Physics of the interior of a spherical, charged black hole with a scalar field

We analyze the physics of nonlinear gravitational processes inside a spherical charged black hole perturbed by a self-gravitating massless scalar field. For this purpose we created an appropriate numerical code. Throughout the paper, in addition to investigation of the properties of the mathematical singularities where some curvature scalars are equal to infinity, we analyze the properties of the physical singularities where the Kretschmann curvature scalar is equal to the Planckian value. Using a homogeneous approximation we analyze the properties of the spacetime near a spacelike singularity in spacetimes influenced by different matter contents, namely, a scalar field, pressureless dust and matter with ultrarelativistic isotropic pressure. We also carry out full nonlinear analyses of the scalar field and geometry of spacetime inside black holes by means of an appropriate numerical code with adaptive mesh refinement capabilities. We use this code to investigate the nonlinear effects of gravitational focusing, mass inflation, matter squeeze, and these effects dependence on the initial boundary conditions. It is demonstrated that the position of the physical singularity inside a black hole is quite different from the positions of the mathematical singularities. In the case of the existence of a strong outgoing flux of the scalar field inside a black hole it is possible to have the existence of two null singularities and one central $r=0$ singularity simultaneously.

Figure 1

Metric coefficients [plot (a)] and Kretschmann scalar [plot (b)] versus $r$ for the case of dust with ${ϵ}_0=0.03$.

Figure 2

Critical value $r_c$ as a function of ${ϵ}_0$ for a dust case, b scalar case and c ultrarelativistic gas.

Figure 3

$r$ versus $t$ for the scalar case for various ${ϵ}_0$. (a) Line a: ${ϵ}_0=0.0001$; b: ${ϵ}_0=0.001$; c: ${ϵ}_0=0.0025$ and d: ${ϵ}_0=0.005$. (b) Line a: ${ϵ}_0=0.010$; b: ${ϵ}_0=0.025$; c: ${ϵ}_0=0.050$ and d: ${ϵ}_0=0.250$.

Figure 4

Mass function versus $r$ for the scalar case for various ${ϵ}_0$. (a) Line a: ${ϵ}_0=0.0001$; b: ${ϵ}_0=0.001$; c: ${ϵ}_0=0.0025$ and d: ${ϵ}_0=0.005$. (b) Line a: ${ϵ}_0=0.010$; b: ${ϵ}_0=0.025$; c: ${ϵ}_0=0.050$ and d: ${ϵ}_0=0.250$.

Figure 5

Metric functions (a)–(c) and Kretschmann scalar (d) versus $r$ for the scalar case for various ${ϵ}_0$. (a) ${ϵ}_0=0.0001$; (b) ${ϵ}_0=0.01$; (c) ${ϵ}_0=0.05$; (d) Kretschmann scalar for a: ${ϵ}_0=0.0001$, b: ${ϵ}_0=0.01$ and c: ${ϵ}_0=0.05$.

Figure 6

Metric functions [plot (a)] and Kretschmann scalar [plot (b)] versus $r$ for the case of ultrarelativistic gas for ${ϵ}_0=0.02$.

Figure 7

$T_{vv}$ and $T_{uu}$ for the simple compact pulse, case: $\Delta =1.0$, $A=0.05$. (a) Lines of constant ${log}_{10}(T_{vv})$. Lines are from ${log}_{10}(T_{vv})=-10.05$ to ${log}_{10}(T_{vv})=-0.85$ in $\Delta {log}_{10}(T_{vv})=0.20$ intervals. Thick dotted line marks ${log}_{10}(T_{vv})=-5.45$. Fully drawn thick line marks apparent horizon. (b) ${log}_{10}(T_{vv})$ along $u=26.00$. (c) Lines of constant ${log}_{10}(T_{uu})$. Lines are from ${log}_{10}(T_{uu})=-10.0$ to ${log}_{10}(T_{uu})=-2.00$ in $\Delta {log}_{10}(T_{uu})=0.25$ intervals. Thick dotted line marks ${log}_{10}(T_{uu})=-3.255$. Fully drawn thick line marks apparent horizon. (d) ${log}_{10}(T_{uu})$ along $v=10.00$.

Figure 8

Illustrations of the mass function for the simple compact pulse, case: $\Delta =1.0$, $A=0.05$. (a) IAH and mass function. Line a is the IAH (left and bottom axis). Lines b–d represent the mass function along $u=27.338$, $u=27.533$ and $u=27.884$, respectively (right and bottom axis). (b) Mass function along lines of constant $u$. Separation between lines is $\Delta u=0.40$, bottom line is along $u=24.00$, top line is along $u=30.00$, thick dotted line is along $u=26.00$.

Figure 9

Lines of constant $r$ for the simple compact pulse, cases (a) $\Delta =1.0$, $A=0.05$ and (b) $\Delta =1.0$, $A=0.01$. Fully drawn thick line marks apparent horizon. (a) Lines are from $r=0.680$ (bottom line) to $r=0.660$ (top left line) in $\Delta r=0.001$ increments. Thick dotted line is $r=0.669$. (b) Lines are from $r=0.68710$ to $r=0.68665$ in $\Delta r=5\times {10}^{-5}$ increments. The thick dotted line is $r=0.68690$.

Figure 10

Various contour plots for the simple compact pulse, case: $\Delta =1.0$, $A=0.20$. (a) Lines of constant ${log}_{10}(T_{uu})$. Lines are from ${log}_{10}(T_{uu})=-10.0$ to ${log}_{10}(T_{uu})=0.00$ in $\Delta {log}_{10}(T_{uu})=0.50$ intervals. Thick dotted line marks ${log}_{10}(T_{uu})=-1.50$ (decreasing “outwards”). Fully drawn thick line marks apparent horizon. (b) ${log}_{10}(T_{uu})$ along $v=15.00$. (c) Lines of constant $r$. Lines are from $r=0.520$ (bottom left line) to $r=0.475$ (top right line) in intervals of $r=0.005$. (d) Lines of constant ${log}_{10}(T_{\theta \theta })$. Lines are from ${log}_{10}(T)=-2.0$ (bottom right line) to ${log}_{10}(T)=32.0$ (top right line) in intervals of $\Delta {log}_{10}(T_{\theta \theta })=2.0$.

Figure 11

Plots for the simple compact pulse, case: $\Delta =1.0$, $A=0.10$. (a) Mass function along lines of constant $u$. From $u=22.40$ (bottom line) to $u=30.00$ (top line) in $\Delta u=0.40$ intervals. Thick dotted line is along $u=26.00$. (b) Lines of constant ${log}_{10}(T_{vv})$. Lines are from ${log}_{10}(T_{vv})=-10.0$ to ${log}_{10}(T_{vv})=-0.25$ in $\Delta {log}_{10}(T_{vv})=0.25$ intervals. Thick dotted line marks ${log}_{10}(T_{vv})=-4.75$. Fully drawn thick line marks apparent horizon. Closely spaced lines near $v\approx 15$ inside apparent horizontal marks local minima.

Figure 12

Plots for the simple compact pulse, case: $\Delta =2.0$, $A=0.20$. (a) Mass function along $v=20.00$. (b) ${log}_{10}(T_{uu})$ along $v=20.00$.

Figure 13

Mass function and Kretschmann scalar along lines of constant $u$ for the simple compact pulse, case: $\Delta =2.0,A=0.2$. Separation between lines is $\Delta u=0.40$. Lines are from $u=19.40$ (lowest right line) to $u=29.80$ (near vertical line in lower left corner). Thick dotted line is along $u=24.60$. (a) Mass function versus $v$. (b) Kretschmann scalar versus $v$. Dotted horizontal line marks line of Planckian curvature.

Figure 14

$v$ versus $r$ along lines of constant $u$ for amplitudes for the simple compact pulse. Separation between lines of constant $u$ is $\Delta u=0.40$. (a) Case: $\Delta =2$, $A=0.010$. From $u=23.00$ (rightmost line) to $u=29.80$ (leftmost line). (b) Case: $\Delta =2$, $A=0.125$. From $u=21.40$ (rightmost line) to $u=29.80$ (leftmost line). (c) Case: $\Delta =2$, $A=0.180$. From $u=20.20$ (rightmost line) to $u=29.80$ (leftmost line). (d) Case: $\Delta =2$, $A=0.200$. From $u=19.80$ (rightmost line) to $u=29.80$ (bottom leftmost line).

Figure 15

Lines of constant $r$ for two different simple compact pulses. Thick dotted lines in the right part of the figures represent the line of Planckian curvature. The bottom fully drawn thick line marks OAH. (a) Case: $\Delta =2.0$, $A=0.18$. Lines are from $r=0.45$ (bottom left line) to $r=0.03$ (top right line) in intervals of $\Delta r=0.01$. (b) Case: $\Delta =2.0$, $A=0.20$. Lines are from $r=0.42$ (bottom left line) to $r=0.20$ (top right line) in intervals of $\Delta r=0.01$. The top right thick line mark $r=0$ singularity.

Figure 16

Contour lines for double sine pulse of form of Eq. (42), case $\Delta =2.0,A=0.25$. Thick bottom line is AH, thick upper line is $r=0$. (a) Lines of constant ${log}_{10}(T_{vv})$. Lines are from ${log}_{10}(T_{vv})=-10.0$ to ${log}_{10}(T_{vv})=2.0$ in intervals of $\Delta {log}_{10}(T_{vv})=0.50$. (b) Lines of constant $r$. From $r=3.00$ (bottom line) to $r=0.10$ (top right thin line) in $\Delta r=0.10$ intervals. The thick dotted line marks $K=K_{\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{k}}$.

Figure 17

Contours for the double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$. The thick top right line marks $r=0$. (a) Lines of constant ${log}_{10}(T_{uu})$. Lines are from ${log}_{10}(T_{uu})=-10.00$ to ${log}_{10}(T_{uu})=3.00$ in $\Delta {log}_{10}(T_{uu})=0.25$ intervals. The thick dotted line marks ${log}_{10}(T_{uu})=1.50$, decreasing leftwards. (b) Lines of constant ${log}_{10}(T_{vv})$. Lines are from ${log}_{10}(T_{vv})=-10.0$ to ${log}_{10}(T_{vv})=3.00$ in $\Delta {log}_{10}(T_{vv})=0.25$ intervals. The thick dotted line marks ${log}_{10}(T_{vv})=1.50$, decreasing downwards.

Figure 18

Lines of constant $r$ for the double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$. The thick top right line marks $r=0$. Lines are from $r=2.50$ (bottom left line) to $r=0.10$ (top right thin line) in intervals of $\Delta r=0.10$. Thick dotted line marks $K=K_{\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{k}}$.

Figure 19

Mass function for the double-flux case double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$ Kretschmann scalar for the double-flux case double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$. (a) Mass function along lines of constant $v$. Lines are from $v=7.04$ (bottom right) to $v=19.04$ (top left) in $v=1.0$ intervals. (b) Mass function along lines of constant $u$, corresponding to the symmetrical equivalent of the lines in (a).

Figure 20

Kretschmann scalar for the double-flux case double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$. (a) Kretschmann scalar along lines of constant $v$. Lines are from $v=7.04$ (bottom right) to $v=19.04$ (top left) in $v=1.0$ intervals. The horizontal dashed line is $K=K_{\mathrm{P}\mathrm{l}\mathrm{a}\mathrm{n}\mathrm{c}\mathrm{k}}$. (b) Kretschmann scalar along lines of constant $u$, corresponding to the symmetrical equivalent of the lines in (a).

Figure 21

$u$ and $v$ versus $r$ for the double-flux case double-flux case based on the simple compact pulse: $\Delta =2.0$, $A=0.25$. (a) $u$ vs $r$ along lines of constant $v$. Lines are from $v=7.04$ (rightmost) to $v=19.04$ (bottom left) in $v=1.0$ intervals; (b) $v$ vs $r$ along lines of constant $u$, corresponding to the symmetrical equivalent of the lines in (a).

Figure 22

Schematic of the points used in the numerical scheme.

Figure 23

Demonstration of the convergence (without AMR) of $r$, $\Phi$, $\sigma$, $C_1$ and $C_2$ along an outgoing ray $u=23.00$. (a) Line a: N(r); line b: ${\psi }_N(\Phi )$ and line c: ${\psi }_N(\sigma )$. Line d shows a line with a slope of minus two. (b) Line a: $\chi (C_1)$ and line b: $\chi (C_2)$. Line c shows a line with a slope of minus two.

Figure 24

$C_1$ (a) and $C_2$ (b) along an outgoing ray $u=23.00$ for resolutions. Line a: $N=100$; line b: $N=200$; line c: $N=400$ and line d: $N=800$.

Figure 25

Mass function as a function of $v$ for the resolutions. Line a: $N=200$; line b: $N=400$; line c: $N=800$; and line d: $N=1600$.

Figure 26

Plots illustrating convergence (with AMR) as function of splitting criteria along outgoing ray at $u=24.60$. (a) line a: $\chi (C_1)$ and line b: $\chi (C_2)$. (b) line a: $\psi (r)$; line b: $\psi (\Phi )$ and line c: $\psi (\sigma )$.

Figure 27

$\Phi$ (a) and $\sigma$ (b) as functions of $r$ along the outgoing ray at $u=24.60$. Error tolerance intervals are for line a: $S_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}={10}^{-4}$; line b: $S_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}={10}^{-5}$; line c: $S_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}={10}^{-6}$ and line d: $S_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}={10}^{-7}$. In all cases $S_{\mathrm{d}\mathrm{e}\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}={10}^{-2}\times S_{\mathrm{s}\mathrm{p}\mathrm{l}\mathrm{i}\mathrm{t}}$.