruitful for an understanding of valence. 

Because hydrogen has more than two energy levels, it actually emits electromagnetic radiation at more than one frequency. Bohr s formulation accounted for all of hydrogen s observed emissions. Bohr published his new atomic structure in 1913. According to Albert Einstein, the Bohr model of the atom was one of the greatest discoveries. ... 

We begin with an abstract of the physics that underlies the kinetics of bond dissociation and structural transitions in a liquid environment. Developed from Einstein s theory of Brownian motion, these well-known concepts take advantage of the huge gap in time scale that separates rapid thermal impulses in liquids (< 10 s) from slow processes in laboratory measurements (e.g. from 10 s to min in the case of force probe tests). Three equivalent formulations describe molecular kinetics in an overdamped liquid environment. The first is a microscopic perspective where molecules behave as particles with instantaneous positions or states x(t) governed by an overdamped Langevin equation of motion,... 

There has also been some controversy as to how the experiment affected the development of special relativity. Einstein commented that the experiment had only a negligible effect on the formulation of his theory. Clearly it was not a starting point for him. Yet the experiment has been repeated by others over many years, upholding the original results in every case. Even if special relativity did not spring directly from its results, the Michelson-Morley experiment has convinced many scientists of the accuracy of Einstein s theory and has remained one of the foundations upon which relativity stands. 

Planck s revolutionary idea about energy provided the basis for Einstein s explanation of the photoelectric effect in 1906 and for the Danish physicist Niels Bohr s atomic model of the hydrogen atom in 1913. Their success, in turn, lent support to Planck s theories, for which he received the Nobel Prize in physics in 1918. In the mid-1920s the combination of Planck s ideas about the particle-like nature of electromagnetic radiation and Erench physicist Louis de Broglie s hypothesis of the wavelike nature of electrons led to the formulation of quantum mechanics, which is still the accepted theory for the behavior of matter at atomic and subatomic levels. 

Kaluza and Klein managed to formulate a unified theory of gravitation and electromagnetism in terms of Einstein s field equations in five-dimensional space, but with the metric tensor defined to be independent of the fourth space dimension. Without this restriction, solution of the equations in apparent 5D vacuum ... 

It is important to emphasise that the relations obtained from Eq. (1.15) do not lead to a unique relation between the mass and the rest mass. The reason is quite deep since it involves two principal problems. First, one needs to account for the commensuration between the conjugate operators and second to unite the formulation with respect to particles with rest mass mo 0 and mo = 0. The latter is a blessing in disguise since, as we know, Einstein s law of general relativity predicts that a photon deviates twice as much as estimated by Newton s classical theory. 

In conclusion, we emphasise the following points (i) we have re-derived a previously obtained operator array formulation, which in its complex symmetric form permits a viable map of gravitational interactions within a combined quantum-classical structure (ii) the choice of representation allows the implementation of a global superposition principle valid both in the classical as well as the quantum domain (iii) the scope of the presentation has focused on obtaining well-known results of Einstein s theory of general relativity particularly in connection with the correct determination of the perihelion motion of the planet Mercury (iv) finally, we have obtained a surprising relation with Godel s celebrated incompleteness theorem. 

During the three productive years of a postdoctoral stay in Mark s Laboratory, Robert extended Einstein s equation (originally derived for linear stress gradient) to parabolic Poiseuille flow. There were excursions with Eirich into kinetic theory and viscosity of gaseous paraffins, as well as viscosity, surface tension, and heat of vaporization correlations of chain molecular fluids. The latter made use of the recently formulated transition-state theory of Eyring, Polanyi, and Wigner. 

Pointing up is formulating a problem. Osborn considered this step to be critically important for the final results and underlined his belief with Einstein s classical theory, saying that... 

This is the basic principle of Einstein s special theory of relativity. However, Einstein was not content with the apparent absolute status conferred to accelerating frames by the behaviour of bodies within them. Einstein sought a general principle of relativity that would require all frames of reference, whatever their relative state of motion, to be equivalent for the formulation of the general laws of nature. In his popular exposition of 1916, Einstein explains this by describing the experiences of an observer... 

The starting point of most relativistic quantum-mechanical methods is the Dirac equation, which is the relativistic analog of the Schrodinger equation. Before Dirac s formulation, an obvious way of starting relativistic quantum mechanics would be the Einstein energy expression... 

Relativistic quantum chemistry is the relativistic formulation of quantum mechanics applied to many-electron systems, i.e., to atoms, molecules and solids. It combines the principles of Einstein s theory of special relativity, which have to be obeyed by any fundamental physical theory, with the basic rules of quantum mechanics. By construction, it represents the most fundamental theory of all molecular sciences, which describes matter by the action, interaction, and motion of the elementary particles of the theory. In this sense it is important for physicists, chemists, material scientists, and biologists with a molecular view of the world. It is important to note that the energy range relevant to the molecular sciences allows us to operate with a reduced and idealized set of "elementary" particles. "Elementary" to chemistry are atomic nuclei and electrons. In most cases, neither the structure of the nuclei nor the explicit description of photons is required for the theory of molecular processes. Of course, this elementary level is not always the most appropriate one if it comes to the investigation of very large nanometer-sized molecular systems. Nevertheless it has two very convenient features ... 

The mechanics of a deformable body treated here is based on Newton s laws of motion and the laws of thermodynamics. In this Chapter we present the fundamental concepts of continuum mechanics, and, for conciseness, the material is presented in Cartesian tensor formulation with the implicit assumption of Einstein s summation convention. Where this convention is exempted we shall denote the index thus ( a). 

Using the definitions given, Einstein s classical expression for the viscosity of a dilute suspension of rigid spheres is formulated as... 

For convenience, humans consider their living space to be Euclidean, and even the most advanced cosmologies are still formulated in such terms. In order to operate in Euclidean space, it is necessary to separate the mathematically equivalent variables of Einstein s equation into a universal time variable and the three familiar variables of coordinate space. This operation destroys the 4D field equations and no longer provides any insight into the nature of matter. For this reason, the existence of matter is added to physical theory as an ad hoc postulate—the prescription of classical Newtonian mechanics. 

Mooney [37] has developed a formulation of the viscosity of suspensions of spheres based on a functional equation, which must be satisfied if the final viscosity is to be independent of the sequence of stepwise additions of partial volume fractions of the spheres to the suspension. Further Einstein s Eq. 2.7 is considered valid for dilute suspensions, Mooney argues for a monodisperse system of spheres that... 

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