We complete our previous investigations concerning the structure and the stability of “ isothermal ” spheres in general relativity .
This concerns objects that are described by a linear equation of state , P = q \epsilon , so that the pressure is proportional to the energy density .
In the Newtonian limit q \rightarrow 0 , this returns the classical isothermal equation of state .
We specifically consider a self-gravitating radiation ( q = 1 / 3 ) , the core of neutron stars ( q = 1 / 3 ) , and a gas of baryons interacting through a vector meson field ( q = 1 ) .
Inspired by recent works , we study how the thermodynamical parameters ( entropy , temperature , baryon number , mass-energy , etc ) scale with the size of the object and find unusual behaviours due to the non-extensivity of the system .
We compare these scaling laws with the area scaling of the black hole entropy .
We also determine the domain of validity of these scaling laws by calculating the critical radius ( for a given central density ) above which relativistic stars described by a linear equation of state become dynamically unstable .
For photon stars ( self-gravitating radiation ) , we show that the criteria of dynamical and thermodynamical stability coincide .
Considering finite spheres , we find that the mass and entropy present damped oscillations as a function of the central density .
We obtain an upper bound for the entropy S and the mass-energy M above which there is no equilibrium state .
We give the critical value of the central density corresponding to the first mass peak , above which the series of equilibria becomes unstable .
We also determine the deviation from the Stefan-Boltzmann law due to self-gravity and plot the corresponding caloric curve .
It presents a striking spiraling behaviour like the caloric curve of isothermal spheres in Newtonian gravity .
We extend our results to d -dimensional spheres and show that the oscillations of mass-versus-central density disappear above a critical dimension d _ { crit } ( q ) .
For Newtonian isothermal stars ( q \rightarrow 0 ) , we recover the critical dimension d _ { crit } = 10 .
For the stiffest stars ( q = 1 ) , we find d _ { crit } = 9 and for a self-gravitating radiation ( q = 1 / d ) we find d _ { crit } = 9.96404372... very close to 10 .
Finally , we give simple analytical solutions of relativistic isothermal spheres in two-dimensional gravity .
Interestingly , unbounded configurations exist for a unique mass M _ { c } = c ^ { 2 } / ( 8 G ) .