  In most theories of gravity involving torsion, the source for torsion is the
intrinsic spin of matter. Since the spins of fermions are normally randomly
oriented in macroscopic bodies, the torsion generated is normally negligible.
However, in a recent paper, Mao et al. point out that there is a class of
theories in which the angular momentum of macroscopic spinning bodies generates
a significant amount of torsion. They argue that by the principle of action
equals reaction, one would expect the angular momentum of test bodies to couple
to a background torsion field, and therefore the precession of the GPB
gyroscopes should be affected in these theories by the torsion generated by the
Earth. We show that in fact the principle of action equals reaction does not
apply to these theories. We examine in detail a generalization of the
Hayashi-Shirafuji theory suggested by Mao et al. called
Einstein-Hayashi-Shirafuji theory. There are a variety of different versions of
this theory, depending on the precise form of the coupling to matter chosen for
the torsion. We show that for any coupling to matter that is compatible with
the spin transport equation postulated by Mao et al., the theory has either
ghosts or an ill-posed initial value formulation. These theoretical problems
can be avoided by specializing the parameters of the theory and in addition
choosing the standard minimal coupling to matter of the torsion tensor. This
yields a consistent theory, but one in which the action equals reaction
principle is violated, and in which the angular momentum of the gyroscopes does
not couple to the Earth's torsion field. Thus, the Einstein-Hayashi-Shirafuji
theory does not predict a detectable torsion signal for Gravity Probe B. There
may be other torsion theories which do.
