Assuming SO(3)-spherical symmetry, the 4–dimensional Einstein equation reduces to an equation conformally related to the field equation for 2–dimensional gravity following from the Lagrangian L = |R|1/3. Solutions for 2–dimensional gravity always possess a local isometry because the traceless part of its Ricci tensor identically vanishes. Combining both facts, we get a new proof of Birkhoff’s t...

The Einstein equation of the general relativity (1) postulates a balance between the spacetime curvature and the energy-momemtum. Initially, Einstein introduced the cosmological constant (CC) Λ into the equation to make the steady state cosmology. When the expansion of the Universe was discovered, Einstein dismissed CC as his greatest blunder. Recent observations of the type Ia supernovae [1,2]...

Hence .0V is a quantity which can only increase with the time and may be regarded as the entropy of the universe. This quantity proves to be equivalent to the expression that Lenz took for the entropy of the Einstein universe, but, in general, the expression for the entropy of a system in a gravitational field does not reduce to so simple a form. 1 Tolman, R. C., these PROCzSDINGs, 14, No. 3, 2...

A Bose-Einstein condensate (BEC) was only experimentally realised in 1995. Research into nonlinear phenomenon (such as vortices) that exist in a BEC is thus in its infancy and a full understanding of how vortices move and interact is still lacking. A Bose-Einstein condensate is described by the GrossPitaevskii (GP) equation, a semi-classical non-linear PDE, given in terms of a wavefunction. In ...

We quantize a semiclassical system defined by the Hamiltonian obtained from the asymptotic self-similar solution of the Gross-Pitaevskii equation for a trapped Bose-Einstein condensate with a linear gain term. On the basis of a Schrödinger equation derived in a space of ellipsoidal parameters, we analytically calculate the quantum mechanical and thermal variance in the ellipsoidal parameters fo...

We apply Christodoulou’s framework, developed to study the Einstein-scalar field equations in spherical symmetry, to the linear wave equation in de Sitter spacetime, as a first step towards the Einstein-scalar field equations with positive cosmological constant. We obtain an integro-differential evolution equation which we solve by taking initial data on a null cone. As a corollary we obtain el...

We characterize the Einstein tensor as the only, up to a constant factor, 2covariant tensor naturally constructed from a semiriemannian metric which is divergence-free and invariant under constant rescalings of the metric. Since these two conditions are also satisfied by the energy-impulse tensor of a relativistic space-time, we discuss how this theorem and its variations can be used to derive ...

Many of the technical complications associated with the general theory of relativity ultimately stem from the nonlinearity of Einstein's equation. It is shown here that an appropriate choice of dynamical variables may be used to eliminate all such nonlinearities beyond a particular order: Both Landau-Lifshitz and tetrad formulations of Einstein's equation are obtained that involve only finite p...

We consider the nonlinear Schrodinger equation with the nonlinearity management which describes Bose-Einstein condensates under Feshbach resonance. By using an averaging theory, we derive the Hamiltonian averaged equation and compare it with other averaging methods developed for this problem. The averaged equation is used for analytical approximations of nonlinearity-managed solitons.

‎The notion of quasi-Einstein metric in physics is equivalent to the notion of Ricci soliton in Riemannian spaces‎. ‎Quasi-Einstein metrics serve also as solution to the Ricci flow equation‎. ‎Here‎, ‎the Riemannian metric is replaced by a Hessian matrix derived from a Finsler structure and a quasi-Einstein Finsler metric is defined‎. ‎In compact case‎, ‎it is proved that the quasi-Einstein met...