Exact mathematical Formula that connect 6 dimensionless physical constants

In this paper are a new formula for the Planck length ℓpℓ and a new formula for the Avogadro number NA. Also 9 Mathematical formulas that connect dimensionless physical constants. The 6 dimensionless physical constants are the Proton to Electron Mass Ratio μ,the Fine-structure constant α,the ratio Ν1 of electric force to gravitational force between electron and proton,the Avogadro number NA,the Gravitational coupling constant αG for the electron and the gravitational coupling constant αG(p) of proton.

Dynamical analysis of Brans–Dicke universe with inverse power-law effective potential

We study Brans–Dicke cosmology with an inverse power-law effective potential. By using dynamical analyses, we search for fixed points corresponding to the radiation-like matter and dark energy-dominated era of our Universe, and the stability of fixed points is also investigated. We find phase space trajectories which are attracted to the stable point of the dark energy-dominated era from unstable fixed points like matter-dominated era of the Universe. The dark energy comes from effective potentials of the Brans–Dicke field, whose variation (related to the time-variation of the gravitational coupling constant) is shown to be in good agreement with observational data.

Holographic subregion complexity of a (1+1)-dimensional $p$-wave superconductor

We analyze the holographic subregion complexity in a three-dimensional black hole with vector hair. This three-dimensional black hole is dual to a (1+1)-dimensional $p$-wave superconductor. We probe the black hole by changing the size of the interval and by fixing $q$ or $T$. We show that the universal part is finite across the superconductor phase transition and has competitive behaviors different from the finite part of the entanglement entropy. The behavior of the subregion complexity depends on the gravitational coupling constant divided by the gauge coupling constant. When this ratio is less than the critical value, the subregion complexity increases as temperature becomes low. This behavior is similar to that of the holographic (1+1)-dimensional $s$-wave superconductor [M. K. Zangeneh, Y. C. Ong, and B. Wang, Phys. Lett. B 771, 130 (2014)]. When the ratio is larger than the critical value, the subregion complexity has a non-monotonic behavior as a function of $q$ or $T$. We also find a discontinuous jump of the subregion complexity as a function of the size of the interval. The subregion complexity has a maximum when it wraps almost the entire spatial circle. Due to competitive behaviors between the normal and condensed phases, the universal term in the condensed phase becomes even smaller than that of the normal phase by probing the black hole horizon at a large interval. This implies that the condensate formed decreases the subregion complexity as in the case of the entanglement entropy.

4-Index theory of gravity and its relation with the violation of the energy–momentum conservation law

Recently, a 4-index generalization of the Einstein theory has been proposed by Moulin [F. Moulin, Eur. Phys. J. C 77, 878 (2017)]. Using this method, we find the most general 2-index field equations derivable from the Einstein–Hilbert action. The application of Newtonian limit, the role of gravitational coupling constant and the effects of the properties of ordinary energy–momentum tensor in obtaining a 4-index gravity theory have been studied. We also address the results of building Weyl free 4-index gravity theory. Our study displays that both the Einstein and Rastall theories can be obtained as the subclasses of a 4-index gravity theory which shows the power of 4-index method in unifying various gravitational theories. It is also obtained that the violation of the energy–momentum conservation law may be allowed in 4-index gravity theory, and moreover, the contraction of 4-index theory generally admits a non-minimal coupling between geometry and matter field in the Rastall way. This study also shows that, unlike the Einstein case, the gravitational coupling constant of 4-index Rastall theory generally differs from that of the ordinary 2-index Rastall theory.

What if Newton’s Gravitational Constant Was Negative?

In this work, we seek a cosmological mechanism that may define the sign of the effective gravitational coupling constant, G. To this end, we consider general scalar-tensor gravity theories as they provide the field theory natural framework for the variation of the gravitational coupling. We find that models with a quadratic potential naturally stabilize the value of G into the positive branch of the evolution and further, that de Sitter inflation and a relaxation to General Relativity is easily attained.

Nonstandard fractional exponential Lagrangians, fractional geodesic equation, complex general relativity, and discrete gravity

Nonstandard Lagrangians are generating functions of different equations of motion. They have gained increasing importance in many different fields. In fact, nonstandard Lagrangians date back to 1978, when Arnold entitled them “non-natural” in his classic book, Mathematical Methods of Classical Mechanics (Springer, New York. 1978). In applied mathematics, most dynamical equations can be obtained by using generating Lagrangian functions (e.g., power-law and exponential Lagrangians), which has been shown by mathematicians, who have also demonstrated that there is an infinite number of such functions. Besides this interesting field, the topic of fractional calculus of variations has gained growing importance because of its wide application in different fields of science. In this paper, we generalize the fractional actionlike variational approach for the case of a nonstandard exponential Lagrangian. To appreciate this new approach, we explore some of its main consequences in Einstein’s general relativity. Some results are revealed and discussed accordingly mainly the transition from general relativity to complex relativity and emergence of a discrete gravitational coupling constant.

NEWTON'S GRAVITATIONAL COUPLING CONSTANT FROM A QUANTUM OF AREA

A previous calculation of Newton's gravitational coupling constant G is generalized. This generalization makes it possible to have "atoms of two-dimensional space" with an integer dimension d atom of the internal space, where the case d atom = 1 is found to be excluded. Given the quantum of area l2, the final formula for G is inversely proportional to the logarithm of the integer d atom . The generalization used may be interpreted as a modification of the energy equipartition law of the microscopic degrees of freedom responsible for gravity, suggesting some form of long-range interaction between these degrees of freedom themselves.

IS GRAVITY AN INTRINSICALLY QUANTUM PHENOMENON? DYNAMICS OF GRAVITY FROM THE ENTROPY OF SPACE–TIME AND THE PRINCIPLE OF EQUIVALENCE

The two surprising features of gravity are (a) the principle of equivalence and (b) the connection between gravity and thermodynamics. Using principle of equivalence and special relativity in the local inertial frame, one could obtain the insight that gravity must possess a geometrical description. We show that, using the same principle of equivalence, special relativity and quantum theory in the local Rindler frame one can obtain the Einstein–Hilbert action functional for gravity and thus the dynamics of the space–time. This approach, which essentially involves postulating that the horizon area must be proportional to the entropy, uses the local Rindler frame as a natural extension of the local inertial frame and leads to the interpretation that the gravitational action represents the free energy of the space–time geometry. As an aside, one also obtains a natural explanation as to: (i) why the covariant action for gravity contains second derivatives of the metric tensor and (ii) why the gravitational coupling constant is positive. The analysis suggests that gravity is intrinsically holographic and even intrinsically quantum mechanical.

FINITE GRAND UNIFIED THEORIES AND INFLATION

A class of finite GUT's in curved space–time is considered in connection with the cosmological inflation scenario. It is confirmed that the use of the running scalar-gravitational coupling constant in these models helps realizing a successful chaotic inflation. The analyses are made for some different sets of the models.