Tensor Of Gravitational Field Research Articles

Separation of Variables Method in Solving Partial Differential Equations and Investigating the Relationship between Gravitational Field Tensor and Energy-Momentum Tensor in Einstein's Theory of Gravity

This research delves into the study of partial differential equations (PDEs) and gravitational fields in spacetime. It focuses on solving PDEs using the Separation of Variables method and explores the relationship between the gravitational field tensor and the energy-momentum tensor, leading to the final equation for the gravitational field tensor. The research also investigates Einstein's theory of gravity and the energy-momentum tensor integral, providing the general solution for the gravitational potential and its implications. Additionally, the mean integration of the gravitational wave tensor is analyzed, yielding an expression for the tensor strain of gravitational waves over an infinitely long period. The components of the gravitational wave tensor and their effect on gravitational sources are examined. Furthermore, the propagation of electromagnetic fields in spacetime is studied using the Retarded Green's Function. The primary objectives of this research are to understand and explore mathematical techniques for solving PDEs and analyzing gravitational fields and their interactions in spacetime. The integration of multiple theoretical concepts related to PDEs, gravitational fields, and electromagnetic fields enhances our understanding of fundamental physics principles. This contributes to the advancement of theoretical physics and opens avenues for potential practical applications, such as gravitational wave detection and electromagnetic field propagation in complex media. In conclusion, this research provides valuable insights into fundamental physics principles and fosters a deeper understanding of their interconnections and implications

Statistical Formulation of Background Independence in Manifestly-Covariant Quantum Gravity Theory

The notion of background independence is a distinguished feature that should characterize the conceptual foundation of any physically-acceptable theory of quantum gravity. It states that the structure of the space-time continuum described by classical General Relativity should possess an emergent character, namely, that it should arise from the quantum-dynamical gravitational field. In this paper, the above issue is addressed in the framework of manifestly-covariant quantum gravity theory. Accordingly, a statistical formulation of background independence is provided, consistent with the principle of manifest covariance. In particular, it is shown that the classical background metric tensor determining the geometric properties of space-time can be expressed consistently in terms of a suitable statistical average of the stochastic quantum gravitational field tensor. As an application, a particular realization of background independence is shown to hold for analytical Gaussian solutions of the quantum probability density function.

Background Independence and Gauge Invariance in General Relativity Part 2—Covariant Quantum Gravity

Background independence is often being claimed as the characteristic property of several current and past models of Quantum Gravity. In actual fact, such a notion has a wider connotation and must be rooted into the validity of the general covariance principle, demanding its logical connection with the notions of manifest covariance and (quantum) gauge invariance. In fact, as we intend to show here, it involves (a) the existence of a well-defined, albeit arbitrary, classical background space-time; and (b) the suitable realization of a dynamical equation for the related background metric field tensor, referred to as quantum-modified Einstein tensor field equation, which actually determines it in a suitable functional setting. Remarkably, it is proved that in the context of the theory of Covariant Quantum Gravity (CQG-theory), recently developed by Cremaschini and Tessarotto (2015–2022), background independence implies that such an equation “emerges” rigorously from the same CQG-theory. This follows in terms of a stochastic quantum expectation value evaluated with respect to the corresponding characteristic quantum PDE. It is shown that an analogous emergence property applies also to the background metric field tensor in terms of stochastic fluctuations of the corresponding underlying quantum tensor of gravitational field. These results warrant the consistent validity of background independence for the prescription of the space-time metric tensor in CQG-theory.

Space-Time Second-Quantization Effects and the Quantum Origin of Cosmological Constant in Covariant Quantum Gravity

Space-time quantum contributions to the classical Einstein equations of General Relativity are determined. The theoretical background is provided by the non-perturbative theory of manifestly-covariant quantum gravity and the trajectory-based representation of the related quantum wave equation in terms of the Generalized Lagrangian path formalism. To reach the target an extended functional setting is introduced, permitting the treatment of a non-stationary background metric tensor allowed to depend on both space-time coordinates and a suitably-defined invariant proper-time parameter. Based on the Hamiltonian representation of the corresponding quantum hydrodynamic equations occurring in such a context, the quantum-modified Einstein field equations are obtained. As an application, the quantum origin of the cosmological constant is investigated. This is shown to be ascribed to the non-linear Bohm quantum interaction of the gravitational field with itself in vacuum and to depend generally also on the realization of the quantum probability density for the quantum gravitational field tensor. The emerging physical picture predicts a generally non-stationary quantum cosmological constant which originates from fluctuations (i.e., gradients) of vacuum quantum gravitational energy density and is consistent with the existence of quantum massive gravitons.

An almost Einsteinian theory of gravitation

A new version of relativistic theory of gravitation is proposed, in which the GR is supplemented with two postulates. Owing to these postulates it turns possible to introduce into consideration a covariant energy-momentum tensor of gravitational field. Additionally the influence of Universe on the gravitating system under consideration is taken into account. The equations describing the gravitation field in the framework of proposed tensor mono-metric theory of gravitation are derived. In its simplest version the proposed theory contains nine free parameters. At a particular choice of these parameters the theory is reduced to GR with a cosmological term.

Impossibility of gravitational collapse

In the framework of the relativistic theory of gravity, we show that a universal mechanism for stopping the process of gravitational compression of a body with large mass with its subsequent radial expansion appears because of the gravitational field tensor. This excludes the gravitational collapse and the possibility of black hole formation.

The Principle of Least Action in Covariant Theory of Gravitation

The integral equations for calculation of metric, equations of substance motion, as well as equations for gravitational and electromagnetic fields in covariant theory of gravitation are obtained by means of variation of action functional. In covariant form stress-energy tensor of gravitational field, strength tensor of gravitational field and 4-current of mass are determined. The meaning of the cosmological constant and its relation to the components of energy density in action functional are explained. The results obtained prove the validity of Mach's principle, assuming that gravitation effects are due to the flows of gravitons. The idea that metric can be entirely determined by variables describing fields’ properties is substantiated.

A NON- STATIC COSMOLOGICAL MODEL IN THE VECTOR MODEL FOR GRAVITATIONAL FIELD

In this paper, based on the vector model for gravitational field we obtained the modified Friedman equations, which were similar to the classical Friedman equations but were added a term of energy – momentum tensor of gravitational field. Non- static flat cosmological model in this model was similar to General Theory of Relativity (GTR) ‘s model but the expansive rate in the vacuum age was difference with General Theory of Relativity ’s model.

A covariant gravitational field equation including the contribution of gravitational field

Using the four-leg metric tensor λ(α)μ, a gravitational field 4-vector potential for index μ is defined as ω(α)μ≡-cλ(α)μ, and a covariant gravitational field equation that includes the gravitational field contribution is proposed as Rμν-gμνR/2+Λgμν=8πG(T(Ⅰ)μν+T(Ⅱ)μν)/c4, where Λ is Einstein's cosmic constant， T(Ⅰ)μν and T(Ⅱ)μν are energy-momentum tensor of pure matter part and pure gravitational field part, respectively. The covariant energy-momentum tensor of gravitational field that belongs to the part of the gravitational source can be constructed as T(Ⅱ)μν=c2(D(α)μρDρν(α)-gμνD(α)τγDτγ(α)/4)/4πG, where D(α)μν≡ω(α)μ;ν-ω(α)ν;μ. The static spherically symmetric gravitational field, the missing mass and the gravitational field quantization are discussed.

The rejection of the ricci tensor in Einstein's first tensorial theory of gravitation

Albert Einstein came very close to the final form of the equations describing the gravitational field during his collaboration with Marcel Grossmann a fecund period which began in August 1912 and lasted nine months. Nevertheless, the theory published in their joint paper1 showed an "ugly dark spot," as Einstein wrote to Lorentz2 : contrary to initial expectations, the field equations were not generally covariant. Einstein struggled hard to iron out what he considered a major flaw, and eventually succeeded in November 1915, when he presented the final version of his general theory of relativity in a number of communications to the Prussian Academy of Science. Accepted reconstructions of the genesis of the Entwurf paper generally take in the following steps: Einstein & Grossmann initially made the right choice, taking the Ricci tensor into account as the gravitational field tensor, but rejected it later in the paper due to the difficulties encountered in recovering the Newtonian limit in the case of a weak and static gravitational field, making do with equations which were covariant with respect to linear transformations of coordinates. Moreover, in a paragraph entitled "Remarks," added at the end of the Mathematical part (the paper has a "Physical part" and a "Mathematical part," written by Einstein and Grossmann, respectively), Einstein gave what he believed was a proof that all generally covariant equations for the gravitational field were physically unacceptable. Interpretation of the way Einstein accounted for his failure to obtain the correct Newtonian limit and for the limited covariance of the equations is still a moot point among historians of physics. Some believe that Einstein's difficulties were of an essentially mathematical nature, while others stress their physical nature, viewed in terms of Einstein's conception of static gravitational fields as it stood in 1912.

Nelson's stochastic quantization of a free linearized gravitational field and its Markovian structure

It is shown that by applying Nelson's stochastic quantization scheme to a free linearized gravitational field tensor, one can associate with the resulting stochastic system a stochastic tensor field which coincides with the ‘space’ part of the Riemannian tensor in Euclidean space-time. However, such a stochastic field fails to satisfy the Markov property. Instead, it satisfies the reflection positivity. The Markovian structure of the stochastic fields associated with the electromagnetic field is also discussed.