Einstein field equations

The term k in this metric is a constant that determines the spacial curvature of the cosmology. For k = 1 the cosmology is a closed spherical universe, for k = 0 the cosmology is flat, and for k - — 1 the cosmology is open. The Einstein field equations give a constraint equation and a dynamical equation for the rate the radius changes with time. If we define a velocity as v = (R/R)H(t)r, where H (t) is the Hubble parameter, a constant locally, the constraint equations is... 

ON THE GENERAL SOLUTIONS OF HILBERT-EINSTEIN FIELD EQUATIONS IN VACUUM... 

Abstract The Hilbert-Einstein field equations describe the structure of space-time in the general theory of relativity. Their simplified form, in a vacuum, was discovered in 1913 by Marcel Grossmann in two joint publications with Einstein, who, for his part, rejected this idea and presented another hypothesis... He reverted to Grossmann s idea only two years later. 

In a vacuum the Hilbert-Einstein field equation reduces to... 

On the general solutions of Hilbert-Einstein field equations in vacuum 327... 

This procedure leads to the Einstein gravitational field equations, one form of which, without cosmological term, is... 

Therefore, this is a statement of our fundamental hypothesis, specifically, that the topology of the vacuum defines the field equations through group and gauge field theory. Prior to the inference and empirical verification of the Aharonov-Bohm effect, there was no such concept in classical electrodynamics, the ether having been denied by Lorentz, Poincare, Einstein, and others. Our development of 0(3) electrodynamics in this chapter, therefore, has a well-defined basis in fundamental topology and empirical data. In the course of the development of... 

It is a matter of history that Einstein was greatly influenced by Mach s ideas as expressed in the latter s The Science of Mechanics (see, e.g., Ref. 19) and believed that they were incorporated in his field equations as long as space was closed [20], The modem general relativistic analysis gives detailed quantitative support to this latter view, showing how Mach s principle can be considered to arise as a consequence of the field equations when appropriate conditions are specified on an initial hypersurface in a closed evolving universe. In fact, in... 

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. 

It should be mentioned that in physics other equations are called Einstein equations such as the relationship between mass and energy, and the field equation of Einstein. 

Entropy in an isolated system increases dS/dt> 0 until it reaches equilibrium dS/dt = 0, and displays a direction of change leading to the thermodynamic arrow of time. The phenomenological approach favoring the retarded potential over the solution to the Maxwell field equation is called the time arrow of radiation. These two arrows of time lead to the Einstein-Ritz controversy Einstein believed that irreversibility is based on probability considerations, while Ritz believed that an initial condition and thus causality is the basis of irreversibility. Causality and probability may be two aspects of the same principle since the arrow of time has a global nature. 

Equation (2) is recognized as the four equations of electromagnetism modified by a wave-like scalar field. Equation (1) represents the 10 Einstein equations of general relativity, equated to energy and momentum derived from the fifth dimension. In short, KK theory is a unified account of gravity, electromagnetism and a scalar field. Kaluza s case, 744 = — 2 = — 1, together with the identification 1... 

A Simplified Presentation of Einstein s Unified Field Equations. 

Planck-power input is initially as positive-cosmological-constant A. Positivity of A is required for positivity of Planck-power input, and constancy of A is required for constancy of Planck-power input. Constancy of A is requisite for A to be most simply encompassed within Einstein s field equations [65-67], besides correlating with constancy of Planck-power input. Positive-cosmological-constant A also implies, or at least is consistent with, constant H (t) = Hq at all cosmic times r, and hence a fixed size of our L-region, with its boundary... 

The Kerr solution (1963) of the Einstein s vacuum field equations describes the gravitational field of a material source at rest having mass and angular momentum. The angular momentum determines a physically significant di-... 

The quaternion field equations (46a,b) are then the factorization of Einstein s tensor field equations ... 

We have seen in this section that the factorization of Einstein s symmetric, second-rank tensor field equations (10 relations) to a quaternion form (16 relations) not only yields the gravitational and electromagnetic manifestations of matter in a unified field theory but also reveals a feature of quantum mechanics. In particular, it was found that in the flat-space approximation to the curved-space representation in general relativity, the time component of the electromagnetic four-current density corresponds in a one-to-one way with the probability density of quantum mechanics. Its integration over all of space in this limit is found to be unity. 

The relative importance of migration and diffusion can be gauged by comparing Ud with the steady-state migrational velocity, u, for an ion of mobility Wj in an electric field (Section 2.3.3). By definition, v = where % is the electric field strength felt by the ion. From the Einstein-Smoluchowski equation, (4.2.2),... 

This calculation precludes development of the Einstein diffusion equation for forced diffusion in the presence of a gravitational field. The coefficient of (ge — gA) in equation (25-77) for the diffusional mass flux of species A can be evaluated via thermodynamics. The extensive Gibbs free energy of a one-phase binary mixture with 3 degrees of freedom requires four independent variables for complete description of this thermodynamic state function. Hence, ( (T, p, N/, Nb) is postulated where Ni represents the mole numbers of species i, and the total differential of is... 

If the superposition of point-source solutions is determined by a Green function , then by analogy with Wheeler-Feynman absorber theory the total interaction results from a symmetrical combination of retarded and advanced Green functions. On dehning these Green functions to be compatible with space-time geometry (Riemann tensor) the interaction is shown to be consistent with Einstein s field equations. 

Expanding universe cosmologies, in contrast, are based on a special metric that assumes a imiversal time, independent of the curved spatial manifold, as a solution of Einstein s gravitational field equations. In reality this is a second-generation generalization of Einstein s static-universe cosmology, once assumed to be the only possible solution of the field equations on a cosmic scale. However, as emphasized by Fuller and Wheeler (1962) ... 

These equalities cannot lead to a sensible solution. From the first equality it follows that i o should be inhnite, which implies that the 3-space degenerates into ordinary Euclidean 3-space but the second equality suggests p = 0, meaning that the universe is empty. To remedy this absurdity Einstein modified the field equations, as in (4.11), by introducing the so-called cosmological constant A, to read ... 

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