Non-singular quantum-inspired gravitational collapse

We consider general relativistic homogeneous gravitational collapses for dust and radiation. We show that replacing the density profile with an effective density justified by some quantum gravity framework leads to the avoidance of the final singularity. The effective density acts on the collapsing cloud by introducing an isotropic pressure, which is negligible at the beginning of the collapse and becomes negative and dominant in the strong-field regime. Event horizons never form and therefore the outcome of the collapse is not a black hole, in the sense that there are no regions causally disconnected from future null infinity. Apparent horizons form when the mass of the object exceeds a critical value, disappear when the matter density approaches an upper bound and gravity becomes very weak (asymptotic-freedom regime), form again after the bounce as a consequence of the decrease in the matter density, and eventually disappear when the density becomes too low and the matter is radiated away. The possibility of detecting radiation coming from the high-density region of a collapsing astrophysical object in which classically there would be the creation of a singularity could open a new window to experimentally test theories of quantum gravity.

Figure 1

Dust collapse models. Left panel: The density $\rho$ in the classical model (solid line), the density $\rho$ in the quantum-gravity-inspired collapse model with $\gamma =1$ (dashed line), and the effective density ${\rho }_{\mathrm{eff}}$ in the quantum-gravity-inspired collapse model with $\gamma =1$ (dashed-dotted line). Right panel: Plot of $a(t)$ in the classical case (solid line) and in the semiclassical model with $\gamma =1$ (dashed line) and $\gamma =2$ (dotted-dashed line). Near the initial time, the semiclassical model has a behavior close to the dust. $a$ either reaches a minimum at $t=t_{\mathrm{cr}}$ and then grows for $t>t_{\mathrm{cr}}$ ($\gamma =1$), or approaches asymptotically a minimum value ($\gamma =2$). Here, $M_0=1$ and ${\rho }_{\mathrm{cr}}=3000$. See the text for details.

Figure 2

Dust collapse models. Left panel: The mass profile $M=M_0$ (solid line) and $M_{\mathrm{eff}}$ for $\gamma =1$ (dashed line). At $t=t_{\mathrm{cr}}$, the effective mass vanishes and thus the spacetime is flat; we are in a regime of asymptotic freedom. Right panel: In order to investigate the breakdown of the weak energy condition, we plot $\rho$ for the classical dust case (solid line) and ${\rho }_{\mathrm{eff}}+p_{\mathrm{eff}}$ for $\gamma =1$ (dashed line). Here, $M_0=1$ and ${\rho }_{\mathrm{cr}}=3000$. See the text for details.

Figure 3

The apparent horizon curve $r_{\mathrm{ah}}(t)$ for the classical dust model (solid line) and the semiclassical model for $\gamma =1$ (dashed line). If the boundary of the cloud is taken smaller than $r_{\min }$ (dashed-dotted line) there are never trapped surfaces forming in the quantum-inspired model. At the time of the bounce, when $\rho ={\rho }_{\mathrm{cr}}$ and gravity is turned off, the spacetime is flat, and there is no horizon. This is a regime of asymptotic freedom.

Figure 4

Radiation collapse models. Left panel: The density $\rho$ in the classical model (solid line), the density $\rho$ in the quantum-inspired collapse model with $\gamma =1$ (dashed line), and the effective density ${\rho }_{\mathrm{eff}}$ in the quantum-inspired collapse model with $\gamma =1$ (dashed-dotted line). Right panel: Plot of $a(t)$ in the classical case (solid line) and in the semiclassical models with $\gamma =1$ (dashed line) and $\gamma =2$ (dotted-dashed line). Near the initial time, the semiclassical model has a behavior close to the classical radiation. $a$ either reaches a minimum at $t=t_{\mathrm{cr}}$ and then grows for $t>t_{\mathrm{cr}}$ ($\gamma =1$), or approaches asymptotically a minimum value ($\gamma =2$). Here, $M_0=1$ and ${\rho }_{\mathrm{cr}}=3000$. See the text for details.

Figure 5

Radiation collapse models. Left panel: The mass profile $M(t)=M_0/a$ for classical radiation (solid line) and for the semiclassical model (dashed line), together with the effective mass profile $M_{\mathrm{eff}}$ (dotted-dashed line). At $t=t_{\mathrm{cr}}$, in the semiclassical model $M(t)$ reaches a maximum, while the effective mass vanishes: the spacetime is flat and we are in a regime of asymptotic freedom. Right panel: $\rho +p$ for the classical radiation case (solid line) and ${\rho }_{\mathrm{eff}}+p_{\mathrm{eff}}$ for $\gamma =1$ (dashed line). In the latter case, the weak energy condition does not hold in the strong-field limit. Here, $M_0=1$ and ${\rho }_{\mathrm{cr}}=3000$. See the text for details.

Figure 6

The apparent horizon curve $r_{\mathrm{ah}}(t)$ for the classical radiation model (solid line) and the semiclassical model for $\gamma =1$ (dashed line). If the boundary of the cloud is taken smaller that $r_{\min }$ (dashed-dotted line) there are no trapped surfaces forming in the quantum-inspired model. At the time of the bounce, when $\rho ={\rho }_{\mathrm{cr}}$ and gravity is turned off, the spacetime is flat and there is no horizon. This is a regime of asymptotic freedom.