Relativistic stars with a linear equation of state: analogy with classical isothermal spheres and black holes

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 = qǫ, so that the pressure is proportional to the energy density. In the Newtonian limit q → 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 dcrit(q). For Newtonian isothermal stars (q → 0), we recover the critical dimension dcrit = 10. For the stiffest stars (q = 1), we find dcrit = 9 and for a self-gravitating radiation (q = 1/d) we find dcrit = 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 Mc = c /(8G).