# Interior Dynamics of Neutral and Charged Black Holes in f(R) Gravity

^{1}

^{2}

^{*}

## Abstract

**:**

## 1. Introduction

#### 1.1. Mass Inflation

- (1)
- Neutral collapse: neutral scalar collapse toward a black hole formation.
- (2)
- Neutral scattering: neutral scalar collapse in a (neutral) Schwarzschild geometry.
- (3)
- Charge scattering: neutral scalar collapse in a (charged) Reissner-Nordström geometry. In this process, the scalar field is scattered by the inner horizon of a Reissner-Nordström black hole.

#### 1.2. New Results

- (1)
- Type I: spacelike scattering. When the scalar field is very strong, the inner horizon can contract to zero volume rapidly, and the central singularity becomes spacelike. The dynamics near the spacelike singularity is similar to that in neutral collapse.
- (2)
- Type II: null scattering. When the scalar field is intermediate, the inner horizon can contract to a place close to the center or reach the center. For each variable (the metric elements and physical scalar field), the spatial and temporal derivatives are almost equal. In the case of the center being reached, the central singularity is null. This type has two stages: early/slow and late/fast. In the early stage, the inner horizon contracts slowly, and the scalar field also varies slowly. In the late stage, the inner horizon contracts quickly, and the dynamics is similar to that in the spacelike case.
- (3)
- Type III: critical scattering. This case is on the edge between the above two cases. The central singularity becomes null.
- (4)
- Type IV: weak scattering. When the scalar field is very weak, the inner horizon contracts but not much. Then the central singularity remains timelike.
- (5)
- Type V: tiny scattering. When the scalar field is very tiny, the influence of the scalar field on the internal geometry is negligible.

## 2. Framework

#### 2.1. Action

#### 2.2. $f\left(R\right)$ Theory

#### 2.3. Coordinate System

**Figure 1.**Initial and boundary conditions for charge scattering. Initial slice is at $t=0$. Definition domain for x is $[-{x}_{b}\phantom{\rule{3.33333pt}{0ex}}{x}_{b}]$. $u=(t-x)/2$ and $v=(t+x)/2$.

- (1)
- In the line element Equation (26), one coordinate is timelike and the rest are spacelike. This is a conventional setup. It is more convenient and more intuitive to use this set of coordinates. For the set of coordinates described by Equation (25), in the equations of motion, many terms are mixed derivatives of u and v; while for the set of coordinates described by Equation (26), in the equations of motion, spatial and temporal derivatives are usually separated.
- (2)
- We set initial conditions close to those in the Reissner-Nordström metric. Consequently, with the terms related to the scalar fields being removed, we can test our code by comparing the numerical results to the analytic ones in the Reissner-Nordström case conveniently. Moreover, by comparing dynamics for scalar collapse to that in the Reissner-Nordström case, we can obtain intuitions on how the scalar fields affect the geometry.
- (3)
- The interactions between scalar fields and the geometry are local effects. In References [29,30], the space between the inner and outer horizons are compactified into finite space. This overcompactification, at least to us, makes it a bit hard to understand the dynamics. In the configuration that we choose, the space is partially compactified, and the picture of charge scattering turns out to be simpler.

**Figure 2.**Contour lines for r defined by Equation (28) in a Reissner-Nordström geometry with $m=1$ and $q=0.7$. Although the exact inner horizon is at regions where $uv$ and $({t}^{2}-{x}^{2})$ are infinite, r can be very close to the inner horizon $r={r}_{-}$ even when $uv$ and $({t}^{2}-{x}^{2})$ take moderate values.

#### 2.4. $f\left(R\right)$ Model

**Figure 3.**Potentials for $f\left(R\right)$ models. $\chi \equiv {f}^{\prime}$ and ${\chi}_{p}\equiv \chi /(1+\chi )$. $U\left(\chi \right)$ and $V\left(\chi \right)$ are the potentials in the Jordan and Einstein frames and can be obtained from Equation (9) and Equation (23), respectively. (

**a**) and (

**b**) are for the Hu-Sawicki model Equations (30), $f\left(R\right)=R-D{R}_{0}R/(R+{R}_{0})$, while (

**c**) and (

**d**) are for the combined model Equations (79), $f\left(R\right)=R-D{R}_{0}R/(R+{R}_{0})+\alpha {R}^{2}$. $D=1.2$, ${R}_{0}={10}^{-5}$, and $\alpha =1$.

## 3. Numerical Setup for Charge Scattering

#### 3.1. Field Equations

#### 3.2. Initial Conditions

#### 3.3. Boundary Conditions

#### 3.4. Discretization Scheme

#### 3.5. Tests of Numerical Code

**Figure 5.**Tests of numerical code for charge scattering. (

**a**) Numerical vs analytic results for a Reissner-Nordström black hole. $m=1$, $q=0.7$, and $\Delta x=\Delta t={10}^{-4}$. The slice is for $(x=3\Delta x,t=t)$. This is a special case of charge scattering with contributions of scalar fields being set to zero. Numerical and analytic results match well at an early stage, while at a later stage gravity and electric field become stronger, the numerical evolutions have a time delay, compared to analytic solutions; (

**b**) Numerical tests for the $\left\{tx\right\}$ constraint Equation (41) and the evolution of ψ on the slice $(x=x,t=0.65)$. They are both second-order convergent.

## 4. Neutral Scalar Collapse

#### 4.1. Numerical Setup

#### 4.2. Black Hole Formation

**Figure 6.**Evolutions in neutral collapse for the Hu-Sawicki model Equation (30), $f\left(R\right)=R-D{R}_{0}R/(R+{R}_{0})$. In (

**a**)–(

**d**), the time interval between two consecutive slices is $10\Delta t=0.05$. (

**e**) and (

**f**) are for the apparent horizon and the singularity curve of the formed black hole.

#### 4.3. Asymptotic Dynamics in the Vicinity of the Central Singularity of the Formed Black Hole

#### 4.4. Mass Inflation

**Figure 7.**(color online) Dynamics on the slice $(x=1,t=t)$ in neutral collapse for the Hu-Sawicki model Equation (30). (

**a**)–(

**c**): dynamical equations for r, η, and σ. (

**b**) Near the central region, due to accumulation, the scalar field ϕ is strong. σ is positive. As a result, in the equation of motion for η (53), the terms $2{e}^{-2\sigma}$ and $16\pi {e}^{-2\sigma}{r}^{2}V$ are negligible. The equation is reduced to ${\eta}_{,tt}\approx {\eta}_{,xx}$. (

**d**) $ln\left({m}_{EF}\right)=alnr+b$, $a=-1.6438\pm 0.0008$, $b=-0.998\pm 0.003$. $ln\left({m}_{JF}\right)=alnr+b$, $a=-2.385\pm 0.002$, $b=-3.443\pm 0.009$.

- (1)
- Values of σ. Due to the backreaction of the scalar fields on the geometry, σ in small-x regions is greater than in large-x regions. In fact, σ is positive in small-x regions, while negative in large-x regions. (See Figure 6b). Our numerical results of the parameter C in $\varphi \approx Cln\xi $ are $C\approx 0.18$ at $x=1$ and $C\approx 0.07$ at $x=2$. Then we have $4\pi {C}^{2}\approx 0.41>1/4$ and $\sigma >0$ at $x=1$, while $4\pi {C}^{2}\approx 0.06<1/4$ and $\sigma <0$ at $x=2$. (See Equation (72)).
- (2)
- Equation of motion for η (53). For positive σ, the terms $2{e}^{-2\sigma}$ and $16\pi {e}^{-2\sigma}{r}^{2}V$ in Equation (53) are negligible, compared to the other two. Then Equation (53) is reduced to ${\eta}_{,tt}\approx {\eta}_{,xx}$. (See Figure 7b). However, for negative σ, the term $2{e}^{-2\sigma}$ is important, and Equation (53) is reduced to ${\eta}_{,tt}\approx -2{e}^{-2\sigma}$. (See Figure 8b).
- (3)

**Figure 8.**(color online) Dynamics on the slice $(x=2,t=t)$ in neutral collapse for the Hu-Sawicki model. (

**a**)–(

**c**): dynamical equations for r, η, and σ. (

**b**) At large-x regions, the scalar field ϕ is weak, and σ is negative. As a result, in the equation of motion for η (53), the term $2{e}^{-2\sigma}$ is important. The equation is reduced to ${\eta}_{,tt}\approx -2{e}^{-2\sigma}$. (

**d**) $lnm=alnr+b$, $a=-0.2673\pm 0.0008$, $b=0.217\pm 0.003$.

## 5. Neutral Scalar Scattering

#### 5.1. A Dark Energy $f\left(R\right)$ Singularity Problem

**Figure 9.**(color online) Results for neutral scalar collapse in a Schwarzschild geometry for the Hu-Sawicki model. (

**a**) and (

**b**): evolutions of r and ${f}^{\prime}$. The time interval between two consecutive slices is $30\Delta t=0.15$. (

**c**) evolutions of r and ${f}^{\prime}$ on the slice $(x=-2,t=t)$. (

**d**) dynamical equation for ϕ on the slice $(x=-2,t=t)$. ϕ is mainly accelerated by the spatial derivative ${\varphi}_{,xx}$ and the geometrical term $-2{r}_{,t}{\varphi}_{,t}/r$. Eventually, ϕ and ${f}^{\prime}$ approach 0 and 1, respectively. Then the Ricci scalar becomes singularity, and the simulation stops.

- (1)
- Schwarzschild geometry: $m=1$.
- (2)
- Physical scalar field: ${\psi (x,t)|}_{t=0}=a\xb7exp\left[-{(x-{x}_{0})}^{2}/b\right]$, $a=0.08$, $b=1$, and ${x}_{0}=4$.
- (3)
- $f\left(R\right)$ model: $f\left(R\right)=R-D{R}_{0}R/(R+{R}_{0})$, $D=1.2$, and ${R}_{0}={10}^{-5}$.
- (4)
- Scalar degree of freedom: ${\varphi (x,t)|}_{t=0}=a+b\xb7exp\left[-{(x-{x}_{0})}^{2}/c\right]$, $a=-0.02$, $b=-0.1$, $c=1$, and ${x}_{0}=-2$.
- (5)
- Grid. Spatial range: $x\in [-10\phantom{\rule{3.33333pt}{0ex}}10]$. Grid spacings: $\Delta x=\Delta t=0.005$.

#### 5.2. Avoidance of the Singularity Problem

**Figure 10.**(color online) Results for neutral scalar collapse in a Schwarzschild geometry in a combined $f\left(R\right)$ model Equation (79). (

**a**) and (

**b**): evolutions of r and ${f}^{\prime}$. The time interval between two consecutive slices is $30\Delta t=0.15$. (

**c**) evolutions of r and ${f}^{\prime}$ on the slice $(x=-2,t=t)$. (

**d**) dynamical equation for ϕ on the slice $(x=-2,t=t)$.

## 6. Results for Charge Scattering

- (1)
- Reissner-Nordström geometry: $m=1$, and $q=0.7$.
- (2)
- Physical scalar field: ${\psi (x,t)|}_{t=0}=a\xb7exp\left[-{(x-{x}_{0})}^{2}/b\right]$, $a=0.08$, $b=1$, and ${x}_{0}=4$.
- (3)
- $f\left(R\right)$ model: $f\left(R\right)=R-D{R}_{0}R/(R+{R}_{0})$, $D=1.2$, and ${R}_{0}={10}^{-5}$.
- (4)
- Scalar degree of freedom: ${\varphi (x,t)|}_{t=0}={\varphi}_{0}$, with ${V}^{\prime}\left({\varphi}_{0}\right)=0$.
- (5)
- Grid. Spatial range: $x\in [-10\phantom{\rule{3.33333pt}{0ex}}10]$. Grid spacings: $\Delta x=\Delta t=0.005$ for Section 6.1 and Section 6.2, and $\Delta x=\Delta t=0.0025$ for Section 6.3 and Section 6.4.

#### 6.1. Evolutions

#### 6.1.1. Outline

- (1)
- Metric components: r and σ. They contribute as gravity.
- (2)
- Scalar fields: ϕ and ψ. They contribute as self-gravitating fields.
- (3)
- Electric field and $V\left(\varphi \right)$. As implied in Equation (33), they are repulsive forces. However, the numerical results show that in charge scattering, compared to the contributions from other quantities, the contribution from $V\left(\varphi \right)$ is negligible.

**Figure 11.**Evolutions in charge scattering. (

**a**)–(

**d**): evolutions of r, σ, ${f}^{\prime}$, and ψ. The time interval between two consecutive slices is $30\Delta t=0.15$. (

**e**) and (

**f**) are for the apparent horizon and the singularity curve of the black hole. When the scalar field is strong enough (around $x=2$), the inner horizon can be pushed to the center, and the central singularity becomes spacelike. When the scalar field is weak enough (e.g., $-4<x<-1$), the inner horizon does not change much. At the intermediate state (e.g., $0<x<1.5$), the inner horizon contracts to zero, and the central singularity becomes null. The results for the inner horizon, especially for $x>2$, are not that accurate. We are aware that the inner horizon is actually at infinity, while r still can be very close to ${r}_{-}$ when x and t take moderate values.

- (1)
- Type I: spacelike scattering. When the scalar field is very strong, the inner horizon can contract to zero volume rapidly, and the central singularity becomes spacelike. Sample slice: $(x=1.5,t=t)$ in Figure 11. See Section 6.2.
- (2)
- Type II: null scattering. When the scalar field is intermediate, the inner horizon can contract to a place close to the center or reach the center. For each variable, the spatial and temporal derivatives are almost equal. In the case of the center being reached, the central singularity becomes null. This type has two stages: early/slow and late/fast. In the early stage, the inner horizon contracts slowly, and the scalar field also varies slowly. In the late stage, the inner horizon contracts quickly, and the dynamics is similar to that in the spacelike case. Sample slice: $(x=0.5,t=t)$ in Figure 11. See Section 6.3 and Section 6.4.
- (3)
- Type III: critical scattering. This case is on the edge between the above two cases. When the central singularity is reached, it becomes null. Sample slice: $(x=1.4225,t=t)$ in Figure 11. Due to the similarity to that in general relativity discussed in Reference [41], details on this type of scattering in $f\left(R\right)$ gravity are skipped in this paper.
- (4)
- (5)
- Type V: tiny scattering. When the scalar field is very tiny, the influence of the scalar field on the geometry is negligible. Sample slice: $(x=-3,t=t)$ in Figure 11.

#### 6.1.2. Causes of Mass Inflation and Evolutions

#### 6.1.3. Locations of Horizons

#### 6.2. Spacelike Scattering

- (1)
- Motion of r. The quantity r does not decelerate much when it crosses the inner horizon of the given Reissner-Nordström black hole, and it can approach the center. (See Figure 12).
- (2)
- Nature of the central singularity. The central singularity converts from timelike into spacelike.
- (3)
- The dynamics in spacelike scattering is similar to that in strong, neutral scalar collapse. The quantity σ takes large positive values, such that in the vicinity of the central singularity, compared to other terms, the term ${e}^{-2\sigma}{q}^{2}/{r}^{2}$ in the equation for r Equation (33) and the term ${e}^{-2\sigma}{q}^{2}/{r}^{4}$ in the equation for σ Equation (35) are negligible. As a result, in the vicinity of the central singularity, the dynamics is similar to that in strong, neutral scalar collapse as expressed by Equations (67)–(70). (See Figure 7 and Figure 13). Then the quantities r, σ, ϕ, and m take similar forms as those in neutral collapse. (See Figure 14).In both strong, neutral scalar collapse and spacelike scattering, the equation of motion for η is reduced to ${\eta}_{,tt}\approx {\eta}_{,xx}$. (See Figure 7b and Figure 13b).Since ${\varphi}_{,t}$ is negative, the term $\sqrt{2/3}\kappa {\varphi}_{,t}{\psi}_{,t}$ in Equation (37) functions as a friction force for ψ. Consequently, compared to ϕ, ψ grows slowly. As the singularity is approached, ψ even approaches constant values. (See Figure 14c,d). As shown in Figure 13e, near the singularity, two major sets of terms can be expressed as$${r}_{,t}{\psi}_{,t}\approx {r}_{,x}{\psi}_{,x},\phantom{dd}{\varphi}_{,t}{\psi}_{,t}\approx {\varphi}_{,x}{\psi}_{,x}$$Alternatively,$$\frac{{\psi}_{,t}}{{\psi}_{,x}}\approx \frac{{r}_{,x}}{{r}_{,t}}\approx \frac{{\varphi}_{,x}}{{\varphi}_{,t}}$$
- (4)
- Growth of mass function.In spacelike scattering, the equation of motion for ϕ can be simplified as$${\sigma}_{,tt}\approx 4\pi {\varphi}_{,t}^{2}$$Consequently, with the results obtained in Section 4.3, σ has the following asymptotic solution:$$\sigma \approx Bln\xi +{\sigma}_{0}\approx -4\pi {C}^{2}ln\xi +{\sigma}_{0}$$$$\begin{array}{cc}\hfill m& =\frac{r}{2}\left[1+\frac{{q}^{2}}{{r}^{2}}+{e}^{2\sigma}({r}_{,t}^{2}-{r}_{,x}^{2})\right]\hfill \\ & \approx \left[\frac{1}{8}(1-{K}^{2}){A}^{3}{e}^{2{\sigma}_{0}}\right]{\xi}^{2B-\frac{1}{2}}\hfill \\ & \approx \left[\frac{1}{8}(1-{K}^{2}){A}^{4(-B+1)}{e}^{2{\sigma}_{0}}\right]{r}^{4B-1}\hfill \\ & \approx \left[\frac{1}{8}(1-{K}^{2}){A}^{4(4\pi {C}^{2}+1)}{e}^{2{\sigma}_{0}}\right]{r}^{-16\pi {C}^{2}-1}\hfill \end{array}$$Numerical results show that, for the sample slice $(x=1.52,t=t)$, near the central singularity, the slope of the singularity curve K is about $0.04$. As shown in Figure 14d, we linearly fit the numerical results of m via$$lnm\approx alnr+b$$obtaining$$a=-16.937\pm 0.003,\phantom{dd}b=-29.04\pm 0.01$$Fitting numerical results for σ according to Equation (87) and combining Equations (88) and (89), we obtain$$\begin{array}{cc}\hfill {a}_{\mathrm{analytic}}& =4B-1=17.0\pm 0.8\hfill \end{array}$$$$\begin{array}{c}\hfill \\ \hfill {b}_{\mathrm{analytic}}& =ln\left[\frac{1}{8}(1-{K}^{2}){A}^{4(-B+1)}{e}^{2{\sigma}_{0}}\right]=-28\pm 4\hfill \end{array}$$Similarly, fitting numerical results for ϕ according to $\varphi \approx Cln\xi $, we obtain$$\begin{array}{cc}\hfill {a}_{\mathrm{analytic}}& =-16\pi {C}^{2}-1=18.06\pm 0.01\hfill \end{array}$$$$\begin{array}{cc}\hfill {b}_{\mathrm{analytic}}& =ln\left[\frac{1}{8}(1-{K}^{2}){A}^{4(4\pi {C}^{2}+1)}{e}^{2{\sigma}_{0}}\right]=-28\pm 4\hfill \end{array}$$One can see that the above three sets of results match well.

**Figure 12.**Evolutions along the slice $(x=1.52,t=t)$ in one spacelike scattering process. (

**a**) evolutions of r and σ. (

**b**) evolutions of ${f}^{\prime}$, ψ, and m. In this case, the scalar fields are so strong, such that r can rapidly cross the line of $r={r}_{-}$ and approach zero. Other variables (σ, ${f}^{\prime}$, ψ, and m) also evolve rapidly after r has crossed the line of $r={r}_{-}$.

#### 6.3. The Late/fast Stage of Null Scattering

**Figure 13.**(color online) Dynamics along the slice $(x=1.52,t=t)$ near the spacelike, central singularity. Near the singularity, we can approximately rewrite the original equations of motion for r, η, σ, ϕ, and ψ as follows. (

**a**) $r{r}_{,tt}\approx -{r}_{,t}^{2}$. (

**b**) ${\eta}_{,tt}\approx {\eta}_{,xx}^{2}$. (

**c**) ${\sigma}_{,tt}\approx 4\pi {\varphi}_{,t}^{2}$. (

**d**) ${\varphi}_{,tt}\approx -2{r}_{,t}{\varphi}_{,t}/r$. (

**e**) ${r}_{,t}{\psi}_{,t}\approx {r}_{,x}{\psi}_{,x}$ and ${\varphi}_{,t}{\psi}_{,t}\approx {\varphi}_{,x}{\psi}_{,x}$. Then we have ${\psi}_{,t}/{\psi}_{,x}\approx {r}_{,x}/{r}_{,t}\approx {\varphi}_{,x}/{\varphi}_{,t}$.

**Figure 14.**Solutions along the slice $(x=1.52,t=t)$ near the spacelike, central singularity. (

**a**) $lnr\approx aln\xi +b$, $a=0.5032\pm 0.0001$, $b=-0.3033\pm 0.0008$. $\sigma \approx aln\xi +b$, $a=-4.0\pm 0.2$, $b=-10\pm 2$. (

**b**) $\varphi \approx aln\xi +b$, $a=0.5827\pm 0.0002$, $b=-1.294\pm 0.002$. $lnm\approx alnr+b$, $a=-16.937\pm 0.003$, $b=-29.04\pm 0.01$. (

**c**) and (

**d**): $\psi (x,t)$ asymptotes to constant values as the central singularity is approached.

**Figure 15.**(color online) Dynamics along the slice $(x=0.5,t=t)$ in null scattering. (

**a**)–(

**f**): dynamical equations for r, η, σ, ϕ, and ψ.

**Figure 16.**Evolutions along the slice $(x=0.5,t=t)$ in null scattering. (

**a**)–(

**f**): evolutions for r, σ, f, ψ, m, and $|1-{K}^{2}|$. At the early stage of mass inflation, $1.2<t<2$, r varies slowly; while at the late stage, $t>2$, r varies rapidly toward zero.

**Figure 17.**Solutions near the null central singularity along the slice $(x=0.5,t=t)$ in null scattering. (

**a**) $lnr\approx aln\xi +b$, $a=0.4421\pm 0.0008$, $b=-0.887\pm 0.002$. $\sigma \approx aln\xi +b$, $a=-5.15\pm 0.01$, $b=27.26\pm 0.02$. $\varphi \approx aln\xi +b$, $a=0.583\pm 0.002$, $b=0.465\pm 0.005$. (

**b**) ψ approaches a constant value $0.4$. $lnm\approx alnr+b$, $a=-25.2\pm 0.1$, $b=17.9\pm 0.3$.

#### 6.4. The Early/slow Stage of Null Scattering

**Figure 18.**(color online) Constraint equations and solutions along the slice $(x=0.5,t=t)$ at the early/slow stage of null scattering. (

**a**) and (

**b**): constraint Equations (41) and (42). (

**c**) ${r}_{,t}\approx a{(t+b)}^{c}+d$, $a=(-1.67\pm 0.01)\times {10}^{-3}$, $b=-0.6622\pm 0.0008$, $c=7.897\pm 0.005$, $d=(-2.610\pm 0.003)\times {10}^{-4}$. $\sigma \approx aln(t+b)+c$, $a=34.39\pm 0.03$, $b=-0.245\pm 0.001$, $c=3.02\pm 0.04$. $\varphi \approx a{(t+b)}^{c}+d$, $a=(-9.7\pm 2.3)\times {10}^{-4}$, $b=-0.71\pm 0.03$, $c=8.3\pm 0.2$, $d=-0.133340\pm 0.000005$. (

**d**) $\psi \approx a{(t+b)}^{c}+d$, $a=(5.0\pm 0.1)\times {10}^{-3}$, $b=-0.362\pm 0.006$, $c=5.40\pm 0.02$, $d=(1.19\pm 0.02)\times {10}^{-3}$. $ln(1-{K}^{2})\approx at+b$, $a\approx -8.73\pm 0.01$, $b\approx 12.71\pm 0.02$. $lnm\approx aln(t+b)+c$, $a=77.77\pm 0.02$, $b=-0.1994\pm 0.0004$, $c=-15.80\pm 0.03$.

- (1)
- ${r}_{,t}\approx a{(t+b)}^{c}+d$, $a=(-1.71\pm 0.01)\times {10}^{-3}$, $b=(-5.877\pm 0.008)\times {10}^{-1}$, $c=7.880\pm 0.005$, $d=(-2.639\pm 0.003)\times {10}^{-4}$.
- (2)
- $\sigma \approx aln(t+b)+c$, $a=34.37\pm 0.03$, $b=(-1.72\pm 0.01)\times {10}^{-1}$, $c=3.06\pm 0.04$.
- (3)
- $\varphi \approx a{(t+b)}^{c}+d$, $a=(-2.7\pm 0.6)\times {10}^{-3}$, $b=(-7.6\pm 0.3)\times {10}^{-1}$, $c=7.4\pm 0.2$, $d=(-1.3336\pm 0.0001)\times {10}^{-1}$.
- (4)
- $\psi \approx a{(t+b)}^{c}+d$, $a=(5.1\pm 0.1)\times {10}^{-3}$, $b=(-2.91\pm 0.06)\times {10}^{-1}$, $c=5.38\pm 0.02$, $d=(1.21\pm 0.02)\times {10}^{-3}$.
- (5)
- $ln(1-{K}^{2})\approx at+b$, $a\approx -8.73\pm 0.01$, $b\approx 12.05\pm 0.02$.
- (6)
- $lnm\approx aln(t+b)+c$, $a=77.69\pm 0.02$, $b=(-1.268\pm 0.004)\times {10}^{-1}$, $c=-15.65\pm 0.03$.

## 7. Weak Scalar Charge Scattering

- (1)
- Physical scalar field: ${\psi (x,t)|}_{t=0}=a\xb7exp\left[-{(x-{x}_{0})}^{2}/b\right]$, $a=0.03$, $b=1$, and ${x}_{0}=4$.
- (2)
- Grid. Spatial range: $x\in [-12\phantom{\rule{3.33333pt}{0ex}}12]$. Grid spacings: $\Delta x=\Delta t=0.002$.

**Figure 19.**Evolutions for charge scattering with a weak scalar field. (

**a**)–(

**d**): evolutions for r, ${f}^{\prime}$, and ψ. The time interval between two consecutive slices is $120\Delta t=0.24$. The central singularity is not approached.

**Figure 20.**(color online) Dynamics and evolutions on the slice $(x=4,t=t)$ in weak scattering. (

**a**)–(

**e**): dynamical equations for r, η, σ, ϕ, and ψ. (

**f**) evolutions of r and m.

## 8. Dark Energy $f\left(R\right)$ Singularity Problem in Charge Scattering

- (1)
- Grid spacings: $\Delta x=\Delta t=0.0025$.
- (2)
- Physical scalar field ψ: ${\psi (x,t)|}_{t=0}=a\xb7exp\left[-{(x-{x}_{0})}^{2}/b\right]$, $a=0.05$, $b=1$, and ${x}_{0}=2$.
- (3)
- Scalar degree of freedom ϕ: ${\varphi (x,t)|}_{t=0}=a\xb7exp[-{(x-{x}_{0})}^{2}]+{\varphi}_{0}$, with ${V}^{\prime}\left({\varphi}_{0}\right)=0$, $a=-0.05$, and ${x}_{0}=-2$.
- (4)

**Figure 21.**(color online) A singularity problem in charge scattering for the Hu-Sawicki model Equation (30), $f\left(R\right)=R-D{R}_{0}R/(R+{R}_{0})$. (

**a**) and (

**b**): evolutions of r and ${f}^{\prime}$. The time interval between two consecutive slices is $20\Delta t=0.01$. (

**c**) evolutions of r, ${f}^{\prime}$, and ${f}_{,t}^{\prime}$ on the the slice $(x=-2,t=t)$. (

**d**) and (

**e**): dynamical equation for ϕ on the slice $(x=-2,t=t)$. In (

**e**), as the inner horizon is approached, ${\varphi}_{,tt}\approx -2{r}_{,t}{\varphi}_{,t}$. This equation describes a positive feedback, since $-2{r}_{,t}/r$ is positive. As a result, when ${\varphi}_{,t}$ is positive, ϕ can be accelerated to zero rapidly. Correspondingly, ${f}^{\prime}$ goes to 1 as plotted in (

**b**) and (

**c**), and the Ricci scalar R becomes singular. Then the simulation breaks down as shown in (

**a**).

**Figure 22.**(color online) Avoidance of the singularity problem in charge scattering for the combined model Equation (109), $f\left(R\right)=R-D{R}_{0}R/(R+{R}_{0})+\alpha {R}^{2}$. (

**a**) and (

**b**): evolutions of r and ${f}^{\prime}$. The time interval between two consecutive slices is $60\Delta t=0.15$. (

**c**) dynamical equation for ϕ on the slice $(x=-2,t=t)$. (

**d**) evolution of m on the slice $(x=-2,t=t)$.

## 9. Summary

#### 9.1. Computational Issues

- (1)
- The Jordan frame vs the Einstein frame. The field equations for $f\left(R\right)$ gravity in the Jordan frame are more complex than those in general relativity. Therefore, for ease of computation, we transform $f\left(R\right)$ gravity from the Jordan frame into the Einstein frame, in which the formalism can be formally treated as Einstein gravity coupled to a scalar field.
- (2)
- $dudv$ vs. $(-d{t}^{2}+d{x}^{2})$ in double-null coordinates. In the studies of mass inflation, the $dudv$ format of the Kruskal-like coordinates, $d{s}^{2}=4{e}^{-2\sigma}dudv+{r}^{2}d{\Omega}^{2}$, is usually used. In the field equations, many terms are mixed derivatives of u and v, e.g., ${r}_{,uv}$. In this paper, we used the $(-d{t}^{2}+d{x}^{2})$ format instead, $d{s}^{2}={e}^{-2\sigma}(-d{t}^{2}+d{x}^{2})+{r}^{2}d{\Omega}^{2}$, with $u=(t-x)/2=\text{const}$ and $v=(t+x)/2=\text{const}$. In the $(t,x)$ line element, one coordinate is timelike, and the rest are spacelike. We are used to this setup. It is more convenient and more intuitive to use this set of coordinates. Moreover, for the $(t,x)$ choice, spatial and temporal derivatives are usually separated, e.g., $({r}_{,tt}-{r}_{,xx})$.We set the initial conditions close to those in a Reissner-Nordström geometry. With this setup, it is convenient to test the code. Removing the terms related to the scalar fields, we can test our code by comparing the numerical results to the analytic ones in a Reissner-Nordström geometry. Moreover, by comparing numerical results for charge scattering to the dynamics in the Reissner-Nordström geometry, we can obtain intuitions as to how the scalar fields affect the geometry.
- (3)
- Cauchy horizon: infinite or local regions? As implied by Equation (28), the exact inner horizon $r={r}_{-}$ is at the regions where $uv$ and $({t}^{2}-{x}^{2})$ are infinite. However, r still can be very close to the inner horizon even when $uv$ and $({t}^{2}-{x}^{2})$ take moderate values. Consequently, at regions where $uv$ and $({t}^{2}-{x}^{2})$ take some moderate values, the scalar fields and the inner horizon still can have strong interactions, resulting in mass inflation.

#### 9.2. Physical Issues

- (1)
- Scalar collapse in $f\left(R\right)$ gravity vs scalar collapse in general relativity. In scalar collapse, the scalar degree of freedom $\varphi (\equiv \sqrt{3/2}ln{f}^{\prime}/\sqrt{8\pi G})$ plays a similar role as a physical scalar field in general relativity. Regarding the physical scalar field in the $f\left(R\right)$ case, when ${\varphi}_{,t}$ is negative (positive), the physical scalar field is suppressed (magnified) by ϕ.
- (2)
- The inner horizon in a Reissner-Nordström black hole vs the central singularity in a Schwarzschild black hole. These two share some similarities.For Reissner-Nordström and Schwarzschild black holes, throughout the whole spacetime, the Misner-Sharp mass function is constant. When a scalar field impacts the inner horizon of a Reissner-Nordström black hole, the scalar field can modify the geometry in the vicinity of the inner horizon significantly, especially on ${r}_{,t}$. The inner horizon contracts and mass inflation takes place. In neutral scalar collapse toward a Schwarzschild black hole formation, the scalar field can also modify the geometry in the vicinity of the central singularity dramatically, especially on the metric component σ [20]. Then mass inflation also happens.The Belinskii, Khalatnikov, and Lifshitz (BKL) conjecture is an important result on dynamics in the vicinity of a spacelike singularity [76,77,78,79]. The first statement of this conjecture is that as the singularity is approached, the dynamical terms dominate the spatial terms in the field equations. In other words, the way gravity changes over time is more important than the variation of the gravitational field from one location to the next [79]. We would like to say that, to a large extent, later evolutions in a strong gravitational field largely erase away the initial information on the connections between neighboring points. As discussed in Reference [20] and also in this paper, in double-null coordinates, using the above argument, one can interpret the following behaviors displayed in numerical simulations: near the central singularity of a Schwarzschild black hole and also near the inner horizon of a Reissner-Nordström black hole, there are$$\frac{{\psi}_{,x}}{{\psi}_{,t}}\approx \frac{{r}_{,x}}{{r}_{,t}}<1$$$${\varphi}_{,tt}\approx -\frac{2{r}_{,t}}{r}{\varphi}_{,t}$$In this paper, it was shown that Equation (110) can explain the causes of mass inflation, while Equation (111) can explain the dark energy $f\left(R\right)$ singularity problem in collapse.The second and third statements of the BKL conjecture are that i) the metric terms will dominate the matter field terms, while the matter field may not be negligible if it is a scalar field; ii) the dynamics of the metric components and the matter fields is described by the Kasner solution. These two statements were confirmed in simulations of neutral scalar collapse in $f\left(R\right)$ gravity in Reference [20] and in general relativity in Reference [41]. The second statement was also verified in charge scattering in this paper. However, the third statement on Kasner solution may not apply to the dynamics near the inner horizon in charge scattering.
- (3)
- Compact stars vs black holes in $f\left(R\right)$ gravity. The internal structure of compact stars is usually in an equilibrium state and is static. In $f\left(R\right)$ gravity, inside compact stars, the scalar degree of freedom, ${f}^{\prime}$, can be coupled to the energy density of the stars and then is not very free to move. However, due to strong gravity, the internal structure of black holes is dynamical. ${f}^{\prime}$ and the matter fields are decoupled. As a result, ${f}^{\prime}$ is more free to move than in the compact stars case. It can keep increasing or decreasing until singularities are met.
- (4)
- Dark energy $f\left(R\right)$ singularity problem: cosmology (or static compact objects) vs black hole physics. In dark energy $f\left(R\right)$ gravity, the Ricci scalar R can be singular in both cosmology and black hole physics. We consider a homogeneous cosmological model. Using the flat Friedmann-Robertson-Walker metric,$$d{s}^{2}=-d{t}^{2}+{a}^{2}\left(t\right)d{x}^{2}$$$$\ddot{{f}^{\prime \prime}}+3H\dot{{f}^{\prime}}+U\left({f}^{\prime}\right)+\frac{8\pi}{3}T=0$$$$-{\varphi}_{,tt}+{\varphi}_{,xx}+\frac{2}{r}(-{r}_{,t}{\varphi}_{,t}+{r}_{,x}{\varphi}_{,x})={e}^{-2\sigma}\left[{V}^{\prime}\left(\varphi \right)+\frac{1}{\sqrt{6}}\kappa {T}^{\left(\psi \right)}\right]$$As the central singularity of a Schwarzschild black hole or the inner horizon of a Reissner-Nordström black hole is approached, the above equation can be simplified as Equation (111), and gravity from the black hole, $-2{r}_{,t}{\varphi}_{,t}/r$, can cause a similar singularity problem as in cosmology or static compact stars.
- (5)
- The combined $f\left(R\right)$ model and the ${R}^{2}$ model: singular or non-singular? At the center of a Schwarzschild black hole and in the very early Universe, the tidal forces are singular, and general relativity fails. Taking into account quantum-gravitational effects, Starobinsky obtained an ${R}^{2}$ model, $f\left(R\right)=R+\alpha {R}^{2}$. This model has a non-singular de Sitter solution, which is unstable both to the past and to the future [42,43]. In Section 5, scalar collapse in a Schwarzschild geometry for the combined model (a combination of dark energy model and ${R}^{2}$ model) was explored. A new Schwarzschild black hole, including a new central singularity, can be formed. Moreover, under certain initial conditions, ${f}^{\prime}$ and R can be pushed to infinity as the central singularity is approached. In Reference [73], scalar collapse in flat geometry for the ${R}^{2}$ model was simulated. Similar results were obtained. Namely, the classical singularity problem, which is present in general relativity, remains in collapse in these models.
- (6)
- Inside vs. outside black holes: local vs global. Throughout the whole spacetime of stationary Schwarzschild and Reissner-Nordström black holes, the Misner-Sharp mass function is equal to the black hole mass. For a gravitational collapsing system, at asymptotic flat regions, the mass function describes the total mass of the dynamical system. However, in this system, near the central singularity of a Schwarzschild black hole or near the inner horizon of a Reissner-Nordström black hole, the dynamics is local. Then the mass function does not provide global information on the mass of the collapsing system.

## Acknowledgments

## Author Contributions

## Conflicts of Interest

## A. Equations of Motion for a Physical Scalar Field and ${f}^{\prime}$ in the Einstein Frame

## B. Reissner-Nordström Metric in Kruskal-Like Coordinates

#### B.1. Patch I: ${r}_{-}<r<+\infty $

#### B.2. Patch II: $0<r<{r}_{+}$

## C. Einstein Tensor and Energy-Momentum Tensor for a Massive Scalar Field

## References

- Burko, L.M.; Ori, A. (Eds.) Internal Structure of Black Holes and Spacetime Singularities; Institute of Physics Publishing: Bristol, UK, 1998.
- Brady, P.R. The Internal Structure of Black holes. Prog. Theor. Phys. Suppl.
**1999**, 136, 29–44. [Google Scholar] [CrossRef] - Berger, B.K. Numerical Approaches to Spacetime Singularities. Living Rev. Relativ.
**2002**, 5, 1. [Google Scholar] [CrossRef] - Joshi, P.S. Gravitational Collapse and Spacetime Singularities; Cambridge University Press: Cambridge, UK, 2007. [Google Scholar]
- Henneaux, M.; Persson, D.; Spindel, P. Spacelike Singularities and Hidden Symmetries of Gravity. Living Rev. Relativ.
**2008**, 11, 1. [Google Scholar] [CrossRef] - Price, R.H. Nonspherical Perturbations of Relativistic Gravitational Collapse. I. Scalar and Gravitational Perturbations. Phys. Rev. D
**1972**, 5, 2419–2438. [Google Scholar] [CrossRef] - Clifton, T.; Ferreira, P.G.; Padilla, A.; Skordis, C. Modified gravity and cosmology. Phys. Rep.
**2012**, 513, 1–189. [Google Scholar] [CrossRef] - Sotiriou, T.P.; Faraoni, V. f(R) Theories Of Gravity. Rev. Mod. Phys.
**2010**, 82, 451–497. [Google Scholar] [CrossRef] - Felice, A.D.; Tsujikawa, S. f(R) Theories. Living Rev. Relativ.
**2010**. [Google Scholar] [CrossRef] - Nojiri, S.; Odintsov, S. D. Unified cosmic history in modified gravity: From F(R) theory to Lorentz non-invariant models. Phys. Rept.
**2011<**