An approximate application of quantum gravity to the rotation problem

Abstract

{Arbitrary initial conditions allow solutions of Einstein's field equations for General Relativity to have arbitrarily large relative rotation of matter and inertial frames. The ``Rotation Problem'' is to explain why the measured relative rotation rate is so small. As it turns out, nearly any reasonable theory of quantum gravity can solve the rotation problem by phase interference.} Even as early as {}{about a quarter of a second after the initial simgularity, quantum cosmology would limit the cosmologies that contribute significantly to a path integral calculation to have relative rms rotation rates less than about} {{}}{$10^{-51}$ radians per year.} Those calculations are based on using 50 e-foldings during inflation. For 55 or 60 e-foldings, the cosmologies contributing significantly to the path integral would have even smaller relative rotation rates. In addition, although inflation dominates the calculation, even if there had been no inflation, the cosmologies contributing significantly to the path integral would have relative rotation rates less than about {}{$10^{-32}$ radians per year at about a quarter of a second after the initial singularity.} These calculations are insensitive to the details of the theory of quantum gravity because the main factor depends only on the size of the visible universe, the Planck time, the free-space speed of light, the Hubble parameter, and the number of e-foldings during inflation. These calculations use the Einstein-Hilbert action in quantum gravity, {{}}{including} large-scale relative rotation of inertial frames and the matter distribution, in which each ``path'' is a cosmology with a different rms relative rotation rate. The calculations include inflation for 50, 55, and 60 e-foldings, and for values of the dependence of relative rotation rate on cosmological scale factor $a$ as $a^{-m}$ for various values of $m$. The calculation shows that the action is an extremum at zero rms relative rotation rate.