Semiclassical gravitational collapse of a radially symmetric massless scalar quantum field

We present a method to study the semiclassical gravitational collapse of a radially symmetric scalar quantum field in a coherent initial state. The formalism utilizes a Fock space basis in the initial metric, is unitary and time reversal invariant up to numerical precision. It maintains exact compatibility of the metric with the expectation values of the energy momentum tensor in the scalar field coherent state throughout the entire time evolution. We find a simple criterion for the smallness of discretization effects, which is violated when a horizon forms. As a first example, we study the collapse of a specific state in the angular momentum $l=0$ approximation. Outside the simulated volume, it produces a Schwarzschild metric with $r_s\sim 3.5{\ell }_p$. We see behavior that is compatible with the onset of horizon formation both in the semiclassical and corresponding classical cases in a regime where we see no evidence for large discretization artifacts. In our example setting, we see that quantum effects accelerate the possible horizon formation and move it radially outward. We find that this effect is robust against variations of the radial resolution, the time step, the volume, the initial position and shape of the inmoving state, the vacuum subtraction, the discretization of the time evolution operator, and the integration scheme of the metric. We briefly discuss potential improvements of the method and the possibility of applying it to black hole evaporation. We also briefly touch on the extension of our formalism to higher angular momenta but leave the details and numerics for a forthcoming publication.

Figure 1

Three snapshots of the gravitational collapse of a spherically symmetric scalar field. The plots show, for three different asymptotic observer times, the radial Hamiltonian density of the scalar field (or its expectation value in the case of the quantum field) and the ratio of local Schwarzschild radius to radius $r_s/r$ for both the classical and semiclassical case. It is clearly visible that quantum effects increase the peak of the forming horizon as well as shift its location radially outward.

Figure 2

Time evolution of the Hamiltonian density $\mathfrak{h}$ and the local $r_s/r$ for our default run. We can clearly see the onset of horizon formation. In the semiclassical case, the horizon formation is both more pronounced and happening at larger $r$.

Figure 3

Time evolution of the metric parameters $\alpha$ and $a$ for our default run. The sharper and more radially outward onset of horizon formation in the semiclassical case is clearly visible.

Figure 4

Time evolution of the vacuum contributions to the densities $\mathfrak{h}$ and $\mathfrak{p}$. For all times shown, the shape of ${\mathfrak{p}}^0$ consists of a negative bump interior to the forming horizon and a positive bump outside of it, indicating an additional energy influx toward the horizon forming region from both the inside and the outside. Correspondingly, the vacuum contribution to the energy density ${\mathfrak{h}}^0$ is positive in the horizon forming region and negative just outside it. The dip in ${\mathfrak{h}}^0$ at the original position of the bump $r=9$ is an effect of the vacuum subtraction procedure.

Figure 5

A more detailed look at the relevant region of the $t=12$ panel of Fig. 4. Horizontal lines indicate the peak position of the total Hamiltonian density $\mathfrak{h}$. It is evident that quantum effects increase the peak height of $\mathfrak{h}$ and shift it outward while slightly decreasing its value in the immediate vicinity.

Figure 6

Contour plot the function $\mathrm{sign}(\mathfrak{h})\ln (1+{10}^4|\mathfrak{h}|)$ of the Hamiltonian density $\mathfrak{h}$ for the time evolution of our default system. The shaded area on top corresponds to the unsafe zone $t\gtrsim 14.8$. The black dashed diagonal lines correspond to $\mathrm{d}r=\pm \mathrm{d}t$ and indicate the trajectory of a radially infalling massless test particle on a fictitious flat background metric.

Figure 7

Time evolution of the final state vacuum contribution to the Hamiltonian density ${\mathfrak{h}}_f^0$ for three different times (top) and a detail from $t=12$ (bottom). Horizontal lines indicate the peak position of the total Hamiltonian density $\mathfrak{h}$. From the perspective of the final state vacuum, quantum effects deplete the energy density inside the forming horizon and enhance it outside.

Figure 8

Comparison of ${\mathfrak{h}}^0$ with ${\mathfrak{h}}_f^0$. In each case, we plot the function $\mathrm{sign}(x)\ln (1+{10}^4|x|)$ where $x$ is either ${\mathfrak{h}}^0$ or ${\mathfrak{h}}_f^0$. The shaded area on top corresponds to the unsafe zone $t\gtrsim 14.8$. The black dashed diagonal lines correspond to $\mathrm{d}r=\pm \mathrm{d}t$ and indicate the trajectory of a radially infalling massless test particle on a fictitious flat background metric.

Figure 9

Comparison between two different radial discretizations $N_r=800$, 1600 of the same physical system at different times $t$. The top panel represents a time that is in the safe zone of both discretizations, while the middle panel corresponds to a time that is out of the safe zone of the coarser $N_r=800$ discretization, while still being inside the safe zone of the finer $N_r=1600$ case. The bottom panel shows a time that is out of the safe zone even for the finer discretization.

Figure 10

Comparison of different radial discretizations at $t=12$. One can see that for $N_r=100$, 200, this time is outside the safe zone, while it is still inside for the finer discretizations.

Figure 11

Comparison of the algorithmically relevant quantity ${\max }_i(h_i)$ for different radial discretizations. The horizontal line is at 0.08, corresponding to the criterion (44).

Figure 12

Comparison of the physically relevant quantity ${\max }_i({\widehat{\mathfrak{h}}}_i)$ for different radial discretizations.

Figure 13

Hamiltonian density $\mathfrak{h}$ and its vacuum contribution ${\mathfrak{h}}^0$ at time $t=12$ for the semiclassical case and various discretizations of the derivative operator. The labels correspond to the number of forward (f) and backward (b) points included in the discretized derivative.

Figure 14

Comparison of the vacuum contribution to the Hamiltonian density ${\mathfrak{h}}^0$ between the classical and semiclassical cases for three different discretizations of the derivative operator. Note that the important qualitative features, i.e., the different peak positions for the classical and semiclassical cases and the enhancement of $\mathfrak{h}$ by vacuum contributions in the peak region, are clearly present for all choices of the discretization.

Figure 15

Same as Fig. 13 for two different radial discretizations corresponding to $N_r=400$ (top) and $N_r=1600$ (bottom).

Figure 16

Comparison of results for different time steps at $t=12$.

Figure 17

Comparison of results with different boundary conditions at $t=12$.

Figure 18

The same plot as Fig. 17 but for a larger discretized volume $r_{N_r}=12$.

Figure 19

Comparison of our default system of radial extent $r_{N_r}=10$ at $t=12$ with two systems of larger radial extent $r_{N_r}=12$, 14 that are otherwise identical.

Figure 20

The same as Fig. 19 but with absorbing boundary conditions.

Figure 21

Comparison of the semiclassical evolution with different initial scalar field bump shapes at $t=2$.

Figure 22

The same as Fig. 21 but at a later time $t=12$.

Figure 23

Comparison of different starting positions of the scalar field. Results are compared at times where the respective systems have reached the same peak position of $\mathfrak{h}$ than the reference $t=12$ for our default system. Fields that started farther out show a more pronounced peak of both $\mathfrak{h}$ and its vacuum contribution ${\mathfrak{h}}^0$.

Figure 24

Comparison of two vacuum subtraction procedures at time $t=2$. The pronounced difference in ${\mathfrak{h}}^0$ between the two procedures is confined to the region of the initial bump position.

Figure 25

The same as Fig. 24 but at a later time $t=12$.

Figure 26

Comparison of the time evolution with our two radial integration schemes at $t=12$.

Figure 27

Some of the singular vectors $U_{·k}$ of the operator $q^0$ for our default system and their singular values ${\omega }_k$. On the left-hand side, we plot every 50th mode starting from the mode with the lowest singular value and the mode with the highest singular value. On the right-hand side, we zoom in on three interesting regions for which we plot every mode. Note that the spacing of the modes decreases for increasing $\omega$ except for the few modes with the highest $\omega$ that are concentrated in the region of the original bump position.

Figure 28

Time evolution of the singular values $\overline{\omega }$ of $\overline{q}$. Starting from the highest, we plot every 10th $\overline{\omega }$. The shaded area on the right marks the unsafe region $t>t_{\mathrm{safe}}$.

Figure 29

The “inside” and “outside” components of the singular vectors of $\overline{q}$. We plot the same representative sample of motes as in Fig. 28 with the same color coding.

Figure 30

The mode separation parameter $s$ (45) versus $t$.

Figure 31

Cross-check of how constant the effective Schwarz-schild radius $r_s=r_{N_r}-d_{N_r}$ is on the outermost shell.

Figure 32

Summary of the collapse for our default case with the finest radial discretization $N_r=1600$. The top panel shows the Position of the maximum of $r_s/r$ versus $t$ where $r_s$ is the local effective Schwarzschild radius of the metric. The middle panels shows the corresponding maximum value of $r_s/r$ and the bottom panel shows the mode separation parameter $s$ (45). The shaded region for large $t$ corresponds to $t>t_{\mathrm{safe}}$. We can clearly see that quantum effects enhance the potential horizon formation and move it radially outward. We also see that in the semiclassical case, our algorithm ultimately produces a horizon, which however happens in the region that we deem to be dominated by discretization artifacts.