New Formulation of the Axially Symmetric Gravitational Field Problem

Abstract

M'&10 3fo. In fact, the situation is worse than this:The mass of an eventual condensation is M(M'AB/1V, or 3f(10 Mo. We therefore come to the conclusion that the time available is too short for the formation of baryon inhomogeneities, and that if inhomogeneities are the explanation of large-scale aggregations of matter, then they are an integral part of the universe from the beginning of its expansion.The case for density Quctuations is certainly no better. For a galactic mass of %=10" Mo, we have from (45), p«10 ' g™ (since M'))M), and the instability hypothesis leads to the improbable conclusion that the foundations of galactic structure are laid down in the radiation era. If we accept the primordialstructure hypothesis and assume that density inhomogeneities exist from the earliest moments, we are still in difhculty because small amplitude fluctuations are not ampli6ed, 30 and the density inhomogeneity is of the order 10' times greater than the required compositional inhomogeneity."Nor are they dissipated. Any dissipation mechanism (such as Misner's (Ref. 4) g in the lepton or hadron eras causes only the very shortwavelengths of ) (ct to decay.The 6eld equations governing the gravitational Geld of a uniformly rotating axially symmetric source are reformulated in terms of a simple variational principle. The new formalism affords a concise unified derivation of the solutions discovered by Weyl and Papapetrou, and permits a simple derivation of the Kerr metric in terms of prolate spheroidal coordinates. More complex solutions are identi6ed by applying perturbation theory.

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“…, θ 1 ), where by θ kl we denote the difference θ k −θ l . Solutions to (2) constructed by the Bethe ansatz can naturally be made compatible with (3). In this note we mainly focus on the recursive equations (4) and (5).…”

“…In [1] we proposed a bootstrap approach to describe the quantum theory descending from the Ernst equation of general relativity [2]. In upshot, the quantum theory is described in terms of matrix elements e.g.…”

“…Equation (2) results in the diagonalization of the operator T , for a fixed number of arguments N . It is basically the familiar transfer matrix…”

Solutions of Einstein's field equations related to Jacobi's inversion problem

“…for the complex function f (ρ, ζ) called the Ernst potential [10], [11]. The operators and ∇ have their usual three-dimensional meaning applied to an axisymmetric function depending on the cylindrical coordinates ρ and ζ:…”

Geroch—Kinnersley—Chitre Group for Dilaton—Axion Gravity

“…Important insights in establishing the integrability of the two-dimensional Einstein and Einstein-Maxwell (EM) theories were due to the works of Neugebauer and Kramer [6], [7] in which the relevance of the three-dimensional σ-models on symmetric spaces has been discovered. A concise complex potential representation given by Ernst [8] for both Einstein and Einstein-Maxwell theories provided new structures particularly useful in this research. Different proofs of the complete integrability of two-dimensional reductions of the Einstein and EM systems were given in the late 70-ths by Harrison [9], Maison [10], Belinskii and Zakharov [11], and Hauser and Ersnt [12].…”