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 $\lambda$, 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.

1. Introduction

The scale-invariant vacuum (SIV) paradigm aims to respond to a fundamental principle expressed by Dirac [1]: “It appears as one of the fundamental principles in Nature that the equations expressing basic laws should be invariant under the widest possible group of transformations”. Our objective is to explore whether, in addition to Galilean, Lorentz invariance and general covariance, some effects of scale invariance are also present in our low-density Universe. This question is fully justified, since scale invariance is present in Maxwell’s equations and general relativity (GR) in empty space without cosmological constants, charges, and currents.

The presence of matter tends to kill scale invariance [2]. Thus, the question arises of how much matter in the Universe is necessary to suppress scale invariance. Would one single atom in the Universe be enough to kill scale invariance? The question of the quantum particle content and the corresponding conformal energy–momentum tensor that may arise from a conformal scalar field metric transformation was first studied by Parker [3]. A key result of this study was that there is no gravitationally induced particle creation in conformally flat space-times when the mass of the scalar field can be neglected in the conformal wave equation. Here, we develop a study of the SIV theory in two stages, first by imposing an extremum of the variations of the action for the scale factor $\lambda$ to derive the general scale-invariant equations, and second by considering a scale-invariant vacuum state of the Universe as a conformally flat space-time background with zero curvature R and zero associated mass term, to arrive at a specific SIV expression for the scale factor $\lambda \left(t\right)=t_0/t$. The two stages are also consistent with the case of non-zero mass, where the general equations factor into GR equations without a cosmological constant term and equation determining $\lambda$ that absorbs any non-zero cosmological constant term (cf. Section 3). Our study on the causal connexions in models of expanding universes indicates that scale invariance is certainly forbidden in cosmological models with densities above the critical density ${\varrho }_{\mathrm{c}}=3H_0^2/\left(8\pi G\right)$ [4]. This result is in agreement with the equations of scale-invariant cosmological models, since they show the absence of possible expansion solutions for ${\Omega }_{\mathrm{m}}\ge 1$ [5].

For universe models with ${\Omega }_{\mathrm{m}}<1$, the question remains open, since scale-invariant cosmological models do have solutions (rather close to the $\Lambda$CDM). In the range of possible ${\Omega }_{\mathrm{m}}$ between 0 and 1, the higher the density, the smaller the effects from scale invariance. For ${\Omega }_{\mathrm{m}}=0.2$ to 0.3, these limited effects are nevertheless sufficient to drive a significant acceleration of the expansion of the universe. Of course, the observations will decide whether the effects of scale invariance are effectively present or not.

Indeed, there is little hope of convincing theoretical astrophysicists about new developments in gravitation theory if these are not resting on a well-established action principle. In order to try to fulfill this requirement, we revisit the known results and aim at a detailed demonstration of the scale-invariant field equation from an action principle in the line of former developments of cotensor calculus and action by Dirac [1]. For a more modern treatment of the scale-invariant gravity idea see [6], which is based on Cartan’s formalism, with plenty of practice with differential-forms, and along a more traditional abstract scalar field approach, which due to its abstractness seems to have stayed disconnected from observational tests. Here, our approach is more traditional, physically motivated, and with as little general abstraction as possible. The interest in such an undertaking is first to obtain a complete derivation of the scale-invariant field equation by imposing an extremum of the action for small changes $\delta g_{\mu \nu }$. Second, we also explore the consequences of an extremum in the action for small variations $\delta \lambda$ of the scale factor. Interestingly enough, in this case, the action principle leads to a well-defined form of the scale factor $\lambda$, corresponding to the gauging condition [5] based on the statement that the macroscopic empty space should be scale-invariant. This gauging condition replaces the one based on the “large number hypothesis” originally proposed by Dirac and also used by Canuto and his collaborators to express $\lambda$ in the scale-invariant framework [1,7,8].

Several positive results for the SIV theory have already been obtained: on basic cosmological tests [5], on the growth of density fluctuations without the need of dark matter [9], and on the clustering of galaxies and galactic rotation [10]. In particular, MOND [11,12] was shown to be a good approximation of SIV theory when the scale factor is taken as a constant. Such an approximation applies with an accuracy better than 1% for dynamical timescales shorter than about 400 Myr [13], encompassing the typical rotation time of spiral galaxies.

After a brief recall of some basics of the cotensor calculus, Section 2 gives the action principle and a resulting demonstration of the scale-invariant field equations; the detailed steps are given in Appendix A. Section 3 shows that the action principle also leads to an expression of the scale factor identical to that derived from our gauging condition [5]. In Section 4, we examine some consequences of the scale-invariant cosmological equations, Newtonian approximation, the two body-problem, and Kepler’s third law. Some of these properties were previously obtained in a timescale suited for cosmological models; here, we give them in a form more appropriate for observational studies with current time units (years or seconds). Section 5 contains the conclusions.

2. The Scale-Invariant Field Equation

The Einstein field equation of general relativity (GR) can be derived from the extremum of an action $I=I_{\mathrm{G}}+I_{\mathrm{M}}$ containing a gravitational and a matter term [14,15], where

$$I_{\mathrm{G}}=\left[\frac{c^4}{16\pi G}\right]\int \sqrt{-g}R\left(x\right)d^{4}x,$$

(1)

with R being the curvature scalar. Often, the multiplicative constants in the bracket are omitted through choosing an appropriate units system, and we do so below; however, they are needed after Equation (15), when we define the energy–momentum tensor. The stationarity of the action I classically leads to Einstein theory. As usual, the functional integrand is determined up to a total derivative of a smooth function, which often is required to vanish at the integration boundary; that is, often at infinity. Terms in $R^2$, $R^3$, etc. can be added to R. Such extensions lead, in particular, to the family of $f\left(R\right)$ theories [16,17].

2.1. Definitions: Weyl Integrable Geometry and Dirac Co-Tensors

The so-called Weyl integrable geometry (WIG) [1,8,18] is a particular case of Weyl’s geometry (WG) [19], which was initially developed to express electromagnetism using a space-time property. WIG now forms an appropriate framework for studying scale invariance. Weyl’s geometry (WG), just like in GR, is endowed with a quadratic form $d{s}^2=g_{\mu \nu }d{x}^{\mu }d{x}^{\nu }$. In addition, the length ℓ of a vector $a^{\mu }$ is determined by a scale factor $\lambda \left(x^{\mu }\right)$ and this also applies to the line element $ds$,

$${\ell }^2={\lambda }^2\left(x^{\mu }\right)g_{\mu \nu }{a}^{\mu }{a}^{\nu },\mathrm{and}d{s}^{\prime }=\lambda \left(x^{\mu }\right)ds.$$

(2)

This last equation relates the line element $ds$ in WG space to $d{s}^{\prime }$ in another space (here, $ds$ is in a WG space; while $d{s}^{\prime }$ is in Einstein GR space, which is a particular WIG space). Thus, the two considered spaces (GR and WIG) are conformally equivalent using the metric conformal transformation $g_{\mu \nu }^{\prime }={\lambda }^2{g}_{\mu \nu }$. Note that one has to distinguish between scale coordinate transformations, conformal coordinate transformations, and conformal transformations of the metric [3]. Usually, the primed quantities will be in a GR frame.

A key property concerns the transport of a vector from a point $P_1\left(x^{\mu }\right)$ to a nearby point $P_2(x^{\mu }+d{x}^{\mu })$. During such a transport, the length of the vector changes as follows:

$$d\ell =\ell {\kappa }_{\nu }d{x}^{\nu }.$$

(3)

Here, ${\kappa }_{\nu }$ is called the coefficient of metrical connection. In GR, the coefficient of metrical connection is ${\kappa }_{\nu }^{\prime }=0$. The specificity of WIG [8], with respect to the classical WG [19], is that the metrical connection ${\kappa }_{\nu }$ is the gradient of a scalar field $\phi$, namely

$${\kappa }_{\nu }=-{\phi }_{,\nu }\mathrm{with}\phi =ln\lambda ,i.e.,{\kappa }_{\nu }=-\frac{\mathsf{\partial }ln\lambda }{\mathsf{\partial }{x}^{\nu }}.$$

(4)

This implies that ${\mathsf{\partial }}_{\nu }{\kappa }_{\mu }={\mathsf{\partial }}_{\mu }{\kappa }_{\nu }$. Thus, the parallel displacement of a vector along a closed loop in WIG does not change its length; as a consequence, the length change of a vector does not depend on the path followed. Weyl’s original theory was withdrawn due to a criticism by Einstein [20], who pointed out that the properties of atoms would then depend on their past world lines. Thus, the atoms of an element in an electromagnetic field could not show sharp lines.

The general covariance of GR requires tensorial expressions. Scale invariance demands further developments: the cotensors. Many mathematical tools for Weyl’s geometry [19] have been applied and developed by [1,21]. A quantity $Y$, scalar, vector, or tensor, from which one obtains a scale-invariant GR object $Y^{\prime }$, and which upon a scale transformation changes like $Y\to Y^{\prime }={\lambda }^n\left(x\right)Y$, is said to be a coscalar, covector, or cotensor of power $\Pi \left(Y\right)=n$; one speaks of scale covariance. For $n=0$ (scale invariance), one has an inscalar, invector, or intensor. The ordinary derivative $Y_{,\mu }$ and the covariant derivative $Y_{;\mu }$ are not necessarily co-covariant (a definition by Dirac) or invariant, but derivatives with such properties can be defined. As an example, let us perform a derivative and a change of scale for scalar S; using (3) we have

$$\begin{array}{c}\hfill S_{,\mu }^{\prime }={\left({\lambda }^{n}S\right)}_{\mu }={\lambda }^n{S}_{\mu }+n{\lambda }^{n-1}{\lambda }_{\mu }S={\lambda }^n(S_{\mu }-n{\kappa }_{\mu }S).\end{array}$$

(5)

The co-covariant derivative $S_{*\mu }$ of a coscalar S is thus

$$S_{*\mu }=S_{\mu }-n{\kappa }_{\mu }S,$$

(6)

which is a covector of power n (ordinary derivatives are denoted by $S_{,\mu }$ or just by $S_{\mu }$ when there is no obviously possible confusion). Co-covariant derivatives preserve scale covariance with the same power. First and second co-covariant derivatives of vectors and tensors can also be defined. Operations on covectors and cotensors can be performed. A brief summary of cotensor calculus was given by Canuto et al. [8]. Many useful expressions can also be found in Dirac [1], as well as in Bouvier & Maeder [18] for geodesics, isometries, and killing vectors. (We limit the presentation here to what is really needed).

2.2. The Scale-Invariant Action Principle

We note that in current scalar–tensor theories of gravitation, often aimed at studying scale invariance, a new field $\phi$ determined by the specific aims of the proponents is usually introduced in the action, in addition to the choice for $\lambda$; see for example Clifton et al. [22] and Ferreira & Tattersall [23], as well as Parker [3]. This field may be a multiplier of the curvature scalar R [15], of functions and derivatives of $\phi$. In WIG, we identify the field $\phi$ with the conformal scale factor $\lambda$ of Equations (2) and (3), thus $\phi =\lambda$ ($\lambda$ being of power $\Pi \left(\lambda \right)=-1$ since ${\lambda }^{\prime }=1={\lambda }^{-1}\lambda$). Since the conformal scale factor $\lambda$ appears in the covariant derivatives, it is not expected to have a particular particle content in the same sense that the Levi–Civita connection coefficients are not expected to give rise to a particular particle content. Thus, no new field is introduced, and the theory, having no additional degree of freedom, is indeed much more constrained than current scalar–tensor theories. The SIV theory is a particular case of scalar-theories, or rather “a cotensor theory“, being based on cotensoral expressions, satisfying both covariance and scale invariance.

The corresponding curvature scalar ${}^{*}R$ is of power $\Pi (*R)=-2$ (meaning that ${}^{*}R$ behaves like ${\lambda }^2$). In order to make the action $I_{\mathrm{G}}$ an inscalar, we take ${\lambda }^2{}^{*}R$ (power −4) in the expression of the action, since $\sqrt{-g}$ is of power 4. Just like scalar–tensor theories, the action contains derivatives of the field; of course, only terms of quartic power are possible, due to the necessary global invariance. This can lead to two additional possible terms of power $\Pi =-4$ multiplied by constants $c_1$ and $c_2$ in the action (7). The action writes [1],

$$\delta I_{\mathrm{G}}=\delta \int \left(-{\lambda }^2{}^{*}R\left(x\right)+c_1{\lambda }^{*\mu }{\lambda }_{*\mu }+c_2{\lambda }^4\right)\sqrt{-g}{d}^{4}x,$$

(7)

where the curvature scalar ${}^{*}R$ is [1],

$${}^{*}R={}^{*}{R}_{\mu }^{\mu }=R-6{\kappa }_{;\mu }^{\mu }+6{\kappa }^{\mu }{\kappa }_{\mu },$$

(8)

where R refers to the curvature scalar of GR. The symbol “;” expresses the usual covariant derivatives related to $g_{\mu \nu }$. This expression was given in Equation (89.2) by Eddington [21] and later by Parker [3], where one has to map $\Omega$ to $\lambda$. In order to match the expression derived by Parker [3] with the original expression by Dirac [1], one has to use the conformally transformed metric ${\tilde{g}}_{\mu \nu }$ to raise and lower indexes for the * objects on the left hand side, while in the second equality using covariant derivatives `;’ based on $g_{\mu \nu }$ which corresponds to the Parker view point. That is, ${}^{*}{R}_{\mu }^{\mu }={\lambda }^{-2}{g}^{\mu \nu }{R}_{\mu \nu }\left(\tilde{g}\right)=R-6{\kappa }_{;\mu }^{\mu }+6{\kappa }^{\mu }{\kappa }_{\mu }.$ This way one recovers Equation (25) in Parker’s paper upon identifying $\lambda$ with $\Omega$ and moving $\Omega$ to the r.h.s of the last equality). As such, the above action applies to the vacuum, since the matter contribution is not yet accounted for. The action should be stationary for arbitrary small variations of $g_{\mu \nu }$ and of $\lambda$.

In the expression ${\lambda }^2{}^{*}R$, the second term on the right hand side (r.h.s.), i.e., ${\lambda }^2{\kappa }_{;\mu }^{\mu }$, can be related to

$${\left({\lambda }^2{\kappa }^{\mu }\right)}_{;\mu }={\lambda }^2{\kappa }_{;\mu }^{\mu }+2\lambda {\kappa }^{\mu }{\lambda }_{\mu }.$$

(9)

We also note that, thanks to Equation (6) with $n=-1$,

$${\lambda }^{*\mu }{\lambda }_{*\mu }=g^{\mu \nu }{\lambda }_{*\mu }{\lambda }_{*\nu }={\lambda }^{\mu }{\lambda }_{\mu }+{\lambda }^2{\kappa }^{\mu }{\kappa }_{\mu }+2\lambda {\kappa }^{\mu }{\lambda }_{\mu }.$$

(10)

One can now look at the first two terms in the action integral:

$$\begin{array}{c}\hfill -{\lambda }^2{}^{*}R+c_1{\lambda }^{*\mu }{\lambda }_{*\mu }=-{\lambda }^{2}R-6{\lambda }^2{\kappa }^{\mu }{\kappa }_{\mu }+6{\left({\lambda }^2{\kappa }^{\mu }\right)}_{;\mu }\\ \hfill -12\lambda {\kappa }^{\mu }{\lambda }_{\mu }+c_1{\lambda }^{\mu }{\lambda }_{\mu }+c_1{\lambda }^2{\kappa }^{\mu }{\kappa }_{\mu }+2c_1\lambda {\kappa }^{\mu }{\lambda }_{\mu }.\end{array}$$

(11)

We choose the constant $c_1$ to be $c_1=6$, in order to reproduce the usual kinetic term for a scalar field $\phi$ related to small deviations near $\lambda =1$; that is, $\lambda \approx 1+\phi$; to be touched upon in the next section. Then, the above expression simplifies to

$$-{\lambda }^2{}^{*}R+6{\lambda }^{*\mu }{\lambda }_{*\mu }=-{\lambda }^{2}R+6\left({\lambda }^2{\kappa }^{\mu }\right){}_{;\mu }+6{\lambda }^{\mu }{\lambda }_{\mu }.$$

(12)

We note that $\left({\lambda }^2{\kappa }^{\mu }\right){}_{;\mu }\sqrt{-g}={\left({\lambda }^2{\kappa }^{\mu }\sqrt{-g}\right)}_{,\mu }$ (since ${\Gamma }_{\alpha \mu }^{\alpha }=\frac{\mathsf{\partial }ln\sqrt{-g}}{\mathsf{\partial }{x}^{\mu }}$).

This is an exact differential and may be eliminated from the action integral [24]. The action is then [3]:

$$\delta I_{\mathrm{G}}=\delta \int (-{\lambda }^{2}R+6{\lambda }^{\mu }{\lambda }_{\mu }+c_2{\lambda }^4)\sqrt{-g}{d}^{4}x.$$

(13)

The variations of the action can be studied with respect to small variations of both $\delta g_{\mu \nu }$ and $\delta \lambda$. The details of the calculations are given in the Appendix A. The extremum with respect to $\delta g_{\mu \nu }$ leads to the expression of the field equation below with the properties of covariance, as in GR with scale invariance in addition.

2.3. The Scale-Covariant Field Equation

The development of the scale-covariant field equation in a vacuum with a cosmological constant is presented in the Appendix A, it gives Equation (A16),

$$\begin{array}{c}\hfill R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R-2g^{\mu \nu }\frac{\left({\lambda }^{\rho }\right){}_{;\rho }}{\lambda }+2\frac{\left({\lambda }^{\mu }\right){}^{;\nu }}{\lambda }\\ \hfill +g^{\mu \nu }\frac{{\lambda }^{\alpha }{\lambda }_{\alpha }}{{\lambda }^2}-4\frac{{\lambda }^{\mu }{\lambda }^{\nu }}{{\lambda }^2}+{\lambda }^2{\Lambda }_{\mathrm{E}}{g}^{\mu \nu }=0.\end{array}$$

(14)

We perform the following identification $c_2=2{\Lambda }_{\mathrm{E}}$, and $\Lambda ={\lambda }^2{\Lambda }_{\mathrm{E}}$, where ${\Lambda }_{\mathrm{E}}$ is the cosmological constant of GR (not necessarily that of the Einstein static universe) and $\Lambda$ the corresponding cosmological constant in WIG space. (The notation ${\Lambda }_{\mathrm{E}}$ is used to avoid any confusion with $\Lambda$ in SIV). The above equation applies in general; thus, apart from $\Lambda$, it forms the first member of the field equation when matter-energy is present. Aside from the absent mass term $g_{\mu \nu }{\mu }_0^2{\lambda }^2$, the $\lambda$ dependent part of the above expression (14) is practically the same modified energy-momentum tensor as discussed by Parker [3]. Therefore, if there is any particle content related to $\lambda$, it can be viewed as a energy–momentum tensor of such matter that causes the gravitational geometry of the spacetime, in accord with the Einstein GR view point.

If one desires, then the integration measure in (13) could be set to be ${\lambda }^2\sqrt{-g}{d}^{4}x$, resulting in the usual Hilbert–Einstein Lagrangian density for gravity ${\mathcal{L}}_G=R$, while the remaining terms can be identified with the matter Lagrangian ${\mathcal{L}}_M\sim {\phi }^{\mu }{\phi }_{\mu }+{\tilde{c}}_2{e}^{2\phi }$, where $\phi =log\lambda$. Such a treatment results in a stress–energy tensor ${\Lambda }_{\mu \nu }=\frac{2}{\sqrt{-g}}\frac{\delta I_M}{\delta g^{\mu \nu }}$ of power $\Pi \left({\Lambda }_{\mu \nu }\right)=-2$ as derived by Parker [3]. This is consistent with our construction of the overall action as a conformally invariant object. Thus, from the scaling of $\Pi (\sqrt{-g}=+4),\mathrm{and}\Pi \left(g^{\mu \nu }\right)=-2$ one obtains $\Pi \left({\Lambda }_{\mu \nu }\right)=-4+2=-2$ coming from the denominator in ${\Lambda }_{\mu \nu }$.

If there is any additional matter, then there will be a general matter action $I_M=\int {\mathcal{L}}_M\sqrt{-g}{d}^{4}x$, which via standard considerations would result in the stress–energy tensor $T_{\mu \nu }=\frac{2}{\sqrt{-g}}\frac{\delta \sqrt{-g}{\mathcal{L}}_M}{\delta g^{\mu \nu }}$ or simply $T_{\mu \nu }=\frac{2}{\sqrt{-g}}\frac{\delta I_M}{\delta g^{\mu \nu }}$. If the matter action is reparametrization-invariant, as needed to understand why there are only electromagnetic and gravitational classical long-range forces [25], then one would again obtain $\Pi \left(T_{\mu \nu }\right)=-2$. As the name suggests, reparametrization invariance is related to the freedom of choosing any reasonable parametrization for a process under consideration. In particular, this could be accomplished via $\lambda$ being only time-dependent but not space-dependent.

This means that there is a large class of matter models that can result in a stress–energy tensor that is of power $\Pi \left(T_{\mu \nu }\right)=-2$. Since, $\Pi \left(\lambda \right)=-1$, then one can construct a scale-invariant stress–energy tensor of zero power by considering $T_{\mu \nu }^{\prime }={\lambda }^{-2}{T}_{\mu \nu }$.

Therefore, in the presence of matter, the variation $\delta I_{\mathrm{M}}$ of the matter action can be considered to take the form:

$$\delta I_{\mathrm{M}}=-\left[\frac{1}{2}\right]\int T_{\mu \nu }\delta g^{\mu \nu }{\lambda }^2\sqrt{-g}{d}^{4}x,$$

(15)

which contains scale-invariant energy–momentum tensor $T_{\mu \nu }$. Equation (15) is consistent with the scale invariance of $T_{\mu \nu }$, since $\Pi \left({\lambda }^2\right)=-2,\Pi (\sqrt{-g}=+4),\Pi \left(g^{\mu \nu }\right)=-2$.

The scale invariance of the momentum–energy tensor has some implications for densities and pressures, if we consider, for example, the typical case of a perfect fluid [8],

$$T_{\mu \nu }=T_{\mu \nu }^{\prime }\Rightarrow (p+\varrho )u_{\mu }{u}_{\nu }-g_{\mu \nu }p=(p^{\prime }+{\varrho }^{\prime })u_{\mu }^{\prime }{u}_{\nu }^{\prime }-g_{\mu \nu }^{\prime }{p}^{\prime }.$$

(16)

There, the velocities $u^{\mu }$ and $u_{\mu }^{\prime }$ transform like

$$\begin{array}{ccc}\hfill u^{\prime \mu }& =& \frac{d{x}^{\mu }}{d{s}^{\prime }}={\lambda }^{-1}\frac{d{x}^{\mu }}{ds}={\lambda }^{-1}{u}^{\mu },\hfill \\ \hfill \mathrm{and}{u}_{\mu }^{\prime }& =& g_{\mu \nu }^{\prime }{u}^{\prime \nu }={\lambda }^2{g}_{\mu \nu }{\lambda }^{-1}{u}^{\nu }=\lambda u_{\mu }.\hfill \end{array}$$

(17)

The contravariant and covariant components of a vector have different powers, and their covariant derivatives are also different. The energy–momentum tensor scales like

$$(p+\varrho )u_{\mu }{u}_{\nu }-g_{\mu \nu }p=(p^{\prime }+{\varrho }^{\prime }){\lambda }^2{u}_{\mu }{u}_{\nu }-{\lambda }^2{g}_{\mu \nu }{p}^{\prime },$$

(18)

with implications for p and $\varrho$ [8],

$$p=p^{\prime }{\lambda }^2\mathrm{and}\varrho ={\varrho }^{\prime }{\lambda }^2.$$

(19)

Thus, the invariance of the stress–energy tensor $T_{\mu \nu }$ implies that the pressure and density are coscalars of power $\Pi \left(p\right)=\Pi \left(\rho \right)=-2$.

We now express the sum $\delta I=\delta I_{\mathrm{G}}+\delta I_{\mathrm{M}}=0$, see Equations (1) and (15),

$$\begin{array}{c}\hfill \delta I=\left[\frac{c^4}{16\pi G}\right]\int d^{4}x\sqrt{-g}{\lambda }^2\left\{R^{\mu \nu }-\frac{1}{2}R{g}^{\mu \nu }-2g^{\mu \nu }\frac{\left({\lambda }^{\rho }\right){}_{;\rho }}{\lambda }+2\frac{\left({\lambda }^{\mu }\right){}^{;\nu }}{\lambda }\right.\\ \hfill \left.+g^{\mu \nu }\frac{{\lambda }^{\alpha }{\lambda }_{\alpha }}{{\lambda }^2}-4\frac{{\lambda }^{\mu }{\lambda }^{\nu }}{{\lambda }^2}+{\lambda }^2{\Lambda }_{\mathrm{E}}{g}^{\mu \nu }+\left[\frac{8\pi G}{c^4}\right]T^{\mu \nu }\right\}\delta g_{\mu \nu }.\end{array}$$

(20)

We have used the fact that, for any tensor $A^{\alpha \beta }$, one has $A^{\alpha \beta }d{g}_{\alpha \beta }=-A_{\alpha \beta }d{g}^{\alpha \beta }$. The constant terms are indicated in brackets. Thus, in the final equation the energy–momentum tensor is preceded by the constant $\left[\frac{8\pi G}{c^4}\right]$. It is also scale-invariant as the vacuum contribution. The constant of gravity G is here kept as a true constant.

We note that $\Lambda$ is a coscalar of power $\Pi (\Lambda )=-2$, consistently with the previous results for p and $\varrho$. This correspondence ensures the scale invariance of the second member of the field equation,

$$\begin{array}{c}\hfill R^{\mu \nu }-\frac{1}{2}{g}^{\mu \nu }R-2g^{\mu \nu }\frac{\left({\lambda }^{\rho }\right){}_{;\rho }}{\lambda }+2\frac{\left({\lambda }^{\mu }\right){}^{;\nu }}{\lambda }+g^{\mu \nu }\frac{{\lambda }^{\alpha }{\lambda }_{\alpha }}{{\lambda }^2}-4\frac{{\lambda }^{\mu }{\lambda }^{\nu }}{{\lambda }^2}=\\ \hfill -\frac{8\pi G}{c^4}{T}^{\mu \nu }-\Lambda g^{\mu \nu }.\end{array}$$

(21)

Hereafter, following the general practice, we use $c=1$. In the above equation, the first member is written in terms of derivatives of $\lambda$, we can also write it in terms of of ${\kappa }_{\nu }=-{\lambda }_{\nu }/\lambda$, noting, for example, that ${\kappa }_{\nu ;\mu }=-\frac{{\lambda }_{\nu ;\mu }}{\lambda }+\frac{{\lambda }_{\nu }{\lambda }_{\mu }}{{\lambda }^2}$. This gives, for example, in the covariant form,

$$\begin{array}{c}\hfill R_{\mu \nu }-\frac{1}{2}{g}_{\mu \nu }R-{\kappa }_{\mu ;\nu }-{\kappa }_{\nu ;\mu }-2{\kappa }_{\mu }{\kappa }_{\nu }+2g_{\mu \nu }{\kappa }_{;\alpha }^{\alpha }-g_{\mu \nu }{\kappa }^{\alpha }{\kappa }_{\alpha }=\\ \hfill -8\pi G{T}_{\mu \nu }-\Lambda g_{\mu \nu }.\end{array}$$

(22)

This equation is similar to the one obtained through development of the Ricci tensor in the cotensor calculus, see Canuto et al. [8]. There, $R_{\mu \nu }$ and R are the usual terms of GR. This equation is the fundamental equation of the gravitational field, with the properties of both general covariance and scale invariance. The terms of the action with variations in $\delta \lambda$ are discussed below in relation with the “gauge fixing” that determines a particular functional form of $\lambda$, based on a physical assumption about the vacuum. Notice that this is a physically justified result and therefore it is not a gauge choice for a standard gauge symmetry model. Usually, any gauge choice would result in the same description of a physical system and, in this respect, the gauge choice is irrelevant, besides allowing us to perform the computation more easily.In the SIV paradigm, the choice of $\lambda$ has physical implications that are manifested in the equation of motion for the matter particles, as seen by the presence of $\kappa$-terms in (31) below.

3. The Action Principle and the SIV Gauge

To apply the field equation of GR, it is necessary to specify the form of the metric, which defines the geometrical properties. In SIV theory, an additional condition is needed to fix the functional form of $\lambda$. Dirac [1] and Canuto et al. [8] chose the so-called “large number hypothesis”. The related numerical coincidences have received a variety of interpretations, among which is the anthropic one [26]. Our choice is to adopt the following condition for the physical gauge [5]: “The macroscopic empty space is scale invariant, homogeneous, and isotropic”. The equation $p=-\varrho c^2$ for the vacuum allows the medium density to remain constant for an adiabatic expansion or contraction [27], which implies that changes in the spatial scales of the empty space do not modify its properties. Thus, consistently, the only possible dependence, if any, of the scale factor $\lambda$ is a dependence on time.

3.1. The Scale-Invariant Vacuum (SIV) Gauge for $\lambda$

Above, the extremum of the action was considered with respect to small variations of $\delta g_{\mu \nu }$, we can also consider the extremum with respect to variations of $\lambda$. Collecting the terms with $\delta \lambda$ in Equations (A5), (A13) and (A14) for the vacuum contributions to the action integral, we obtain as a condition for a stationary action

$$(-12{\lambda }_{;\alpha }^{\alpha }-2\lambda R+4c_2{\lambda }^3)\sqrt{-g}\delta \lambda =0.$$

(23)

Making the identification $c_2=2{\Lambda }_{\mathrm{E}}$, we obtain

$$6{\lambda }_{;\alpha }^{\alpha }+\lambda R=4{\lambda }^3{\Lambda }_{\mathrm{E}}.$$

(24)

By identifying $\lambda$ with $\varphi$, this is the same as Equation (35) derived by Parker [3], but here the mass term ${\mu }_0^2\varphi$ is missing. The curvature scalar R is different from zero, as expected in de Sitter space-time. However, the de Sitter space-time is conformal to the flat Minkowski space; for the following particular condition, these two spaces are even strictly identical [5],

$$\frac{3{\lambda }^{-2}}{{\Lambda }_{\mathrm{E}}{t}^2}=1.$$

(25)

In this case, since in Minkowski space $R=0$, the action principle results in the expression

$$6{\lambda }_{;\alpha }^{\alpha }=4{\lambda }^3{\Lambda }_{\mathrm{E}},\mathrm{which}\mathrm{gives}6\ddot{\lambda }=4{\lambda }^3{\Lambda }_{\mathrm{E}}.$$

(26)

The gauging conditions based on the hypothesis of the homogeneity and isotropy of the empty space impose the following two conditions [5]:

$$\begin{array}{c}\hfill 3\frac{{\dot{\lambda }}^2}{{\lambda }^2}={\lambda }^2{\Lambda }_{\mathrm{E}}\mathrm{and}2\frac{\ddot{\lambda }}{\lambda }-\frac{{\dot{\lambda }}^2}{{\lambda }^2}={\lambda }^2{\Lambda }_{\mathrm{E}}.\end{array}$$

(27)

Upon taking the trace of (22), one derives the first equation above while using the vacuum assumption $R_{\mu \nu }=T_{\mu \nu }=0$. We can easily verify that these are equivalent to Equation (26). Introducing the first (27) into the second one, we obtain

$$2\frac{\ddot{\lambda }}{\lambda }-\frac{1}{3}{\lambda }^2{\Lambda }_{\mathrm{E}}={\lambda }^2{\Lambda }_{\mathrm{E}}\mathrm{and}\mathrm{thus}6\ddot{\lambda }=4{\lambda }^3{\Lambda }_{\mathrm{E}}.$$

(28)

The solution to this differential equation, based on the first equation in (27), is very simple:

$$\lambda =\sqrt{\frac{3}{{\Lambda }_{\mathrm{E}}}}\frac{1}{t},$$

(29)

Noticeably, this last relation is the same as the above Equation (25), which was necessary to assume the identity of the Minkowski and de Sitter spaces. This verifies the consistency of the above expression for $\lambda$. It is also remarkable that these equations imply ${\Lambda }_{\mathrm{E}}=$ const, consistently with the properties of the Einstein’s cosmological constant. However, there is a deeper connection to the coscalar power of $\Lambda$; that is, it is a coscalar of power 2, resulting in $\Lambda {\lambda }^2={\Lambda }^{\prime }={\Lambda }_{\mathrm{E}}$. Equation (29) implies a specific relation between the current age of the universe and ${\Lambda }_{\mathrm{E}}$ when choosing units such that ${\lambda }_0=1$ in the current epoch.

Now, we can go back and take another viewpoint of (22), this time by adopting the functional form of $\lambda =t_0/t$ (29). Such a choice will remove all the $\kappa$ terms, along with the $\Lambda$ term, and will result in the standard Einstein GR equations $R_{\mu \nu }-g_{\mu \nu }R/2=-8\pi G{T}_{\mu \nu }$ without a cosmological constant; thus, any matter content is possible after removing the $\Lambda$ term through the selection of the SIV gauge (29) for the scale factor $\lambda \left(t\right)$.

Equations (26) and (27) also mean that ${\Lambda }_{\mathrm{E}}$, or the energy density of the vacuum (since ${\Lambda }_{\mathrm{E}}=8\pi G{\rho }_{\mathrm{vac}}/c^2$), is a function of the time-variations of $\lambda$; in particular, the first of (27) illustrates this relation well. Thus, the energy density of the vacuum may be expressed as the gradient of a scalar function $\psi$ [4],

$${\varrho }_{\mathrm{vac}}=\frac{1}{2}C{\dot{\psi }}^2,\mathrm{with}\psi =-\frac{\dot{\lambda }}{\lambda },$$

(30)

with $C=\frac{3}{4\pi G}$ and c = 1 (a reminder to the reader). According to [4], the above $\psi$ may play the role of a “rolling field” during the inflation.

In standard cosmological models based on GR (e.g., Friedmann and $\Lambda$CDM models), the constant ${\Lambda }_{\mathrm{E}}$ is not a direct function of the matter content. Thus, through the above relations, the same applies to the form of the scale factor $\lambda \sim 1/t$, which just like ${\Lambda }_{\mathrm{E}}$ is independent from the matter content in the Universe, generally expressed by ${\Omega }_{\mathrm{m}}$. However, the matter content may drastically limit the range of possible variations in the above t parameter (thus, limiting the range of $\lambda$-variations), but it does not modify the functional dependence $\lambda \left(t\right)$, see Section 4.1.

For now, we conclude that the action principle applied to the vacuum leads to the same expression of the scale factor as obtained by the abovementioned gauging condition. This gives support to the above gauging condition, relating $\lambda$ and ${\Lambda }_{\mathrm{E}}$, and to its significance. We note that there are also positive implications for the well-known cosmological constant problem [4].

3.2. The Geodesics from an Action Principle

To study dynamics, we also need an equation of motion. The equation of geodesics in Weyl’s geometry was first derived by Dirac [1]. It obeys the following equation:

$$\frac{d{u}^{\alpha }}{ds}+{\Gamma }_{\mu \nu }^{\alpha }{u}^{\mu }{u}^{\nu }-{\kappa }_{\mu }{u}^{\mu }{u}^{\alpha }+{\kappa }^{\alpha }{u}_{\mu }{u}^{\mu }=0,$$

(31)

It is customary to chose the parametrization such that $u_{\mu }{u}^{\mu }=\pm 1$, depending on the signature of the metric used. The geodesic of a free particle can be obtained from the condition that the following action is minimum [18],

$$\delta I={\int }_{P_1}^{P_2}\delta \left(\lambda ds\right)={\int }_{P_1}^{P_2}\delta d{s}^{\prime }=0.$$

(32)

This is a one-dimensional problem of an unknown function $x^{\mu }$ of time. The above condition means, in fact, that the corresponding path is an extremum, i.e., in the Riemann space it is a minimum, while for pseudo-Riemannian space with Lorentzian metric, it is a maximum. The application of this action principle confirms the above equation of the geodesics.

4. Basic Applications of the Scale-Invariant Dynamics

In this section, we present what we feel is the minimally viable demonstration of the SIV paradigm and the relevant equations. After all, this paper is titled the Action Principle for Scale Invariance and its Applications (Part I), which is the first part dedicated to mathematical formalism. We are in the process of preparing an upcoming manuscript (Part II), where we plan to discuss specific results related to the early Universe; that is, applications to inflation, Big Bang nucleosynthesis, and the growth of the density fluctuations within the SIV; while, in the late time Universe, the applications of the SIV paradigm are related to scale-invariant dynamics of galaxies, MOND, dark matter, and the dwarf spheroidals, where one can find MOND to be a peculiar case of the SIV theory, as well as possible SIV effects in the Earth–Moon system. These represent about six different applications, which would have made this paper too large for a normal paper, so here we only focus on two key demonstrations of the SIV paradigm: scale invariant cosmology as the initial motivation and the equations of motion as pertaining to Kepler’s third law.

4.1. Cosmological Constraints

The main consequences of the field and geodesic equations for the dynamics are now drawn in view of future comparisons with observations. The field equation with the FLWR metric, together with the gauging conditions (27), leads to the cosmological equations [5],

$$\begin{array}{c}\hfill \frac{8\pi G\varrho }{3}=\frac{k}{a^2}+\frac{{\dot{a}}^2}{a^2}+2\frac{\dot{a}\dot{\lambda }}{a\lambda },\end{array}$$

(33)

$$\begin{array}{c}\hfill -8\pi Gp=\frac{k}{a^2}+2\frac{\ddot{a}}{a}+\frac{\dot{a^2}}{a^2}+4\frac{\dot{a}\dot{\lambda }}{a\lambda }\end{array}$$

(34)

$$\begin{array}{c}\hfill \mathrm{leading}\mathrm{to}-\frac{4\pi G}{3}\left(3p+\varrho \right)=\frac{\ddot{a}}{a}+\frac{\dot{a}\dot{\lambda }}{a\lambda },\end{array}$$

(35)

$$\begin{array}{c}\hfill \mathrm{with}\varrho a^{3(1+c_s^2)}{\lambda }^{1+3c_s^2}=\mathrm{const}.\end{array}$$

(36)

The last expression is the conservation law, with a sound velocity $c_s^2=0$ for a dust model and $c_s^2=1/3$ for the radiative era. If $\lambda$ is a constant ($\lambda =1$), the derivatives of $\lambda$ vanish and one is brought back to the Friedmann equations. The extra term in Equations (33)–(35) represents an additional acceleration in the direction of motion. For an expanding Universe, this extra force produces an accelerated expansion. For a contraction, the additional term favors collapse, cf. the growth of density fluctuations [9]. Analytical solutions for the flat SIV models with $k=0$ were found for the matter era [28],

$$a\left(t\right)={\left[\frac{t^3-{\Omega }_{\mathrm{m}}}{1-{\Omega }_{\mathrm{m}}}\right]}^{2/3},\mathrm{thus}{t}_{\mathrm{in}}={\Omega }_{\mathrm{m}}^{1/3}.$$

(37)

These equations allow flatness for different values of ${\Omega }_{\mathrm{m}}$, unlike the Friedmann models. The timescale t is $t_0=1$ at present, with $a\left(t_0\right)=1$. The graphical solutions are relatively close to the corresponding $\Lambda$CDM models [5], the deviations being larger at very low ${\Omega }_{\mathrm{m}}$-values. The initial time for $a\left(t_{\mathrm{in}}\right)=0$ is $t_{\mathrm{in}}={\Omega }_{\mathrm{m}}^{1/3}$; the dependence in 1/3 produces a rapid increase in $t_{\mathrm{in}}$ for increasing ${\Omega }_{\mathrm{m}}$. For ${\Omega }_{\mathrm{m}}=0,0.01,0.1,0.3,0.5$, the values of $t_{\mathrm{in}}$ are 0, 0.215, 0.464, 0.669, and 0.794, respectively. This strongly reduces the possible range of $\lambda \left(t\right)=(t_0/t)$ (which varies between $1/t_{\mathrm{in}}$ and $1/t_0=1$) for increasing ${\Omega }_{\mathrm{m}}$. For ${\Omega }_{\mathrm{m}}\ge 1$, there are no possible scale-invariant models, consistent with causality relations in an expanding Universe [4].

While t varies between $t_{\mathrm{in}}$ and $t_0=1$ at present, the usual timescale $\tau$ in years or seconds varies from ${\tau }_{\mathrm{in}}=0$ and ${\tau }_0=13.8$ Gyr at present [29]. The relation between these age units is $\frac{\tau -{\tau }_{\mathrm{in}}}{{\tau }_0-{\tau }_{\mathrm{in}}}=\frac{t-t_{\mathrm{in}}}{t_0-t_{\mathrm{in}}}$, expressing that the age fractions of an event are the same. This gives

$$\begin{array}{c}\hfill \tau ={\tau }_0\frac{t-t_{\mathrm{in}}}{1-t_{\mathrm{in}}}\mathrm{and}t=t_{\mathrm{in}}+\frac{\tau }{{\tau }_0}(1-t_{\mathrm{in}}),\end{array}$$

(38)

$$\begin{array}{c}\hfill \mathrm{with}\frac{d\tau }{dt}=\frac{{\tau }_0}{1-t_{\mathrm{in}}},\mathrm{and}\frac{dt}{d\tau }=\frac{1-t_{\mathrm{in}}}{{\tau }_0}.\end{array}$$

(39)

For increasing ${\Omega }_{\mathrm{m}}$, the timescale t is squeezed, since $t_{\mathrm{in}}={\Omega }_{\mathrm{m}}^{1/3}$ is increasing, which reduces the range of $\lambda$ variations. Both timescales are evidently linearly related. Thus, for a given ${\Omega }_{\mathrm{m}}$, the derivatives are constant, which is useful for connecting the time variations, since $\frac{dX}{d\tau }=\frac{dX}{dt}\frac{dt}{d\tau }$.

4.2. Newton-like Dynamics

The weak-field approximation is obtained from the above geodesics, with the following assumptions: $g_{ij}=-{\delta }_{ij}$, $g_{00}=1+\frac{\Phi }{c^2}$, ${\Gamma }_{00}^i=\frac{1}{c^2}{\mathsf{\partial }}^i\Phi$, ${\mathsf{\partial }}^i=g^{i\alpha }\frac{\mathsf{\partial }}{\mathsf{\partial }{x}^{\alpha }}$, with $\Phi =-GM/r$ being the Newtonian potential. In addition, for slow motions, $u^i\to \frac{{\upsilon }^i}{c}$ and $u^0\to 1$. The geodesic Equation (31) becomes

$$\frac{1}{c^2}\frac{d{\upsilon }^i}{dt}+\frac{1}{c^2}{\mathsf{\partial }}^i\Phi -\kappa \frac{{\upsilon }^i}{c^2}=0,$$

(40)

where $\kappa =-\frac{\dot{\lambda }}{\lambda }=\frac{1}{t}$. This leads to the modified Newton equation in t-scale [10,30],

$$\frac{d^2\mathbf{r}}{d{t}^2}=-\frac{G_{t}M}{r^2}\frac{\mathbf{r}}{r}+\kappa \left(t\right)\frac{d\mathbf{r}}{dt},$$

(41)

Equation (36) imposes for a dust Universe the relation $\varrho a^3\lambda =const.$, implying that the mass of an object is a coscalar of power $\Pi \left(M\right)=1$, thus $M=M\left(t\right)=M\left(t_0\right)\frac{t}{t_0}$. In current units, according to (38), this becomes

$$M\left(\tau \right)=M\left({\tau }_0\right)(t_{\mathrm{in}}+\frac{\tau }{{\tau }_0}(1-t_{\mathrm{in}}).$$

(42)

A variation in mass is also a common situation in Special Relativity, where masses change for velocities tending to c. Noticeably, the potential $\Phi =GM/r$ of an object appears as a more fundamental quantity, being scale-invariant throughout the evolution of the Universe.

We emphasize that this (possibly surprising) mass variation is also consistent in scale invariance with the Lagrangian definition of mass in mechanics, where the mass appears as a proportionality factor between the Lagrangian $\mathcal{L}$ and ${\upsilon }^2$ the square of the velocities [24]. As shown in Appendix B, the Lagrangian definition, together with the action principle in scale invariance and the need for preserving the uniformity of the space-time, imposes that the masses are of power $\Pi =1$, thus leading to a variation with time.

In a non-empty Universe, the effects of mass changes are rather limited. As an example, for ${\Omega }_{\mathrm{m}}=0.3$, the mass at the Big Bang was $M\left(t_{\mathrm{in}}\right)=0.6694M\left(t_0\right)$, since $M\left(t_{\mathrm{in}}\right)={\Omega }_{\mathrm{m}}^{1/3}M\left(t_0\right)$. Over the last 400 Myr the variations were smaller than 1%. This relative “constancy” of the mass over long periods is what has allowed us to show that MOND theory is a valid approximation of the present SIV theory over the rotation period of spiral galaxies [13].

Equation (41) is expressed in variable t, where the age of the Universe is $(1-t_{\mathrm{in}})$. One has $\frac{d^{2}r}{d{t}^2}=\frac{d^{2}r}{d{\tau }^2}{\left(\frac{d\tau }{dt}\right)}^2$, and the constant $G_t$ in t-units becomes $G_t{\left(\frac{dt}{d\tau }\right)}^2=G$ in current units. The above equation becomes at present ${\tau }_0$,

$$\frac{d^2\mathbf{r}}{d{\tau }^2}=-\frac{GM}{r^2}\frac{\mathbf{r}}{r}+\frac{{\psi }_0}{{\tau }_0}\frac{d\mathbf{r}}{d\tau },\mathrm{with}{\psi }_0=1-t_{\mathrm{in}}.$$

(43)

The additional term on the right is an acceleration in the direction of the motion, we call it the dynamical gravity, which is usually very small, since ${\tau }_0$ is very large. This term, proportional to the velocity, favors collapse during a contraction, and produces an outwards acceleration in an expansion. The parameter ${\psi }_0$ only applies at present. For ${\Omega }_{\mathrm{m}}=0,0.05,0.10,0.20,0.30,1$, one has ${\psi }_0=1,0.632,0.536,0.415,0.331,0.$ In other epochs $\tau$, instead of ${\psi }_0$ in Equation (43), one has the numerical factor $\psi$

$$\psi \left(\tau \right)=\frac{t_0-t_{\mathrm{in}}}{\left(t_{\mathrm{in}}+\frac{\tau }{{\tau }_0}(t_0-t_{\mathrm{in}})\right)}\left(=\frac{t_0-t_{\mathrm{in}}}{t}\right).$$

(44)

We now study the time $\tau$ variations in the $\psi$ in the current time units. Let us express the derivative of the numerical factor $\psi$ given in Equation (44),

$$\frac{d\psi }{d\tau }=\frac{d\psi }{dt}\frac{dt}{d\tau }=-\frac{{(1-t_{\mathrm{in}})}^2}{t^2{\tau }_0}=-\frac{{\psi }^2}{{\tau }_0},\mathrm{and}\frac{\dot{\psi }}{\psi }=-\frac{\psi }{{\tau }_0}.$$

(45)

Moreover, since one has $\kappa =-\dot{\lambda }/\lambda =1/t$, this implies $\dot{\kappa }=-{\kappa }^2$.

4.3. The Two-Body Problem within the SIV Paradigm

To fully appreciate the dynamical effects of scale invariance, it is appropriate to examine the two-body problem [10,30]. In these references, the equations were given in the timescale t. Here, we better give them in the $\tau$-scale (years, seconds). The equation of motion (41) can be written in plane polar coordinates r and $\phi$

$$\begin{array}{c}\hfill \ddot{r}-r{\dot{\phi }}^2=-\frac{GM}{r^2}+\frac{\psi }{{\tau }_0}\dot{r},\end{array}$$

(46)

$$\begin{array}{c}\hfill r\ddot{\phi }+2\dot{r}\dot{\phi }=\frac{\psi }{{\tau }_0}r\dot{\phi }.\end{array}$$

(47)

The term $\kappa \left(t\right)$ in t-time is replaced by $\kappa \left(\tau \right)=\psi \left(\tau \right)/{\tau }_0$ in $\tau$-units. The integration of this last equation gives the equivalent law of angular momentum conservation

$$\frac{\psi }{{\tau }_0}{r}^2\dot{\phi }=L=\mathrm{const}.$$

(48)

for $\frac{d\psi }{d\tau }$ see Equation (45). Here, and in the rest of the text, $\phi$ is the angular coordinate; that is $\phi \in \left[0,2\pi \right]$. L is a scale-invariant quantity. This means that $r^2\dot{\phi }$ increases like ${\tau }_0/\psi$, i.e., linearly with t. After some manipulations using Equation (48) to develop Equation (46) [10], we obtain the equivalent of the Binet equation,

$$\frac{d^{2}u}{d{\phi }^2}+u=\frac{GM}{L^2}{\left(\frac{\psi }{{\tau }_0}\right)}^2,\mathrm{where}u=1/r.$$

(49)

with an additional parenthesis. Its general solution is a conic

$$r=\frac{r_c}{1+ecos\phi },\mathrm{with}{r}_c=\frac{L^2}{GM}{\left(\frac{{\tau }_0}{\psi }\right)}^2.$$

(50)

$r_c$ is a particular solution for $\frac{d^{2}u}{d{\phi }^2}=0$: the radius of a circular orbit ($e=0$). For an elliptical orbital, the expression of the semi-major axis a is

$$a=\frac{r_c}{1-e^2}.$$

(51)

From (50), we verify that a, like $r_c$, scales with t (since $\psi \sim 1/t$); this means, for example, that elliptical orbits slightly spiral outwards, with a constant eccentricity.

According to (48), the orbital velocity $\upsilon$ of a circular motion behaves as follows:

$${\upsilon }^2={\left(r_c\dot{\phi }\right)}^2=\frac{L^2}{r_c^2}{\left(\frac{{\tau }_0}{\psi }\right)}^2.$$

(52)

This is a scale invariant quantity, since $r_c\sim t$ and $\psi \sim 1/t$. Now, we can express $L^2$ with (50) and obatin the usual expression:

$${\upsilon }^2=\frac{GM}{r_c},$$

(53)

The scaling of M and $r_c$ with t confirms the scale invariance ${\upsilon }^2$, which remains constant during the orbital expansion. In a subtle interplay, the tangential acceleration of the “dynamical gravity” exactly compensates the usual slowing down due to orbital expansion. This is also consistent with the time increase in $r^2\dot{\phi }$ (48).

The constancy of ${\upsilon }^2$ during expansion compares with MOND, where in the deep-limit, the orbital velocity becomes independent from radius [12], a key point regarding the flat rotation curves of galaxies. This concordance is not surprising, since MOND appears as an approximation of the Newton-like Equation (43), where the scale factor $\lambda$ may be considered constant [13]; an acceptable approximation over a few ${10}^8$ years.

4.4. Secular Variations of the Orbital Parameters

We now study the time variations of the orbital parameters in the current time units $\tau$. Let us recall the derivative of the numerical factor $\psi$ given in Equation (44). Using (45) and (42), based on (52) and (53) above, the radius $r_c$ of the circular orbit behaves as follows:

$$r_c=\frac{L^2}{GM}{\left(\frac{{\tau }_0}{\psi }\right)}^2=\frac{L^2}{GM\left({\tau }_0\right)}\frac{{\tau }_0^2(t_{\mathrm{in}}+\frac{\tau }{{\tau }_0}(1-t_{\mathrm{in}})}{{(1-t_{\mathrm{in}})}^2}.$$

(54)

With $\dot{r_c}=\frac{L^2{\tau }_0}{GM\left({\tau }_0\right)(1-t_{\mathrm{in}})}$, the relative variation with time $\tau$ of the semi-major axis a given by (51) of an elliptical orbit becomes

$$\frac{\dot{a}}{a}=\frac{\dot{r_c}}{r_c}=\frac{\psi }{{\tau }_0}\mathrm{at}\mathrm{time}{\tau }_0:\psi ={\psi }_0=1-t_{\mathrm{in}}.$$

(55)

As an example, for the Earth–Moon system, the above relation would predict a lunar recession amounting to 0.92 cm/yr [31]. All variable quantities that have a linear dependence on time t (and $\tau$) have a relative variation equal to $\frac{\psi }{{\tau }_0}$. This is the case for the orbital radius or semi-major axis, the mass and the rotation period T. For the masses, we can obtain from Equations (42) and (44)

$$\frac{\dot{M}}{M}=\frac{\psi }{{\tau }_0}.$$

(56)

Let us check the orbital period T, which is equal to $T=\frac{2\pi }{\dot{\phi }}$. From $r_c$ given by (50) and the conservation law (48), we have

$$T=\frac{2\pi r_c^2\psi }{L{\tau }_0}=\frac{2\pi \psi L{r}_c^2}{L^2{\tau }_0}=\frac{2\pi L{r}_c{\tau }_0}{GM\psi }.$$

(57)

Since M and $r_c$ vary in the same way, the period T varies like $1/\psi$; thus, based on Equation (45),

$$\frac{\dot{T}}{T}=-\frac{\dot{\psi }}{\psi }=\frac{\psi }{{\tau }_0}.$$

(58)

Remarkably, the above quantities strictly conserve Kepler’s third law. We have the law of angular momentum conservation (48) and the expression of the radius $r_c$ (50). Substituting L from the first into the second,

$$\begin{array}{c}\hfill \frac{\psi }{{\tau }_0}{r}_c^2\frac{2\pi }{T}=L,\mathrm{and}{r}_c=\frac{L^2}{GM}(\frac{{\tau }_0}{\psi }{)}^2.\\ \hfill \mathrm{we}\mathrm{get}\frac{4{\pi }^2{r}_c^3}{GM{T}^2}=1,\end{array}$$

(59)

where we considered circular orbits of a test particle of negligible mass, to simplify the derivation. If it has a significant mass m, this has to be added to M. The quantities T, M, and $r_c$ all have the same functional dependence on time $\tau$, as illustrated by the same relative variations of the function $\frac{\psi }{{\tau }_0}$. Thus, we have a cubic dependence on the same function, both in the numerator and denominator, implying that at any time $\tau$ Kepler’s third law is maintained.

5. Conclusions

We have revisited the scale-invariant field equations and some of the basic applications of the corresponding geodesic equations. Along the way, we have demonstrated that any scale invariant matter action, or even a weaker symmetry known as reparametrization invariance, naturally results in a “standard” energy–momentum tensor for matter of power $\Pi \left({\rho }_m\right)=-2$ that can be related to a scale-invariant energy–momentum tensor upon augmenting the integration measure $\sqrt{-g}{d}^{4}x$ by the factor ${\lambda }^2$. The example of a familiar ideal fluid was considered, to show the scaling power for energy-density and pressure. The choice of $\lambda$ also makes specific predictions for an additional dynamic gravity term in the geodesics equations. As such, this could be observationally accessed, and therefore this is not just another choice of gauge that does not have observational effects. It may be just the thing we need to get out of the current dark ages, dominated by dark matter and dark energy, and into enlightenment, where the scale invariance and/or reparametrization invariance could be the answer to understanding our Universe without a need for dark stuff.

The justification using an action principle of the scale invariant field equations, of the geodesics equations, and of the gauging condition for the scale factor $\lambda$ strengthens the theoretical basis of the scale-invariant paradigm, for which several observational tests have provided positive support for the theory. The equivalent Newton-like equation, the two-body problem, its secular variations, and the modified angular momentum conservation are given in conventional time units in the current epoch. The constancy of the orbital velocity during secular expansion is an interesting consequence in relation with the flat rotation curves of galaxies and similar problems in very wide binaries [32]. Kepler’s third law remains a rock untouched by scale-invariance effects.