Kiefer C. Quantum gravity

ARROW OF TIME 321
to understand the validity of (10.42) for the universe as a whole. In an early
stage, the universe was a hot plasma in thermal equilibrium. Only the expansion
of the universe and the ensuing redshift of the radiation are responsible for the
fact that radiation has decoupled from matter and cooled to its present value
of about three Kelvin—the temperature of the approximately isotropic cosmic
background radiation with which the night sky ‘glows’. During the expansion, a
strong thermal non-equilibrium could develop, which enabled the formation of
structure.
The second arrow is described by the Second Law of Thermodynamics: for a
closed system entropy does not decrease. The total change of entropy is given by
( ) ( )
dS dS dS
dt = + ,
dt
ext
dt
int
} {{ } } {{ }
dS ext=δQ/T ≥0
so that according to the Second Law, the second term is non-negative. As the
increase of entropy is also relevant for physiological processes, the Second Law
is responsible for the subjective experience of irreversibility, in particular for the
ageing process. If applied to the universe as a whole, it would predict the increase
of its total entropy, which would seem to lead to its ‘heat death’ (‘Wärmetod’).
The laws of thermodynamics are based on microscopic statistical laws which
are time-symmetric. How can the Second Law be derived from such laws? As
early as the nineteenth century objections were formulated against a statistical
foundation of the Second Law. These were, in particular,
• Loschmidt’s reversibility objection (‘Umkehreinwand’), and
• Zermelo’s recurrence objection (‘Wiederkehreinwand’).
Loschmidt’s objection states that a reversible dynamics must lead to an equal
amount of transitions from an improbable to a probable state and from a probable
to an improbable state. With overwhelming probability, the system should
be in its most probable state, that is, in thermal equilibrium. Zermelo’s objection
is based on a theorem by Poincaré, according to which every system comes
arbitrarily close to its initial state (and therefore to its initial entropy) after
a finite amount of time. This objection is irrelevant, since the corresponding
‘Poincaré times’ are bigger than the age of the universe already for systems
with few particles. The reversibility objection can only be avoided if a special
boundary condition of low entropy holds for the early universe. Therefore, for
the derivation of the Second Law, one needs a special boundary condition.
Such a boundary condition must either be postulated or derived from a fundamental
theory. The formal description of entropy increase from such a boundary
condition is done by master equations; cf. Joos et al. (2003). These are equations
for the ‘relevant’ (coarse-grained) part of the system. In an open system, the entropy
can of course decrease, provided the entropy capacity of the environment
is large enough to at least compensate this entropy decrease. This is crucial for