Insights and guidelines on the Cauchy horizon theorems

Here we begin with the Reissner-Nordstrom black hole which is a static, stationary, and charged black hole, with the given metric as shown in the following,

where

where $M$ is the ADM mass and $Q$ is the asymptotically defined electric charge Reissner ; Nordstrom . Since $f(r)=0$ can yield two solutions, one can notice that, if $M>Q$, this metric has two apparent horizons which can be denoted as $r_{+}$ and $r_{-}$ with $r_{+}>r_{-}$, where $r_{+}$ is the outer horizon and $r_{-}$ is the inner horizon.

Let us define that $u$ denotes the in-going null direction and $v$ denotes the out-going null direction. After we maximally extend the spacetime in Kruskal-Szekeres coordinates dInverno:1992gxs , i.e.,

with $r=r(u,v)$, one can notice that there are two directions of the horizons, where $r_{,v}=0$ horizons (red color) are always parallel to the out-going null direction, while $r_{,u}=0$ horizons (blue color) are always parallel to the in-going null direction as depicted in Fig. 1. In addition, there exists a time-like singularity at $r=0$; the inner horizon is the boundary where we cannot decide the geometry from the past data, and hence, this horizon is known as the Cauchy horizon Hawking:1973uf . The outer horizon is the boundary between the interior and the asymptotic infinity, and hence, this horizon is known as the event horizon. So, in the static and charged black hole case, the inner apparent horizon is the Cauchy horizon and the outer apparent horizon is the event horizon.

Here, we list down several common senses of the charged black holes:

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  $r_{,v}=0$ apparent horizon is parallel to the out-going null direction; $r_{,u}=0$ apparent horizon is parallel to the in-going null direction.

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  The $r=0$ singularity is time-like.

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  The inner apparent horizon is the Cauchy horizon; the outer apparent horizon is the event horizon.

However, these common senses are valid only if the metric is static. In the dynamic cases, all three assertions are not valid in general. On the other hand, if one applies such common sense to some proofs about the non-existence of inner horizons, the corresponding results might not be rigorous or even wrong because, in a realistic situation, the metric should be dynamic.

One of the most important issues about the inner horizon is the instability of inner horizons. When there exists an energy flux along the in-going null direction, we can define an observer who is moving along the out-going null direction and measures the energy flux. Near the inner horizon, the observed energy flux is approximately

where $\kappa$ is the surface gravity of the inner horizon Poisson:1990eh . Therefore, the observed energy must be exponentially amplified near the inner horizon. This effect is known as the mass inflation.

Theoretically, we can have a static, stationary, and charged black hole solution that is independent of time. However, in reality, there is no such an eternal object in the universe; every black hole must be generated from gravitational collapses. Therefore, energy fluctuations necessarily exist inside a charged black hole. Thus, the consequence of mass inflation is that, in realistic black holes that are generated from gravitational collapses, there must be a strong back-reaction due to the exponentially increased energy flux Bonanno:1994ma .

Hence, it is unlikely to precisely study the interior structures of a charged black hole from the perturbative analysis on the background of static solution. The only possible way to precisely understand the interior structure is to numerically evolve the corresponding equations of motion. In this paper, we apply the double-null formalism to study the formation of an inner horizon in the gravitational collapse of charged black holes.

In this context, there have been some mathematical investigations and proofs regarding the inner horizon of black holes. In the literature, these investigations are mainly to formulate the no-Cauchy-horizon theorems. For example, we specify several mathematical assertions from the literature as follows:

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  Ref. An:2021plu has shown numerically the Cauchy horizon does not exist in the interior of the scalar-charged hairy black holes with spherical and planar symmetries in the Born-Infield theory.

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  Ref. Cai:2020wrp has shown numerically the Cauchy horizon does not exist in the interior of the scalar charged hairy black holes with spherical and planar symmetries in Einstein-Maxwell-scalar theory of a complex scalar field with an arbitrary form of the scalar potential.

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  Ref. Devecioglu:2021xug has studied the non-existence of the Cauchy horizon for some specific types of black holes analytically in the Einstein-Maxwell-Horndeski theories.

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  Ref. Grandi:2021ajl has studied numerically the non-existence of Cauchy horizon for scalar charged hairy black holes in the 5-dimensions of Einstein-Gauss-Bonnet theory with a massive complex scalar field.

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  Ref. Yang:2021civ has studied the possibility of putting a constraint on the number of horizons for the static black holes which can only possess at most one non-degenerated inner Killing horizon inside the event horizon when they satisfy the strong or null energy condition.

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  Ref. Devecioglu:2023hmn has studied numerically the non-existence of Cauchy horizon for charged black holes with planar configuration in the Einstein-Maxwell-Gauss-Bonnet-scalar theory.

Here we list down some common features in the literature as follows.

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  1. Some authors assumed that the black hole geometry will follow a regular and static solution. If a gravitational collapse occurs and then after a long time has sufficiently passed, a black hole solution will approach a static solution, but this is in the perspective of the outside observer. One can ask whether this assumption is still true for an inside observer.

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  2. Some authors assumed hairs to prove the theorems for some specific cases, i.e., the dynamics of matter fields near the inner horizon. If any inconsistency can arise from the corresponding dynamics, then they can prove the non-existence of the Cauchy horizon. However, the no-hair theorem has not been established near the inner apparent horizon, and thus there is no systematic way to evade the no-hair theorem inside the outer horizon; hence, it would be risky to assume the existence of such a conserved quantity associated with the symmetry of the theory for the inner horizon.

In this paper, we will show some examples that satisfy the following conditions. (1) The inner horizon is formed and approaches a (weak) curvature singularity. Hence, the interior geometry is no longer static or regular. (2) Near the inner horizon, there can be non-trivial field dynamics. This is not surprising because the interior geometry is no longer static. In the next section, we will discuss more issues that go beyond common sense.

Let us consider the most generic double-null coordinates with spherical symmetry:

In the the double-null formalism, one can obtain the equations of motion and solve them numerically based on this metric with appropriate boundary conditions (see Appendix C) Nakonieczna:2018tih . So far, plenty of numerical simulations have been carried out to investigate the dynamics of the formation of inner horizons during the gravitational collapse of the charged black holes using the double-null formalism Hod:1998gy ; Hwang:2011mn ; Sorkin:2001hf ; Hong:2008mw ; Sorkin:2001hf ; Hansen:2014rua ; Hansen:2015dxa ; Nakonieczna:2015umf ; Nakonieczna:2016iof ; Chen:2015nma . In this section, we summarize some important results that are relevant to the issue of the inner horizon by revisiting some of our previous works.

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  The authors investigated the properties of the Cauchy horizon inside the formation of a charged black hole during a gravitational collapse caused by an input pulse consisting of a complex scalar field in the Einstein-Maxwell theory Hod:1998gy ; Hwang:2011mn . The numerical results show that the inner horizon is formed after the gravitational collapse. If we turn off Hawking radiation, the inner horizon becomes a Cauchy horizon.

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  The authors extended the above investigation to study the semi-classical description for the emission of Hawking radiation from a charged black hole Sorkin:2001hf ; Hong:2008mw . The inclusion of Hawking radiation allows that the inner horizon and the Cauchy horizon can be distinguished.

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  The authors generalized the above two investigations from the Einstein gravity to the Brans-Dicke theory that can cover a diverse range of theories, including string-inspired models Hansen:2014rua ; Hansen:2015dxa ; Nakonieczna:2015umf ; Nakonieczna:2016iof , $f(R)$ gravity Hwang:2011kg ; Chen:2015nma , etc. These results show that the inner and outer horizons can be varied dynamically due to the Brans-Dicke field. Also, we can see the existence or non-existence of inner horizons relating couplings between the Maxwell field and the Brans-Dicke field.

Before we proceed to present the above results, we need to define clearly the inner horizon and singularity in the next subsection, since their appearance is highly sophisticated in the double-null simulation.

The Reissner-Nordstrom black hole in the double-null coordinate contains two kinds of inner horizons: $r_{,v}=0$ and $r_{,u}=0$, where the former is parallel to a constant $u$ line and another is parallel to a constant $v$ line. The two horizons appear with the same areal radius, and they are well coincide with the causally defined Cauchy horizon.

However, we need to distinguish three kinds of horizons for the dynamical cases:

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  H1. $r_{,v}=0$ inner apparent horizon.

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  H2. $r_{,u}=0$ inner apparent horizon.

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  H3. Cauchy horizon (boundary where one can causally determine from the past data).

To define the third type of horizon (H3) precisely, one needs to take into account the subtleties to define the singularity and its properties. Thus, let us grasp several basic concepts about singularity as follows.

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  Type 1: it lies at $r=0$ and the corresponding curvature quantities diverge; this kind of singularity is in the classical sense.

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  Type 2: it lies in $r>0$ and the corresponding curvature quantities diverge; this kind of singularity is also in the classical sense which is so-called a weak singularity Ori:1991zz .

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  Type 3: the radius of singularity is non-vanishing but at the Planck scale $r\simeq\ell_{\mathrm{P}}$ and the corresponding curvature quantities are also the Planck scale.

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  Type 4: the radius of singularity is greater than the Planck scale $r>\ell_{\mathrm{P}}$ but the curvature quantities are at the Planck scale. This kind of singularity is known as the quantum version of the weak singularity.

Therefore, we can classify that Type 1 and 2 are classical singularities, while Type 3 and 4 are defined when there exists the Planck constant $\hbar$ for the quantum regime. Type 1, 2, and 3 will dynamically generate the Cauchy horizon because one cannot extend computations inside the singularity; however, the property of Type 4 is less known, because one may extend general relativistic computations inside the singularity. The only problem is that such an extension is inconsistent in terms of quantum gravity except for some extreme cases (e.g., introducing the large $N$-rescaling Yeom:2009zp ; Kim:2013fv ).

In this subsection, we discuss the formation and evaporation process of charged black holes based on Ref. Hong:2008mw . Hence, we begin with the typical model where a complex scalar field is coupled with a $U(1)$ gauge field that is given as follows:

where $\mathcal{R}$ is the Ricci scalar, $\phi$ is a complex scalar field, $A_{\mu}$ is the $U(1)$ gauge field, $e$ is the gauge coupling, and $F_{\mu\nu}=A_{\nu;\mu}-A_{\mu;\nu}$. In the double-null formalism, one can observe a charged black hole that is formed in this model due to the appearance of the input pulse $\phi(u,v)$ at $u=0$ to cause a gravitational collapse to occur in the spacetime,

where the pulse is defined in $0\leq v\leq v_{\mathrm{f}}$, $A=0.25$, and $e=0.1$.

To turn on the Hawking radiation, we introduce the renormalized energy-momentum tensor $\langle\hat{T}_{\mu\nu}\rangle$ in an approximate form Davies:1976ei , where the tensor is proportional to $P\propto N\ell_{P}^{2}$ (the quantity $P$ is precisely defined in Appendix A), where $N$ is the number of scalar field that contribute to the Hawking radiation; this is the natural cutoff if $N$ fields contribute to Hawking radiation Dvali:2012uq . If $P=0$, we turn off Hawking radiation; we choose $P=0.1$ for the evaporating case.

Solving the equations of motion derived from Eq. (6) numerically using the double-null formalism (see Appendices A and C for more details), our numerical results demonstrate that a charged black hole is formed and evaporated during the process of gravitational collapse. Regarding this, we address several important remarks from the numerical results in Hong:2008mw :

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  1. There exist two kinds of singularities, which are Type 1 and Type 2. On the left of Fig. 3, one can see a Type 1 singularity at $r=0$ (thick black curve) and an outer apparent horizon $r_{,v}=0$ (red curve).

  In the late time limit, i.e., as $v$ goes to infinity, there appears the H1 horizon ($r_{,v}=0$) which is parallel to the ingoing null direction. This will eventually approach a Type 2 curvature singularity ($r>0$) due to the mass inflation effect.

  Therefore, the Type 2 singularity coincides with the horizon H1, although it is parallel with constant $v$ lines. This is somehow counter-intuitive with the common sense which has been mentioned in Sec. II. However, this will help to preserve the strong cosmic censorship conjecture, because no observer can penetrate the Cauchy horizon due to the weak singularity.

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  2. If we turn on the Hawking radiation as well as the incoming negative energy flux, the situation becomes more complicated. In the left of Fig. 3, there exists the Type 3 singularity $r\simeq\ell_{P}$ and an outer apparent horizon $r_{,v}=0$. At the inner apparent horizon, the H1 horizon ($r_{,v}=0$) and the H2 horizon ($r_{,u}=0$) are overlapped. Hence, as an observer penetrates inside the inner apparent horizon, the observer will experience that the areal radius increases, which is analogous to a wormhole structure (see more details in Hwang:2010im ). Due to the existence of the Type 3 singularity, there appears the H3 Cauchy horizon which is the future affected region from the singularity. Beyond the H1/H2 inner apparent horizon, the curvature rapidly increases, approximately $m\sim e^{+M^{2}}$, where $m$ is the Misner-Sharp mass Hong:2008mw ; hence, the gray-colored region represents the super-Plankian region which goes beyond the scope of the semi-classical description. Therefore, the gray-colored region (inside the H1 and H2 inner horizons) corresponds to Type 4 singularity ($r>\ell_{P}$).

  To summarize, there appears both Type 3 singularity and Type 4 singularity. Type 3 singularity generates the Cauchy horizon, while it is hidden by the Type 4 singularity. On the other hand, if one can trust the computation beyond the Type 4 singularity, e.g., assuming a large number of matter fields ($\sim e^{M^{2}}$, see Yeom:2009zp ; Kim:2013fv ), the strong cosmic censorship might be violated even with the semi-classical effects Poisson:1997my . The Type 4 singularity very well coincides with an inner apparent horizon, although there is an ambiguity in defining the exact location of the singularity because it depends on the choice of the cutoff.

Therefore, if we only define the apparent horizon using H1, one may neglect the important and interesting behaviors relating to the inner horizon and the Cauchy horizon. We can conclude that in dynamical situations, H1, H2, and H3 horizons are distinguishable; also, Type 1, Type 2, Type 3, and Type 4 singularities all appear in different situations.

As we have mentioned, Fig. 3 confirms that the interior of a black hole does not remain as the interior of the static solution. The inner horizon becomes the $r_{,v}=0$ horizon and is parallel to the in-going null direction, which is inconsistent with the static solution. Moreover, in the late time ($v\rightarrow\infty$) limit, the inner horizon becomes a curvature singularity, and hence, the regularity condition is also not satisfied.

Some literature heavily relied on the assumption that there may exist hairs associated with the matter fields to establish some theorems about the non-existence of Cauchy horizons. In this subsection, we focus on the cases in which some hairs may help to remove the Cauchy horizon, but surprisingly there exists a counter-example that there is the Cauchy horizon with a scalar hair. Here, a crucial factor to support the counter-example is the existence of scalar hair at the outer horizon, although the non-trivial scalar hair can exist on the inner horizon.

To investigate this issue, we consider the following model with a non-minimally coupled scalar field Hansen:2014rua ; Hansen:2015dxa ; Nakonieczna:2015umf ; Nakonieczna:2016iof :

where $\Phi$ is the Brans-Dicke field, $\omega$ is the Brans-Dicke parameter, $V(\Phi)$ is the potential of the Brans-Dicke field, and $\beta$ is a constant. Similarly, to observe the formation of the inner horizon for charged black hole during the gravitational collapse using the double-null formalism we send an input pulse of a charged scalar field ($0\leq v\leq v_{\mathrm{f}}$) at $u=0$, where it is given by

For simplicity, we fix $e=0.3$ and $\beta=0$ or $\beta=1$.

In this model, we can consider the Brans-Dicke field which is non-minimally coupled with the Maxwell field. Solving the equations of motion derived from Eq. (8) numerically in the double-null formalism (see Appendices B and C for more details), we demonstrate our numerical results from Hansen:2014rua ; Hansen:2015dxa ; Nakonieczna:2015umf ; Nakonieczna:2016iof in the following.

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  1. In Fig. 5, we considered $\beta=0$ and $\omega=1000$. The large value of $\omega$ implies that the dynamics of the Brans-Dicke field can be switched off effectively and this result is consistent with Einstein gravity.

  Interestingly, the dynamics of the real and imaginary parts of the complex scalar field do not disappear in the large $v$ limit. We find that the derivative of a complex scalar field with respect to $u$ might vanish but might not vanish with respect to $v$. Therefore, the dynamics of a non-trivial complex scalar field can exist in the inner horizon.

  If we turn on the Hawking radiation, one can still see the dynamics of the scalar fields that penetrate beyond the inner apparent horizon: see Fig. 15 of Hong:2008mw .

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  2. We considered $\beta=0$ in the left of Fig. 5. The minimal coupling between the Maxwell field and the Brans-Dicke field in the Einstein frame does not form Brans-Dicke hair on the outer horizon.

  Inside the outer apparent horizon, the appearance of $r_{,v}=0$ (red curve) and $r_{,u}=0$ (light blue curve) horizons are highly sophisticated because we perform the calculations in the Jordan frame. Hence, both H1 and H2 can appear in various places in the Jordan frame.

  In the $v\rightarrow\infty$ limit, there exists a Type 2 singularity, but there exists a gradient of the Brans-Dicke field along the $u$ direction. Therefore, one can interpret that there exists the Cauchy horizon in the $v\rightarrow\infty$ limit, and at the same time, there exists a non-trivial scalar field dynamics along the Cauchy horizon.

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  3. In the right of Fig. 5, we considered the $\beta=1$ case. The non-minimal coupling between the Brans-Dicke field and the Maxwell field gives rise to the existence of the Brans-Dicke hair along the outer apparent horizon.

  One interesting observation is that there is no inner apparent horizon or Type 2 singularity inside the outer horizon. So, this implies that if there exists a hair at the outer horizon, the inner Cauchy horizon structure disappears, which is very different from pure Einstein gravity. Of course, at this moment, this is just a phenomenological observation and we need further mathematical investigations.

Therefore, the scalar hair at the outer apparent horizon plays a more important role than the gradient of the (scalar or vector) field along the inner apparent horizon to determine the existence of the Cauchy horizon.