  In this paper, we demonstrate how the Lyapunov exponents close to zero of a
system of many hard spheres can be described as Goldstone modes, by using a
Boltzmann type of approach. At low densities, the correct form is found for the
wave number dependence of the exponents as well as for the corresponding
eigenvectors in tangent-space. The predicted values for the Lyapunov exponents
belonging to the transverse mode are within a few percent of the values found
in recent simulations, the propagation velocity for the longitudinal mode is
within 1%, but the value for the Lyapunov exponent belonging to the
longitudinal mode deviates from the simulations by 30%. For higher densities,
the predicted values deviate more from the values calculated in the
simulations. These deviations may be due to contributions from ring collisions
and similar terms, which, even at low densities, can contribute to the leading
order.
