  The Kerr black hole is stationary and axisymmetric, which leads to
conservation of energy and azimuthal angular momentum along the orbits of free
test particles in its vicinity, but also to conservation laws for the evolution
of continuum matter fields. However, the Kerr space-time possesses an
additional "hidden symmetry" which exhibits itself in an unexpected conserved
quantity along geodesics known as the Carter constant. We investigate the
possibility of using this hidden symmetry to obtain conservation laws and other
identities which could be used to test astrophysical simulations of the
evolution of matter fields near spinning black holes. After deriving such
identities, we set up a simple numerical toy model on which we demonstrate how
they can detect the violations of evolution equations in a numerical
simulation. Even though one of the expressions we derive is in the form of a
conservation law, we end up recommending an equivalent but simpler expression
that is not in the form of a conservation law for practical implementation.
