**6.2. Extreme-Relativistic Scaling Laws**

The above summary describes the qualitative features of a roughly
homogeneous expanding fireball. Surprisingly similar scaling laws
exists also for inhomogeneous fireballs
[229]
as well as for relativistic winds
[53].
Consider a spherical fireball with
an arbitrary radial distribution of radiation and matter. Under
optically thick conditions the radiation and the relativistic leptons
(with energy density *e*) and the matter (with baryon mass density
) at each
radius behave like a single fluid, moving with the
same velocity. The pressure, *p*, and the energy density, *e*, are
related by *p* = *e* / 3. We can express the relativistic
conservation equations of baryon number, energy and momentum using
characteristic coordinates: *r* and
*s* *t* -
*r* as
[229]:

(16) |

(17) |

(18) |

where *u* = *u*^{r} =
(^{2}
- 1)^{1/2}, and we use units in which *c* = 1 and the
mass of the particles *m* = 1. The derivative
/
*r*
now refers to constant *s*, i.e. is calculated along a characteristic
moving outward at the speed of light. After a short acceleration
phase we expect that the motion of a fluid shell will become highly
relativistic
( >>
1). If we restrict our attention to the
evolution of the fireball from this point on, we may treat
^{-1}
as a small parameter and set
*u*, which is
accurate to order *O*(^{-2}).
Then, under a wide range of conditions the quantities
on the right-hand sides of Eqs. 16-18 are significantly
smaller than those on the left. When we neglect the right hand sides
of Eqs. 16-18 the problem becomes effectively only *r*
dependent. We obtain the following conservation
laws for each fluid shell:

(19) |

A scaling solution that is valid in both the radiation-dominated and
matter-dominated regimes, as well as in the transition zone in
between, can be obtained by combining the conserved quantities in
Eq. 19 appropriately. Let *t*_{0} be the time and
*r*_{0} be the radius at which a fluid shell in the
fireball first becomes ultra-relativistic, with
few. We label various
properties of the shell at this time by a subscript 0, e.g.
_{0},
_{0},
and *e*_{0}. Defining the auxiliary quantity *D*, where

(20) |

we find that

(21) |

These are parametric relations which give *r*,
, and *e*
of each fluid shell at any time in terms of the
of the
shell at that time. The parametric solution 21 describes both the
radiation-dominated and matter-dominated phases of the fireball within
the frozen pulse approximation. That is as long as the fireball does
not spread due to variation in the velocity.