# Violation of the Dominant Energy Condition in Geometrodynamics

## Abstract

**:**

## 1. Introduction

## 2. Violation of the Isotropic Dominant Energy Condition in Einstein’s Geometrodynamics

## 3. Violation of the Energy-Dominant Condition in the Geometrodynamics with Logunov Constraints

**,**${g}_{\mu \nu}$ is the metric tensor of the effective Riemannian space, $g$ is the determinant of the metric tensor of the effective Riemannian space, ${\gamma}_{\sigma \mu}^{\nu}$ are the Christoffel symbols of Minkowski space where the nonzero components in the spherical coordinates $r,\vartheta ,\psi $ are ${\gamma}_{22}^{1}=-r,{\gamma}_{33}^{1}=-r{\mathrm{sin}}^{2}\left(\vartheta \right),{\gamma}_{33}^{2}=-\mathrm{sin}\left(\vartheta \right)\mathrm{cos}\left(\vartheta \right)$

**,**${\gamma}_{12}^{2}={\gamma}_{13}^{3}=\frac{1}{r},{\gamma}_{23}^{3}=ctg\left(\vartheta \right)$. Then for the Logunov metric, the equations of Logunov’s connections have the form $\frac{\partial}{\partial {x}^{0}}\left[\frac{{a}^{6}}{{N}^{2}}\right]=0,\frac{\partial}{\partial r}\left[{r}^{2}\sqrt{1-k{r}^{2}}\right]=\frac{2r}{\sqrt{1-k{r}^{2}}}$, where it follows that ${N}^{2}={\left(\frac{a}{{a}_{0}}\right)}^{6}$,$k=0$, i.e., ${g}_{00}={\left({g}_{ik}\right)}^{2}$. In the theory of gravity with Logunov’s connections, two components ${g}^{00}$ and ${g}^{ik}$ of the metric tensor are dependent. That is why, with Logunov’s connections, variation of the action leads to Equation (9).

- In this approach, the field equations of the theory are derived from the conditional extremum of the action with constraints ${D}_{\mu}{\tilde{g}}^{\mu \nu}=0$ due to which in «RTG with Logunov constraints» instead of two Friedmann Equations (1) and (2), one Equation (9) arises$$\left(\frac{{a}^{\u2033}}{a}\right)+2{\left(\frac{{a}^{\prime}}{a}\right)}^{2}=4\pi G\left({\epsilon}_{\phi}-{p}_{\phi}\right)$$
- This equation is more general as any solution of Friedman equations is the solution of Equation (9). However, not every solution of Equation (9) is the solution of Equations (1) and (2). Formally, Equation (9) is a linear combination of Equations (1) and (2). For example, for a vacuum-like medium with the state equation ${\epsilon}_{\phi}+{p}_{\phi}=0,{\epsilon}_{\phi}={U}_{0},{p}_{\phi}=-{U}_{0},{U}_{0}0$, the solution of Equation (9) has the form$$a\left(t\right)={a}_{0}\sqrt[3]{\mathrm{cosh}\left(3{H}_{0}t\right)}$$
- Since ${\epsilon}_{\phi}-{p}_{\phi}=U\left(\phi \right)$, Equation (9) shows that evolution of the scaling factor is determined only by the potential energy density, a scalar field, and does not depend on both the kinetic energy density and the rest energy density of the quantum vacuum. Thus, the values of the kinetic energy density and the rest energy density of the quantum vacuum ${\epsilon}_{v}=\frac{1}{4{\pi}^{2}}{\displaystyle \underset{0}{\overset{M}{\int}}{k}^{2}\sqrt{{k}^{2}+{m}^{2}}}dk\approx \frac{{M}^{4}}{16{\pi}^{2}}$ do not influence Universe evolution. We estimate the vacuum energy using Einstain’s formula ${k}_{0}=\sqrt{{k}^{2}+{m}^{2}}$ that does not involve the vacuum energy in the potential form. This explains the fact that astronomical observations and theoretical calculations demonstrate significant difference in the value of the cosmological constant.
- In the case of inhomogeneity of the Newtonian type $\left(\Delta \phi =0\right)$, the metric remains homogeneous.

## 4. Radiation of a Compact Scalar Field Configuration

## 5. Conclusions

## Funding

## Acknowledgments

## Conflicts of Interest

## References

- Friedman, Y. Relativistic Gravitation Based on Symmetry. Symmetry
**2020**, 12, 183. [Google Scholar] [CrossRef] [Green Version] - Friedman, Y.; Scarr, T.; Gootvilig, D.H. The pre-potential of a field propagating with the speed of light and its dual symmetry. Symmetry
**2019**, 11, 1430. [Google Scholar] [CrossRef] [Green Version] - Dymnikova, I.G. Dark energy and space-time symmetry. Universe
**2017**, 3, 20. [Google Scholar] [CrossRef] [Green Version] - Dymnikova, I.G.; Dobosz, A.; Soltysek, B. Lemaitre dark energy model singled out by the holographic principle. Gravit. Cosmol.
**2017**, 23, 28. [Google Scholar] [CrossRef] - Lasukov, V.V. An atomic model of the Big-Bang. Rus. Phys. J.
**2012**, 55, 1157. [Google Scholar] [CrossRef] - Lasukov, V.V. The Newton primordial atom in superspace-time. Int. J. Geom. Methods Mod. Phys.
**2016**, 13, 1650020. [Google Scholar] [CrossRef] - Dymnikova, I.G. Regular rotating de Sitter–Kerr black holes and solitons. Class. Quantum Gravity
**2016**, 33, 145010. [Google Scholar] [CrossRef] - Gliner, E.B. Inflation universe and the vacuum-like state of physical medium. Usp. Fiz. Nauk.
**2002**, 172, 221. [Google Scholar] [CrossRef] - Dymnikova, I.G.; Korpusik, M. Thermodynamics of regular cosmological black holes with the de Sitter interior. Entropy
**2011**, 13, 1967–1991. [Google Scholar] [CrossRef] - Rubakov, V.A. A null energy condition and its violation. Usp. Fiz. Nauk.
**2014**, 184, 137. [Google Scholar] [CrossRef] - Misner, C.; Thorne, K.; Wheeler, J. Gravitation (Freeman, San Francisco, 1973); Freeman: San Francisco, CA, USA, 1973; Available online: http://www.berlinet.de/enderlein/Krr (accessed on 30 December 2019).
- Starobinsky, A. New type of isotropic cosmological models without singularity. Phys. Lett. B
**1980**, 91, 99. [Google Scholar] [CrossRef] - Dymnikova, I.G. Vacuum nonsingular black hole. Gen. Relativ. Gravit.
**1992**, 24, 235. [Google Scholar] [CrossRef] - Dymnikova, I.G. The cosmological term as a source of mass. Class. Quantum Gravity
**2002**, 19, 725. [Google Scholar] [CrossRef] [Green Version] - Hartle, J.; Hawking, S. Wave function of the Universe. Phys. Rev.
**1983**, 28, 2960. [Google Scholar] [CrossRef] - Misner, C.W.; Wheele, J.A. Classical physics as geometry. Ann. Phys.
**1957**, 2, 525. [Google Scholar] [CrossRef] - Wheeler, J.A. On the nature of quantum geometrodynamics. Ann. Phys.
**1957**, 2, 604. [Google Scholar] [CrossRef] - Altshuller, B.L.; Barvinskiy, A.O. Quantum cosmology and physics of transitions with a change of the space-time signature. Usp. Fiz. Nauk.
**1996**, 166, 46. [Google Scholar] - Linde, A.D. Elementary Particle Physics and Inflationary Cosmology; Nauka: Moscow, Russia, 1990; Available online: https://arxiv.org/abs/hep-th/0503203 (accessed on 30 December 2019).
- Peebles, P.J.E. Principles of Physical Cosmology; Princeton University Press: London, UK, 1993. [Google Scholar]
- Vilenkin, A. Predictions from quantum Cosmology. Phys. Rev. Lett.
**1995**, 74, 846. [Google Scholar] [CrossRef] [Green Version] - Vilenkin, A.; Ford, L.R. Gravitational effects upon cosmological phase transitions. Phys. Rev. D
**1982**, 26, 1231. [Google Scholar] [CrossRef] - Vilenkin, A. The birth of inflationary Universes. Phys. Rev. D
**1983**, 27, 2848. [Google Scholar] [CrossRef] - Weinberg, S. Anthropic bound on the cosmological constant. Phys. Rev. Lett.
**1987**, 59, 2607. [Google Scholar] [CrossRef] [PubMed] - Gibbons, G.W.; Hawking, S.W. Cosmological event horizons, thermodynamics, and particle creation. Phys. Rev. D
**1977**, 15, 2738. [Google Scholar] [CrossRef] [Green Version] - Linde, A. Chaotic inflation. Phys. Lett. B
**1983**, 129, 177. [Google Scholar] [CrossRef] - Linde, A. New inflationary universe scenario: a possible solution of the horizon, flatness, homogeneity, isotropy and primordial monopole problems. Phys. Lett. B
**1982**, 108, 389. [Google Scholar] [CrossRef] - Witt, B.S. Quantum theory of gravity. I. The canonical theory. Phys. Rev. D
**1967**, 160, 1113. [Google Scholar] - Witt, B.S. Quantum theory of gravity. II. The manifestly covariant theory. Phys. Rev. D
**1967**, 162, 1195. [Google Scholar] - Vilenkin, A. Quantum creation of Universes. Phys. Lett. B
**1982**, 117, 25. [Google Scholar] [CrossRef] - Penrose, R. Gravitational collapse and space-time singularities. Phys. Rev. Lett.
**1965**, 14, 57. [Google Scholar] [CrossRef] [Green Version] - Lasukov, V.V. Scalar-gravitational Hamiltonian in the relativistic gravitation theory with Logunov relations. Rus. Phys. J.
**2013**, 56, 9. [Google Scholar] [CrossRef] - Logunov, A.A.; Mestvirishvili, M.A. Relativistic Theory of Gravitation; Nauka: Moscow, Russia, 1989. [Google Scholar]
- Logunov, A.A. Relativistic Theory of Gravitation; Nauka: Moscow, Russia, 2012. [Google Scholar]
- Gershtein, S.S.; Logunov, A.A.; Mestvirishvili, M.A. Gravitational field self-limitation and its role in the Universe. Usp. Fiz. Nauk.
**2006**, 176, 1207. [Google Scholar] [CrossRef] - Bekenstain, J.D. Black holes and the second law. Nuovo. cim. Lett.
**1972**, 4, 737. [Google Scholar] [CrossRef] - Bekenstain, J.D. Black holes and entropy. Phys. Rev. D
**1973**, 7, 2333. [Google Scholar] [CrossRef] - Bekenstain, J.D. Generalized second law of thermodynamics in black hole physics. Phys. Rev. D
**1974**, 9, 292. [Google Scholar] [CrossRef] [Green Version] - Hawking, S.W. Particle creation by black holes. Math. Phys.
**1975**, 43, 199. [Google Scholar] [CrossRef] - Lasukov, V.V. Quantum-field approach in classical physics and geometrodynamics. Rus. Phys. J.
**2019**, 61, 566. [Google Scholar] [CrossRef] - Lasukov, V.V. Mach’s principle in the atomic formulation and the classical potential vacuum. Rus. Phys. J.
**2014**, 57, 783. [Google Scholar] [CrossRef] - Ginzburg, V.L. On some advances in physics and astronomy over the past three years. Usp. Fiz. Nauk.
**2002**, 172, 213. [Google Scholar] [CrossRef]

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Lasukov, V.
Violation of the Dominant Energy Condition in Geometrodynamics. *Symmetry* **2020**, *12*, 400.
https://doi.org/10.3390/sym12030400

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Lasukov V.
Violation of the Dominant Energy Condition in Geometrodynamics. *Symmetry*. 2020; 12(3):400.
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**Chicago/Turabian Style**

Lasukov, Vladimir.
2020. "Violation of the Dominant Energy Condition in Geometrodynamics" *Symmetry* 12, no. 3: 400.
https://doi.org/10.3390/sym12030400