Frozen-in tetrad field strengths in general relativity: characterization, existence, and tetrad dependence

The exterior derivatives of the one-forms of an orthonormal tetrad have recently been proposed as gravitational field strengths that are frozen into a timelike congruence when an ideal Ohm-type condition holds, with conserved fluxes and helicity. We study this condition as an equation for the tetrad and the congruence. Its general local solution is a sum of exact one-forms and one-forms that are constant along the congruence. If the transport field has constant tetrad components, as for adapted tetrads, it is a Killing field of constant norm and the only vacuum solution is flat. For geodesic congruences the third Lie derivative of the metric vanishes and the tidal tensor is constrained by the kinematics, which excludes the comoving observers of expanding Friedmann-Lemaître-Robertson-Walker universes other than the flat Milne universe and all irrotational geodesic congruences of curved vacuum spacetimes. Spherically symmetric tetrads of the Schwarzschild spacetime and tetrads of the Kerr spacetime that are invariant under its Killing flows are excluded for all observers. Nevertheless, every pp-wave admits an ideal tetrad, and we give exact examples in the Levi-Civita spacetime and in a cosmological model. Finally, the fluxes and the helicity, which is the translational Chern-Simons form of the tetrad, change under local Lorentz transformations with the spacetime and the observers fixed.