Weyl transformation: A dynamical degree of freedom in the light of Dirac’s Large Number hypothesis

In Einstein’s Field Equation (EFE), the geometry of the spacetime is connected with the matter distribution. The geometry or the gravitational sector deals with classical macroscopic objects involving gravitational units while the matter sector can be better described by quantum theory involving atomic units. It has been argued by Bisabr [ arXiv:gr-qc/1904.09336 ] that there exists an epoch-dependent conversion factor between these two unit systems present in two different conformal frames, i.e. the conformal factor is epoch-dependent. We argue that the conformal transformation (CT) is a dynamical degree of freedom describing it’s possible relevance in inflation in context to the graceful exit problem, dynamics of the cosmological constant [Formula: see text] and justify the argument in the light of consequences of Dirac’s Large Number hypothesis (LNH).

Dirac's large number hypothesis: A journey from concept to implication

Large dimensionless numbers, arising out of ratios of various physical constants, intrigued many scientists, especially Dirac. Relying on the coincidence of large numbers, Dirac arrived at the revolutionary hypothesis that the gravitational constant [Formula: see text] should vary inversely at the cosmic time [Formula: see text]. This hypothesis of Dirac, known as the Large Number Hypothesis (LNH), sparked off many speculations, arguments and new ideas in terms of applications. Works done by several authors with LNH as their basic platform are extensively reviewed in this work. Relationship between some of those works are pointed out here elaborately. Possibility of time variations of physical constants other than [Formula: see text] as well as large numbers in various realm of physical and biological sciences are also discussed.

NEW LARGE NUMBER HYPOTHESES AND GEOMETRICAL RELATIONS ON SUBSTRUCTURES OF THE UNIVERSE

In analogy to Dirac’s large number hypothesis (LNH), we propose the “less large number hypothesis (LLNH)” of NG~NS where NG and NS are the total number of galaxies in the universe and the average total number of stars in a galaxy since both of them are of the order of 1011. Also suggested are the two “geometrical LLNH relations” of NG~(RU/RG)2 and NS~RG/RS where RU, RG and RS are the “radius of the universe”, the average radius of a galaxy and that of a star respectively. Furthermore, it is pointed out that the “even less large number hypothesis (ELLNH)” of n1~n2~n3 may also work for n1≡RU/DG, n2≡DG/RG and n3≡RG/DS where DG and DS are the average distance between two neighboring galaxies and that between two neighboring stars since all of them are of the order of 103.