  In deep supercooled liquids, particles get trapped in transient cages made up
of neighbouring particles. Here we define a cage from a geometrical quantity,
free volume, such that the free volume of a particle is the cage volume. First
we show that the relationship between the average cage volume and the
structural relaxation time questions the existence of glass transition in hard
sphere systems. Our observation suggests that the cage volume is zero at the
transition. Further we show that cage rearrangements are strongly coupled to
the single particle squared displacements. Additionally a cage can rearrange by
losing its neighbours with almost no change in particle displacements. The
picture presented here also supports the complex scenarios of relaxation,
dynamic heterogeneity and cooperative rearrangement.
