Gravitational collapse of Hagedorn fluids

We consider a toy model for the relativistic collapse of a homogeneous perfect fluid that takes into account an equation of state for high density matter, in the form of a Hagedorn phase, and semiclassical corrections in the strong field. We show that collapse reaches a critical minimum size and then bounces. We discuss the conditions needed for the collapse to halt and form a compact object. We argue that implications of models such as the one presented here are of great importance for astrophysics as they show that black holes may not be the only final outcome of collapse of very massive stars.

Figure 1

Plot of $p_0/\rho$ as a function of $k\in [-1,1]$. The weak energy condition $\rho +p\ge 0$ is satisfied in regions (I) and (IV) and violated in regions (II) and (III). The solid line is given by $-k$, while the dashed line is given by $-k/(k+1)$ and for $\rho ={\rho }_0$ represents the region where collapse halts and $a_{\mathrm{cr}}=0$.

Figure 2

Comparison of the scale factor in the four cases of classical or semiclassical collapse with linear EOS or Hagedorn EOS with the parameters chosen as $k=1/3$, $p_0=50$ and ${\rho }_0=1000/3$ and the initial conditions taken as $a(0)=1$, $M_0=1$. (i) The solid line is the semiclassical Hagedorn collapse, (ii) the dotted line is the semiclassical collapse with linear EOS, (iii) the dashed line is the classical Hagedorn collapse, and (iv) the dotted-dashed line is the classical collapse with linear EOS. Note that cases (i) and (ii) lead to a bounce when $a=a_{\mathrm{cr}}$, while cases (iii) and (iv) lead to the formation of a singularity when $a=0$.

Figure 3

Comparison of the apparent horizon in the four cases of classical or semiclassical collapse with linear EOS or Hagedorn EOS with the parameters chosen as $k=1/3$, $p_0=50$ and ${\rho }_0=1000/3$ and the initial conditions taken as $a(0)=1$, $M_0=1$. (i) The solid line is the semiclassical Hagedorn collapse, (ii) the dotted line is the semiclassical collapse with linear EOS, (iii) the dashed line is the classical Hagedorn collapse, and (iv) the dotted-dashed line is the classical collapse with linear EOS. Note that in cases (i) and (ii) the apparent horizon curve $r_{\mathrm{ah}}$ goes to infinity as $a\to a_{\mathrm{cr}}$, while in cases (iii) and (iv) $r_{\mathrm{ah}}\to 0$ as $a\to 0$.

Figure 4

The thick line represents the boundary of the cloud $r_b$; the grey area represents the trapped region. Dashed lines represent the classical apparent horizon (a.h.) and event horizon (e.h.), without semiclassical corrections. After the time of the bounce $t_B$ the cloud expansion is described by the time reversal of the collapsing solution. Left panel: Penrose diagram of homogeneous collapse with semiclassical corrections. Every shell bounces at the same time $t_B$ when the effective mass becomes zero. Right panel: Penrose diagram for the more realistic case of inhomogeneous collapse. As the central shell reaches the critical density, the outer shells still have nonzero effective mass. The total mass of the system decreases until the time of the bounce and then increases again. The outer horizon shrinks to a minimum radius due to the decrease of the effective mass. After the bounce the black hole turns into a white hole.