Josephson Effects in Slowly Rotating Spacetimes

Abstract

We investigate Josephson phenomena in a slowly rotating stationary spacetime, emphasizing the distinct roles of gravitational redshift and rotational frame dragging motivated by [10.1007/JHEP02(2026)006]. Using a covariant formulation based on gauge-invariant phase dynamics and conserved currents within a $3+1$ decomposition, we analyze both AC and DC Josephson effects and interferometric configurations. Restricting attention to linear order in the rotation parameter $a$, and working in the Eulerian/ZAMO frame, we show that in the slow-rotation slicing adopted here the lapse function agrees with its static (Schwarzschild-type) form up to $\mathcal{O}(a^2)$, while rotational effects enter through the shift vector. Consequently, redshift effects on Josephson frequencies and DC critical currents remain unchanged relative to the non-rotating case at $\mathcal{O}(a)$. The AC Josephson relation retains its redshifted structure when expressed in terms of proper voltages and reduces to the standard flat-spacetime form when formulated in terms of asymptotic (Killing-time) observables. Likewise, the DC critical current measured at infinity scales with a single power of the lapse function and is unaffected by rotation at linear order in the absence of azimuthal condensate momentum. Rotational effects become relevant only in configurations sensitive to spatial phase transport or to synchronization with respect to the global time coordinate. In particular, RF-driven interferometric setups can acquire Sagnac-type phase offsets associated with frame dragging, whereas the DC fluxoid constraint remains unshifted at linear order in the present approximation. Our results provide a clean separation between lapse-driven redshift effects and shift-driven rotational contributions in Josephson physics and furnish a consistent framework for superconducting circuits in stationary spacetimes.

Figures (2)

• Figure 1: Frame-dragging-induced phase shift $\Delta\psi_{\rm FD}$ as a function of the frequency $f$ for an equatorial loop around a neutron star (solid blue curve) and around the Earth (dashed red curve). The phase shift is obtained from the operational relation $\Delta\psi_{\rm FD}=\omega\,\Delta t_{\rm FD}=2\pi f\,\Delta t_{\rm FD}$, where $\Delta t_{\rm FD}$ is the Sagnac-type time delay induced by frame dragging (Eq. \ref{['eq:time_delay']}). The neutron-star signal is strongly enhanced by the compactness of the source, whereas the terrestrial phase shift remains negligible over the plotted frequency range. The legends, tick labels, and axes titles are too small. Please enlarge. You can remove the plot title to be consistent with Fig. 3.
• Figure 2: The legends, tick labels, and axes titles are too small. Please enlarge. Comparison between DC and RF-driven SQUID interference envelopes in the slow-rotation regime. The solid curve shows the DC envelope, which depends only on the magnetic flux $\Phi$ and is not translated by frame dragging at linear order. The dashed curve corresponds to the first Shapiro step ($n=1$) with a finite non-gravitational phase offset $\delta$. The dotted curve includes an additional frame-dragging contribution $\Delta\psi_{\rm FD}=\omega_J\Delta t_{\rm FD}$, producing a further translation of the RF envelope. Gravitational redshift affects only the overall amplitude through the lapse function. In the plot legend, the label $\Delta\psi$ denotes the same non-gravitational offset $\delta$.