scholarly journals Classical Variational Theory of the Cosmological Constant and Its Consistency with Quantum Prescription

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 633 ◽  
Author(s):  
Claudio Cremaschini ◽  
Massimo Tessarotto
Keyword(s):  
Variational Principle ◽  
Cosmological Constant ◽  
Lagrange Multiplier ◽  
Proper Time ◽  
Variational Theory ◽  
Field Equations ◽  
Metric Tensor ◽  
Covariant Quantum ◽  
Hamilton Variational Principle ◽  
Extremal Metric

The manifestly-covariant Hamiltonian structure of classical General Relativity is shown to be associated with a path-integral synchronous Hamilton variational principle for the Einstein field equations. A realization of the same variational principle in both unconstrained and constrained forms is provided. As a consequence, the cosmological constant is found to be identified with a Lagrange multiplier associated with the normalization constraint for the extremal metric tensor. In particular, it is proved that the same Lagrange multiplier identifies a 4-scalar gauge function generally dependent on an invariant proper-time parameter s. Such a result is shown to be consistent with the prediction of the cosmological constant based on the theory of manifestly-covariant quantum gravity.

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

Symmetry ◽  
2018 ◽  
Vol 10 (7) ◽  
pp. 287 ◽  
Author(s):  
Claudio Cremaschini ◽  
Massimo Tessarotto
Keyword(s):  
Gravitational Field ◽  
Quantum Gravity ◽  
Cosmological Constant ◽  
Proper Time ◽  
Theoretical Background ◽  
Space Time ◽  
Einstein Equations ◽  
Field Equations ◽  
Metric Tensor ◽  
Covariant Quantum

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.

scholarly journals Variational theory of the Ricci curvature tensor dynamics

2021 ◽  
Vol 81 (11) ◽  
Author(s):  
Claudio Cremaschini ◽  
Jiří Kovář ◽  
Zdeněk Stuchlík ◽  
Massimo Tessarotto
Keyword(s):  
Variational Principle ◽  
Ricci Curvature ◽  
Curvature Tensor ◽  
Hamiltonian Dynamics ◽  
Variational Theory ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Metric Tensor ◽  
Physical Conditions ◽  
Polynomial Expression

AbstractIn this letter a new Lagrangian variational principle is proved to hold for the Einstein field equations, in which the independent variational tensor field is identified with the Ricci curvature tensor $$R^{\mu \nu }$$ R μ ν rather than the metric tensor $$g_{\mu \nu }$$ g μ ν . The corresponding Lagrangian function, denoted as $$L_{R}$$ L R , is realized by a polynomial expression of the Ricci 4-scalar $$R\equiv g_{\mu \nu }R^{\mu \nu }$$ R ≡ g μ ν R μ ν and of the quadratic curvature 4-scalar $$\rho \equiv R^{\mu \nu }R_{\mu \nu }$$ ρ ≡ R μ ν R μ ν . The Lagrangian variational principle applies both to vacuum and non-vacuum cases and for its validity it demands a non-vanishing, and actually also positive, cosmological constant $$\Lambda >0$$ Λ > 0 . Then, by implementing the deDonder–Weyl formalism, the physical conditions for the existence of a manifestly-covariant Hamiltonian structure associated with such a Lagrangian formulation are investigated. As a consequence, it is proved that the Ricci tensor can obey a Hamiltonian dynamics which is consistent with the solutions predicted by the Einstein field equations.

scholarly journals A Theoretical Calculation of the Cosmological Constant Based on a Mechanical Model of Vacuum

2021 ◽  
Author(s):  
Xiao-Song Wang
Keyword(s):  
Cosmological Constant ◽  
Mass Density ◽  
Zero Point ◽  
Momentum Tensor ◽  
Einstein’S Equations ◽  
Field Equations ◽  
Cosmological Constant Problem ◽  
Metric Tensor ◽  
Point Energy ◽  
Einstein's Equations

We suppose that vacuum is filled with a kind of continuously distributed matter, which may be called the $\Omega(1)$ substratum, or the electromagnetic aether. Lord Kelvin believes that the electromagnetic aether must also generate gravity. We also suppose that vacuum is filled with another kind of continuously distributed substance, which may be called the $\Omega(2)$ substratum. Based on a theorem of V. Fock on the mass tensor of a fluid, the contravariant energy-momentum tensors of the $\Omega(1)$ and $\Omega(2)$ substratums are established. Quasi-static solutions of the gravitational field equations in vacuum are obtained. Based on an assumption, relationships between the contravariant energy-momentum tensor of the $\Omega(1)$ and $\Omega(2)$ substratums and the contravariant metric tensor are obtained. Thus, the cosmological constant is calculated theoretically. The $\Omega(1)$ and $\Omega(2)$ substratums may be a possible candidate of the dark energy. The zero-point energy of electromagnetic fields will not appear as a source term in the Einstein's equations. The cosmological constant problem is one of the puzzles in physics. Some people believed that all kinds of energies should appear as source terms in the Einstein's equations. It may be this belief that leads to the cosmological constant problem. The mass density of the $\Omega(1)$ and $\Omega(2)$ substratums is equivalent to that $31.33195$ protons contained in a box with a volume of $1.0 {m}^{3}$.

scholarly journals Coupling of quantum gravitational field with Riemann and Ricci curvature tensors

2021 ◽  
Vol 81 (6) ◽  
Author(s):  
Claudio Cremaschini ◽  
Massimo Tessarotto
Keyword(s):  
Gravitational Field ◽  
Harmonic Oscillator ◽  
Variational Principles ◽  
Theoretical Problem ◽  
Mathematical Framework ◽  
Variational Theory ◽  
Field Equations ◽  
Covariant Quantum ◽  
Desitter Space ◽  
Curvature Tensors

AbstractThe theoretical problem of establishing the coupling properties existing between the classical and quantum gravitational field with the Ricci and Riemann curvature tensors of General Relativity is addressed. The mathematical framework is provided by synchronous Hamilton variational principles and the validity of classical and quantum canonical Hamiltonian structures for the gravitational field dynamics. It is shown that, for the classical variational theory, manifestly-covariant Hamiltonian functions expressed by either the Ricci or Riemann tensors are both admitted, which yield the correct form of Einstein field equations. On the other hand, the corresponding realization of manifestly-covariant quantum gravity theories is not equivalent. The requirement imposed is that the Hamiltonian potential should represent a positive-definite quadratic form when performing a quadratic expansion around the equilibrium solution. This condition in fact warrants the existence of positive eigenvalues of the quantum Hamiltonian in the harmonic-oscillator representation, to be related to the graviton mass. Accordingly, it is shown that in the background of the deSitter space-time, only the Ricci tensor coupling is physically admitted. In contrast, the coupling of quantum gravitational field with the Riemann tensor generally prevents the possibility of achieving a Hamiltonian potential appropriate for the implementation of the quantum harmonic-oscillator solution.

scholarly journals Quantum-Gravity Screening Effect of the Cosmological Constant in the DeSitter Space–Time

Symmetry ◽  
2020 ◽  
Vol 12 (4) ◽  
pp. 531 ◽  
Author(s):  
Claudio Cremaschini ◽  
Massimo Tessarotto
Keyword(s):  
Quantum Gravity ◽  
Cosmological Constant ◽  
Space Time ◽  
Energy Tensor ◽  
Einstein Field Equations ◽  
Field Equations ◽  
Metric Tensor ◽  
Screening Effect ◽  
Periodic Perturbations ◽  
Desitter Space

Small-amplitude quantum-gravity periodic perturbations of the metric tensor, occurring in sequences of phase-shifted oscillations, are investigated for vacuum conditions and in the context of the manifestly-covariant theory of quantum gravity. The theoretical background is provided by the Hamiltonian representation of the quantum hydrodynamic equations yielding, in turn, quantum modifications of the Einstein field equations. It is shown that in the case of the DeSitter space–time sequences of small-size periodic perturbations with prescribed frequency are actually permitted, each one with its characteristic initial phase. The same perturbations give rise to non-linear modifications of the Einstein field equations in terms of a suitable stochastic-averaged and divergence-free quantum stress-energy tensor. As a result, a quantum-driven screening effect arises which is shown to affect the magnitude of the cosmological constant. Observable features on the DeSitter space–time solution and on the graviton mass estimate are pointed out.

scholarly journals A Theoretical Calculation of the Cosmological Constant Based on the Theory of Vacuum Mechanics

2019 ◽  
Author(s):  
Xiao-Song Wang
Keyword(s):  
Cosmological Constant ◽  
Energy Momentum Tensor ◽  
Mass Density ◽  
Zero Point ◽  
Momentum Tensor ◽  
Einstein’S Equations ◽  
Field Equations ◽  
Metric Tensor ◽  
Point Energy ◽  
Einstein's Equations

Lord Kelvin believes that the electromagnetic aether must also generate gravity. Based on a theorem of V. Fock on the mass tensor of an elastic continuum, the contravariant energy-momentum tensor of the $\Omega(1)$ substratum is established. Quasi-static solutions of the gravitational field equations in vacuum are obtained. Based on an assumption, relationships between the contravariant energy-momentum tensor of the $\Omega(1)$ substratum and the contravariant metric tensor are obtained. Thus, the cosmological constant is derived theoretically. The $\Omega(1)$ substratum, or we say the electromagnetic aether, may be a possible candidate of the dark energy. The zero-point energy of electromagnetic fields will not appear as a source term in the generalized Einstein's equations. Some people believed that all kinds of energies should appear as source terms in the Einstein's equations. It may be this belief that leads to the so called cosmological constant problem. The mass density of the electromagnetic aether is equivalent to that $31.33195$ protons contained in a box with a volume of $1.0{m}^{3}$.

Cosmological constant Λ in f(R,T) modified gravity

2016 ◽  
Vol 13 (05) ◽  
pp. 1650058 ◽  
Author(s):  
Gyan Prakash Singh ◽  
Binaya Kumar Bishi ◽  
Pradyumn Kumar Sahoo
Keyword(s):  
Cosmological Model ◽  
Cosmological Constant ◽  
Modified Gravity ◽  
Bianchi Type ◽  
Field Equations ◽  
Type Iii ◽  
Expansion Scalar ◽  
Modified Theory Of Gravity ◽  
Shear Scalar ◽  
Modified Theory

In this paper, we have studied the Bianchi type-III cosmological model in the presence of cosmological constant in the context of [Formula: see text] modified theory of gravity. Here, we have discussed two classes of [Formula: see text] gravity, i.e. [Formula: see text] and [Formula: see text]. In both classes, the modified field equations are solved by the relation expansion scalar [Formula: see text] that is proportional to shear scalar [Formula: see text] which gives [Formula: see text], where [Formula: see text] and [Formula: see text] are metric potentials. Also we have discussed some physical and kinematical properties of the models.

On a non-symmetric theory of the pure gravitational field

1958 ◽  
Vol 54 (1) ◽  
pp. 72-80 ◽  
Author(s):  
D. W. Sciama
Keyword(s):  
Gravitational Field ◽  
Energy Momentum Tensor ◽  
Equations Of Motion ◽  
Rest Mass ◽  
Symmetric Tensor ◽  
Momentum Tensor ◽  
Unified Field Theory ◽  
Field Equations ◽  
Metric Tensor ◽  
Unified Field

ABSTRACTIt is suggested, on heuristic grounds, that the energy-momentum tensor of a material field with non-zero spin and non-zero rest-mass should be non-symmetric. The usual relationship between energy-momentum tensor and gravitational potential then implies that the latter should also be a non-symmetric tensor. This suggestion has nothing to do with unified field theory; it is concerned with the pure gravitational field.A theory of gravitation based on a non-symmetric potential is developed. Field equations are derived, and a study is made of Rosenfeld identities, Bianchi identities, angular momentum and the equations of motion of test particles. These latter equations represent the geodesics of a Riemannian space whose contravariant metric tensor is gij–, in agreement with a result of Lichnerowicz(9) on the bicharacteristics of the Einstein–Schrödinger field equations.

scholarly journals TRAVERSABLE WORMHOLES IN (2+1) AND (3+1) DIMENSIONS WITH A COSMOLOGICAL CONSTANT

1995 ◽  
Vol 04 (02) ◽  
pp. 231-245 ◽  
Author(s):  
M.S.R. DELGATY ◽  
R.B. MANN
Keyword(s):  
Energy Density ◽  
Cosmological Constant ◽  
Positive Energy ◽  
Mass Energy ◽  
Field Equations ◽  
Dimensional Solution ◽  
Partial Stability ◽  
Traversable Wormholes ◽  
Spacetime Curvature ◽  
Radial Tension

Macroscopic traversable wormhole solutions to Einstein’s field equations in (2+1) and (3+1) dimensions with a cosmological constant are investigated. Ensuring traversability severely constrains the material used to generate the wormhole’s spacetime curvature. Although the presence of a cosmological constant modifies to some extent the type of matter permitted [for example it is possible to have a positive energy density for the material threading the throat of the wormhole in (2+1) dimensions], the material must still be “exotic,” that is matter with a larger radial tension than total mass-energy density multiplied by c2. Two specific solutions are applied to the general cases and a partial stability analysis of a (2+1) dimensional solution is explored.

Unified Field Theory

1950 ◽  
Vol 2 ◽  
pp. 427-439 ◽  
Author(s):  
Max Wyman
Keyword(s):  
Field Theory ◽  
Variational Principle ◽  
Electromagnetic Fields ◽  
Linear Connection ◽  
Hamiltonian Function ◽  
Unified Theory ◽  
Unified Field Theory ◽  
Field Equations ◽  
Unified Field

Introduction. In a recent unified theory originated by Einstein and Straus [l], the gravitational and electromagnetic fields are represented by a single nonsymmetric tensor gy which is a function of four coordinates xr(r = 1, 2, 3, 4). In addition a non-symmetric linear connection Γjki is assumed for the space and a Hamiltonian function is defined in terms of gij and Γjki. By means of a variational principle in which the gij and Γjki are allowed to vary independently the field equations are obtained and can be written(0.1)(0.2)(0.3)(0.4)

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