Action Principle for Scale Invariance and Applications (Part I)

Abstract

On the basis of a general action principle, we revisit the scale invariant field equation using the cotensor relations by Dirac (1973). This action principle also leads to an expression for the scale factor λ, which corresponds to the one derived from the gauging condition, which assumes that a macroscopic empty space is scale-invariant, homogeneous, and isotropic. These results strengthen the basis of the scale-invariant vacuum (SIV) paradigm. From the field and geodesic equations, we derive, in current time units (years, seconds), the Newton-like equation, the equations of the two-body problem, and its secular variations. In a two-body system, orbits very slightly expand, while the orbital velocity keeps constant during expansion. Interestingly enough, Kepler’s third law is a remarkable scale-invariant property.

The Scale-Invariant Vacuum Paradigm: Main Results and Current Progress Review (Part II)

“…The sections are about MOND as a peculiar case of the SIV theory [4], local dynamical effects within the SIV as pertained to the lunar recession [7], and our latest study of the Big Bang Nucleosynthesis (BBNS) within the SIV paradigm [8], along with an Appendix A on the error by Banik and Kroupa concerning the Lunar Motion in the SIV theory. While the present paper is very much influenced by our previous 2022 paper [1], as noted in the title note above, its extra labeling of (Part II) refers to its original intent to be a sequel to the paper "Action Principle for Scale Invariance and Applications (Part I)" published in the first part of the special issue "Nature and Origin of Dark Matter and Dark Energy" [9].…”

“…In Section 2.1, we present a concise review of the mathematical framework encompassing Weyl integrable geometry, Dirac co-calculus, and reparametrization invariance. Instead of re-deriving the weak-field SIV results [9], the concept of reparametrization invariance [10] is employed to illustrate the corresponding equations of motion. The relevant discussion on reparametrization invariance is provided in Section 2.2, which addresses the implications of venturing beyond Einstein's general relativity.…”

“…Here, our approach is more traditional, physically motivated and with as little general abstraction as possible. For more mathematical details, we refer the reader to the companion paper on the "Action Principle for Scale Invariance and Applications (Part I)" [9].…”

“…The scale-invariant cosmology equations, first introduced by Dirac in 1973 [19] and re-derived by Canuto in 1977 [20], are based on the corresponding expressions of the Ricci tensor and a relevant extension of the Einstein equations. We have recently revisited the topic to bring it into focus and aligned the SIV paradigm [9]. In what follows, we paint a broad-stroke picture of the equations and their consequences.…”

“…[38][39][40], in an attempt to account for different properties and to generalize them. Here, however, the main point we wanted to emphasize is that the SIV, which rests on a unique and well-defined basis [9], gives an equation of motion ( 30) which, at gravities lower than a 0 , provides results, for which MOND gives a valid approximation, however only over a limited range. However, the theory and the basic equations are different, despite their tangent behaviors over some domain of gravities.…”