Gravity Probe Spin: Prospects for measuring general-relativistic precession of intrinsic spin using a ferromagnetic gyroscope

An experimental test at the intersection of quantum physics and general relativity is proposed: measurement of relativistic frame dragging and geodetic precession using intrinsic spin of electrons. The behavior of intrinsic spin in spacetime dragged and warped by a massive rotating body is an experimentally open question, hence the results of such a measurement could have important theoretical consequences. Such a measurement is possible by using mm-scale ferromagnetic gyroscopes in orbit around the Earth. Under conditions where the rotational angular momentum of a ferromagnet is sufficiently small, a ferromagnet’s angular momentum is dominated by atomic electron spins and is predicted to exhibit macroscopic gyroscopic behavior. If such a ferromagnetic gyroscope is sufficiently isolated from the environment, rapid averaging of quantum uncertainty via the spin-lattice interaction enables readout of the ferromagnetic gyroscope dynamics with sufficient sensitivity to measure both the Lense-Thirring (frame dragging) and de Sitter (geodetic precession) effects due to the Earth.

Figure 1

Sensitivity to general relativistic spin-precession effects in the proposed “Gravity Probe Spin” experiment. The vertical scale on the right is in units of milliarcseconds (mas) per year. The black curve shows the projected uncertainty $\mathrm{\Delta }\mathrm{\Omega }$ in the measurement of the precession frequency $\mathrm{\Omega }$ using a 1-mm radius spherical FG under conditions listed in Table 1. This curve results from two contributions summed in quadrature. First, the short-term statistical uncertainty is dominated by background gas collisions [Eq. (5), dashed gray line]. Second, the long-term uncertainty in the measurement is expected to be dominated by magnetic field drift within the superconducting magnetic shields, here assumed to be linear with rate $3\times {10}^{-26}\mathrm{G}/\mathrm{s}$ (dotted gray line). The blue line and light blue shaded area indicate the level beyond which the measurements are sensitive to the de Sitter effect [13, 51, 52] and the red line and pink shaded area indicate the level beyond which the measurements are sensitive to the Lense-Thirring effect [12, 50], calculated for the GP-B orbit and gyrogravitational ratio $\mathfrak{g}=1$ [Eqs. (2) and (3)]. The green line and light green shaded area show existing experimental constraints on anomalous gravity-induced spin-precession [81, 82, 83].

Figure 2

Conceptual schematic diagram of a “Gravity Probe Spin” experiment. A freely floating spherical FG located within a superconducting shield is in a circular polar orbit. The magnetic field $\mathbf{B}$ (from the frozen flux in the superconducting shields) is oriented parallel to the direction of Earth’s rotation axis ${\mathbf{\Omega }}_E$, both designated to point along $z$. The insert shows the initial orientation of the FG’s magnetic moment and spin $m$ along the $x$ axis. The pick-up coil measures the FG’s magnetization along $x$. This geometry is designed for the detection of the Lense-Thirring effect.

Figure 3

Estimated power spectral density (PSD) of the time-dependent flux signal $\mathrm{\Phi }$ due to a precessing FG that would be measured by a SQUID pick-up coil as in Fig. 2. The plot shows the PSD of a time-domain signal of duration $T=3\times {10}^7\mathrm{s}$ obtained by numerical solution of differential equations based on the model discussed in Sec. pp2. The parameters of the model match those listed in Table 1. The gray dotted line marks the Larmor frequency, ${\mathrm{\Omega }}_B/(2\pi )$, the gray dot-dashed line marks the nutation frequency, ${\mathrm{\Omega }}_1/(2\pi )$, and the red dashed line marks the second harmonic of the orbital frequency, $v/(\pi R)$. In order to enhance visualization, for this plot we choose $\mathfrak{g}={10}^7$ for the Lense-Thirring effect, just below the present experimental constraints (Fig. 1 in the main text).

Figure 4

The black curve shows the PSD of the time-dependent flux signal $\mathrm{\Phi }$ under the same conditions and assumptions as in Fig. 3. The blue curve, vertically offset for easier comparison, shows the PSD of the time-dependent flux signal $\mathrm{\Phi }$ for the case where the gyrogravitational ratio $\mathfrak{g}=0$. The dashed red line marks the second harmonic of the orbital frequency, $v/(\pi R)$, and prominent signals at sidebands shifted by the Larmor frequency are indicated by the red arrows at $v/(\pi R)\pm {\mathrm{\Omega }}_B/(2\pi )$. Note also sidebands at $v/(\pi R)\pm {\mathrm{\Omega }}_B/\pi$.

Figure 5

PSD of the difference in time-dependent flux signal with $\mathfrak{g}=1$ between two gyroscopes. The gyroscopes situated in opposite external magnetic fields along the $z$ axis. The conditions and assumptions are the same as in Fig. 3. The dashed red line marks the second harmonic of the orbital frequency, $v/(\pi R)$, and prominent signals at sidebands shifted by the Larmor frequency are indicated by the red arrows at $v/(\pi R)\pm {\mathrm{\Omega }}_B/(2\pi )$.

Figure 6

Conceptual schematic diagram of a “Gravity Probe Spin” experiment similar to that shown in Fig. 2 except that the orbit is elliptical and the magnetic field $\mathbf{B}$ is directed along the $y$-axis, perpendicular to the orbital plane. This geometry is designed for the detection of the de Sitter effect.

Figure 7

PSD of the difference in time-dependent flux signal with $\mathfrak{g}=1$ between two gyroscopes. The gyroscopes are situated, respectively, in external magnetic fields along the $y$ axis with equal magnitudes and opposite directions. The FG is modeled for the duration of ${10}^6\mathrm{s}$ in a polar elliptical orbit as indicated in Fig. 6, with ellipticity of 0.3. The dashed red line marks the first harmonic of the orbital frequency, ${\omega }_{\mathrm{orb}}/(2\pi )$, and prominent signals at sidebands shifted by the Larmor frequency are indicated by the red arrows at ${\omega }_{\mathrm{orb}}/(2\pi )\pm {\mathrm{\Omega }}_B/(2\pi )$.