Gurzadyan theorem

In cosmology, the Gurzadyan theorem, proved by Vahe Gurzadyan,[1] states the most general functional form for the force satisfying the condition of identity of the gravity of the sphere and of a point mass located in the sphere's center. This theorem thus refers to the first statement of Isaac Newton’s [2] shell theorem (the identity mentioned above) but not the second one, namely, the absence of gravitational force inside a shell.[3]

The theorem had entered and its importance for cosmology outlined in several papers [4][5] as well as in shell theorem.

The formula and the cosmological constant

The formula for the force derived in [1] has the form

${\displaystyle F=-{\frac {GMm}{r^{2}}}+{\frac {\Lambda c^{2}mr}{3}},}$

where ${\displaystyle G}$ and ${\displaystyle \Lambda }$ are constants. The first term is the familiar law of universal gravitation, the second one corresponds to the cosmological constant term in general relativity and McCrea-Milne cosmology.[6] Then the field is force-free only in the center of a shell but the confinement (oscillator) term does not change the initial ${\displaystyle O(4)}$ symmetry of the Newtonian field. Also, this field corresponds to the only field possessing the property of the Newtonian one: the closing of orbits at any negative value of energy, i.e. the coincidence of the period of variation of the value of the radius vector with that of its revolution by ${\displaystyle 2\pi }$ (resonance principle) .

Consequences: cosmological constant as a physical constant

Einstein named the cosmological constant as a universal constant, introducing it to define the static cosmological model.[7][8] Einstein has stated:[9] “I should have initially set ${\displaystyle \lambda =0}$ in Newton's sense. But the new considerations speak for a non-zero ${\displaystyle \lambda }$, which strives to bring about a non-zero mean density ${\displaystyle \rho _{0}}$ of matter.” This theorem solves that contradiction between “non-zero ${\displaystyle \lambda }$” and Newton's law.

From this theorem the cosmological constant ${\displaystyle \Lambda }$ emerges as additional constant of gravity along with the Newton's gravitational constant ${\displaystyle G}$. Then, the cosmological constant is dimension independent and matter-uncoupled and hence can be considered even more universal than Newton's gravitational constant.[10]

For ${\displaystyle \Lambda }$ joining the set of fundamental constants ${\displaystyle (G,c,\hbar )}$, the gravitational Newton's constant, the speed of light and the Planck constant, yields

${\displaystyle [c]=LT^{-1},\quad [G]=M^{-1}L^{3}T^{-2},\quad [\hbar ]=ML^{2}T^{-1},\quad [\Lambda ]=L^{-2},}$

and a dimensionless quantity emerges for the 4-constant set ${\displaystyle (G,\Lambda ,c,\hbar )}$ [11]

${\displaystyle I={\frac {c^{3a}}{\Lambda ^{a}G^{a}\hbar ^{a}}},}$

where ${\displaystyle a}$ is a real number. Note, no dimensionless quantity is possible to construct from the 3 constants ${\displaystyle G,c,\hbar }$.

This within a numerical factor, ${\displaystyle a=1}$, coincides with the information (or entropy) of de Sitter event horizon [12]

${\displaystyle I_{dS}=3\pi {\frac {c^{3}}{\Lambda G\hbar }},}$

and the Bekenstein Bound [13]

${\displaystyle I_{BB}={\frac {3\pi c^{3}}{\Lambda G\hbar ln2}}.}$

Rescaling of physical constants

Within the Conformal Cyclic Cosmology [14][15] this theorem implies that, in each aeon of an initial value of ${\displaystyle \Lambda }$, the values of the 3 physical constants will be eligible for rescaling fulfilling the dimensionless ratio of invariants with respect to the conformal transformation [11]

${\displaystyle {\tilde {g}}_{\mu \nu }=\Omega ^{2}g_{\mu \nu },}$

Then the ratio yields

${\displaystyle {\frac {Q_{dS}}{Q_{p}}}=m({\frac {c^{3}}{\hbar G\Lambda }})^{n}=mI^{n},\quad m,n\in \mathbb {R} ,}$

for all physical quantities in Planck (initial) and de Sitter (final) eras of the aeons, remaining invariant under conformal transformations.

Inhomogeneous Fredholm equation

This theorem, in the context of nonlocal effects in a system of gravitating particles, leads to the inhomogeneous Dirichlet boundary problem for the Poisson equation [16]

${\displaystyle \Delta \Phi ({\bf {x}})=ANG_{3}S_{3}^{2}{\bigg (}\int _{y\in [0,\infty ]}\exp {\big (}-y^{2}/(2\theta ){\big )}y^{2}dy{\bigg )}\cdot \exp(-\Phi /\theta )-{\frac {c^{2}\Lambda }{2}},}$

where ${\displaystyle R_{\Omega }}$ is the radius of the region, ${\displaystyle A,\theta ,R_{\Omega }\in {\mathbb {R} }^{1}}$. Its solution can be expressed in terms of the double layer potential, which leads to an inhomogeneous nonlinear Hammerstein integral equation for the gravitational potential ${\displaystyle U({\bf {x}})={\widetilde {\lambda }}{\widehat {\mathfrak {G}}}({U})+\alpha (\theta ,\Lambda ){\bf {x}}^{2},~~{\widehat {\mathfrak {G}}}({U})\equiv \int _{\Omega '}{\mathcal {K}}(|{\bf {x}}-{\bf {x}}'|)\exp {\big (}-{U}({\bf {x}}'){\big )}d{{\bf {x}}'},}$

${\displaystyle U({\bf {x}})\equiv (\Phi ({\bf {x}})-C_{0})/\theta ,~~{\widetilde {\lambda }}\equiv {\widetilde {\lambda _{II}}}(\theta )\equiv {\frac {\lambda _{I}}{\theta }}\exp(-C_{0}/\theta ),~~\alpha (\theta ,\Lambda )=-{\frac {\Lambda c^{2}}{12\theta }}.}$

This leads to a linear inhomogeneous 2nd kind Fredholm equation

${\displaystyle \phi ({\bf {x}})=\lambda ^{(0)}\int _{\Omega '}{\mathcal {K}}(|{\bf {x}}-{\bf {x}}'|)\phi ({\bf {x}}')d{\bf {x}}'+{\widehat {\beta }}({\bf {x}}),}$

${\displaystyle {\widehat {\beta }}({\bf {x}})\equiv -\lambda ^{(0)}\int _{\Omega '}{\mathcal {K}}(|{\bf {x}}-{\bf {x}}'|)\alpha |{\bf {x}}'|^{2}d{\bf {x}}'-\alpha |{\bf {x}}|^{2},}$

${\displaystyle {U}({\bf {x}})={U}_{0}-\phi ({\bf {x}}),~~|{\phi }|\ll {U}_{0};~~~\lambda ^{(0)}\equiv -{\widetilde {\lambda }}\exp(-U_{0}).}$

Its solution can be expressed in terms of the resolvent ${\displaystyle \Gamma }$ of the integral kernel and the non-linear (repulsive) term ${\displaystyle \phi ({\bf {x}})=-{\widehat {\beta }}({\bf {x}})+\lambda ^{(0)}\sum _{\bf {n}}\langle -{\widehat {\beta }}({\bf {x}}),\phi _{\bf {n}}\rangle \phi _{\bf {n}}\lambda _{\bf {n}}^{-1}+(\lambda ^{(0)})^{2}\sum _{\bf {n}}\langle -{\widehat {\beta }}({\bf {x}}),\phi _{\bf {n}}\rangle \phi _{\bf {n}}\lambda _{\bf {n}}^{-1}(\lambda _{\bf {n}}-\lambda ^{(0)})^{-1}=}$

${\displaystyle -{\widehat {\beta }}({\bf {x}})+\lambda ^{(0)}\int _{\Omega }\underbrace {{\bigg (}{\mathcal {K}}({\bf {x}},{\bf {x}}')+\lambda ^{(0)}\sum _{\bf {n}}\phi _{\bf {n}}({\bf {x}})\phi _{\bf {n}}({\bf {x}}')\lambda _{\bf {n}}^{-1}(\lambda _{\bf {n}}-\lambda ^{(0)})^{-1}{\bigg )}} _{\Gamma ({\bf {x}},{\bf {x}}',\lambda ^{(0)})}(-{\widehat {\beta }}({\bf {x}}))d{\bf {x}}'.}$

Observational indications

The dynamics of groups and clusters of galaxies are claimed to fit the theorem,[10][17] see also.[18] The possibility of two Hubble flows, a local one, determined by that formula, and a global one, described by Friedmannian cosmological equations was stated in.[19]

References

1. Gurzadyan, Vahe (1985). "The cosmological constant in McCrea-Milne cosmological scheme". The Observatory. 105: 42–43. Bibcode:1985Obs...105...42G.
2. Newton, Isaac (1687). Philosophiae Naturalis Principia Mathematica. London. pp. Theorem XXXI.
3. Markoutsakis, M. (2021). Geometry, Symmetries, and Classical Physics: A Mosaic. CRC Press. ISBN 978-0367535230.
4. Vedenyapin, V.V.; Fimin, N.N.; Chechetkin, V.M. (2021). "The generalized Friedmann model as a self-similar solution of Vlasov–Poisson equation system". European Physical Journal Plus. 136 (6): 670. Bibcode:2021EPJP..136..670V. doi:10.1140/epjp/s13360-021-01659-7.
5. Chardin, G.; Debois, Y.; Manfredi, G.; Miller, B.; Stahl, C. (2021). "MOND-like behavior in the Dirac–Milne universe: Flat rotation curves and mass versus velocity relations in galaxies and clusters". Astronomy and Astrophysics. 652: 16. arXiv:2102.08834. Bibcode:2021A&A...652A..91C. doi:10.1051/0004-6361/202140575.
6. McCrea, W.H.; Milne, E.A. (1934). "Newtonian Universes and the curvature of space". Quarterly Journal of Mathematics. 5: 73. doi:10.1093/qmath/os-5.1.73.
7. Einstein, A. (1917). "Kosmologische Betrachtungen zur allgemeinen Relativitätstheorie". Koniglich Preussische Akad. Der Wissenschaften. 142.
8. Einstein, A. (1918). "Bemerkung zu Herrn Schrödingers Notiz "Über ein Lösungssystem der allgemein kovarianten Gravitationsgleichungen", 3 Mär 1918". Phys. Z. 19: 165.
9. Einstein, A. (1917). "Letter to Michele Besso". The Collected Papers of Albert Einstein. 8 (308). Princeton University Press, 1998: 404.
10. Gurzadyan, V.G.; Stepanian, A. (2018). "Two fundamental constants of gravity unifying the dark matter and the dark energy". European Physical Journal C. 78: 632. arXiv:1807.11705. doi:10.1140/epjc/s10052-018-6117-1.
11. Gurzadyan, V.G.; Stepanian, A. (2019). "Cosmological constant as a fundamental constant". European Physical Journal Plus. 134 (3): 98. arXiv:1901.10299. doi:10.1140/epjp/i2019-12532-3.
12. Gibbons, G.W.; Hawking, S.W. (1977). "Cosmological event horizons, thermodynamics, and particle creation". Phys. Rev. D. 15 (10): 2738. Bibcode:1977PhRvD..15.2738G. doi:10.1103/PhysRevD.15.2738.
13. Bekenstein, J.D. (1981). "Universal upper bound on the entropy-to-energy ratio for bounded systems". Phys. Rev. D. 23 (2): 287. Bibcode:1981PhRvD..23..287B. doi:10.1103/PhysRevD.23.287.
14. Penrose, R. (2010). "Cycles of Time: An Extraordinary New View of the Universe". Bodley Head, London. Bibcode:2010cten.book.....P.
15. Gurzadyan, V.G.; Penrose, R. (2013). "On CCC-predicted concentric low-variance circles in the CMB sky". European Physical Journal Plus. 128 (2): 22. arXiv:1302.5162. Bibcode:2013EPJP..128...22G. doi:10.1140/epjp/i2013-13022-4. S2CID 55249027.
16. Gurzadyan, V.G.; Fimin, N.N.; Chechetkin, V.M. (2023). "Cosmic voids and the kinetic analysis.II. Link to Hubble tension". Astronomy and Astrophysics. 672: A95. arXiv:2303.03194. Bibcode:2023A&A...672A..95G. doi:10.1051/0004-6361/202346139.
17. Gurzadyan, V.G. (2019). "On the common nature of dark matter and dark energy: galaxy groups". European Physical Journal Plus. 134 (1): 14. arXiv:1712.10014. Bibcode:2019EPJP..134...14G. doi:10.1140/epjp/i2019-12418-4.
18. Chardin, G.; et al. (2021). "MOND-like behavior in the Dirac-Milne universe -- Flat rotation curves and mass/velocity relations in galaxies and clusters". Astronomy and Astrophysics. 652: 91. arXiv:2102.08834. Bibcode:2021A&A...652A..91C. doi:10.1051/0004-6361/202140575.
19. Gurzadyan, V.G.; Stepanian, A. (2021). "Hubble tension and absolute constraints on the local Hubble parameter". Astronomy and Astrophysics. 653: A145. arXiv:2108.07407. Bibcode:2021A&A...653A.145G. doi:10.1051/0004-6361/202141736.