Evidence for an Effective Gravitational Constant from the Laboratory Measurements

Abstract

We consider, within the framework of an improved $5D$ Kaluza-Klein theory (KK$\psi$), the set of twenty most precise values ever published of Earth-based gravitational constant measurements. We take into account the possibility of a jump discontinuity into the geomagnetic potential in connection with the spin crossover of the iron (III) complex ions migrating from the Earth’s outer core into the deep Earth mantle as well as the spin crossover of the copper (III) complex ions migrating downwards within either the Earth’s crust or deep in the Earth’s mantle. Moreover, it turns out that the boundary conditions assigned to both internal, $\phi$, and external, $\psi$, scalar fields are sensitive to the aforementioned jump discontinuity too. Again, the fits to the experimental data are found in good agreement with the predictions of the linearized KK$\psi$ theory. An estimate around the vacuum expectation values $\psi=v$ and $\phi=1$ of the second derivatives ${\partial^{2}f_{\textrm{EM}}}/{\partial\phi^{2}}(v,1)$, ${\partial^{2}f_{\textrm{EM}}}/{\partial\phi\partial\psi}(v,1)$ and ${\partial^{2}f_{\textrm{EM}}}/{\partial\psi^{2}}(v,1)$ of the coupling function, $f_{\textrm{EM}}(\psi,\phi)$, of the electromagnetic field (EM) with $\psi$ and $\phi$, is provided for the first time, thereby enabling the determination of the product $v\times f_{\textrm{EM}}(v,\phi)$ and $v\times f_{\textrm{EM}}(\psi,1)$ in the second-order approximation from the experimental data.

Appendices

APPENDIX A

As one knows, the central ion of a complex ion can make a spin crossover (or spin transition/spin equilibrium) from a low spin state to a high spin state (or conversely) by absorbing (or emitting) a photon [34–36] or by migrating from a high-pressure area to a low-pressure area (or conversely). So, let us consider a distribution of identical monocinetic ions that constitute a current $I$. We argue that these charged carriers, by making a pressure induced spin crossover at the Earth’s core-mantle boundary, will be subject to a drag force,

$$\vec{F}={-}\frac{1}{2}\frac{\left|V_{0}\right|I}{c}\vec{v},$$

(A.1)

while creating in turn a jump discontinuity equal to $\pm V_{0}$ in the geomagnetic potential⁵ As such, this implies an infinitesimal work⁶ equal to

$$\delta W=\pm\frac{1}{2}V_{0}\frac{v^{2}}{c}dQ.$$

(A.2)

Hence the expression of the infinitesimal variation of the internal energy is

$$dU=\sum_{i}{Y_{i}dX_{i}}+\mu dN\pm\frac{1}{2}\frac{v^{2}}{c}V_{0}dQ$$

$${}+\mu^{\prime}dN^{\prime}\pm\frac{1}{2}\frac{v^{\prime 2}}{c}V_{0}dQ^{\prime},$$

(A.3)

and consequently, the expression of the infinitesimal variation of the total energy

$$dE_{\textrm{total}}=d\left[N(E_{c}+E_{p})+N^{\prime}(E_{c}^{\prime}+E_{p}^{\prime})\right]+dU$$

$${}=Nd(E_{c}+E_{p})+N^{\prime}d(E_{c}^{\prime}+E_{p}^{\prime})$$

$${}+\sum_{i}{Y_{i}dX_{i}}+\left[\mu+E_{p}+\frac{1}{2}(mc\pm qV_{0})\frac{v^{2}}{c}\right]dN$$

$${}+\left[\mu^{\prime}+E_{p}^{\prime}+\frac{1}{2}(mc\pm qV_{0})\frac{v^{\prime 2}}{c}\right]dN^{\prime},$$

(A.4)

where $E_{c}=\frac{1}{2}mv^{2}$, $E_{c}^{\prime}=\frac{1}{2}mv^{\prime 2}$, $dQ=qdN$, $dQ^{\prime}=qdN^{\prime}$, $Y_{i}$ is the intensive quantity associated to the extensive quantity $X_{i}$, $N$ and $N^{\prime}$ denote the numbers of particles of mass $m$ and electric charge $q$ all identical up to the spin state, where the prime (respectively the absence of prime) stands for the population of those particles either in the high spin state (respectively. the low spin state) which undergoes a pressure-induced spin-crossover (see [36–39]) thereby increasing the number of population in the low spin state (respectively, the high spin state). Furthermore, $dE_{c}=0$ and $dE_{c}^{\prime}=0$, for ions in uniform motion. By neglecting the variation of the altitude, Eq. (A.4) reduces to

$$dE_{\textrm{total}}=\sum_{i}{Y_{i}dX_{i}}$$

$${}+\left[\mu+E_{p}+\frac{1}{2}(mc^{2}\pm qcV_{0})(\frac{v^{2}}{c^{2}})\right]dN$$

$${}+\left[\mu^{\prime}+E_{p}^{\prime}+\frac{1}{2}(mc^{2}\pm qcV_{0})(\frac{v^{\prime 2}}{c^{2}})\right]dN^{\prime}$$

(A.5)

Moreover, for an isolated system in thermodynamic equilibrium, $\mu^{\prime}=\mu$, $N^{\prime}+N=\textrm{const}$, the total energy $E_{\textrm{total}}$ is conserved, and all $X_{i}$ (including the entropy) are assumed constant. Consequently, since $E_{p}^{\prime}=E_{p}$ at the boundary where the spin crossover occurs, Eq. (A.5) reduces in turn to

$$(mc\pm qV_{0})(v^{2}-v^{\prime 2})=0.$$

(A.6)

Fig. 5

The curve $v\times f_{\textrm{EM}}(\psi,1)$ versus $\psi$ in the second-order approximation, derived from the experimental data.

Let us emphasize that Eq. (A.6) holds only if either the cations are subject to no drag force and no spin crossover, in which case $v^{\prime}=v$ with respect to the terrestrial reference frame, or otherwise $mc\pm qV_{0}=0$ during the spin crossover and $v^{\prime}\neq v$ in accordance with the observed velocity gradient. This means that the spin transition of iron (III) cations from the high spin state to the low spin state (or converserly) in the deep mantle of the Earth above (or below) a certain pressure and temperature, will also involve a jump discontinuity of the geomagnetic potential of an amount, $V_{0}$, such that

$$V_{0}=\pm\frac{mc}{q}.$$

(A.7)

We argue that the above formula corresponds solely to a spin transition $\Delta s=\pm 1$. For $\Delta s\neq\pm 1$, it generalizes either to

$$V_{0}={-}\Delta s\frac{mc}{q},$$

(A.8)

or to

$$\Delta V=\frac{mc}{q}\Delta s.$$

(A.9)

Peculiarly, when the spin crossover does not occur, $\Delta V=V_{0}=0$ since $\Delta s=0$. When $n$ various species undergo the spin crossover at the same time at a point, according to the principle of superposition, Eq. (A.8) rewrites as

$$V_{0}=-\sum_{i=1}^{i=n}{\Delta s_{i}\frac{m_{i}c}{q_{i}}},$$

(A.10)

where $\Delta s_{i}=s_{i}({\textrm{final}})-s_{i}({\textrm{initia}}l)$, hence $\Delta s_{i}= s_{i}({\textrm{low}})-s_{i}({\textrm{high}})$ for a spin transition from high spin to low spin and $\Delta s_{i}=s_{i}(h{\textrm{igh}})-s_{i}({\textrm{low}})$ for a spin transition from low to high spin.

APPENDIX B

Let us recall that the (pseudo)-scalar or vector electromagnetic potentials have no direct physical meaning in classical physics because of their arbitrary choice due to the gauge invariance. However, things change drastically in quantum mechanics, as can be seen in the Aharonov–Bohm effect [40]. Throughout, based on quantum-mechanical considerations not requiring any quantum gravity treatment still lacking hitherto, as a first attempt we provide a semiclassical justification of the relation between jump discontinuities in the geomagnetic potential and subsequently in the scalar fields $\psi$ and $\phi$ through Eqs. (8, (9). Indeed, the vector potential $\vec{A}$ created in space by a magnetic dipole moment $\vec{M}$ reads (see Section 3)

$$\vec{A}=\frac{\mu_{0}}{4\pi r^{3}}\vec{M}\times\vec{r}=\vec{M}\times\frac{\vec{E}}{Nqc^{2}},$$

(B.1)

where $\vec{E}=\dfrac{Nq\vec{r}}{4\pi\epsilon_{0}r^{3}}$ denotes the electric field created radially by a set of $N$ identical central ions of electric charge $q$ and mass $m$ at a point defined by the position vector $\vec{r}$ within a domain $(\Omega)$ of volume $V_{\Omega}$. Now, the magnetic flux $\Phi$ reads

$$\Phi=\iint{\vec{B}\cdot\vec{dS}}=\oint{\vec{A}\cdot\vec{dl}}$$

$${}=\oint{\bigg{(}\vec{M}\times\frac{\vec{E}}{Nqc^{2}}\bigg{)}\vec{dl}}=\frac{ME\sin\Theta}{Nqc^{2}}l,$$

(B.2)

where we have set $\Theta=(\vec{M};\vec{r})$. Moreover, the pseudoscalar geomagnetic potential reads (see Section 3)

$$V=\frac{\mu_{0}}{4\pi r^{3}}\vec{M}\cdot\vec{r}=\vec{M}.\frac{\vec{E}}{Nqc^{2}}=\frac{ME\cos\Theta}{Nqc^{2}}.$$

(B.3)

Now, the standard deviations

$$\sigma_{V}=\left(\int\limits_{-\pi}^{\pi}{\iiint_{\Omega}{(V-\bar{V})^{2}\frac{d^{3}r}{V_{\Omega}}}\frac{d\Theta}{2\pi}}\right)^{1/2}$$

and

$$\sigma_{\Phi}=\left(\int\limits_{-\pi}^{\pi}{\iiint_{\Omega}{(\Phi-\bar{\Phi})^{2}\frac{d^{3}r}{V_{\Omega}}}\frac{d\Theta}{2\pi}}\right)^{1/2},$$

respectively, of $V$ and $\Phi$ reduce to

$$\sigma_{V}=\dfrac{M\left\langle E\right\rangle}{\sqrt{2}Nqc^{2}},\quad\sigma_{\Phi}=\frac{M\left\langle E\right\rangle}{\sqrt{2}Nqc^{2}}l$$

on account that both means $V$ and $\Phi$ are equal to zero, where we have set

$$\left\langle E\right\rangle=\left(\iiint_{\Omega}{E\frac{d^{3}r}{V_{\Omega}}}\right)^{1/2}.$$

So, by combining Eqs. (B.2) and (B.3), it follows

$$\sigma_{V}=\frac{\sigma_{\Phi}}{l}.$$

(B.4)

Analogously to the case of the Aharonov–Bohm effect, the standard deviation of the fluctuating magnetic flux associated to random jump discontinuities in the geomagnetic potential due to the spin crossover of identical central ions of mass $m$, should be quantified through circular loops of radius $l/(2\pi)=\hbar/(mc)$ (Compton wavelength of a particle of mass $m$). Thus,

$$\sigma_{\Phi}=k\frac{h}{q},$$

(B.5)

and accordingly,

$$\sigma_{V}=k\frac{mc}{q},$$

(B.6)

where $k$ is an integer.