After the 2019 redefinition of the SI, the kilogram is, via the fixed Planck constant, explicitly the unit of inertial mass, and it is realized in practice by two primary routes: atom counting in an enriched 28Si sphere by the X-ray crystal density (XRCD) method, and the Kibble balance. Mana and Schlamminger (Metrologia 59, 043001, 2022) showed that only atom counting realizes inertial mass without assuming the universality of free fall, while the Kibble-balance result under ordinary circumstances presumes equivalence of inertial and gravitational mass; they argued that agreement between the two realizations can therefore be interpreted as a test of the equivalence principle, bounding the Eötvös-parameter difference between the gravimeter's falling body and the silicon sphere. That interpretive claim, however, was never converted into a published quantitative bound, and the physical reach of this route was assessed only qualitatively. The open problem is to establish quantitatively what equivalence-principle test the XRCD-versus-Kibble agreement constitutes: a formal measurement model over the international kilogram-realization comparisons, an explicit numerical upper limit on the Eötvös-parameter difference, and a rigorous determination of the sensitivity floor of this electromagnetic-versus-gravitational comparison and of whether it can ever be physically meaningful.

Background

The inertial mass mim_i of a body fixes the force needed to accelerate it, mi=F/am_i = F/a; its gravitational mass mgm_g fixes the force of Newtonian gravitation on it, F=GmgMg/r2F = G m_g M_g / r^2. The equivalence principle identifies the two, and implies the universality of free fall (UFF): in a given gravitational field all bodies fall with the same acceleration, independent of composition. Writing mg(X)/mi(X)=1+η(X)m_g(X)/m_i(X) = 1 + \eta(X) for a material XX, where η\eta is the Eötvös parameter, experiments can only constrain differences Δη(X,Y)=η(X)−η(Y)\Delta\eta(X,Y) = \eta(X) - \eta(Y). Torsion-balance null experiments bound Δη\Delta\eta for pairs of test materials at the level of a few parts in 101310^{13}, and the MICROSCOPE space mission reached the order of 10−1510^{-15}; both compare gravitational and inertial accelerations of freely suspended or orbiting bodies.

Since 20 May 2019 the kilogram is defined through the fixed numerical value of the Planck constant hh. Because the stipulated mass defect between the two hyperfine ground states of the 133Cs atom is exactly hνCs/c2h\nu_{\mathrm{Cs}}/c^2 — a consequence of E=hνE = h\nu and E=mc2E = mc^2, where the mass in E=mc2E = mc^2 is the inertial (rest) mass of special relativity — the kilogram is now, in principle, the unit of inertial mass. Two primary methods realize it. In the X-ray crystal density (XRCD) method, a quasi-spherical single crystal of enriched 28Si is measured; the number of atoms NSi=8V/a03N_{\mathrm{Si}} = 8V/a_0^3 follows from the sphere volume VV and the cubic lattice parameter a0a_0, and the mass follows from NSiN_{\mathrm{Si}} times the atomic mass. The absolute atomic-mass scale is anchored to hh through atom-recoil and spectroscopic determinations of h/mh/m (inertial quantities) and relative mass spectrometry in Penning traps (cyclotron frequencies, again inertial); gravity plays no role. In a Kibble balance, the weight of a kilogram artifact is balanced by an electromagnetic force on a coil in a magnetic field (weighing mode), and the voltage induced when the same coil moves at known velocity is measured (moving mode); Josephson and quantum-Hall effects (KJ=2e/hK_J = 2e/h, RK=h/e2R_K = h/e^2) tie the electrical quantities to hh. The local gravitational acceleration gg is measured with a free-fall gravimeter, which observes the kinematic acceleration aa of a falling body, so that g=(mi(ff)/mg(ff)) ag = (m_i(\mathrm{ff})/m_g(\mathrm{ff}))\,a for that body. Consequently the Kibble balance determines

mKB(K)=mi(K) 1+η(K)1+η(ff),m_{\mathrm{KB}}(K) = m_i(K)\,\frac{1+\eta(K)}{1+\eta(\mathrm{ff})},

which equals the inertial mass of the weighed artifact KK only under equivalence-principle assumptions; under ordinary circumstances (falling body different from the weighed artifact, UFF assumed for the former) it delivers the artifact's gravitational mass.

The international comparison of kilogram realizations CCM.M-K8.2019 (Kibble balances at BIPM, KRISS, NIST, NRC, a joule balance at NIM, and XRCD determinations at NMIJ and PTB, compared through 1 kg transfer standards at the BIPM) established the agreement of the two routes at the few-parts-in-10810^8 level and initiated dissemination of the kilogram from a consensus value. Mana and Schlamminger observed that this comparison embodies, in more than one step, the weighing of the same object by atom counting and by Kibble balance, so that agreement yields a constraint on Δη(ff, 28Si)\Delta\eta(\mathrm{ff},\,{}^{28}\mathrm{Si}); they assessed the reach of this constraint only qualitatively and did not derive a bound from the comparison data. They nevertheless argued the comparison should not be dismissed, for two unique features: it rests on the absolute weighing of a 1 kg body, and it compares an electromagnetic with a gravitational acceleration, whereas Eötvös-like experiments compare gravitational with inertial accelerations.

Problem Statement

Establish, quantitatively, what test of the equivalence principle the agreement between the two primary realizations of the Planck-constant-based kilogram constitutes. Since atom counting by the XRCD method realizes the inertial mass of a 28Si sphere without assuming the universality of free fall, while the Kibble-balance result under ordinary circumstances presumes equivalence of inertial and gravitational mass, the agreement of the two realizations bounds the Eötvös-parameter difference Δη(ff, 28Si)\Delta\eta(\mathrm{ff},\,{}^{28}\mathrm{Si}) between the free-fall gravimeter's dropping body and the silicon sphere. The problem is to: (1) construct the formal measurement model that maps the published results and full uncertainty structure of the international comparisons of kilogram realizations (CCM.M-K8.2019 and its successors) — including the multi-step transfer chains, vacuum-to-air transfers, and correlated uncertainty components — onto a rigorous upper limit on Δη\Delta\eta, making every reliance on the equivalence principle explicit (the gravimeter chain, intermediate transfer artifacts, and the conditions under which the Kibble balance yields inertial rather than gravitational mass); (2) evaluate the numerical bound implied by currently published comparison data; and (3) determine the sensitivity floor of this route, i.e. the realization-comparison uncertainty, if any, at which the test becomes physically meaningful — a test counting as physically meaningful when its projected bound on Δη\Delta\eta is competitive with the existing torsion-balance and space null tests of the universality of free fall at the 10−1310^{-13} to 10−1510^{-15} level — taking into account that it compares an electromagnetic with a gravitational acceleration and involves the absolute weighing of a 1 kg artifact.

References

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