Number Of Field Equations Research Articles

Generating exact polytropes in non-conservative unimodular geometries

The trace-free Einstein equations contain one equation less than the complete field equations. In a static and spherically symmetric spacetime, the number of field equations is thus reduced to two. The equation of pressure isotropy of general relativity, however, is preserved thus showing that any known perfect fluid spacetime is a suitable candidate for the trace-free scenario. The extra freedom in imposing two constraints may now be exploited to include polytopes, something that is difficult in general relativity. The point here is that using any known exact solution one can find a polytropic star for various values of the polytropic index. One arrives at Tolman–Oppenheimer–Volkoff type equations and can study their solutions explicitly. Two examples of well-known stellar distributions that generate polytropes with physically reasonable behaviour are discussed. These models are regular, exhibit a sound speed that is never superluminal and are adiabatically stable in the sense of Chandrasekhar. We investigate a compactness measure confirming that our results are consistent with some observational data.

Formulation of thermo-hydro-mechanical coupling behavior of unsaturated soils based on hybrid mixture theory

Thermo-Hydro-Mechanical (THM) coupling processes in unsaturated soils are very important in both theoretical researches and engineering applications. A coupled formulation based on hybrid mixture theory is derived to model the THM coupling behavior of unsaturated soils. The free-energy and dissipative functions for different phases are derived from Taylor’s series expansions. Constitutive relations for THM coupled behaviors of unsaturated soils, which include deformation, entropy change, fluid flow, heat conduction, and dynamic compatibility conditions on the interfaces, are then established. The number of field equations is shown to be equal to the number of unknown variables; thus, a closure of this coupling problem is established. In addition to modifications of the physical conservation equations with coupling effect terms, the constitutive equations, which consider the coupling between elastoplastic deformation of the soil skeleton, fluid flow, and heat transfer, are also derived.

A Strange Feature of Einstein's most Recent Generalized Field Theory

Einstein and Straus1 developed a unified field theory in which the field variables consist of sixteen elements gμv (μ,v = 1, 2, 3, 4) of a non-symmetric tensor connected by a number of field equations. Schrodinger2 also propounded a similar theory in which the field equations are nearly the same except that they contain additional terms involving a parameter λ, which takes a role analogous to that of the cosmological constant. Einstein's field equations will follow from Schrodinger's equation if we put λ = 0. Though there exists at present some confusion regarding the interpretation of the field variables, yet the most plausible and apparently most natural interpretation is to take the symmetric part of the tensor as the metric and the antisymmetric part as the electromagnetic tensor. The solution of Einstein's field equation for the case can be obtained by writing λ = 0 for corresponding solutions of Schrodinger's field equations obtained by Papapetrou3, and will be as follows: It is very natural to interpret m as mass and e as charge in this case.