Tensor-Tensor Theory of Gravitation

Astrophysics and Space Science, 2015

A certain vector-tensor (VT) theory of gravitation was tested in previous papers. In the background universe, the vector field of the theory has a certain energy density, which is appropriate to play the role of vacuum energy (cosmological constant). Moreover, this background and its perturbations may explain the temperature angular power spectrum of the cosmic microwave background (CMB) obtained with WMAP (Wilkinson Map Anisotropy Probe), and other observations, as e.g., the Ia supernova luminosities. The parametrized post-Newtonian limit of the VT theory has been proved to be identical to that of general relativity (GR), and there are no quantum ghosts and classical instabilities. Here, the stationary spherically symmetric solution, in the absence of any matter content, is derived and studied. The metric of this solution is formally identical to that of the Reissner-Nordströmde Sitter solution of GR, but the role of the electrical charge is played by a certain quantity Γ depending on both the vector field and the parameters of the VT theory. The black hole and cosmological horizons are discussed. The radius of the VT black hole horizon deviates with respect to that of the Kottler-Schwarzschildde Sitter radius. Realistic relative deviations depend on Γ and reach maximum values close to 30 per cent. For large enough Γ values, there is no any black hole horizon, but only a cosmological horizon. The radius of this

Researchgate , 2019

We consider a (4+D)-dimensional Friedmann-Robertson-Walker type universe having complex scale factor R + iR I , where R is the scale factor corresponding to the usual 4-dimensional Universe while iR I is that of D-dimensional space. It is then compared with (4+D)-dimensional Kaluza-Klein Cosmology having two scale factors R and a(= iR I). It is shown that the rate of compactification of higher dimension depends on extra dimension 'D'. The Wheeler-DeWitt equation is constructed and general solution is obtained. It is found that for D = 6 (i.e. in 10 dimension), the Wheeler-DeWitt equation is symmetric under the exchange of R I R. I. Introduction: In 1915 Einstein published the general theory of relativity. He expected the universe to be 'closed' and to be filled with matter. Now, if we go outside the gravitating sphere, we see the gravitation would be weaker and weaker. According to Einstein's theory of general relativity, the matter-space-time cannot be separated by any cost. Thus, outside the Einstein's universe, where real time cannot be defined, the corresponding space (although, the matter belongs to another phase) must be measured as imaginary. Thus the space-time of the universe is actually a complex space-time. Here we consider the real space-time (unfolded space-time) for Einstein and imaginary space-time (folded space-time) for us. We found a relation between folded and unfolded space-time of the universe by using Wheeler De-Witt equation. The generalized solution for the Einstein field equations for a homogeneous universe was first presented by Alexander Friedmann. The Friedmann equation for the evolution of the cosmic scale factor R(t) which represents the size of the universe, is Differentiating the above equation with respect to time t and since the total matter in a given expanding volume is unchanged, that means = constant. We have, i.e

Arxiv preprint gr-qc/0403050, 2004

In the following GR9 (1980) Conference abstract, we briefly considered a covariant reformulation of the GR from three points of view: geometrodynamics, Lagrange-Euler field theory, and gauge field theory. From a perspective of geometrodynamics, a definition of a reference frame as a differential manifold with an affine connection results in separation of the respective contributions of inertial and gravitational fields represented by the affine connection and the nonmetricity tensor within the Levi-Civita connection of GR. Resulting decomposition of Riemannian, Ricci and Einstein curvature tensors into affine and nonmetric parts allows recasting of Einstein's field equations in a form invariant with respect to the choice of a reference frame, wherein the gravity is described by nonmetricity of space-time. A covariant Lagrangian is proposed leading to the same field equation. All three approaches ultimately lead to the same fully covariant theory of gravitation with a covariant tensor of energy-momentum of the gravitational field and differential and integral conservation laws. The role

Foundations of Relativistic Gravithermodynamics. Publishing house “TVORY”, Vinnytsia, 2022

The cardinal difference between relativistic gravithermodynamics (RGTD) and general relativity (GR) is that in RGTD the extranuclear thermodynamic characteristics of matter are used in the tensor of energy-momentum to describe only its quasi-equilibrium motion. For the description of the inertial motion in RGTD only the hypothetical intranuclear gravithermodynamic characteristics of matter are used. Exactly this fact allows avoid the necessity of non-baryonic dark matter in the Universe in principle. Evolutionary self-contraction of microobjects of lower layers of gravithermodynamically bonded matter outpaces the similar self-contraction of its upper layers. This is the exact reason of the curvature of intrinsic space of matter. That is why gravitational field itself should be primarily considered as the field of spatial inhomogeneity of evolutionary decreasing of the size of matter microobjects in the background Euclidean space of expanding Universe. In correspondence to this the gravitational field itself is the field of spatial inhomogeneity of gravithermodynamic state of dense matter of compact astronomical objects, as well as of strongly rarefied gas-dust matter of space vacuum. And, therefore, the gravitational field fundamentally cannot exist without matter. That is why it is not an independent form of matter. It is shown that equations of the gravitational field of GR should be considered as equations of spatially inhomogeneous gravithermodynamic state of only utterly cooled down matter. This matter can only be the hypothetical substances such as ideal gas, ideal liquid and the matter of absolutely solid body. The real matter will be inevitably cooling down for infinite time and never will reach the state that is described by the equations of gravitational field of the GR. Only conditional identity of inertial mass of moving matter to its gravitational mass only by gravity-quantum clock, which is located in the point, from which the matter started its inertial motion, and due to the usage of corrected value of gravitational constant in its pseudo-centric intrinsic frame of reference of spatial coordinates and time, is justified. This is related to the equivalence of inertial mass of matter to the Hamiltonian of its inert free energy, while the gravitational mass of matter is equivalent to the Lagrangian of its ordinary rest energy. The identity of the multiplicative component of the Gibbs free energy to the ordinary rest energy of matter, which is equivalent to its gravitational mass, is substantiated. It was proved that values of thermodynamic internal energy and Gibbs free energy of matter of inertially moving body are equal in all global gravithermodynamic frames of references of spatial coordinates and time (GT-FR) that are also inertially moving relatively to matter. The temporal invariance of not only the momentum but also of Lagrangian of ordinary rest energy and of equivalent to it gravitational mass of matter is justified. And that is precisely why there is the temporal invariance and a conformal Lorentz-invariance of thermodynamic potentials and parameters in examined modification of transformations of the special theory of relativity (SR). Conformal relativistic transformations of increments of metrical spatial segments and metrical temporal intervals (instead of increments of coordinates and coordinate time of SR) were received. Exactly this fact allows avoid not only the twins paradox when twins are inertially moving but also the necessity of the dark energy in the Universe. It is also shown that the tensor of energy-momentum of matter (right side of the gravitational field equation) should be formed not being based on external thermodynamic parameters, but being based exactly on the intranuclear gravithermodynamic parameters. The canonical differential equation of the dynamic gravitational field (that corresponds to matter, which moves in this field only by inertia) is examined. It is this equation that determines the spatial distribution of the average density of mass of matter in the quasi-equilibrium state of its motion by inertia. It is shown that at the edge of the galaxy the excessively strong ordinary (unreduced) centrifugal pseudo-forces of inertia are compensated mainly by centripetal pseudo-forces of evolutionary self-contraction of matter in the background Euclidean space of expanding Universe, and not by the weak gravitational pseudo-forces at the edge of the galaxy. The standard value of the average density of matter gravitational mass at the edge of a galaxy is determined by the cosmological constant Λ and the difference between unity and the maximum value of the parameter bc. And it is a nonzero standard value, despite the local gravitational radius at the edge of a galaxy takes the zero value. It is shown that the decrease in the strength of the gravitational field of flat galaxies is inversely proportional to the radial distance, and not its square, and that the increase in the gravitational "constant" is inversely proportional to the fourth degree of the limit velocity of matter motion. Consequently, the presence of dark non-baryonic matter in the Universe is unnecessary. Of course, bodies free fall in gravitational field is an original realization of their tendency to increase the evolutionary self-contraction of microobjects of their matter, and the realization of the tendency of the whole gravitationally bonded inhomogeneous matter to the minimum of the integral values of its inert free energy and thermodynamic Helmholtz free energy. Bodies that fall accelerate independently in spatially inhomogeneous medium of the outer space or atmosphere. Such bodies transform their continuously released intra-atomic energy into kinetic energy. It is shown that in case of bodies’ free fall the gravitational deceleration of the rate of their intrinsic time is completely compensated by the motion due to isotropic all-round conformal gauge self-contraction of the size of falling bodies in the background Euclidean space of the Universe. Clocks that fall free are inertially moving and, therefore, continue to count time at the same rate as when they were in the state of rest. Similarly, the rate of time of astronomical body is not changed in the process of its motion in elliptical orbit. The dilatation of intrinsic time of distant galaxies is also absent, which points on the fact that Etherington identity does not correspond to reality. The fact that Hubble’s redshift is linearly dependent on transversal comoving distance instead of luminosity distance is justified. It is shown that mentioned above fact corresponds to astronomical observations. According to this the presence of dark energy in the Universe is also unnecessary. For the collective gravithermodynamic Gibbs microstates the connection between all thermodynamic potentials and parameters of matter have been found. This connection is realized with the help of several wave functions that can take arbitrary values with certain probability. The quantum equation of gravitational field have been found, the solutions of which set the spatial distribution of gravitational radius of matter in its every new gravithermodynamic state with the polynomial function with the next more high degree. The indicator of the degree of this function of continuously cooling down matter can successively take only integer and semi-integer values. That is why the process of cooling down of the whole RGTD-bonded matter is the quantum process that is caused by its spontaneous transition to the polynomial function with more high value of degree and, therefore, to the next quantized collective gravithermodynamic state.

2009

The Gravitational, Electromagnetic, Weak & the Strong force are here brought together under a single roof via an extension of Reimann geometry to a new geometry (coined Reimann-Hilbert Space); that unlike Reimann geometry, preserves both the length and the angle of a vector under parallel transport. The affine connection of this new geometry-the Reimann-Hilbert Space, is a tensor and this leads us to a geodesic law that truly upholds the Principle of Relativity. The geodesic law emerging from the General Theory of Relativity (GTR) is well known to be in contempt of the Principle of Relativity which is a principle upon which the GTR is founded. The geodesic law for particles in the GTR must be formulated in special (or privileged) coordinate systems i.e. gaussian coordinate systems whereas the Principle of Relativity clearly forbids the existence of special (or privileged) coordinate systems in manner redolent of the way the Special Theory of Relativity forbids the existence of an absolute (or privileged) frame of reference. In the low energy regime and low spacetime curvature the unified field equations derived herein are seen to reduce to the well known Maxwell-Procca equation, the none-abelian nuclear force field equations, the Lorentz equation of motion for charged particles and the Dirac Equation. Further, to the already existing four known forces, the theory predicts the existence of yet another force. We have coined this the super-force and this force obeys S U(4, 4) gauge invariance. Furthermore, unlike in the GTR, gravitation is here represented by a single scaler potential, and electromagnetic field and the nuclear forces are described by the electromagnetic vector potential (A µ) which describes the metric tensor i.e. g µν = A µ A ν. From this (g µν = A µ A ν), it is seen that gravity waves may not exist in the sense envisaged by the GTR.