Quantum superposition principle and gravitational collapse: Scattering times for spherical shells

A quantum theory of spherically symmetric thin shells of null dust and their gravitational field is studied. In Nucl. Phys. B603, 555 (2001), it has been shown how superpositions of quantum states with different geometries can lead to a solution of the singularity problem and black hole information paradox: the shells bounce and re-expand and the evolution is unitary. The corresponding scattering times will be defined in the present paper. To this aim, a spherical mirror of radius $R_m$ is introduced. The classical formula for scattering times of the shell reflected from the mirror is extended to quantum theory. The scattering times and their spreads are calculated. They have a regular limit for $R_m\to 0$ and they reveal a resonance at $E_m=c^4{R}_m/2G$. Except for the resonance, they are roughly of the order of the time the light needs to cross the flat space distance between the observer and the mirror. Some ideas are discussed of how the construction of the quantum theory could be changed so that the scattering times become considerably longer.

Figure 1

Penrose-like diagram of the space-time $\overline{\mathcal{M}}$. The shaded region lies inside the mirror with the radius $R=R_m$. The ingoing-shell trajectory defined by $Y=V=v$ starts at past lightlike infinity ${\mathcal{I}}_{-}$, becomes an outgoing-shell trajectory $X=U=u$ at the mirror and ends up at future lightlike infinity ${\mathcal{I}}_{+}$. The region outside the shell is denoted by $\mathcal{E}$, that inside the outgoing (ingoing) shell by ${\mathcal{F}}_{+}({\mathcal{F}}_{-})$. The arrows show the directions in which the double-null coordinates $U$ and $V$ (or $X$ and $Y$ ) increase.

Figure 2

The diagram on the right hand side displays the Penrose diagram of the outgoing Ref. [5] shell space-time. The shell trajectory is given by $U=u$. The shaded region is cut out and glued with the corresponding shaded region taken from the ingoing space-time on the left hand side. Here, the shell trajectory is given by $Y=v$. The rectangles outside of the shell are isometric.

Figure 3

Schematic diagram of the trajectory of the shell bouncing at the mirror at the radius $R=R_m$. Three embedding hyper-surfaces $\Sigma$ are drawn. ${\Sigma }_1$intersects the ingoing shell, ${\Sigma }_2$ the outgoing one. ${\Sigma }_3$ goes through the point where the shell hits the mirror.

Figure 4

The expected scattering time $\frac{1}{R_o}⟨\varphi |\widehat{s}(R_o)|\varphi ⟩$ for box wave packets is plotted as a function of the dimension-free mean energy $\overline{q}$. Plots are shown for packets with constant energy widths $w=0.01$ and $w=0.0001$. The mean scattering time has a peak near the critical energy at $\rho =0.01$. The peak is more distinctive when the packet is narrower.

Figure 5

The construction of the 3-surface ${\Sigma }_{-}^{\prime \prime }$. The dotted lines are the 3-surfaces of constant external and internal Schwarzschild time. The space-time above the shell is Schwarzschild and below it is Minkowski one. The observer trajectory is not shown, but if it crossed the shell at $T_{-}>T_m$, the scattering time would be negative.