Regular black holes and boson stars in semiclassical gravity

TL;DR

This work develops a canonical semiclassical gravity framework in which a quantum scalar field backreacts on classical geometry in spherical symmetry. By replacing matter terms with their expectation values in the effective constraints $H_{ m eff}=H_G+\langle H_M\rangle=0$ and $C_{ m eff}=C_G+\langle C_M\rangle=0$, and solving the static sector with a fixed quantum state up to a phase, the authors derive five coupled ODEs for $\mathcal{N}=N^2$, $\mathcal{N}^r$, and $\langle V(\phi)\rangle$, sourced by $\langle\phi'^2\rangle$ and $\langle P_\phi^2\rangle$. Numerical solutions produce boson stars and regular black holes that are asymptotically flat, de Sitter, or anti-de Sitter, with localized matter sources and a cosmological-constant-like asymptote given by $\langle V(\phi)\rangle\to\lambda$. The study shows that standard nonsingular black hole proposals (e.g., Bardeen, Hayward) do not arise in this framework, and reveals features such as quantum-hair tails extending beyond horizons. Overall, the method captures full backreaction without computing $\langle T_{ab}\rangle$ on a fixed background and opens paths to axisymmetric and dynamical extensions, while highlighting limitations in resolving curvature singularities within this semiclassical setting.

Abstract

We use a Hamiltonian version of the semiclassical Einstein equation to study classical gravity coupled to a quantum scalar field with potential in spherical symmetry. The system is defined by effective constraints where the matter terms are replaced by their expectation values in a quantum state. For the static case, we find numerically that the resulting equations admit asymptotically flat, de Sitter, and anti-de Sitter boson star and regular black hole solutions. We also show that the Bardeen and Hayward nonsingular black hole proposals (and some generalizations thereof) are not solutions to our equations.

Figures (3)

• Figure 1: Asymptotically flat boson stars:$\rho$ and $\langle V\rangle$ are the expectation values of the scalar field density and potential; $K$ is the Kretschmann scalar; and the apparent horizon function $|\nabla r|^2$ has no roots. (The red curves are for parameter values $A = 1.5, \alpha = 1, u_{1} = 0, B\approx0.898, \beta = 1, u_{2} = 0$ and boundary data ${\cal N}(0) = 1, {\cal N}^{r}(0) = 0, \langle V(\phi)\rangle(0) = -0.1$; the blue ones are for $A = 2, \alpha = 1, u_{1} = 0, B\approx1.28, \beta = 1, u_{2} = 0$ and ${\cal N}(0) = 1, {\cal N}^{r}(0) = 0, \langle V(\phi)\rangle(0) = -0.5$.)
• Figure 2: Asymptotically flat black holes:$\rho$ and $\langle V\rangle$ are the expectation values of the scalar field density and potential; $K$ is the Kretschmann scalar; $|\nabla r|^2$ shows an inner horizon at $r\approx0.1$ and an outer one at $r\approx0.4$. (The red curves are for the parameters $A = 230, \alpha = 10, u_{1} = 0, B\approx498.663, \beta = 10, u_{2} = 0$ and boundary data ${\cal N}(0) = (0.02)^{2}, {\cal N}^{r}(0) = 0, \langle V(\phi)\rangle(0) = 1000$; the blue ones are for $A = 350, \alpha = 10, u_{1} = 0, B\approx659.851, \beta = 10, u_{2} = 0$ and ${\cal N}(0) = (0.01)^{2}, {\cal N}^{r}(0) = 0, \langle V(\phi)\rangle(0) = 2000$.
• Figure 3: Regular dS and AdS black holes: the upper row shows the apparent horizon function with roots at the outer and inner horizons; the lower row shows the mass function with cosmological constant (\ref{['Mfn']}). (All data are the same as those used for Fig. \ref{['Fig:bh1']} except that $B = 503.5$ (red) and $B = 670$ (blue) for the dS case and that $B = 495$ (red) and $B = 658.5$ (blue) for the AdS case.)