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In this problem we consider a spherical shell of an ideal gas that expands at ultra - relativistic speeds. Such model has been previously considered in order to describe the early stages of GRBs[1]

The evolution of such shell is determined by the conservation laws. Since both the inner and outer surfaces of the shell move at velocities very close to the speed of light, then the width of the shell (in the lab frame) remains the same. The volume in the lab frame therefore increases as the radius squared $V \propto r^2$. In the co - moving frame the volume would be greater by a Lorentz factor $V' \approx \gamma V \propto \gamma r^2$. The reason only a single Lorentz factor is added is that only Lorentz contraction affects only one dimension. The number density in the co moving frame varies according to

$n \propto V^{-1} \propto r^{-2} \gamma^{-1}$

Entropy is also conserved, therefore the energy in the co moving frame varies as

$e \propto V^{-\eta} \propto r^{-2 \eta} \gamma^{-\eta}$

Where $\eta$ denotes the adiabatic index (usually $\frac{4}{3}$). The last conservation law is the conservation of momentum. This one is a bit tricky. In Newtonian coasting, since mass and momentum are conserved, the velocity remains the same. On the other hand, in the relativistic case, as the shell cools adiabatically, the effective mass / inertia (i.e. ratio between force and acceleration) decreases, so the shell actually accelerates, even though no external force is acting on it. Quantitatively, the effective mass density is the enthalpy $h = e + p$ where $p$ is the pressure. The total effective mass is obtained by multiplying by the co moving volume $M \propto M \gamma r^2$, so the total radial momentum is proportional to $M \beta \gamma \propto h \gamma^2 \beta r^2$. This means that the enthalpy varies as

$h \propto \gamma^{-2} \beta^{-1} r^{-2}$.

In the ultra relativistic limit $h \approx e$ and $\beta \rightarrow 1$. Combining the relations above yields

$\gamma \propto r$

$n \propto r^{-3}$

$e \propto r^{-4}$

References Edit

1. T. Piran, A. Shemi, R. Narayan, Hydrodynamics of Relativistic Fireballs, MNRAS vol. 263, No. 4 page 861 (1993)