Charged black holes in string-inspired gravity: I. Causal structures and responses of the Brans-Dicke field

We investigate gravitational collapses of charged black holes in string-inspired gravity models, including dilaton gravity and braneworld model, as well as f(R) gravity and the ghost limit. If we turn on gauge coupling, the causal structures and the responses of the Brans-Dicke field depend on the coupling between the charged matter and the Brans-Dicke field. For Type IIA inspired models, a Cauchy horizon exists, while there is no Cauchy horizon for Type I or Heterotic inspired models. For Type IIA inspired models, the no-hair theorem is satisfied asymptotically, while it is biased to the weak coupling limit for Type I or Heterotic inspired models. Apart from string theory, we find that in the ghost limit, a gravitational collapse can induce inflation by itself and create one-way traversable wormholes without the need of other special initial conditions.


Introduction
To investigate the theory of everything which explains gravity and quantum theory in a consistent way, some candidates are proposed [1,2] and one of the most competitive ideas is string theory [3].
String theory can explain not only the finite quantum interactions of gravitons, but also explain the possible modifications of gravity and matter sectors [4]. Such modified gravity models or additional matter contents are useful to study various topics, for example, the inflationary and late time cosmology [5], black hole physics and applications to AdS/CFT [6] or astrophysical applications [7], and so on. Moreover, if string theory is the correct model of quantum gravity, then such a modification should explain various traditional problems of gravity, e.g., the singularity problem of a black hole [8] or cosmology [9].
One of typical modification via string theory is the existence of the dilaton field [10]. This is in fact the same as the Brans-Dicke field [11] that controls the strength of the couplings of interactions. The kinetic term and the potential term of the Brans-Dicke field can be changed by the conformal transformation from the string frame (or the Jordan frame) to the Einstein frame and it is equivalent to just a canonical scalar field model in the Einstein frame [12]. However, if the model couples to the other interactions, then after the conformal transformation, the matter sector is no longer canonical. One of interesting examples is the gauge field. In Section 2, we summarize that there are various types of string effective actions and they generate different types of interactions between the dilaton field and the gauge field.
In this paper, we want to investigate responses of the Brans-Dicke field that depend on couplings between the Brans-Dicke field and the gauge field. Naively speaking, they will form charged black holes. Already there are a lot of papers that investigate stationary solutions [13]. However, the dynamics will be complicated in general for realistic gravitational collapses. For such cases, we expect complicated phenomena such as -Causal structure. In general string frame (Jordan frame), the null energy condition can be violated and hence the dynamics of locally defined horizons [14] can be complicated. In addition, the existence of the central singularity and the Cauchy horizon curvature singularity [15] should be affected by the dynamics of the Brans-Dicke field.
-Response of the Brans-Dicke field. Already some solutions are known that violates the assumptions of the no-hair theorem [17,13]. However, can it be possible to form scalar hairs from the gravitational collapses? For neutral black holes, it was investigated by [16], but the responses due to the charge can give quite different phenomena. This may related to the destabilization of the moduli/dilaton field that should be controlled by a proper potential and may cause observational consequences [7].
In this paper, we mainly focus on these two behaviors. We necessarily need to introduce numerical techniques. At the authors best knowledge, the best technique to investigate inside the black hole is the double-null formalism [18]. There are some previous works regarding string-inspired models using the double-null formalism [19,20] or the other methods [21], as well as the authors [22,23,24].
In this paper, we will search more wide range of parameters. For example, we will cover not only directly string-inspired models, but also indirectly string-inspired models (f (R) inspired model [25]) as well as a natural generalization of parameters, e.g., the ghost limit.
In fact, this issue contains a huge number of parameters. For example, -Model dependence: Type I, Type II, Heterotic, or braneworld inspired models, etc.
-Potential dependence: whether there is a potential that stabilizes the moduli or dilaton field or not [7].
Although some issues were already investigated by the authors or others, still there remain a lot of interesting and various topics. In this paper, we cannot include all issues because of the limited length of the paper, and only focus on the basic model dependence, with no dilaton potential, asymptotic flat, spherical symmetry, and four dimensions. This is why this paper is subtitled: "I.
Causal structures and responses of the Brans-Dicke field." To investigate more general features of mass inflation, we need to investigate the cases with different potentials as well as various parameters. Furthermore, it can be interesting to change the asymptotic cosmological constant and different symmetries. The variation of dimensions and their semi-classical responses will be also interesting to study. These issues will be investigated in future papers.
Apart from the physical issues, this paper is in itself technically worthwhile and important, since this is a quite generalized and complicated implementation of the double-null formalism. We are sure that there are many more variations to be made from this numerical technique and that this paper is an important step towards further useful or realistic string-inspired models. We hope that this paper can be a leading paper that will introduce the upcoming papers.
This paper is organized as follows; In Section 2, we summarize various models inspired from string theory. In Section 3, we implement our model to the double-null formalism. In Section 4, we especially summarize the results in terms of two issues: causal structures (e.g., dynamics of local horizons and existence of Cauchy horizons) and responses of the Brans-Dicke field (e.g., check of the no-hair theorem and the direction of coupling bias). In Section 5, we summarize and discuss future issues. In Appendices, we comment on the consistency and convergence of our numerical scheme and present a catalog of simulations for various initial conditions.

General review of motivations
In this paper, we consider the following action (c = G = = 1): where Φ is the Brans-Dicke field, R is the Ricci scalar, ω is the Brans-Dicke coupling, F µν is the kinetic term of the gauge field and β is the coupling between the Brans-Dicke field and the gauge field. Especially ω and β depend on the models. Next, we illustrate some of the models and motivations; Dilaton gravity Every low energy effective action of string theory contains the following sector that includes the dilaton field [10]: where d is the space dimensions, λ s is the length scale of string units, and φ is the dilaton field.
This can be transformed by redefining φ, such as where G d+1 is the d + 1 dimensional gravitation constant. Then, we recover the Brans-Dicke theory with the ω = −1 limit.
In general, there will be higher loop corrections to the low energy effective action. For example, in a model from Heterotic string theory compactified on a Z N orbifold [29], this can be well approximated by the correction to the Brans-Dicke coupling ω: where κ is a positive constant of order one which is related to anomaly coefficients. Therefore, the coupling should be field dependent. Hence, quite various range of ω can be covered by string-inspired models.
Couplings to the gauge field When we further consider the coupling to the gauge field, the parameter β may depend on the models [4]. We illustrate detailed examples; Type IIA The effective action (bosonic sector) of the Type IIA is expanded by where H 3 is the field strength tensor of the NS-NS two-form field B 2 , F 2 is the field strength tensor of the R-R one form field form field. Therefore, after the dimensional reduction, it is reasonable to obtain the following effective action: and after the redefinition of dilaton, we obtain the gravitational sector by and the two-form field term becomes Therefore, Type IIA corresponds to the ω = −1 and β = 0 case.
Type I The effective action of Type I is expanded by whereH 3 is the mixed contribution of the R-R two-form A 2 and of the matrix valued oneform A 1 , F 2 is the field strength tensor of A 1 with the gauge symmetry by the SO(32) group.
Therefore, after the dimensional reduction and the symmetry breaking to U (1), we can reduce the action by and after the field redefinition, we obtain the same gravitational sector plus the two-form field Therefore, Type I corresponds to the ω = −1 and β = 0.5 case.
Heterotic The effective action of the Heterotic theory is expanded by whereH 3 is the mixed contribution of the NS-NS two-form field B 2 and of the matrix valued one-form A 1 , F 2 is the field strength tensor of A 1 with the gauge symmetry by the SO(32) or E 8 × E 8 group. Therefore, after the dimensional reduction and symmetry breaking to U (1), we can reduce the action by and after the field redefinition, we obtain the same gravitational sector plus the two-form field Therefore, Heterotic theory corresponds to the ω = −1 and β = 1 case.
Braneworld models In the Randall-Sundrum model [30], they introduced two branes, where two branes are connected by a warp factor. In the end, one side has a positive tension and the other side has a negative tension. According to Garriga and Tanaka [31], one can calculate the effective action on branes in the weak field limit as a Brans-Dicke-type theory and we obtain the ω where s is the distance between branes, l = −6/Λ is the length scale of the anti de Sitter space, and the sign ± denotes the sign of the tension. These braneworld inspired models can allow various range of ω near −3/2.
f (R) gravity In general, string theory can introduce various higher curvature terms. If we mainly focus on the Ricci scalar sectors, then we obtain so-called the f (R) gravity. The action of f (R) gravity is We can modify this action to the scalar-tensor type model by introducing an auxiliary field ψ.
Then, the gravity sector changes the form [32] as follows where we have a constraint ψ = R. From this, by defining a new field Φ ≡ f (ψ), we can rewrite the action where Now we obtain the Brans-Dicke theory of the ω = 0 limit with a potential of the Brans-Dicke field, though we ignore the potential term in this paper.
3 Model for charged black holes

Brans-Dicke theory with U (1) gauge field
The prototype action of the Brans-Dicke theory with a U (1) gauge field becomes where φ is a complex scalar field 1 with a gauge coupling e, A µ is a gauge field, and F µν = A ν;µ −A µ;ν .
The Einstein equation becomes as follows: where the Brans-Dicke part of the energy-momentum tensors are and the matter part of the energy-momentum tensors are where Here, we define (25) and T H µν is the renormalized energy-momentum tensor to include Hawking radiation, while in this paper, we will not consider this semi-classical sector.
The field equations are as follows: where

Implementation to double-null formalism
We use the double-null coordinates assuming spherical symmetry, where u is the retarded time, v is the advanced time, dΩ 2 = dθ 2 + sin 2 θdϕ 2 , where θ and ϕ are angular coordinates.
We introduce auxiliary variables, following the notation of [24]: The metric function α, the radial function r, the Brans-Dicke field Φ and a complex scalar field s ≡ √ 4πφ, and define The Einstein tensor is then given as follows: Also, we can obtain the energy-momentum tensors for the Brans-Dicke field part and the scalar field part: where we will define q(u, v) ≡ 2r 2 a ,v /α 2 as the charge function.
To implement the double-null formalism into our numerical scheme, it is convenient to represent all equations as first order differential equations. Note that The Einstein equations for α ,uv , r ,uv , and the field equation for Φ are then coupled as follows (we define X ≡ Φ β X for any quantity X): where After solving these coupled equations, we can write all equations: including the field equations where Equations (50) and (58) are evolution equations for α ,uv , r ,uv , Φ ,uv and s ,uv , which can be solved with the same integration scheme used in the authors' previous papers [22,26]. The gauge field a can be evolved using the same numerical scheme via Equation (57) or we can evolve a and q via Equations (53) and (54). We have found the latter method to be more robust for this study and have used it for our integration via a standard 4th order Runge-Kutta scheme. Tests of the consistency and convergence for our implementation of the present set of equations are presented in Appendix A. First off, we have the gauge freedom to choose r along the initial null surfaces. Here, we choose r(u, 0) ,u = r u0 < 0 and r(0, v) ,v = r v0 > 0 such that the radial function for an in-going observer decreases and that for an out-going observer increases. In addition,we assume that the effective gravitational constant G = 1/Φ is asymptotically unity, hence we can set Φ(u, 0) = Φ(0, v) = 1.

Initial conditions and free parameters
In-going null direction We use an in-going shell-shaped scalar field such that its interior is not affected by the shell. Thus, we can simply choose s(u, 0) = 0, α(u, 0) = 1, q(u, 0) = 0 and a(u, 0) = 0. Then, the Misner-Sharpe mass function should vanish at u = v = 0. To satisfy this, it is convenient to choose r ,u (u, 0) = −1/2, r ,v (0, v) = 1/2. We also choose the radial function at the initial point u = v = 0 to be r(0, 0) = r 0 = 20, thereby fully determining the radial function along the initial null segments. These initial conditions are in agreement with the constraint equations (Equation (51)) and thus completes the assignment of initial conditions along the in-going null segment.
Out-going null direction We can choose an arbitrary function for s(0, v) to induce a collapsing, in-going pulse. In this paper, we use for v i ≤ v ≤ v f and otherwise s(0, v) = 0. Next, we can integrate constraint equation Equation (52) to determine α(0, v) on the u = 0 surface, simultaneously with integrating Equations (53) and (54) for determining q(0, v) and a(0, v).
We have now fully specified the five dynamic variables (α, r, Φ, s, a) along the both in-and outgoing null segments in a manner consistent with the constraint equations (Equations (51) and (52)).

Causal structures and dynamics of Brans-Dicke field
In this section, we report the possible causal structures and responses of the Brans-Dicke field for neutral and charged black holes of various string-inspired models from numerical calculations by using the double-null formalism.

Neutral black holes
In this paper, we want to study charged black holes. Before we start that topic, let us briefly summarize the case of neutral black holes; Figure 1 shows causal structures of neutral black holes for β = 0, Figure 2 shows neutral black holes for β = 0.5 and Figure 3 shows neutral black holes for β = 1.
Let us briefly look at the Brans-Dicke field equation for the neutral limit: Therefore, if ω > −3/2, then since the term T M − 2βL EM is positive, it pushes the Brans-Dicke field to the strong coupling limit (Φ < 1) and vice versa. If ω < −3/2 then it behaves oppositely. The β = 1 case is exceptional and in this case, there is no back-reactions to the Brans-Dicke field and hence there is no dependence on ω (four figures in Figure 3 are the same, bar numerical errors).
As we observed in the previous paper [16], if ω > −3/2, then during the gravitational collapse, the Brans-Dicke field is pushed to Φ < 1 limit and after the collapse, the pushed field is repulsed   and approaches the stationary limit. On the other hand, if ω < −3/2, then Brans-Dicke is pushed to Φ > 1 limit during gravitational collapses and approaches the stationary limit. Because of this response, there appears not only the apparent horizon r ,v = 0, but also r ,u = 0 for ω > −3/2. In addition, the apparent horizon can show strong oscillations for ω < −3/2 and perhaps it can cause a Cauchy horizon as a naked singularity. As 1 − β decreases, the back-reaction decreases. This is clear, if we observe the location of r ,u = 0 horizons or the oscillations of the apparent horizon r ,v = 0 and compare with the β = 0 case and the β = 0.5; they tend to disappear, where these phenomena are the typical behaviors which are due to the Brans-Dicke field.
The oscillations of the apparent horizon r ,v = 0 depends on the number of peaks of the initial collapsing field (see (B) and Figure 20 in Appendix B, and compare with Figure 5 of [16]). This is one difference from [16] and this paper. The location of r ,u = 0 horizon appears and increases as the dynamics of the Brans-Dicke field increases, e.g., as ω approaches −3/2 or 1 − β increases, as denoted in (C). For ω < −3/2, the typical shape is (E), while the apparent horizon and the singularity can meet as was observed in [16] and the causal structure can be (F). For all cases, when the dynamics of the Brans-Dicke field is suppressed, the causal structure approaches (D) and follows that of Einstein gravity.

Charged black holes
Now we turn on the gauge coupling parameter e = 0.3. Note that the case of δ = 0.5 is the case with maximum asymptotic charge (for Einstein gravity, it is demonstrated in [24]), since the phase between the real part and the imaginary part become maximum. However, in general it does not mean that extreme (or naked) black holes could be formed (this is also demonstrated by [24]). The excessive charge will be repelled outside; and then, either excessive charge is repelled to future infinity or forms a stable charge cloud outside the horizon [26]. So, in terms of gravitational collapses, δ = 0.5 is the only way to legally give the largest amount of charge.
Case β = 0: Figure 5 shows charged black holes with β = 0. The obvious feature is the existence of a Cauchy horizon inside the outer apparent horizon. The singularity is always spacelike. However, in the limit v → ∞, there will be nonzero distance (in terms of u-coordinate) between the outer apparent horizon and the singularity. This indicates the existence of the Cauchy horizon. These global behaviors are the same as the case of charged black holes in Einstein gravity. In left of  One important difference is the complicated horizon behaviors inside the apparent horizon. This is not difficult to explain, since there is dynamics of the matter field as well as the Brans-Dicke field inside the horizon and these cause the complicated dynamics of horizons. In Einstein gravity, if we do not turn on the semi-classical effects, then there is no complicated horizon dynamics inside the horizon; But, if we turn on the semi-classical effects, then the null energy condition is violated and hence the inside horizon structure becomes complicated [24]. Also, for modified gravity models such as f (R) gravity (in the Jordan frame), similar behaviors were also observed [25].
Case β = 0.5: Figure 6 shows the case of β = 0.5. As we compare with that of β = 0, there are two clear differences.
First, for ω > −3/2, the Cauchy horizon disappears. In other words, there is a spacelike singularity and it approaches the apparent horizon in the v → ∞ limit. This was observed by [20] (for dilaton limit in the Einstein frame) and our results (various ω in the Jordan frame) are consistent with the previous one. This issue will be clarified as we study the details of the response of Brans-Dicke field in the following section. In terms of the α function, there is a signature of mass inflation, for example right of Figure 7 for e = 0.3 and ω > −3/2 (as well as in Figure 15 in Appendix B), since there appears a region where α approaches zero. However, this is not enough to generate a Cauchy horizon as was in β = 0 cases. Secondly, for ω < −3/2, there can be a r ,u = 0 horizon. Of course, this in itself is not so surprising, but the point is that the horizon r ,u = 0 will eventually form a cosmological horizon.
This was not observed in gravitational collapses of neutral matters. Therefore, we can state that if ω < −3/2, then a charged gravitational collapse can induce inflation and a baby universe. If we interpret the r ,u = 0 horizon as a throat of a (one-way traversable) wormhole, then we can make a one-way traversable wormhole using a charged gravitational collapse. Of course, we know that if we consider the limit ω < −3/2, then the Brans-Dicke field becomes a ghost and hence it can allow a wormhole or a baby universe [33]. However, what we show is that such objects can be obtained by gravitational collapse of charged matter field, i.e., in an extremely simple way, while previously this was obtained from very complicated initial settings [34].
Case β = 1: Figure 8 shares a lot of qualitative properties with the case of β = 0.5. In the neutral limit, there was no similar behaviors for ω < −3/2 limit. Therefore, this makes it more clear that the existence of the inflating region genuinely comes from the effects of charge.
Importantly, it is noted that for the case of ω = −1 (dilaton gravity with Heterotic theory), there exists a spacelike singularity. There were some discussions that extreme black holes can be regular [13]. However, if the black hole is formed from a gravitational collapse, then we have to conclude that such a regular black hole state is impossible to obtain.  β = 0.5 or 1 cases, the gravitational collapse can induce inflation ((I) of Figure 9). If the end point of r ,u = 0 in the v → ∞ limit (green dot in (I)) does not meet the spacelike future infinity, there will be a Cauchy horizon in the v → ∞ limit.

Responses of the Brans-Dicke field
The Brans-Dicke field equation is Note that we can approximately present this as follows: The kinetic terms mainly contribute during the gravitational collapse (in our simulations, v < v f = 20), while the charge terms will contribute after the gravitational collapse (in our simulations, v > v f = 20) and outside the apparent horizon. For outside the black hole, the latter term will remain since the charge is a conserved quantity, while the former term will disappear as the gravitational collapse ends.
If we keep this in mind, then for β = 0 case, there will be essentially no effects after the gravitational collapse and the outside the horizon. In other words, the Brans-Dicke hair will decay as time goes on and approach a stationary limit, Figure 10 shows such behaviors. On the other hand, in the β = 1 limit, there will be almost no back-reaction to the Brans-Dicke field during the gravitational collapse; Only after the gravitational collapse has finished will the Brans-Dicke field be affected. Figure 12 shows such behaviors very clearly.
If 0 < β < 1, then one can see both effects at the same time. Figure 11 shows the behaviors; One can see the contrast between during (0 < v < 20) and after (v > 20) the gravitational collapse, which is different from those of β = 0 or β = 1.
Now let us think about the directions of the Brans-Dicke fields, whether it becomes greater or less than one. For ω > −3/2, if the term T M − 2βL EM is positive, then it pushes the Brans-Dicke field to the strong coupling limit (Φ < 1); If it is negative, then it pushes to the weak coupling limit (Φ > 1). To summarize: -Effects of kinetic terms: If β < 1 and ω > −3/2, then the kinetic terms always push the Brans-Dicke field to the strong coupling limit.
-Effects of charge terms: If β > 0 and ω > −3/2, then the charge terms always push the Brans-Dicke field to the weak coupling limit.
β < 0: All effects push the Brans-Dicke field to the strong coupling limit. As time goes on, the Brans-Dicke hair will be formed in the strong coupling direction.
β = 0: The Brans-Dicke field is pushed to the strong coupling limit, but only dynamically. As time goes on, the Brans-Dicke hair will disappear in the stationary limit.
-0 < β < 1: The Brans-Dicke field is pushed to the strong coupling limit dynamically. On the other hand, as time goes on, the Brans-Dicke hair will be formed in the weak coupling direction.
β = 1: The Brans-Dicke field does not affected by gravitational collapse. However, as time goes on, the Brans-Dicke hair will be formed in the weak coupling direction.
-1 < β: All effects push the Brans-Dicke field to the weak coupling limit.

Discussion
In this paper, we have investigated charged black holes of string-inspired gravity models. Two parameters are important: The Brans-Dicke parameter ω and the coupling between the Brans-Dicke field and the gauge field β. This model covers lots of string-inspired models: The dilaton gravity ω = −1 with Type IIA (β = 0), Type I (β = 0.5), Heterotic (β = 1), some braneworld inspired models ω > −3/2 and f (R) inspired models ω = 0. In addition, we investigated the ghost limit ω < −3/2. We used numerical techniques with double-null formalism to numerically solve the dynamic equations.
We focused on the causal structures and responses of the Brans-Dicke field, especially reactions via charges. For usual non-ghost limit ω > −3/2, the direction of the response of the Brans-Dicke field, whether biased to strong coupling limit or weak coupling limit, is determined by β. If β = 1, then there is no response of the Brans-Dicke field during the gravitational collapse; If β = 0, then there is no response of the Brans-Dicke field via charges after a black hole is formed. This means that after the gravitational collapse, as long as β > 0, the Brans-Dicke field will be biased to the weak coupling limit.
This is very crucial in order to understand the nature of charged black holes for β > 0. One important point is that the black hole will have Brans-Dicke hair in general. The other important point is that the Brans-Dicke field is biased to the weak coupling limit and this will screen the charge inside the horizon. Therefore, this explains the absence of the Cauchy horizon, while it exists for the β = 0 case. However, we did not try to control the Brans-Dicke field using a potential and hence there remains further points to be clarified, we will study these issues in a future paper II, which will focus on mass inflation and dependence on parameters and potentials. This is at least the most obvious next issue to study, but there are many other interesting parameters and issues to persue based on this paper.
Although not a string-inspired model, the ω < −3/2 and β > 0 cases are also interesting in the sense that a charged matter collapse can induce inflation, while the outside still does not inflate. So, as long as the theory allows for ghosts, the formation of a (at least one-way traversable) wormhole does not require complicated initial conditions. Of course, the existence of the ghost may allow complicated instabilities and this should be carefully investigated further. A detailed study of this issue is also postponed to a future paper.
To see not only outside, but also inside the black hole and to study the detailed relations between gravity and various field contents, probably the best way is to use the double-null formalism. The authors hope that this paper will be an important step to prepare for upcoming investigations of diverse and various interesting topics, including the interests of theoretical gravity, string theory, holography, astrophysical applications, as well as cosmology, by varying matter, dimensions, gravity model, and symmetry, etc.
where x i N denotes the dynamic variable x at the i-th grid point of simulation with resolution N and where x i HighRes denotes the dynamic variable of the same i-th point for a simulation with the highest numerical resolution done by us. Obviously, this expression only makes sense for those i points that coincide in all simulations.
The first six plots in Figure 13 show the relative convergence, ξ, for the dynamic variables α, r, Φ, Re s, Im s and a, respectively. The lines in the figures are marked by their numerical resolution measured in terms of the most coarse resolution N 0 , the high resolution simulation used to calculate expression Equation (63), has a numerical resolution of 32 times the base resolution, i.e., N HighRes = 32N 0 .
From these figures, it is clear that the dynamic variables are converging for simulations of increasing resolution. Furthermore, since we plot the relative convergence of the dynamic variables, we see that the relative change between the two highest resolution simulations show that the variables change on the order of 1 % or less, which we considered a quite acceptable.
It is, however, not enough to demonstrate that the simulations are converging, they must also converge to a 'physical ' solution, i.e., the residuals of the constraint equations (Equations (51) and (52)) must converge to zero. To demonstrate this, we calculate the relative convergence of the constraint equation residuals, (relative to the Einstein-tensor) in a similar way to Equation (63): where C i N denotes the residual of the constraint equation (C uu or C vv ), at the i-th point for simulation with resolution N and where G i HighRes denotes the corresponding Einstein-tensor component (G uu or G vv respectively) at the same point.
The relative convergence of the residuals of the constraint equations are demonstrated in the bottom two plots of Figure 13, where it is seen that they converge towards zero for higher resolution simulations. This indicates that not only are the numerical solutions converging for simulations of higher resolution, but that they are indeed converging towards a physical solution.
Finally, it is emphasized that the results presented in this Appendix are not the only convergence tests that we have performed, they are merely representative of typical convergence behavior. For all results presented in this paper, we have performed a large number of simulations with varying resolutions to ensure that the results had converged to their physical solution.

Appendix B. Catalog of metric function and energy-momentum tensors
In this appendix, we summarize the metric function α and energy-momentum tensor components T uu and T vv . Figures 14, 15, and 16 are plots of α for β = 0, 0.5, and 1, respectively. For typical situations of mass inflation, α → 0 exponentially, and hence almost all curvature quantities (Ricci scalar, Kretchmann scalar, etc.) behaves α −n (n > 1) and hence curvatures exponentially diverge. are plots of T vv for β = 0, 0.5, and 1, respectively. Note that around the apparent horizon r ,v = 0, and hence, for an out-going observer, if T vv > 0, then the r ,v = 0 horizon changes the sign of r ,v from + to −, while T vv < 0, changes the sign of r ,v from − to +. Also, around the apparent horizon r ,u = 0, and hence, for an in-going observer, if T uu > 0, then the r ,u = 0 horizon changes the sign of r ,u from + to −, while T uu < 0, changes the sign of r ,u from − to +. For all figures, these rules are always satisfied even with complicated horizon dynamics. This is also a simple check of the consistency of our simulations.