Gradient conformal killing vectors and exact solutions

We establish a connection between conformally related Einstein spaces and conformai killing vectors (CKV). We begin with the conformal map and prove that (a) under the conformal mapping¯g _ik=^–2g_ik, the necessary and sufficient condition for the tracefree part of the Ricci tensor (S _ik=R_ik–(R/4)g _ik) to remain invariant is that _i is a CKV ofg _ik, and (b) the most general form for for conformally flat Einstein space, which is the de Sitter space, is composed of three terms each of which alone represents a flat space. The existence of gradient CKV (GCKV) is examined in relation to vacuum and perfect fluid spacetimes.     

Further consequences of Einstein locality

The results of a previous paper for the properties of the correlation functionP() at small angles following from Einstein locality are extended to the general case whereP(0)–1. Experimental investigations are suggested to check the validity of some theorems on the behavior of the correlation functionP() which are deduced directly from Einstein locality.     

Einstein and Brans–Dicke Frames in Multidimensional Cosmology

Inhomogeneous multidimensional cosmological models with a higher-dimensional space-time manifold _{0 i=1} ⁿ M_i (n 1) are in stigated under dimensional reduction to a D ₀-dimensional effective non-minimally coupled -model which generalizes the familiar Brans–Dicke model. The general form of the Einstein frame representation of multidimensional solutions known in the Brans–Dicke frame is given with respect to cosmic synchronous time. As an example, the transformation is demonstrated explicitly for the generalized Kasner solutions where it is shown that solutions in the Einstein frame show no inflation of the external space although they can undergo deflation after the cosmic synchronous time inversion.     

The Inverse Spectral Method for Colliding Gravitational Waves

The problem of colliding gravitational waves gives rise to a Goursat problem in the triangular region 1 x < y 1 for a certain 2 × 2 matrix valued nonlinear equation. This equation, which is a particular exact reduction of the vacuum Einstein equations, is integrable, i.e. it possesses a Lax pair formulation. Using the simultaneous spectral analysis of this Lax pair we study the above Goursat problem as well as its linearized version. It is shown that the linear problem reduces to a scalar Riemann–Hilbert problem, which can be solved in closed form, while the nonlinear problem reduces to a 2 × 2 matrix Riemann–Hilbert problem, which under certain conditions is solvable.     

The Relativistic Composite-Velocity Reciprocity Principle

Gyrogroup theory [A. A. Ungar, Found. Phys. 27, 881–951 (1997)] enables the study of the algebra of Einstein's addition to be guided by analogies shared with the algebra of vector addition. The capability of gyrogroup theory to capture analogies is demonstrated in this article by exposing the relativistic composite-velocity reciprocity principle. The breakdown of commutativity in the Einstein velocity addition of relativistically admissible velocities seemingly gives rise to a corresponding breakdown of the relativistic composite-velocity reciprocity principle, since seemingly (i) on one hand, the velocity reciprocal to the composite velocity uv is –(uv) and (ii) on the other hand, it is (–v)(–u). But (iii) –(uv)(–v)(–u). We remove the confusion in (i), (ii), and (iii) by employing the gyrocommutative gyrogroup structure of Einstein's addition and, subsequently, present the relativistic composite-velocity reciprocity principle with the Thomas rotation that it involves.     

A Canonical Null Gauge for the Einstein Equations

Associated with any choice of outgoing null (characteristic) coordinate, we construct a null tetrad which is determined uniquely by purely local information. Unlike other well-known characteristic gauge conditions [1–4], this canonical tetrad is determined at any point solely by the metric and its first derivatives at that point. The tetrad leads to a radial coordinate which is also uniquely determined by purely local information. These properties greatly simplify the structure of the hypersurface Einstein equations and may be used to directly compare metrics with compatible null foliations.     

Nonlinear lagrangians and Einstein spaces

It is shown that, for a Riemannian spaceV _d of dimensiond, solutions of the equation ((–g)^1/2 R ⁿ)/g^ab = 0 forn = (1/4)(d+2) may be interpreted as (d + 1)-dimensional Einstein spaces.     

The Boltzmann Equation for Bose–Einstein Particles: Velocity Concentration and Convergence to Equilibrium

Long time behavior of solutions of the spatially homogeneous Boltzmann equation for Bose–Einstein particles is studied for hard potentials with certain cutoffs and for the hard sphere model. It is proved that in the cutoff case solutions as time converge to the Bose–Einstein distribution in L¹ topology with the weighted measure , where for temperature and for T<T_c. In particular this implies that if T<T_c then the solutions in the velocity regions (with ) converge to a unique Dirac delta function (velocity concentration). All these convergence are uniform with respect to the cutoff constants. For the hard sphere model, these results hold also for weak or distributional solutions. Our methods are based on entropy inequalities and an observation that the convergence to Bose–Einstein distributions can be reduced to the convergence to Maxwell distributions.     

An Einstein-Gödel universe

The metric for the standard static Einstein model of the universe can be expressed in coordinates for which a congruence of spacelike world lines of the model will be twisting. A method of shifting the twist has been devised by which the twist on spacelike world lines is shifted onto the timelike world lines. The model universe then becomes Gödel's model. A combined Einstein-Gödel model containing a parameter is obtained. Switching from +1 to –1 will effect the shift of twist in the world lines and lead from the Einstein model to the Gödel model.     

On the Appeal to a Pre-Established Harmony Between Pure Mathematics and Relativity Physics

Soon after its appearance in 1905, the Einsteinian relativity with its relativistically admissible 3-velocities was recognized by Vladimir Variak in 1908 as the realization in physics of the hyperbolic geometry of Bolyai and Lobachevski. At the same time, however, during the years 1907–1909 Minkowski reformulated the Einsteinian relativity in terms of a space of 4-velocities that now bears his name. As a result, the special theory of relativity that we find in the mainstream literature is not the one originally formulated by Einstein but, rather, the one reformulated by Minkowski. Thus, in particular, one of the most powerful ideas of Einstein in 1905, the Einstein addition of relativistically admissible 3-velocities that need not be parallel, is unheard of in most texts on relativity physics. Following our recently published book, Beyond the Einstein Addition, Law and its Gyroscopic Thomas Precession: The Theory of Gyrogroups and Gyrovector Spaces [1], the aim of this article is to employ the principle of pre-established harmony between mathematics and physics to demonstrate that the original Einsteinian relativity, as opposed to the Minkowskian relativity, is the legitimate formulation of special relativity whose time has returned.     

Bianchi type V perfect fluid cosmologies

Bianchi type V solutions of the Einstein equations are studied using the Hamiltonian approach. Explicit expressions depending on a single quadrature are given for the metric components in the general orthogonal perfect fluid case. It is shown that the quadrature can be evaluated in terms of elementary or elliptic integrals when the parameter in the equation of statep=(–1) takes the values 1, 10/9, 4/3, 14/9, 5/3, 2.     

Electrical properties of p-type ZnGeP2 irradiated by H+ ions

The properties of p-type ZnGeP₂ [p₀=(5–10)·10¹⁰ cm^–3, ₀=(2–5)·10^–7 (·cm)^–1], irradiated with H⁺ ions [E=5 MeV, T_irr=300 K, D=(1·10¹²–1.7·10¹⁶) cm^–2] are studied. An increase in the resistivity (to gr_max - 5·10¹¹ ·cm) and subsequent reduction in for large currents of H⁺ ions ( - 9·10⁸ ·cm for D - 1.7·10¹⁶ cm^–2), is observed in irradiated crystals. The resistivity of irradiated p-type ZnGeP₂ is found to be very sensitive to hydrostatic pressure [(4–5)·10^–5 bar^–1]. The annealing of radiation defects in the temperature interval (20–600) °C is examined.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 10, pp. 91–93, October, 1991.     

Exciton transfer between localized states in ZnO quantum well structures

The dynamics of photocreated excitons in a CdZnO/MgZnO quantum well (QW) was studied by comparing the experimental photoluminescence (PL) data with the results of Monte Carlo simulations of the exciton hopping. The temperature-dependent PL linewidth was found to be in reasonable agreement with the model of exciton hopping, with an additional inhomogeneous broadening (Γ) accounted for. The simulation analysis revealed fluctuations of the band potential to be 20 meV with an additional inhomogeneous broadening of , and a crossover from a non-thermalized to thermalized exciton energy distribution at about 100 K. In addition, a Bose–Einstein distribution like temperature dependence of the exciton energy in the wells was extracted using the data on the PL peak position.     

Discontinuities of solutions of the Einstein equations. II

Right zero-vectors of the characteristic matrix of the Einstein equations are constructed on isotropic cones. Relationships for the discontinuities of two derived functions of the field in the and surfaces are indicated. Quantities describing the weak discontinuities of solutions of the gravitational field equations are constructed.Translated from Izvestiya Vysshikh Uchebnykh Zavedenii, Fizika, No. 2, pp. 38–41, February, 1982.     

Xanthopoulos Theorem in the Kaluza–Klein Theory

It is shown that if _AB is an exact solution of the Einstein vacuum field equations in 4 + 1 dimensions, R^ _AB = 0, and l _A is a null vector field, then _AB + l _A l _B is also an exact solution of the Einstein equations R^ _AB = 0 if and only if the perturbation l _A l _B satisfies the Einstein equations linearized about _AB. Then, making use of the Kaluza–Klein approach, it is shown that this result allows us to obtain exact solutions of the Einstein–Maxwell equations (possibly coupled to a scalar field) by solving a system of linear equations.     

The reinterpretation of wave mechanics

The author begins by recalling how he was led in 1923–24 to the ideas of wave mechanics in generalizing the ideas of Einstein's theory of light quanta. He made himself at that time a concrete physical picture of the coexistence of waves and particles and, in 1927, attempted to give them precise form in his theory of the double solution. As other ideas prevailed at the time, he abandoned the development of his conception. But for the past twenty years, once again convinced, like Einstein, that present-day quantum mechanics is only a statistical theory and does not give a true picture of physical reality, he has again taken up his old ideas and developed them considerably. He has in particular introduced an element of randomness into the theory and has thus attained to a hidden thermodynamics of particles, the results of which appear to be very interesting.     

Vorticity and isotropic singularities

We prove that if a solution of the Einstein field equations with perfect fluid source and law equation of state [p = (–1)] admits an Isotropic singularity, then necessarily the fluid flow is irrotational. This shows the essential equivalence of the seemingly distinct concepts of quasi-isotropic singularities and Friedrnann-like singularities of Lifshitz and Khalatnikov and Eardley, Liang and Sachs, respectively.     

A note on the hydrodynamics of viscous fluids

The cross-product of the velocity and the vorticity in a viscous incompressible fluid is formulated and its properties investigated. When the cross-product is identically null, either the flow is vortex-free, or the velocity and the vorticity are parallel to each other. The second case yields the following important result for three-dimensional flows: if the velocity and the vorticity are related by a position-independent scalar function, that function must be time-independent as well. (English translation of the seventy-five-years old Czech text — Trkal V.: asopis pro pstování mathematiky a fysiky48 (1919) 302–311.)The paper has been translated (by I. Gregora) from Czech original which appeared in asopis pro pstování mathematiky a fysiky, Vol. 48 (1919), pp. 302–311; see a contextualizing account in the preceding paper in this issue, p. 89. The reprints are available on request from the Editorial Office.I express my warm gratitude to Prof. Dr. F. Závika for his valuable comments.     

The effect of the relative position of the bandgap with respect to laser wavelength on the behaviour of DFWM in semiconductor-doped glasses

Degenerated four-wave mixing (DFWM) was realised in five commercial semiconductor-doped glasses (Corning CS3–66, CS3–67, CS3–68, CS3–69, CS3–70). The reflectivity obtained for each glass seems very dependent on the energy excess (E=hv–E _g) and relatively independant on l, the product of the absorption coefficient () by the sample thickness (l). The decay time of the DFWM signal was measured for each glass. The relaxation was found to be different when the photon energy is under or over the gap.     

Continuum and discretum—Unified field theory and elementary constants

Unitary field theories and SUPER-GUT theories work with an universal continuum, the structured spacetime of R. Descartes, B. Spinoza, B. Riemann, and A. Einstein, or a (Machian ^(1–3) ) structured vacuum according the quantum theory of unitary fields (Dirac, ^(4,5) and Heisenberg ^(6–8) ). The atomistic aspect of the substantial world is represented by the fundamental constants which are invariant against all transformations and which depend on nothings (Planck ^(9–11) ). A satisfactory unitary theory has to involve these constants like the mathematical numbers. Today, Planck's conception of the three elementary constants , c, and G may be the key to general relativistic quantum field theory like unitary theory. However, the elementary constants are a question of measurement-theory, also.According to Popper's theory ^(12–16) of induction, such unitary theories are universal explaining theories. The fundamental constants involve the complementarity between the universal statements in unitary theory and the basic statements in the language of classical observables.