Secular Post-NEWTONian Effects for the Restricted Three-Body Problem in TREDER'S Dynamics with Induced Inertia

In the framework of TREDER'S dynamics, in which according to the MACH -EINSTEIN -doctrin the inertial masses are induced by the gravitational interaction with the particles of the cosmos, we calculate the secular post-NEWTON ian effects for the restricted three-body problem. The dominant secular post-NEWTON ian variation of nodes and apsides is shown to be the same as in EINSTEIN'S theory of gravitation. The formulation of the gravodynamics in the HERTZ ian configuration space allows – as in General Relativity – the explanation of the effect as an “geodetic precession” of the lines of nodes and apsides.     

Zur dynamischen Äquivalenz der post-NEWTONSCHEN Näherung für die Planetentheorie in der allgemeinen Relativitätstheorie und in der trägheitsfreien Mechanik mit RIEMANNSchem Gravitationspotential

Proof of the dynamical equivalence of the post-NEWTONian approximations for the LAGRANGAIN representations of planetary motions according to GRT and according to our “RIEMANNian dynamics with MACH EINSTEIN doctrine”. Finally, we discuss the equivalence problem for general post-NEWTONian LAGRANGians of the one-body Problem in celestical mechanics.     

On Expanding MACH-EINSTEIN Universes

The “telescopic” gravodynamics of the MACH -EINSTEIN -Doctrine gives expanding “MACH ian clouds” as world models. The universe becomes a closed system (according to the classical statistical mechanics) if and only if the total energy H of this cloud, i.e. its HAMILTON ian, vanishes. This cloud is the MACH ian equivalent to the EINSTEIN-de SITTER universe of general relativity (or of “NEWTON ian cosmology”). However, the MACH ian cloud with H = o describes a linearly expanding universe.     

Das Korrespondenzprinzip in der Kosmologie I. Der isotrope Kosmos als NEWTONscher Grenzfall der relativistischen Gravitationstheorien

According to the principle of correspondence (in HEISENBERG 's formulation) each general relativistic theory of gravitation must give a NEWTON ian representation for an isotropic cosmos with the ROBERTSON -WALKER metric. Indeed, the FRIEDMANN equations can be interpreted as the expression for the HAMILTON ian H of a closed NEWTON ian system of the cosmic fundamental particles, written in the rest-system of the center of gravity. In this HAMILTON ian H only the relative-coordinates and the relative-velocities of the particles are present and one can write H without absolute quantities but only with MILNE 's relative-quantities. The time-independence of the HAMILTON ian H = 0 is the FRIEDMANN equation. – This NEWTON ian deduction of the FRIEDMANN equation is more general than the relativistic deduction and than MILNE 's deduction for a NEWTON ian fluid, too. In the general NEWTON ian form H the parameter f M of the active mass can be an arbitrary function of the cosmic time t. The choice f = f(t), M = M(t) defines the divers modifications of relativistic cosmology. – In general relativity fM = const and M = const are resulting from EINSTEIN 's equations and from EINSTEIN 's principle of equivalence.     

Das Korrespondenzprinzip in der Kosmologie. II. Klassische und relativistische Darstellungen der Weltmodelle

According to the equivalence between the FRIEDMANN equation of relativistic cosmology and the condition for the time-independence H = o of the HAMILTON ian H of an isotropic particle-system in the NEWTON ian mechanics (which equivalence is proved in the part I of our paper) we construct the corresponding classical HAMILTON ians to the relativistic world-models. Each cosmological model which is resulting from a physically meaningful gravitation theory must give a FRIEDMANN equation as the cosmological formulation of the time-independence condition of the energy H for the corresponding NEWTON ian N-particle system. In general relativity, EINSTEIN's field equations are including EINSTEIN's strong principle of equivalence and are giving the constance f = o and M = o of the gravitation-number f and of the mass M of the universe additional to FRIEDMANN's equation. – In special relativity, we have fM = o and this MILNE -universe is possessing a NEWTON ian and a general relativistic interpretation, too. – However, if the postulate together with the “cosmological principle” other principles about the world structure, too (p. e. MACH'S or DIRAC'S principle or the “perfect cosmological principle” by the steady-state cosmology), then EINSTEIN'S weak principle of equivalence can be fulfilled, only. In these world models the gravity-mass fM becomes a function of the cosmic time t [d/dt(fM) ± o] and this variability of fM is compatible with the constance H = o of the energy H of the NEWTON ian particle-system. For flat three-dimensional cosmological spaces (with H = Ḣ = o) a creation of rest-mass (M > o) is possible. This creation is the pecularity of the steady-state cosmos (with M > o, f = o) and of JORDAN'S cosmos (with M > o, f < o). The MACH -EINSTEIN -doctrine about the perfect determination of the inertia and of the space-time-metric by the cosmic gravitation is founded on the substitution of the NEWTON ian HAMILTON ian by a GAUSS -RIEMANN ian gravitation potential U^*(r_AB' v_AB) (TREDER 1972). Therefore, the FRIEDMANN equation for a universe with MACH'S principle is resulting from the analytical expression of the time-independence of this RIEMANNian potential U* = 0. In the case of such MACH-EINSTEIN's-Universes EINSTEIN'S condition 3fM = c8r between the mass A4 and the radius Y of the universe is valid additional to FRIEDMANN'S equation. For these universes, the EINSTEIN condition determinates the instantaneous value of the gravitation-number f. - The explicite form of the conditions H = o or h' = o gives the equation of motion for the cosmic fundamental particles with attraction and repulsion forces, generally.     

On the post-NEWTONian approximation of theories of gravity

The physical meaning of the parameters of the post-NEWTON ian approximation is considered from a theoretical point of view. It is shown that the post-NEWTON ian approximation of any LORENTZ -covariant theory, which implies POISSON equations for every metric coefficient in the linear approximation, is the member of a three-parametric family of post-NEWTON ian metrics. Some tetrad theories are cited.     

Die Absorption der Schwerkraft und das SEELIGER-YUKAWA-Potential,Akademiker W. A. AMBARZUMJAN zum 65. Geburtstag zugeeignet

In our tetrad-formulation of the gravitation theory a potential-like interaction between the gravitationpotential and the matter-tensor is postulated. In the case of a dust-like matter from this interaction results a screening of the quasi-NEWTON ian potential by the matter-density given by a SEELIGER -YUKAWA -potential with the absorptionconstant approximately. — In the cosmological case from this screening of gravitation a finite effective gravitationradius R of the universe is resulting which radius R is given by the condition for an EINSTEIN -universe 4.     

Polteilchen in der TREDERSchen Tetradentheorie

In the framework of TREDERS ' tetrad theory of gravitation the interaction of pole-particles is considered in post-NEWTON ian approximation. The perihelion shift for the two-body problem is calculated for different matter-field-couplings. In general it depends not only from the sum of the masses but also from the reduced mass. The additional term is connected with the absorption of gravitation.     

Gravitation und weitreichende schwache Wechselwirkungen bei Neutrino-Feldern Gedanken zu einer Theorie der solaren Neutrinos

The strange non-evidence of the solar-neutrino current by the experiments of DAVIS et al. postulates a fundamental revision of the theory of weak interactions and of its relations to gravitation theory. (We assume that the astrophysical stellar models are not completely wrong.) – Our paper is based on PAULI 's grand hypothesis about the connection between weak and gravitational interactions. According to PAULI and BLACKETT the (dimensionless) gravitation constant is the square of the (dimensionless) FERMI -interaction constant and according to the hypotheses of PAULI, DE BROGLIE , and JORDAN the RIEMANN -EINSTEIN gravitational metric g_ik is fusioned by the four independent WEYL ian neutrino fields (β-neutrinos and β-antineutrinos, μ-neutrinos and μ-antineutrinos). This fusion gives four reference tetrads h_i^A(x^l) as neutrino-current vectors, firstly. Then, the metric g_ik is defined by the equation g_ik = η_AB h_i^Ah_η^B according to EINSTEIN 's theory of tele-parallelism in RIEMANN ian space-times. The relation of the gravitation field theory to FERMI 's theory of weak interactions becomes evident in our reference-tetrads theory of gravitation (TREDER 1967, 1971). – According to this theory the coupling of the gravitation potential h_i^A with the matter T_ιⁱ is given by a potential-like (FERMI -like) interaction term. In this interaction term two WEYL spinor-fields are operating on the matter-tensor, simultanously. Therefore, the gravitation coupling constant is PAULI 's square of the FERMI -constant. Besides of the fusion of the RIEMANN -EINSTEIN metric g_ik by four WEYL spinors we are able to construct a conformal flat metric ĝ_ik = ϕ²η_ik by fusion from each two WEYL spinors. (This hypothesis is in connection with our interpretation of EINSTEIN 's hermitian field theory as a unified field-theory of the gravitational metric g_ik and a WEYL spinor field [TREDER 1972].) Moreover, from the reference-tetrads theory is resulting that the WEYL spinors in the “new metric” ĝ_ik are interacting with the DIRAC matter current by a FERMI -like interaction term and that these WEYL spinors fulfil a wave equation in the vacuum. Therefore, we have a long-range interaction with the radiced gravitational constant \documentclass{article}\pagestyle{empty}\begin{document}$ \sqrt {\frac{{tm^2 }}{{hc}}} $\end{document} as a coupling constant. That means, we have a long-range interaction which is 10¹⁸ times stronger than the gravitation interaction. – However, according to the algebraic structure of the conform-flat this long-range interaction is effective for the neutrino currents, only. And for these neutrinos the interaction is giving an EINSTEIN -like redshift of its frequences. The characteristic quantity of this “EINSTEIN shift” is a second gravitation radius â of each body: N = number of baryons, m = mass of a baryon.) This radius â is 10¹⁸ times larger than the EINSTEIN -SCHWARZSCHILD gravitation radius a = fM/c²: But, this big “weak radius” â has a meaning for the neutrinos, only.–The determination of the exterior and of the interior “metrics” ĝ_ik is given by an “ansatz” which is analogous to the ansatz for determination of strong gravitational fields in our tetrads theory. That is by an ansatz which includes the “self-absorption” of the field by the matter. For all celestial bodies the “weak radius” â is much greater than its geometrical dimension. Therefore, a total EINSTEIN redshift of the neutrino frequences v is resulting according to the geometrical meaning of our long-range weak interaction potential ĝ_ik = ϕ²η_ik. That means, the cosmic neutrino radiation becomes very weak and unable for nuclear reactions. Theoretically, our hypothesis means an ansatz for unitary theory of gravitation and of weak interaction. This unitary field theory is firstly based on EINSTEIN 's hermitian field theory and secondly based on our reference-tetrads theory of gravitation.     

Trägheitsfreie Dynamik und HAMILTONsche Bewegungsgleichung

We formulate the canonical equations of motion for particles with an (post-HEWTON ian) interaction potential and the HAMILTON ian form of our MACH ian dynamics without inertia.     

Zur Darstellung der anomalen EINSTEIN-Effekte mit Hilfe einer effektiven Metrik

We discuss the presumed excesses of the gravitooptical EINSTEIN effects of the sun. Supposing hypothetically the significance of the anomalies asserted we try to represent these by means of an effective space-time metric being the formal model of the dielectric properties of vacuum. The effective metric may be chosen such that the nonlinear electromagnetic effects determined by the new metric are compatible with the quantum electrodynamics.     

Singularitäten in der Allgemeinen Relativitätstheorie

“Regular solutions of EINSTEIN 's equations” mean very different things. In the case of the empty-space equations, R_ik = 0, such solutions must be metrics g_ik(x^l) without additionaly singular “field sources” (EINSTEIN 's “Particle problem”). – However the “phenomenological matter” is defined by the EINSTEIN equations R_ik – 1/2g_ikR =–xT_ik itselves. Therefore if 10 regular functions g_ik(x^l) are given (which the inequalities of LORENTZ -signature fulfil) then these g_ik define 10 functions T_ik(x^l) without singularities. But, the matter-tensor T_ik must fulfil the two inequalities T ≥ 0, T ≥ 1/2 T only and therefore the EINSTEIN -equations with “phenomenological matter” mean the two inequalities R ≥ 0, R ≤ 0 which are incompatible with a permanently regular metric with LORENTZ -signature, generally.     

Kaustiken in der TREDERschen Gravitationstheorie und die kosmologische Singularität

We investigate criteria under which the space-times being solutions of TREDER 's tetrad theory of gravitation have caustics. These criteria and some properties of the caustics are compared with the corresponding results in EINSTEIN 's general relativity.     

Die Asymmetrie der kosmischen Zeit und RIEMANNS Gravitationstheorie1)

RIEMANN himself has considered his formulation of the differential geometry of curved spaces as a first step to a unified geometrical theory of “one ether of gravity, electricity and magnetism”. RIEMANN has pointed out that a fundamental point in such a theory of gravitation has to be the asymmetry of its sources: only positive masses exist. – According to RIEMANN this asymmetry of sources to be coupled with an asymmetry of gravitation field equation against the time-reversion t → - t. Therefore, the gravitation field equation is of the type of a continuity-equation of a velocity field v_i˜g^ikθ _k Φ. RIEMANN 's ether is incompressible in empty space-domains: θ _k (g^1/2v^k) = o. But, in domains with a massdensity σ > o it is θg^1/2/θt = −2 kcσ = − 2 kcg^1/2σ₀ (with a universal constant kc). The matter-density defines depressions of the ether. In a general-relativistic approach RIEMANN 's ansatz means that in empty space-time domains the world-geometry is the purely metrical “RIEMANN ian” geometry. However, in domains with a non-vanishing matter-tensor T_μ^v ≠ o the geometry becomes “non-RIEMANN ian” affine connecting and is of the type of WEYL 's geometry or of the “EINSTEIN -CARTAN theories of gravitation”. Especially, RIEMANN 's field equation for the empty space θ _k ((g^1/2g^ikθ _k Φ) = o. is the EINSTEIN equation (-|g_μv|)^1/2 R₀⁰ = o with g₀₀ = - Φ²c^-4.     

On singularities in electrodynamics and gravitational theory

The singularity and particle problems in electrodynamics and gravitational theory are discussed. Comparing different modifications of MAXWELL 's electrodynamics and general theory of relativity, arguments are given which show possibilities of solving these problems by a modification that replaces EINSTEIN 's equations by those of the EINSTEIN -BACH -WEYL -type.     

REDUCE programs for algebraic computation in general relativity

REDUCE programs for algebraic computation in General Relativity are presented which use the metric tensor as input. The programs calculate — according to the user's option — the CHRISTOFFEL symbols, the RIEMANN tensor, the RICCI tensor, the RICCI scalar, the EINSTEIN tensor and the WEYL tensor.     

Inertia-free mechanics and the bi-metric procedure

We consider the bi-metric procedure to implement field-theoretical concepts into the mechanical models developped so far in inertia-free gravodynamics. The main result is the fact, that we are able to hide the absolute time underlying the mechanical models for non-gravitational fields by the equivalence principle, but not for the gravitation theory, which reveals the instantaneous gravitational action in one parameter of the post-Newtonian approximation.     

NEWTONsche Relativ-Mechanik

The NEWTONian mechanics involves the dynamical absoluteness of motions which means the presence of absolute accelerations in NEWTON's equations of motion. Against NEWTON's point of view HUYGENS (and LEIBNIZ and the young KANT and later MACH and POINCARÉ) have postulated that the dynamics of a closed system of n particles P_a is depending on relative kinematical quantities, only. These relative quantities have to be the differences of the absolute kinematical quantities of each particle P_a according to HUYGENS, KANT and POINCARÉ. — We prove that a closed system of N NEWTONian particles with the same atomic masses m (according to the hypothesis of HERTZ) can be described by a HUYGENsian relative dynamics in the rest system of the center of masses. In this reference system the HAMILTONian, the LAGRANGian, the virial, the angular momentum etc. of the N particle system are depending on the difference quantities, only. — However, the absolute dynamical meaning of motion in the NEWTONian mechanics leads to a paradoxon for this relativistic representation. This paradoxon is resulting by an ideal combination of two independing particle systems to one system in the mind. No such paradoxon is existing by a very modification of NEWTON's dynamics with respect to MACH's principle of the relativity of inertia.     

On Equations of Motion for Cross Term Modified Gravitational Field Equations

As proposed by TREDER , possible consequences of a unitary field theory may be described phenomenologically by additional cross terms in EINSTEIN 's equations. The violation of the weak principle of equivalence and potential observable effects are discussed in deriving hydrodynamic EIH equations. Conclusions on gravitational instabilities follow in the quasistatic approximation.     

Mach–poincaré gravodynamics and condensation phenomena in an expanding universe

We discuss the condensation phenomena in a very simple model, which allows for transparent calculations. We compare the condensation times of subsystems decoupled from the cosmological expansion in different approaches to Mach-Poincaré gravodynamics with Newtonian mechanics. In all cases contracting subsystems without inner rotation collapse, only the time-scales differ.