General relativity effects in precision spin experimental tests of fundamental symmetries

A search for the $P$- and $CP(T)$-violating electric dipole moments (EDM) of atoms, particles, and nuclei with sensitivity up to $10^{-15}$ in units of the magnetic dipole moments, allowed by all discrete symmetries, is one of the topical problems of modern physics. According to Sakharov, $CP$ violation is one of the three key criteria of the baryogenesis in the generally accepted paradigm of the Big Bang cosmology. All three criteria are supported by the Standard Model, but it fails to describe quantitatively the observed baryon asymmetry of the Universe. This is regarded as a strong argument in favor of the existence of $CP$-symmetry breaking mechanisms beyond the minimal Standard Model, which can lead to measurable EDMs of atoms, particles, and nuclei. Searches for the EDM via the spin rotation in electric fields are currently underway in dozens of laboratories worldwide. Direct searches for the EDM of charged particles and nuclei are possible only in storage rings (COSY, NICA). After successful studies by the JEDI collaboration at the COSY synchrotron, at the forefront in the field is the search for the proton EDM in an electrostatic storage ring with the proton spin frozen at the magic energy with the projected sensitivity $d_{\rm p}\sim 10^{-29}~e$ cm. A prototype PTR storage ring is proposed as a precursor to such a dedicated storage ring, with the prospect of the frozen proton spin ring becoming a part of the physics at CERN beyond the Large Hadron Collider program. Following a brief introduction to $CP$-violation physics and baryogenesis, the review presents a detailed discussion of significant contributions to the spin dynamics from terrestrial gravity along with new effects of Earth's rotation in ultrasensitive searches for the EDM of charged particles and neutrons. Quite remarkably, for the projected sensitivity to the proton EDM, these false EDM effects can exceed the signal of the proton EDM by one to two orders of magnitude and become comparable to the EDM contribution in experiments with ultracold neutrons. We also discuss the role of a precessing spin as a detector of axion-like dark matter, and consider applications of quantum gravitational anomalies to dense matter hydrodynamics and spin phenomena in noncentral nuclear collisions.