Über ein weiteres Variationsprinzip im Zusammenhang mit kraftfreien Magnetfeldern

Im Anschluß an ein von Woltjer [1] diskutiertes Variationsproblem wird gezeigt: Die Euler-Lagrange-Gleichungen des Variationsproblems mit der Nebenbedingung wo H = rot A gesetzt ist, sind die Differentialgleichungen der kraftfreien Magnetfelder mit variablem α. Die Nebenbedingung läßt sich für alle Felder H erfüllen, die keinen singulären Punkt mit H = 0 in dem betrachteten Volumen V haben. In persuance of a variational principle discussed by Woltjer [1] it is shown that the Euler-Lagrange-equations of the variational problem with the secondary condition where H = rot A are the differential equations of the force-free magnetic fields with a variable scalar α. The secondary condition can be accomplished for all magnetic fields which do not contain singular points with H = 0 in the volume V under consideration.     

Zur unitarisierten Gravitationstheorie mit lang- und kurzrechweitigen Termen (mit ruhmasselosen und schweren Gravitonen)

In our papers, TREDER [1, 2] we have formulated a unified electrodynamics of the fourth order with bi-wave equations for the vector potential A. In this electrodynamics EINSTEIN ian photon and heavy W-mesons are the field quanta. In correspondence to this field theory we are able to formulate a unified theory of gravitation, too. The field equations for the gravitational metrics grr in this theory are corresponding with the EINSTEIN equations of General Relativity in the same way like the electromagnetic bi-wave equations are corresponding with the MAXWELL equations. The metric g_μν is a linear functional of an EINSTEIN ian long-range potential g_μν and of a subatomic short-range potential definierte Materie-Tensor die gemeinsame Quelle für alle drei Felder ist. Dann ist g1_μν, g2_μν und g_μν und es gelten die Funktional-Bedingungen wobei hier g2_μν Feldgleichungen vom “kosmologischen Typ” befriedigt. By these conditions, the short-range interaction becomes a repulsive force and the action of the NEWTON -EINSTEIN ian attraction and of the subatomic repulsion makes the matter point-like (as in the E.-I.-H.-method) but self-consistent. The gravitational metrics g2_μν become regulary. P. e., in the EINSTEIN approximation the field of a point-like mass M is given by a SCHWARZSCHILD     

Gravitationskollaps und Lichtgeschwindigkeit im Gravitationsfeld

An elementary criterion of the stability of a matter sphere against gravitational collapse is given by the circular velocity condition of POINCARÉ : In a space with a spherically symmetric gravitation potential ? (r) and with a spherically symmetric metric gik (e.g., a SCHWARZSCHILD space time) the circular velocity V^* of a particle on the surface r = R of the matter-sphere must be (This condition is a consequence of the virial theorem and of the POINCARÉ theorem.) - However, EINSTEIN 's axiom of causality implies that this velocity V^* must be smaller than the local velocity of light v: V^*2 < v². And this local velocity v is a function of the gravitation potential ?, too: v = v [?]. In the case of NEWTON 's or EINSTEIN 's theory the spherically symmetric gravitation potential is given by the NEWTON ian function ? = fM/r. In the special theory of relativity, we would have v = c (c = EINSTEIN 's fundamental velocity) and grr = 1. Therefore, the specialrelativistic stability condition is R > fMc^?2. - But in the NEWTON ian theory v is depending of the gravitation potential and depends of the boundary condition for the light propagation, also. According to the ansatz of LAPLACE (1799) we have: (emanation-theory of light). But, according to SOLDNER (1801), we have Therefore, we are finding in the case of LAPLACE the same condition R > fMc^?2 as in the SRT. But, in the case of SOLDER 's ansatz non condition for stability is resulting. - In the general relativistic theories the local velocity of light is given by EINSTEIN 's expression According to EINSTEIN 's theory of “static gravitation” (1911/12) we have grr = 1 and therefore the formula and according to the GRT (with - g_ω = grr^?1) we have the formula Therefore, the Hilbert-Laue condition r= R > 3fMc^?2 results as stability condition. From the gravo-optical point of view, in GRT and for the classical ansatz of LAPLACE “black-holes” with bounding states of light result for R ≤ 2fM^?2. But, no “black-holes” are existing according to SOLDNER 's ansatz. However, in GRT each black-hole must be a “collapsar”. But according to the classical theory of LAPLACE we have uncollapsed “black- holes” for the domain .     

On Static Asymptotically Flat Solutions of Fourth Order Gravitational Field Equations

It is shown that each non-flat regular static asymptotically flat solution of the gravitational field equation following from the Lagrangian has in a certain sense positive energy. Further, for a set of parameters including the BACH -EINSTEIN theory some results concerning the full nonlinear behaviour of the solutions of the field equation will be given.     

Über stationäre,rotierende, inkompressible Plasmen mit minimalen inneren Verlusten

The condition that the Ohmic and the friction losses are minimized leads to the assumption that both the magnetic field H and the velocity field v are Trkal-fields. . In the rotationally symmetric, stationary case the boundary conditions and the condition that the toroidal part of v and H should vanish on the boundary, lead to a linear eigenvalue problem for α, β which in case of a rectangular domain easily can be resolved. It follows: .     

Einsteins hermitesche Relativitätstheorie als Unifikation von Gravo- und Chromodynamik

Einstein's Hermitian Theory of Relativity as Unification of Gravo- and Chromodynamics Einstein's Hermitian unified field theory is the continuation of the Riemannian GRG to complexe values with a Hermitian fundamental tensor gμ_v = g_v*μ This complexe continuation of GRG implies the possibility of matter and anti matter with a sort of CPT theorem. — Einstein himself has interpreted his theory as a unification and generalization of the Einstein and Maxwell theory, th. i. of gravodynamics and of electrodynamics. However — according the EIH approximation —, from Einstein's equations no Coulomb-like forces between the charges are resulting (INFELD, 1950). But, the forces between two charges ?_A and ?_B have the form (Treder 1957) It is interesting that such forces are postulated in the classical models of the chromodynamics of the interactions between quarks (for the confinement of their motions. If we interprete the purely imaginary part gμ_v of the hermitian metrics gμ_v=gμ_v+gμ_v as the dual of the field of gluons then, all peculiarities of Einstein's theory become physically meaningful. — Einstein's own interpretation suggests that the both long-range fields, gravitation and electromagnetism, must be unified in a geometrical field theory. However, the potential α/r + ε/2 has a “longer range” than the Coulomb potential ～1, and such an asymptotical potential ～ ε/2 is resulting from Einstein's equations (TREDER 1957). In Einstein's theory there are no free charges with \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_A^n {\varepsilon A} $\end{document}. (Wyman 1950) because the field mass of a charged particle becomes infinite asymptotically: That means, in a chromodynamics we dont's have free quarks. The same divergence are resulting from one-particle systems with non-vanishing total charges: M～ε²r. However, if the total charges vanish because in a domain ～L³ the positive sources are compensated by negative sources, the field masses of the n-charge systems become finite. From the gravitational part of Einstein's equations we get field masses which are the masses measured by observers in distances r ? L. That means, the masses of quark systems with the colour condition \documentclass{article}\pagestyle{empty}\begin{document}$ \sum\limits_A^n {\varepsilon A} $\end{document} are proportional to the linear dimension L of the system.     

Über die mögliche lineare Form von LORENTZ-kovarianten Gravitationstheorien

LORENTZ -covariant theories of gravitation which fulfil EINSTEIN 's weak principle of equivalence and which contain a pure Newtonian theory as an approximation are tensortheories with the linear approximative form for the field equations. In the case of EINSTEIN 's strong principle of equivalence the exact field equations must be the general relativistic EINSTEIN -equations (or the bimetrical EINSTEIN -ROSEN -equations). This follows from the dynamical equations and the BIANCHI identity according to JÁNOSSY and TREDER . However, from NEWTON 's axiom of reaction together with the weak principle of equivalence results that the strong principle of equivalence must be valid for the linear approximation of the field equations with sources. Therefore, the linear approximation of all physically meaningful Lorentz-covariant theories of gravitation is given by the linearized EINSTEIN -equations (with HILBERT -conditions): , that is by the ansatz α = 2. The main point of our arguments is LAUE 's postulate of the self-consistency of perfect static systems of isolated gravitational masses. In the lowest order of approximation this self-consistency is only possible if the gravitational matter-tensor is identical with the special-relativistic energy-momentum-tensor T_μv. LAUE 's postulate is fulfilled exactly for the general relativistic field equations according to the theorems of BIRKHOFF , TOLMAN and EINSTEIN and PAULI .     

The Thermo-Field Emission of Electrons in Arc Discharges

Starting from the conventional theory of thermo-field emission of electrons from metal surfaces, the range of validity and the errors of several approximate emission formulae are examined that are additive or harmonic combinations of the limiting thermionic (Richardson) and field emission (Fowler—Nordheim) equations. An optimization of such kinds of equations results in the simple dependence (for example) (iTF electron current density, T temperature, F field strength, k, A, B etc. are constants), or (dependent on only one variable Y = T + β′F β′ = const.). If the emission equations are applied to arc cathodes, both T and F must be expressed as functions of the ion current (I_i) and the ion current density (j_i); a simple interpolating formula of the electron current density j_e adapted to numerical results in the case of Cu cathodes becomes (ā, B?, C?: constants).     

Nachweis von radikalischen Zersetzungsprodukten in CCl4-Entladungen mit der EPR-Spektroskopie

Among the dissociation products of a r.f. discharge in CCl₄-vapour the radicals CCl₃ and Cl₂- have been identified by EPR-spectroscopy after condensation at 80 K. The gas-phase reaction CCl₃ + O₂ → CCl₃OO has been used to estimate the CCl₃ concentration to be approximately 5 · 10¹³ radicals/cm³ at p = 55 Pa. The life-time of CCl₃-radicals is governed by the rate of desactivation of some exited species (eg. CCl₄*) via . At a pressure p = 55 Pa a half-life of CCl₃-radicals of τ1/2 = 80 ms has been found.     

Gibt es Plancksche Sekunden des Kosmos?

On Planckian Seconds of the Universe According to the Planckian limites for quantum energies the cosmological scenaria break-down for space dimensions (or for life times of the universe) and for Planckian lengths, not till.     

Desorptions- und Reaktionskinetik der Erdalkalien Calcium und Strontium mit Chlor an Wolfram. Teil I: Chemisches Gleichgewicht der Oberflächenreaktion M + Cl = MCl im stationären Zustand (M = Ca,Sr)

Desorption- and Reactionkinetics of the Alkaline Earth Elements Calcium and Strontium with Chlorine on a Tungsten Surface — Part I: Chemical Equilibrium of the Surface Reaction M + Cl ? MCl in the Steady State (M = Ca, Br) Utilizing positive and negative surface ionization the reaction M + Cl = MCl (M = Ca, Sr) was studied at a hot tungsten surface under steady state conditions. Comparing the results obtained either by simultaneous M- and Cl₂ -exposures or by MCl₂ -exposure the existence of chemical equilibrium could be confirmed for the reaction in the temperature interval 1600 K.2000 K; at higher temperatures this equilibrium can be disturbed considerably by the desorption of the reacting components. From the experimental results we obtained under conditions of chemical equilibrium the energy of dissociation of MCl-molecules in the gasphase (D = (3.9 ± 0.15) eV, D = (4.2 ± 0.15)eV) and in the case of a strong disturbance of the equilibrium the difference between the activation energies of desorption and of dissociation of MCl-molecules on the surface (? - D? = (1.6 ± 0.2) eV, ? - D? = (1.4 ± 0.2) eV).     

Vakuum-Lichtgeschwindigkeit und differentielle Aberration

Vacuum Light Velocity and Differential Abberation In the frame work of Newton's theories of gravity and of light Laplace (1796) has deduced that the effective velocity c* of light is dependent on the gravitational potential of its source: Laplace, Olbers, a. o. have demonstrated that this effect implies differential aberrations of the light of different cosmisc sources. - But, Einstein's principles of relativity imply the independence of the velocity of light on its sources. This assertion is the fundamental principle of the Einsteinian theories of relativity. However, there seems to be no direct experimental facts disproving Laplace's formula, till today.     

Secondary Neutral Pion-nucleon Flux Ratio at the Top of the Atmosphere

The CERN Intersecting Storage Ring data on the inclusive reaction p + p → π° + X has been fitted with a phenomenological form where W_c(a,b,z) is the generalized confluent function defined by . M(a, b, z) being the well known Kummer's function. Using this form and primary proton spectrum of Goddard Space Flight Center Group the secondary neutral pion-nucleon flux ratio at the top of the atmosphere has been estimated. The derived result has been compared with those expected from p + p → π^± + X data, via different models.     

Über die Schwere des Lichtes in der Newtonschen und in der Einsteinschen Gravitationstheorie

The gravity theories of Newton and Einstein are giving opposite sentences about the velocity of light in gravitational field. According to the Newtonian theory the velocity v in gravitational field is greater than the velocity c in a field-free space: v > c. According to general relativity theory we have a smaller velocity: v < c. For a spherical symmetric gravitational field Newton's theory gives \documentclass{article}\pagestyle{empty}\begin{document}$ v \approx c\left({1 + \frac{{fM}}{{c^2 r}}} \right) $\end{document} but Einstein's theory of 1911 gives \documentclass{article}\pagestyle{empty}\begin{document}$ v \approx c\left({1 - \frac{{fM}}{{c^2 r}}} \right) $\end{document} and general relativity gives \documentclass{article}\pagestyle{empty}\begin{document}$ v \approx c\left({1 - 2\frac{{fM}}{{rc^2 }}} \right) $\end{document}. Therefore, the radarecho-measurations of Shapiro are the experimentum crucis for Einstein's against Newton's theory.     

Desorptions- und Reaktionskinetik der Erdalkalien Calcium und Strontium mit Chlor an Wolfram. Part II: Kinetics of the Elementary Steps of the Surface Reaction M + Cl ⇌ MCl (M = Ca,Sr)

Desorption- and Reactionkinetics of the Alkaline Earth Elements Calcium and Strontium with Chlorine on a Tungsten Surface — Part II: Kinetics of the Elementary Steps of the Surface Reaction M + Cl ? MCl (M = Ca, Sr) Utilizing pulsed molecular-beam-technique the kinetics of desorption of Strontium, Calcium, and Chlorine as well as that of the molecules SrCl and CaCl, which are formed at the hot tungsten surface, was investigated. Thereby, the following values were obtained for the activation energies of desorption: ? = (3.76 ± 0.05) eV, ? = (3.32 ± 0.07) eV, ? = (4.16 ± 0.05) eV, ? = (4.2 ± 0.3) eV and ? = (3.9 ± 0.3) eV. Combining these results with the steady-state-results from part I [1] the temperature dependency of the rate constants of dissociation and recombination of MCl-molecules at the tungsten surface could be determined. The values obtained for the dissociation energies D of SrCl and CaCl on tungsten are (0.5 ± 0.5) eV and (0.3 ± 0.5) eV, respectively. The molecules are stabilized on the surface by the activation barrier for dissociation D? only, which was found to be (2.8 ± 0.5) eV for SrCl and (2.3 ± 0.5) eV for CaCl.     

Adsorption,Desorption, Dissoziation und Rekombination von SO2 an einer Palladium(111)-Oberfläche

Adsorption, Desorption, Dissociation and Recombination of SO₂ on a Palladium (111) Surface The adsorption, desorption as well as decomposition- and recombination-reactions of SO₂ on Pd(1 1 1) were studied for temperatures T = 160-1200 K using LEED, AES, thermal desorption-mass-spectrometry and molecular beam techniques. At 160 K SO₂ adsorption with an initial sticking coefficient s₀ = 1 is molecular and non-ordered; it is characterized by a precursor state and leads to a saturation coverage Θ ≈ 0,3. Heating up the adlayer SO₂ is the only desorption product, namely directly from (SO₂)_ad in the α-peak (T_max = 240 K) and as the product of recombination of (SO)_ad and O_ad in the β-peak (T_max = 330-370 K). A great part of the oxygen originating from SO₂-dissociation is incorporated into the subsurface region, resulting in an atomic S-adlayer with Θ_S = 1/7 which exhibits a (\documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt 7 {\rm x}\sqrt 7 $\end{document}) R ± 19,1°-superstructure. This structure is also observed, if a 320 K-SO₂-exposure induced (2 × 2)-SO saturation layer with Θ_SO = 0,5 is heated up or if SO, is exposed at T > 500 K, where it corresponds to Θ_S, values of 3/7 and 2/7, respectively. Furthermore the poisoning effect of adsorbed sulfur on the dissociative O₂,-adsorption and the oxidation of sulfur by heating up an O? S-coadsorption layer were studied. As a result the following kinetic parameters (activation energies and frequency factors) were determined: .     

On the Nuclear Energy Generation Rate in a Simple Analytic Stellar Model

This paper is concerned with the hydrostatic equilibrium of a purely gaseous star generating energy at a rate . By the common technique of integration theory of special functions the total net rate of nuclear energy generation is evaluated analytically for a linear density distribution function assumed for a simple stellar model.     

Kinetic Study of the Emission of the First Negative System in an Argon-Nitrogen Plasma at Atmospheric Pressure

Quantitative optical spectroscopy measurements of the emission spectra of the N(B²∑_u,)ν′→X²∑_gν″ transition (first negative system) in an Ar-N₂ microwave discharge at atmospheric pressure have allowed determination of the rate coefficient of the production of N molecules in the B² ∑_u, state with vibrational level ν′ = 0. The N(B²∑_u, ν′) molecules are produced by the reaction in a surface-wave-induced microwave discharge (2450 MHz) sustained in an open-ended dielectric tube. The rate coefficient K (T) has been obtained for ν′ν″ = 0 for different gas temperatures by varying the incident microwave power. The K₀₀(T) values are between 7.10^?10 and 4.10^?10 cm³ s^?1 for the temperature range 2500 to 3450K.     

Reaktionskinetik der Ozonbildung in der Sauerstoffglimmentladung

Hg-light absorption at λ = 2537 ǎ by O₃ has been used to measure the concentration of O₃ in the positive column of an oxygen glow discharge under the nonstationary conditions of the Q-form (tube diameter 3:4 cm). For all gas pressures investigated (p = 7 — 12 torr) the O₃ density (n₃) as a function of the current reaches a remarkable maximum at i ≈ 15 mA. n₃ ^max was found to increase almost linearly with the gas pressure. At i = 80 mA, on the other hand, n₃ increases according to a quadratic relation. In order to understand the dependence of n₃ on the pressure and on the current density (or equivalently on the electron density n_e) a kinetic model previously developed has been used involving the following particles . For example, it can be shown that in the region of the maximum of n₃ one main production process of ozone is the associative detachment reaction . The calculated maximum of the ozone density is higher than the measured one and appears at higher electron density.     

Die Zurückführung der Gleichungen eines rotationssymetrischen,statischen, inkompressiblen Plasmas von unendlicher Leitfähigkeit auf ein Randwertproblem

The equations of a rotationally symmetric, static, incompressible plasma with infinite conductivity are equivalent to an elliptic differential equation for the function where p means the pressure, ? the density, and Φ the potential of the external forces. Moreover this differential equality contains two arbitrary functions of ξ. When ξ_?² + ξ_?² < 0, both arbitrary functions can be computed from the boundary values of Hφ und H⊥ (the component of H , which is perpendicular to the boundary).