Vacuum polarization and stress-energy of a quantum field inside of two-dimensional black holes

TL;DR

This work analyzes quantum effects for a massless minimally coupled scalar field in two-dimensional black hole spacetimes, focusing on vacuum polarization $\langle \phi^2 \rangle$ and the stress-energy tensor $\langle T_{ab} \rangle$ across interior and exterior regions. Using analytic mode solutions in 2D Schwarzschild and a collapsing null-shell geometry, the authors compute $\langle \phi^2 \rangle$ and renormalized $\langle T_{ab} \rangle$ for Boulware, Unruh, Hartle-Hawking, and in states, revealing state-dependent instabilities in $\langle \phi^2 \rangle$ (notably linear-in-$t$ growth for Unruh and in) while showing that the stress-energy remains regular on the future horizon for the Hartle-Hawking and Unruh states. In the collapsing shell case, the in state has no infrared divergence, but exhibits horizon divergences and a discontinuity at the shell, while the Unruh state provides a good late-time approximation to interior quantum effects outside the collapsing matter. The results suggest that, in 4D, backreaction effects may be well captured outside collapsing matter by the Unruh state, motivating further study of interior quantum effects and their observational implications. The work also achieves a detailed extension of exterior Schwarzschild results into the interior region for all states in 2D.

Abstract

Quantum effects are studied in both Schwarzschild spacetime and a spacetime in which a null shell collapses to form a black hole via the vacuum polarization $\langle φ^2 \rangle$ and stress-energy tensor $\langle T_{ab} \rangle$ for a massless minimally-coupled scalar field in two dimensions. For Schwarzschild spacetime, the Boulware, Unruh, and Hartle-Hawking states are considered. For the collapsing null shell spacetime, the \textit{in} vacuum state is used. Instabilities of the Unruh, Hartle-Hawking, and \textit{in} states resulting from the behavior of $\langle φ^2 \rangle$ in the regions inside and outside of the horizon are found. The question of how well the Unruh state for the eternal black hole approximates quantum effects in the interior of a black hole that forms from collapse is addressed.

Figures (7)

• Figure 1: Penrose diagram for a Schwarzschild black hole. The wavy lines are the singularities at $r = 0$. The past and future horizons are labeled $H^\pm$ and past and future null infinity are labeled $\mathscr{I}^{\pm}$. We call the region labeled I, the exterior region and the region labeled II the interior region.
• Figure 2: Penrose Diagram of a black hole that forms from a collapsing null shell in 4D or in 2D when a perfectly reflecting mirror is placed at $r = 0$ and only the region to the right of the mirror is considered. The vertical line corresponds to the spatial point $r = 0$ inside the shell and the wavy horizontal line corresponds to the singularity at $r=0$ outside the shell.
• Figure 3: Plot of $\langle \phi^2\rangle$ as a function of $\frac{r}{M}$. On the right-hand side of the plot, from top to bottom, the curves are for the Boulware (blue), Unruh (red), and Hartle-Hawking (green) states. For the Unruh state, the plot is for the surface $\frac{v}{M} = 10$. The plot covers both the interior and the exterior regions.
• Figure 4: Plots of $\langle \phi^2 \rangle$ as a function of $\frac{t}{M}$. The upper (blue) curve corresponds to the in state with $v_0 = 4$ and the lower (red) curve corresponds to the Unruh state. The left panel is for the interior region at $\frac{r}{M} = 1$ and the right panel is for the exterior region at $\frac{r}{M} = 3$. A careful examination of both plots shows that, as expected, $\langle \phi^2 \rangle_{in}$ is diverging faster than $\langle \phi^2 \rangle_U$ in both regions.
• Figure 5: Plot of $M^2 \langle T_{uu} \rangle_U$ as a function of $\frac{r}{M}$. The plot covers both the interior and exterior regions. Note that $\langle T_{uu} \rangle_U$ diverges at $r = 0$, vanishes at $r = 2M$, and approaches a constant in the limit $r \to \infty$.