---
abstract: 'Non classical rotational inertia observed in rotating supersolid $He^4$ can be accounted for by a gravitomagnetic London moment similar to the one recently reported in rotating superconductive rings.'
author:
- 'C. J. de Matos[^1]'
title: Gravitomagnetic London Moment in Rotating Supersolid $He^4$
---

Non Classical Rotational Inertia (NCRI) was predicted by London 50 years ago [@London]. It was eventually verified experimentally by Hess and Fairbank [@Hess], who set Helium in a suspended bucket into rotation above the critical temperature $T_c$ at which superfluidity sets in, and then cooled it (with the bucket still rotating at angular velocity $\omega$) through $T_c$. They found that, provided $\omega$ is less than a critical value $\omega_c$, the apparent moment of inertia of the helium - that is, the ratio of its angular momentum to $\omega$ - is not given by the classical value $I_{classical}$ but rather by $$I(T)=\frac{L}{\omega}=I_{classical}\Bigg[1-f_s(T)\Bigg]\label{1}$$ where $f_s(T)=\rho^*/\rho$ is the superfluid fraction. In liquid helium, $f_s(T)$ tends to 1 in the zero-temperature limit and to zero when $T=T_c$, $\rho^*$ and $\rho$ being respectively the mass density of superfluid helium, and the mass density of normal helium.

NCRI has recently been observed in rotating supersolid $He^4$ by Kim and Chan [@Kim]. They measured the resonance frequency of a torsional oscillator that contains an annulus of solid $He^4$. Below $230 mK$, the frequency experiences a relative increase that depends on the temperature and drive amplitude and reaches a maximum of about four parts in $10^5$. Having excluded by various control experiments, other explanations, they conclude that the data indicate a change in the moment of inertia of the supersolid, which according to Equ.(\[1\]), corresponds to a maximum supersolid fraction ($f_s(T)=\rho^*_{supersolid \,
He^4}/\rho_{normal \, solid \, He^4}$) of $~0.017$ (see figure \[chan\]).

Tajmar and the author recently [@Tajmar1] [@de; @Matos1] [@Tajmar2] observed a gravitomagnetic London moment, $B_g \,
[Rad/s]$, in rotating superconductive rings. $$B_g=2\omega f_s(T)\label{2}$$ Where $\omega$ is the angular velocity of the ring, and $f_s(T)=\rho^*/\rho$ is the Cooper pairs fraction, $\rho^*$ being the Cooper pairs mass density and $\rho$ the superconductor’s bulk density.

Assuming that a rotating supersolid $He^4$ crystal also exhibits a gravitomagnetic London moment, $B_g$ proportional to the supersolid fraction, its angular momentum would be given by $$L=I_{classical}\Bigg[\omega-\frac{1}{2} B_g\Bigg]\label{3}$$ Doing Equ.(\[2\]) into Equ.(\[3\]) we find back Equ.(\[1\])!

Therefore we conclude that the rotation of supersolid $He^4$ exhibits a gravitomagnetic London moment similar to the one observed in rotating superconductive rings, which can account for the observed NCRI in this physical system. Kim and Chan’s experiment would tend to confirm the existence of the gravitomagnetic London moment in rotating quantum materials.

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London, F., “Superfluids”, wiley New York 1954, Vol II, P. 144

Hess, G. B., Fairbank, W. M., Phys. Rev. Lett., **19** 216 (1967)

Kim, E., Chan, W., Science, **305**, 1941 (2004)

Tajmar, M., Plesescu, F., Marhold, K., de Matos, C.J., “Experimental Detection of the Gravitomagentic London Moment”, 2006, gr-qc/0603033

de Matos, C. J., “Gravitoelectromagnetism and Dark Energy in Superconductors”, to appear in Int. J. Mod. Phys. D, (2007). (also available gr-qc:/0607004)

Tajmar, M., Plesescu, F., Marhold, K. “Measurement of Gravitomagentic and acceleration fields around a rotating superconductor”, 2006, gr-qc/0610015

[^1]: ESA-HQ, European Space Agency, 8-10 rue Mario Nikis, 75015 Paris, France, e-mail: Clovis.de.Matos@esa.int
