  It is argued that the observed Thermodynamic Arrow of Time must arise from
the boundary conditions of the universe. We analyse the consequences of the no
boundary proposal, the only reasonably complete set of boundary conditions that
has been put forward. We study perturbations of a Friedmann model containing a
massive scalar field but our results should be independent of the details of
the matter content. We find that gravitational wave perturbations have an
amplitude that remains in the linear regime at all times and is roughly time
symmetric about the time of maximum expansion. Thus gravitational wave
perturbations do not give rise to an Arrow of Time. However density
perturbations behave very differently. They are small at one end of the
universe's history, but grow larger and become non linear as the universe gets
larger. Contrary to an earlier claim, the density perturbations do not get
small again at the other end of the universe's history. They therefore give
rise to a Thermodynamic Arrow of Time that points in a constant direction while
the universe expands and contracts again. The Arrow of Time does not reverse at
the point of maximum expansion. One has to appeal to the Weak Anthropic
Principle to explain why we observe the Thermodynamic Arrow to agree with the
Cosmological Arrow, the direction of time in which the universe is expanding.
