Exact Solutions and Scalar Fields in Gravity - Instituto Avanzado de ...

Conformal symmetry and deflationary gas universe 251
Both the collision integral and the force describe interactions
within the many-particle system. While C accounts for elastic
binary collisions, we assume that all other microscopic interactions,
however involved their detailed structure may be, can be mapped onto
an effective one–particle force which may be used for a subsequent
self–consistent treatment. Let us restrict ourselves to the class of forces
which admit solutions of Boltzmann’s equation that are of the type of
Jüttner’s distribution function
where and is timelike. For the collision integral
vanishes: Substituting into Eq. (10) we obtain the
condition
For the most general expression of the relevant force projection [9],
the condition (12) decomposes into the “generalized” equilibrium conditions
With the identification and restricting ourselves to the homogeneous
and isotropic case with and we may read
off that a force with
reproduces the previous consistency condition (7) for this case. The
latter is recovered as generalized equilibrium condition for particles in a
force field of the type (13).
One may interpret this feature in a way which is familiar from gauge
theories: Gauge field theories rely on the fact that local symmetry requirements
(local gauge invariance) necessarily imply the existence of
additional interaction fields (gauge fields). In the present context we
impose the “symmetry” requirement (7) (below we clarify in which
sense the modification of the conformal symmetry is again a symmetry).
Within the presented gas dynamical framework this “symmetry”
can only be realized if one introduces additional interactions, here described
by an effective force field Consequently, in a sense, this force
field may be regarded as the analogue of gauge fields.