Obtaining a class of conformally flat pure radiation metrics with cosmological constant using invariant operators

The paper considers a spherically symmetric configuration of the gravitational and electromagnetic fields with allowance to the cosmological constant, and its quantization. After dimensional reduction, the original action is transformed to new variables in the R- and T-regions. The exclusion of the non-dynamic degree of freedom from the obtained action leads to an action for the geodesic in the configuration space, which proves to be conformally flat. We use the Gitman–Tyutin formalism for the obtained dynamical system, which Lagrange function is degenerate. After performing a suitable canonical transformation, the constraints found from the Lagrange function are reduced to the canonical form. Herewith the physical part of the Hamilton function vanishes. To construct quantum theory, we introduce additional physical quantities – charge and mass functions. Since Hamilton operator equals zero, it leads to the fact that the desired wave function of the system obeys only the eigenvalue equations for the mass and charge operators. The solution of these equations leads to continuous charge and mass spectra.

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On m-quasi-Einstein spacetimes

International Journal of Geometric Methods in Modern Physics ◽

10.1142/s0219887821500663 ◽

2021 ◽

pp. 2150066

Author(s):

Absos Ali Shaikh ◽

Shyamal Kumar Hui ◽

Akshoy Patra

Keyword(s):

Vector Field ◽

Energy Density ◽

Viscous Fluid ◽

Cosmological Constant ◽

Ricci Tensor ◽

Killing Vector ◽

Killing Vector Field ◽

Conformally Flat ◽

Matter Distribution ◽

Einstein's Equation

In this paper, we have studied [Formula: see text]-quasi-Einstein spacetimes. Some basic results of such spacetimes are derived. Perfect and viscous fluid [Formula: see text]-quasi-Einstein spacetimes are also studied and the expressions of pressure, cosmological constant and energy density are obtained. We have proved that if the generator [Formula: see text] of an [Formula: see text]-quasi-Einstein spacetime is a Killing vector field, then the spacetime is either conformally flat or of Petrov-type [Formula: see text]. It is also shown that if the function [Formula: see text] of an [Formula: see text]-quasi-Einstein spacetime satisfying Einstein’s equation is harmonic and the matter distribution is perfect fluid, then Segre’ characteristics of the Ricci tensor is [(1,1), 1]. Finally, an example is constructed for the proper existence of such a spacetime.

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Interpreting a conformally flat pure radiation spacetime

Classical and Quantum Gravity ◽

10.1088/0264-9381/15/12/015 ◽

1998 ◽

Vol 15 (12) ◽

pp. 3863-3871 ◽

Cited By ~ 8

Author(s):

J B Griffiths ◽

J Podolský

Keyword(s):

Conformally Flat ◽

Pure Radiation

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Stress-energy tensor for parallel plate on background of conformally flat brane-world geometries and cosmological constant problem

The European Physical Journal C ◽

10.1140/epjc/s2004-02040-y ◽

2004 ◽

Vol 38 (3) ◽

pp. 373-378 ◽

Cited By ~ 13

Author(s):

M. R. Setare

Keyword(s):

Cosmological Constant ◽

Parallel Plate ◽

Brane World ◽

Energy Tensor ◽

Conformally Flat ◽

Stress Energy Tensor ◽

Cosmological Constant Problem ◽

Stress Energy

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The global future stability of the FLRW solutions to the Dust-Einstein system with a positive cosmological constant

Journal of Hyperbolic Differential Equations ◽

10.1142/s0219891615500046 ◽

2015 ◽

Vol 12 (01) ◽

pp. 87-188 ◽

Cited By ~ 14

Author(s):

Mahir Hadžić ◽

Jared Speck

Keyword(s):

Cosmological Constant ◽

Evolution Equations ◽

Equations Of State ◽

Energy Estimates ◽

Positive Cosmological Constant ◽

Small Perturbations ◽

Pure Radiation ◽

State Formula ◽

Elliptic Estimates ◽

Dissipative Terms

We study small perturbations of the Friedman–Lemaître–Robertson–Walker (FLRW) solutions to the dust-Einstein system with a positive cosmological constant in the case that the space-like Cauchy hypersurfaces are diffeomorphic to 𝕋3. We show that the FLRW solutions are nonlinearly globally future-stable under small perturbations of their initial data. In our analysis, we construct harmonic-type coordinates such that the cosmological constant results in the presence of dissipative terms in the evolution equations. Our result extends those of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715; C. Lübbe and J. A. Valiente Kroon, A conformal approach for the analysis of the nonlinear stability of pure radiation cosmologies, Ann. Phys. 328 (2013) 1–25], where analogous results were proved for the Euler–Einstein system under the equations of state [Formula: see text], [Formula: see text]. The dust-Einstein system is the case cs = 0. The main difficulty that we overcome here is that the dust's energy density loses one degree of differentiability compared to the cases [Formula: see text], which introduces many obstacles for closing the estimates. To resolve this difficulty, we commute the equations with a well-chosen differential operator and derive elliptic estimates that complement the energy estimates of [I. Rodnianski and J. Speck, The nonlinear future stability of the FLRW family of solutions to the irrotational Euler–Einstein system with a positive cosmological constant, J. Eur. Math. Soc. 15 (2013) 2369–2462; J. Speck, The nonlinear future stability of the FLRW family of solutions to the Euler–Einstein system with a positive cosmological constant, Selecta Math. 18 (2012) 633–715]. Our results apply in particular to small perturbations of the vanishing dust state containing vacuum regions.

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