On the Cosmic Censorship Conjecture

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On the Cosmic Censorship Conjecture

Sjabbo Schaveling

Student nr. 10001230

Supervisors: Ben Freivogel and Erik Verlinde Executed from June 2013 to Februari 2014 on the

Institute of Theoretical Physics Amsterdam

University of Amsterdam

Faculty of Natural Sciences, Mathematics and Computer Science June 5, 2014

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1 Abstract

The singularity theorems are considered and the prediction of black holes is discussed. In the prediction of black holes, the validity of the (weak) cos-mic censorship conjecture plays a crucial role. Both the weak and the strong cosmic censorship conjectures are formulated and discussed in detail, since they are crucial for the completeness of general relativity (GR). A theorem which proves the validity of both conjectures if the Einstein equations are coupled to the massless scalar field is discussed. Also the Einstein-Vaslov system is briefly discussed.

As an example where both conjectures might break down, the Reissner-Nordstr öm solution is considered. Arguments are made why both con-jectures actually hold true for the Reissner-Nordstr öm solution. It turns out that Schwinger pair production close to the inner horizon alters the causal structure significantly for a certain class of black holes. Also the mass-inflation principle is found to play an important role in the causal structure of the Reissner-Nordstr öm black hole.

2 Populaire Samenvatting (Dutch)

Iedereen heeft wel eens gehoord van zwarte gaten. Het zijn mysterieuze objecten, die zelfs wetenschappers nog niet helemaal begrijpen. De al-gemene relativiteitstheorie van Einstein voorspelt het bestaan van zwarte gaten. Het zijn objecten met een punt van oneindige dichtheid in het mid-den. Het is dit zogenaamde singuliere punt dat alle problemen veroor-zaakt.

Er is op dit moment nog geen theorie die voorspelt hoe zwaartekracht zich op hele kleine schaal en bij grote dichtheden gedraagt. Natuurkundigen zijn al jaren hard op zoek naar deze theorie van kwantumgravitatie, die bij-voorbeeld de singuliere punten in zwarte gaten zou kunnen beschrijven. Helaas is deze theorie nog niet gevonden. Dus op dit moment kunnen natuurkundigen nog niet voorspellen wat er allemaal gebeurt dichtbij zo’n singulier punt. Misschien komt er wel gigantisch veel straling uit, of is het een wormgat naar een ander universum. Wie weet!

Meestal wordt er over zwarte gaten verteld ze een waarnemingshorizon hebben, waar alle invallende materie achter verdwijnt. In het bijzonder wordt er verteld dat er geen licht of andere materie uit een zwart gat komt (op hawkingstraling na). Dit hebben natuurkundigen alleen nog niet be-wezen. De relativiteitstheorie voorspelt zelfs het bestaan van zwarte gaten waar je theoretisch gezien naar toe kunt reizen, de singulariteit kunt beki-jken en weer terug kunt reizen. Rond deze zogemaamde naakte singuliere punten kun je niet voorspellen wat er gebeurt. Er is simpelweg geen

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theo-rie voor.

Omdat natuurkundigen graag voorspellen wat er gaande is in het uni-versum, proberen ze al jaren te bewijzen dat deze zwarte gaten niet in de natuur voorkomen. De eerste die dit probleem voorzag, en het zo-genaamde cosmic censorship vermoeden opstelde, was Roger Penrose in 1969. Cosmic censorship betekent zoveel als kosmische censuur. Dit slaat op het vermoeden dat singuliere punten in de natuur afgeschermd worden van waarnemers. Het cosmic censorship vermoeden is nog steeds niet be-wezen. We weten dus nog steeds niet of deze naakte singuliere punten echt in de natuur voorkomen.

Over deze hypothese, die natuurkundigen al jaren proberen te bewijzen, ga ik het hebben in deze scriptie. Er zullen een paar voorbeelden worden genoemd van naakte singuliere punten, en we zullen zelfs een voorbeeld van een wormgat zien. Helaas komt ook in deze scriptie geen algemeen bewijs van het cosmic censorship vermoeden voor.

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3 Introduction

At the end of the nineteenth century, people (for example Lord Kelvin) were convinced that physics was finished. All that was left for physics were more precise measurements. However two problems in particular kept physics from being finished. The first problem being the velocity of light which appeared as a constant in Maxwell’s theory for electromagnetism, and the second problem being Max Planck’s law for blackbody radiation. Einstein showed that this blackbody spectrum could not be accounted for by classical electromagnetism. He showed that this would lead to an infi-nite energy radiation.

We all know what came next. The first problem led to Einstein’s theory of General Relativity. The second problem led to quantum mechanics. These theories are fundamental to almost all of today’s discoveries in physics. These two ‘problems’ of physics, as Lord Kelvin presumably called them, pushed the limits of physics. They made scientists realize that another the-ory was needed.

Today, physics is more complicated than in the nineteenth century, but problems which push physics forward still exist. On the contrary, physics is far from complete. One area in which physics is considered incomplete, is Einstein’s theory itself: General Relativity (GR). One way in which this in-completeness manifests itself is through Penrose’s cosmic censorship jectures, which remain unproven to date. The weak cosmic censorship con-jecture states that sufficiently far away, no singularities can be observed directly. The strong version states that locally one cannot observe any sin-gularities. However the strong conjecture doesn’t imply the weak conjec-ture, contrary to what the name suggests.

As in any area in physics, general relativity is concerned with the study of differential equations, the Einstein field equations. To study physics, it is of great importance to know the domain of predictability of the theory. It is the strong cosmic censorship conjecture that asserts us that for generic initial conditions, the initial conditions predict the future development of the whole space-time. Hence both conjectures are of great importance for the study of GR.

This thesis will give a brief introduction in the weak and the strong cosmic censorship conjecture. In the first part, both conjectures are stated, with the proper definitions and a comment on the physical meaning and importance of these conjectures. In the second part the Reissner-Nordstr ¨om solution of the Einstein field equations is considered, as a specific example in which both conjectures come into play. Hence illustrating the issues which the conjectures raise, and why they might ‘push the limits of physics’.

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4 Singularity Theorems

Singularity theorems are among the most important results of classical gen-eral relativity. They are fundamental to the study of black holes. Given an energy condition and some other specific starting conditions, singularity theorems state that there will be a singularity somewhere in the past or in the future. The strength of these theorems is that they prove that singulari-ties are not some artifact arising from spherical symmetry, but they predict singularities in general situations. In this section we state singularity the-orems which use the null energy condition, since this is the only condition believed to hold in the universe.[11]

But why are these so called singularities so important? In physics, we wish to have a theory which is valid everywhere. Or at least have a valid the-ory for every region in spacetime. Singularity theorems predict that this will not be the case for the current theory for gravity, Einstein’s general rel-ativity (GR). Under some conditions there will exist geodesics which will cease to exist in a ‘finite proper time’. In other words there is a singularitiy present. This means that we cannot predict the future in the case where a singularity arises. However, not all is lost.

We already know other theories in which singularities exist: Newtonian gravity, classical electrodynamics and fluid dynamics, to name a few ex-amples. However, in al those cases there turned out to be an underlying theory which solved the singularities: GR, QED and the molacular model for fluids. In GR the theory we are looking for is quantum gravity.

Now we could try to find such a theory, which people have been trying. We could also try to prove that given ‘generic’ starting conditions we will always remain in the domain of validity of the theory. Or in other words, that infinitly far away we will not observe any strange effects arising from the underlying theory. This is exactly what the weak cosmic censorship conjectjecture says. The strong cosmic censorship conjecture asserts us that we can locally see no singularities. [13]

We could then try to prove that all the singularities predicted by the sin-gularity theorems are hidden behind a horizon, or are very special cases, which would not occur under normal circumstances. however this is still an open problem in classical GR. To address this problem, we will first have to find an appropriate form to put the cosmic censorship conjecture in. But first we will give some of the theorems which predict the singularities in a spacetime, so called singularity theorems.

The first theorem is a classical singularity theorem, proven by Hawking and Penrose (for an extended discussion of those theorems, see [6].)

Theorem 1. Let(M, g)be a spacetime with a non compact Cauchy surface V, obeying the null energy condition. If V contains a trapped surfaceΣ then at least one of the future directed null normal geodesics fromΣ is incomplete.

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For the exact definitions, I refer to the appendix. It is believed that the null energy condition holds for all matter in the universe[11]. We want to know when a non-compact Cauchy surface and a trapped surface is present. The condition thatM posesses a Cauchy surface is called global hyperbolicity. The conditions for global hyperbolicity are that no naked singularities exist (in the sense of the definition given in the appendix) and that no closed causal curves exist.[16]

The non-compact condition can be replaced by the condition J+(Σ) 6= J+(V), for a Cauchy surface V. This is the condition that there exists some observer which can avoid falling into the singularity. This condition seems physi-cally reasonable, since we assume that the data considered is hyperbolli-cally flat. An hyperbollihyperbolli-cally flat spacetime looks like Minkowski space at infinity. [6]

We also want to know under which circumstances there is a trapped sur-face present. This is a much more delicate matter. Schoen et al. and Beig et al. have given some conditions under which a trapped surface is present.[15][14] Christodoulou has shown that for initial data sufficiently close to the trivial data (Euclidian R3_{), the maximal future development} of the initial data is complete, and has shown that the theorem does ap-parently not apply to this case. An issue raised by Christodoulou is that a trapped surface can also form from starting conditions without a trapped surface. Recently however, Christodoulou solved this problem, by specify-ing for what initial data a closed trapped surface would form. [13] [18] We see that if we assume that we can see no singularities from ‘infinity’, causality and asymptotical flatness, we only have to find the closed trapped surfaces to apply the above theorem. Of course there might be other cases for which singularities will form. The Causality and asymptotical flatness conditions hold true in the universe, as far as we know, since causality is the only reason we can study physics and since we live in a region with small curvature.

That no naked singularities exist is by no means trivial. On the contrary, GR predicts the existence of naked singularities, as we will see in the next sections. But as we will also try to argue in the next few sections, those examples are very special cases. The implications of the existence of naked singularities would be huge: we could in principle observe the effects of the quantum-gravity theorem underlying classical GR. Also the big bang wouldn’t be the only naked singularity in our universe anymore. As some have argued, this might be a great consequence. [22] However, there seems to be a general consensus about the weak cosmic censorship conjecture be-ing true.[2] But not all hope is lost, for the weak cosmic censorship con-jecture (as well as the strong cosmic censorship concon-jecture) has still to be proven. It is important to realize that the strong cosmic censorship conjec-ture doesn’t imply the weak one, as we will see in section 6.

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To say something about singularities in the past, for the null energy con-dition, requires a different kind of theorem. Consider a test particle and a comoving observer on a geodesic with four-velocity vµ_{. For the freely} moving test particle, denote the four-velocity by uµ_{, and let ∂r}µ _{be the} vector which joins the worldlines of the test particle and the observer at equal proper time. Then we can define the generalized hubble parameter as H = ∂ur

∂r =

−vµ(Duµ/dτ

γ2−κ , where γ = uνv

ν_{. We take the parameter τ of the} observer to be the proper time in the timelike case, and some affine param-eter in the null case. Also κ=vµvµ

Now the theorem states that if the average hubble expansion Hav obeys Hav >0 along a null or timelike geodesic, then the geodesic is past incomplete.[19] This theorem has a much simpler form. In particular if we assume inflation, the theorem implies that there must be a singularity (big bang) in the past. The Big Bang provides a ‘counterexample’ to the cosmic censorship conjec-ture, since there are geodesics terminating in the past in the big bang. [20] However, the big bang is not the only naked singularity which is predicted by GR (although there are certain cyclic models which claim that the ‘big bang’ is not ‘really’ a singularity[38]). In the last sections we will give some examples of naked singularities which are predicted by GR. We will argue why the naked singularities are not ‘generic’ in some sense however, and why they do not form a counter example for the weak cosmic censorship conjecture (WCCC) and the strong cosmic censorship conjecture (SCCC). In the next two sections we will give several formulations of the WCCC and the SCCC, among which an alternative formulation in the spherical sym-metric case.

5 Formulation Cosmic Censorship Conjecture

We formulate the weak cosmic censorship conjecture as formulated by Wald[2]. For the exact definitions of used, see the appendix. Wee will try to give an intuitive explanation of the conjecture here.

Theorem 2. Let M be a 3-manifold which, topologically, is the connected sum ofR3_{and a compact manifold. Let}_(H_{, g, k}₎_{be non-singular, asymptotically flat} initial data onMfor a solution to Einstein’s equation with suitable matter (where k denotes the appropriate initial data for the matter). Then generically, the maximal Cauchy Evolution of this data is a space-time, (M,g) which is asymptotically flat at future null infinity, with complete J+.

As has been discussed in the appendix, the completeness of the future null infinity, J+, means that observers sufficiently far away can live their lives without noticing any effects from singularities. What the theorem says, in non-technical terms, is that given singularity free initial data which

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is ‘flat at infinity’, any observer far away won’t observe any effects from naked singularities. Or stated mathematically, the future null infinity does contain its limit points. So if no naked singularities are present initially, any observer far away won’t notice anything from the naked singularities that might possibly arise in the future development of the initial data.

However we want to assure that locally, things are well behaved too. In other words, we want to check that initial data does indeed predict the whole space-time and that it is not the case that some small region (i.e. the naked singularity) is left out of the domain of dependence. In that case the Cauchy surface would still be a Cauchy surface, but the maximal future development of the surface could be embedded into a larger manifold (so with the naked singularity included) for which the surface wouldn’t be a Cauchy surface anymore, since every point in the causal future of the naked singularity is not in the domain of dependence of the surface.

The conjecture that asserts that singularities are well behaved locally is called the strong cosmic censorship conjecture (SCCC). The SCCC was for-mulated by Christodoulou, but he later changed it to the following formu-lation due to a counterexample of Dafermos.[18]

Theorem 3. Generic asymptotically flat initial data, with a suitable matter

con-tent have a maximal future development which is inextendible as a metric with locally square integrable Cristoffel symbols.

Here the square integrability of the Christoffel symbols is a condition sufficient to rule out the existence of weak solutions to the Einstein field equation (weak solutions are solutions for which not all the derivatives of the metric tensor exist). Recently however, Dafermos put forward a con-jecture which states that for small perturbations, the Kerr black hole might provide a counter example to the SCCC (see section 15 of [25]). Since black holes can arise from ‘reasonable matter content’, we then have a counterex-ample.

Originally, both conjectures were formulated by Penrose[22]. Although some definitions needed for the mathematical formulation of the conjec-tures were only properly defined recently, these conjecconjec-tures are recognised to be central to classical GR, since they assure that things are indeed as well behaved as we hope they are.

6 A spherically symmetric massless scalar field

In this section, an alternative way to state both the strong and the weak cen-sorship conjecture at once is given, in the situation of spherical symmetry where only a massless scalar field is present. A massless scalar field is gov-erned by the Klein-Gordon equation, and is also called a linear scalar fied. Of course this doesn’t represent a physical situation since in the universe

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we will seldom have real spherical symmetry, but it gives a nice insight in the workings of cosmic censorship. Also the massless scalar field doesn’t represent real matter, since it has no potential, and no mass. The following formulation of the weak and the strong ccc is due to Christodoulou [13]. The theorem (since it is proven by Christodoulou in this particular case) implies the SCCC and the WCCC for the spherically symmetric massless scalar field.[18]

To understand the formulation given in this section, a few concepts are needed. The theorem makes use of the dimension of a set. The co-dimension of a set is the total co-dimension of the space in which the set is imbedded minus the dimension of the set. Consider the set of all initial data AC, and the set of initial data E for which naked singularities arise. What we want to prove is that E is somehow ‘not generic’. Wald has sug-gested a measure is needed on AC, so that one can say that E has measure zero. [2] If such a measure would be defined, it would certainly have the property that sets with a positive co-dimension have measure zero. Those sets correspond to for example points inR or a line in R2. So one possi-bility is to prove that E has positive co-dimension in AC. This is what the theorem states.

Since spherical symmetry is assumed, a manifold Q can be defined as the quotient manifoldM/SO(3). This is just a fancy way of saying that a rota-tion leaves the space-time invariant. So in this manifold, the radius and the time are the only things that matters. The presence of a massless scalar field is also assumed. Because the spherical symmetry, the initial conditions on Q are fully specified by the radius.

So a possible choice for the ‘space of initial conditions’ AC, which will turn out to work fine, is the space of absolute continuous functions with a finite total variation on the halfline. Although not relevant for the discussion, a function f is said to be absolutely continuous if for every e > 0 there is a

δ > 0 such that if for every finite sequence of pairwise disjoint intervals (xk, yk)with∑k|yk−xk| <δ, we have∑k|f(yk) − f(xk)| < e. In this case functions from the halflineR≥0toR are considered. A finite total variation on the halfline means that

V_∞0(f) =sup P nP−1

∑

i=0 |f(xi+1) − f(xi)| <∞ where P∈ {P= (x1, x2, ..., xnP)|P is a partition of[0,∞)}.

In this paragraph we will only consider starting conditions which specify a spherically symmetric scalar field (a scalar field is a field which is invariant under Lorentz transformations, for example the field of spin zero particles). From the spherical symmetry it follows that a function f ∈ AC completely specifies the initial data for the scalar field. The condition that the total vari-ation of f is finite only says that a restricted class of scalar fields is taken.

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Let R be the set of initial data which leads to a completely regular maxi-mal future development (no singularities), and let S be the complement of R. So S is the set of (spherically symmetric) initial data (we are still talk-ing about the AC-space of initial data) in which a stalk-ingularity arises. Or in other words, the maximal future development of the initial data is incom-plete. Let G ⊂ S be the set of initial data for which no naked singularities arise. By definition, a naked singularity is a singular point p ∈ J−(J+), the causal past of the future null infinity. A more precise notion of a naked singularity is needed. For this a decomposition of the boundary of the max-imal future development of the initial data is needed (this boundary is non empty, since by assumption the maximal future development ofΣ ∈ S is incomplete. So it contains at least one singular point).

We will write i0 for spacelike infinity, J+for future null infinity, and i+for timelike infinity. Now define Γ to be the path of the center of symmetry (the singularity), and define b_Γ the future limit point ofΓ. Now take two spheres originating from b_Γ, S1_Γ and S2_Γ which are defined to be the con-nected parts of the boundary of the future development where the area-radius function r goes to 0 (in technical terms, r extends to a continuous function on Q∪S1,2_Γ which is zero on S1,2_Γ ). CH_Γ is a connected Cauchy horizon which is defined to be the part emanating from b_Γ where r goes to some ˜r > 0 continuously. Obviously, if CH_Γ is non empty, there is a problem, since there is some radius within which an observer can be ‘influ-enced’ by the singularity. The central componentB0of the future boundary originating from b_Γis defined to be S2_Γ∪S1_Γ∪CH_Γ, and the singular bound-aryB =Q\ (Q∪i0∪J+∪i+∪CHi+). Here CH_i+ is a connected outgoing null Cauchy Horizon where r tends to r > 0. Define S to be a nowhere timelike curve for which r goes to 0, and Si+ a connected null component with r to 0. One can see the central component as the future lightcone of the singularity. Let O be a singular point, and letAbe the past boundary of a trapped surface (the boundary is empty if there is no trapped surface present).

If a trapped surfaceT, with past boundaryAforms, then the non central componentB\B₀is the future boundary of the trapped surface. Now if on B0, the future lightcone of the singularity, the radius goes to infinity, we can say that we have a naked singularity. Since then, because B0 is null, we have a null path escaping to infinity. If it is non empty, but the radius doesn’t go to infinity, then we also have a problem, since then the future development is extendible, and doesn’t predict the ‘whole’ space-time. this is a contradiction with the strong censorship conjecture.

To get back to the initial conditions, we see that ifB₀ = O, and if the non-central component and the past boundaryAoriginate from O, so that the trapped surface has a decreasing surface area towards the future, then there is no naked singularity present, in the strong and in the weak sense. So we

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(a) Decomposition of the boundaryBof the future developmentM.

(b) The past boundaryAof the trapped surfaceT.

Figure 1: The decomposition of the boundary of the space-time. Pictures taken from [23].

will take the initial conditions G for which no naked singularities arise to be the set of all initial data for which the maximal future development has

B₀ =O, andA,B\B₀are non-empty and have nonempty intersection with O.

Now we somehow want to prove that in the set of all irregular initial data S (initial data for which a singularity is created), the set E= S\G for which naked singularities arises, is non generic. Or as has been said before, that the set has a positive codimension. It is this statement which the follow-ing theorem expresses. Only in a technical way. Namely, it is shown that through each point in E, one can construct a line that intersects E only once, and for two different points in E, it is shown that the two lines through those points don’t intersect. In this way one can construct a ‘plane’ inside S in which E is only a ‘line’, and thus has positive codimension.

Theorem 4. Let G⊂ S be the set of all initial data for which the maximal future development hasB₀ = O, andA,B \ B₀are nonempty and issue from O. Then E = S\G has the following property. For each initial α0 ∈ E there is a function f ∈ AC, depending on α0, such that the line Lα0 = α0+c f|c∈R in AC is

contained in G and crosses E in only one point, namely α0. Also, the lines Lα0,1

and Lα0,2 do not intersect if α0,1 6=α0,2.

This theorem is proven for the spherically symmetric massless scalar field by Christodoulou in [13]. So we now know that the weak and the strong cosmic censorship conjecture holds in this case. The above discus-sion is just to give an insight in the mathematical framework in which the cosmic censorship conjectures are proven. This framework is very appeal-ing, since one can easily imagine a generalization to the non spherical case. To prove it would be a much harder task. In 2008, the WCCC was proven to hold for a class of initial data in the Einstein-Vlasov system. The Vlasov

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equation describes the evolution of a charged plasma, and hence describes more realistic matter. It is frequently being used by astrophysicists to de-scribe the behavior of galaxies.[41] The proof of this theorem assumes the dominant energy condition however. But despite the energy conditions used, the results of Christodoulou and Andreasson strengthen our intu-ition that the WCCC holds indeed true in physically realistic scenarios. As has been said, there is less evidence that the SCCC holds true.

By the no hair theorem, stationary black holes are determined completely by the angular momentum, the charge and the mass.[42] So it makes sense to study the characteristics of Kerr-Newman black holes. In this thesis, for simplicity, a special case of the Kerr-Newman space-time is studied: the Reissner-Nordstr ¨om solution. This spherically symmetric charged black hole will be considered in the next section. It is known that several diffi-culties arise in the causal structure of the Reissner-Nordstr ¨om black hole [23]. However due to possible quantum effects, the exact nature of these difficulties remains somewhat uncertain.

Recently, Hawking even proposed that due to quantum effects black holes don’t exist at all. Instead collapsing matter eventually reaches a point where an apparent horizon is formed, but no event horizon. The difference be-tween the two is that the apparent horizon is the boundary of the region from which no light rays can escape to future null infinity. It is an hori-zon only in certain co ¨ordinates. An event horihori-zon is defined as the region in which the events cannot affect the ‘external universe’ (this notion is de-fined only in asymptotically flat spaces, for which there is an external uni-verse). Eventually the apparent horizon would disappear and ‘release’ the matter which was held inside the apparent horizon. This matter doesn’t form a singularity in the collapse, so that no information is destroyed.[43] Although this idea would solve the problems of cosmic censorship (no sin-gularities would arise at all), since it has yet to be published at the moment of writing, it will not be considered here.

7 The internal structure of the Reissner-Nordstr ¨om black

hole

In this section we will discuss the internal structure of a RNBH. Examples of naked singularities are given, and later it is argued why these singular-ities are not ‘physical’. The stationary solution with its characteristics will be considered first. To study the behavior of the inner horizon under per-turbations, Schwinger pair production will be discussed. It might provide a mechanism for perturbing the inner horizon (provided that the rate of pair production is faster then the evaporation time of the black hole, and pro-vided that the black hole doesn’t lose its charge before a pair is produced).

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Then the non stationary solution for a charged black hole will be given, and calculate the effect of a perturbation at the inner horizon. It turns out that this perturbation gives rise to a phenomenon called mass inflation. This effect is also studied.

The stationary solution of a charged black hole with charge Q and mass M is given by the following formula. It is assumed that the black hole charge is constant, which, since black holes with a charge much larger than the electron charge are considered here, is approximately true.

ds2= gµνdx µ_dxν ₍₁₎ = −∆(r)dt2+ dr 2 ∆(r)+r 2_d_Ω2 ₍₂₎ ∆(r) =1−2GM r + GQ2 r2

By considering the radii at which grr = 0, we see that at r± = GM±

pG2_M2₋_GQ2 _{there are event horizons (the metric is diagonal, so since} gµν_gνρ ₌

δρµ, grr= _g1_rr). Or with the right terms of c and 4 πe0restored, r± = GM c2 ± s G2_M2 c4 − GQ2 4πe0c4 .

Several values of the parameters Q and M will now be considered. It has been pointed out that it is not possible to overcharge a RNBH by simply dropping charged particles into a black hole (to overcome the coulomb re-pulsion a particle would need a higher energy, and so the mass of the black hole grows more than the charge does if a charged particle is dropped into an extremal, Q2 = 4πe0GM2, black hole), and that the extreme RNBH so-lution is not stable under perturbations.[29]. However, theoretically, these are still interesting cases.

So firstly, assume Q2> 4πe0GM2. As has been said, this is not expected to be a physical situation. However it does point out nicely the characteristics of a naked singularity. In this case the equation∆(r) =0 has no real solu-tions. The metric is entirely smooth, except of course at r=0 where the Ricci Scalar R becomes infinite. If we change co ¨ordinates (see [10] for details) we can draw a conformal diagram for this overcharged RNBH as in figure 2(b).

This conformal diagram is very similar to that of Minkowski space, only with a singlularity at r=0. As it turns out, there are null geodesics which intersect the singularity and reach future null infinity. Hence by the def-inition given in the appendix, this is a naked singularity. To see this, one has to solve the geodesic equation for the RNBH (here we again use natural units): d2_xµ dλ2 +Γ µ ρσ dxρ dλ dxσ dλ =0. (3)

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This yields four equations for the null geodesics, where λ is the affine parameter which parameterizes the null geodesics (the equivalent of the proper time for null geodesics) such that the momentum four-vector is equal to dx_dλµ = pµ_{, the derivative of the null geodesic with respect to λ.} Γµ

ρσ is the Christoffel connection. This connection is needed to define a covariant derivative (a derivative which transforms like a tensor, see [10] chapter 3 for details). It is sufficient to know that the Christoffel symbols can be calculated from the metric (which defines the geometry after all) by the following formula:

Γµ ρσ = 1 2g µν₍ ∂ρgσν+∂σgνρ−∂νgρσ). (4) However, one can obtain the equation of radial motion for a RNBH also by dividing the metric by dλ2 _{on both sides (dτ}2 ₌ _{0 for null curves, which} will be considered here):

ds2 dλ2 = 1 ∆(r) dr dλ 2 −∆(r) dt dλ 2 + r2sin2θ dφ dλ 2 +r2 dθ dλ 2! (5) One can simplify this expression in several ways. First remember that for massless particles, E2−p2 = −gµνdx_dλµdx_dλν =0. Secondly, the Lagrangian for the Reissner-Nordstr ¨om solution can be computed (see chapter 4 of [10] for details) from the Ricci scalar R. Since the metric coefficients are indepen-dent of t and φ, the Lagrangian is also indepenindepen-dent of t and φ. Hence for massive particles we have (notice the dot on the t and the φ, which denotes the derivative with respect to the proper time)

E= −∂L ∂ ˙t =∆(r)m ˙t+ qQ r (6) L= ∂L ∂ ˙φ =r 2_sin2₍ θ)˙φm (7)

are conserved quantities. Here E is the energy and L is the angular mo-mentum of the mass m (a test particle). However, we are considering null geodesics, so q=0. Also, the θ is constant, hence we can assume θ = π

2. For null geodesics, the proper time vanishes, hence we have to use another affine parameter λ, which we may normalize such that pµ ₌ dxµ

dλ. Here the dot denotes the derivative with respect to the affine parameter λ, and we are left with the conserved quantities:

E= ∆(r)˙t (8)

L=r2˙φ. (9)

Using these relations we can simplify 5 to

−E2+ (dr

dλ)

2₊_∆₍_r₎L2

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Since only radial trajectories are considered, L=0. Hence the equation of motion simplifies to

dr

dλ = ±E (11)

r=ri±Eλ. (12)

riis the initial position of the photon. Hence there are null geodesics which end in the singularity and null geodesics which start in the singularity and travel outward to the future null infinity.

Also, an observer could just travel to the singularity and return to the place they came from to tell what they have seen, since (contrary to the Schwarzschild black hole, see the next paragraph) the timelike co ¨ordinate doesn’t switch to being spacelike. The initial co ¨ordinates are valid in the whole space-time. The conformal diagram is displayed in figure 2(b). The case where Q2 = 4πe0GM2(the extremal RNBH) is theoretically also very interesting, due to several properties. See again [10] for details. It is not discussed here.

Now assume Q2 <4πe0GM2. This case is expected to apply to a physically realistic black hole. In this case the metric 2 has three singular radii: r=0, r = r+, and r = r−. r+and r− are called the outer and the inner horizon,

respectively. However, if the solution is extended by switching to so called radiation co ¨ordinates, we see that both horizons are perfectly regular, and in-between r+and r−, the outer and inner horizon, the radius is decreasing

as a function of time.[10] When we get past the inner horizon r−, the radius

no longer decreases as a function of proper time, and hitting the singularity at r=0 is no longer necessary. Moreover, you can cross r−again, and find

that this time the radius is increasing as a function of proper time. When you are spitted out of the outer horizon, you find yourself in a whole other universe. In a conformal diagram the structure of the RNBH looks like fig-ure 2(a).

Let us see how these statements can be derived from the metric in equa-tion 2. The name event horizon is derived from the fact that events which take place inside the horizon cannot affect the outer world. It is not to be confused with the apparent horizon (which in the Reissner-Nordstr ¨om case coincides with the event horizon), which comes from the fact that for an outside observer, events seem to be frozen on this horizon. One can-not see beyond this radius (assuming spherical symmetry) from outside. To see this, consider radial null curves: curves with φ and θ constant and vanishing ds2.

ds2=0= −∆(r)dt2+ dr

2 ∆(r).

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(a) The conformal diagram for the analytically extended RN so-lution with 4πe0GM2>Q2

(b) The conformal diagram for the RN solution with 4πe0GM2<Q2

Figure 2: The conformal diagrams for two different charged black holes. This implies dt dr = ± 1 1−2GM r + GQ2 r2 . (13)

This means that the light, as seen by an observer outside the outer hori-zon, stops moving at the inner and outer horihori-zon, and events seem in-deed to freeze. However this is only an illusion of the co ¨ordinate system used. To see this, we introduce the Eddington-Finkelstein co ¨ordinates for the Schwarzschild solution, a much simpler case. For the charged black hole, the situation is harder in general, because of the r2_{dependence. For a} full treatment, see for example [40].

The Schwarzschild metric is a special case of the Reissner-Nordstr ¨om met-ric, with Q=0. Now define the retarded(outgoing) time u and the advanced (ingoing) time v as

u=t−r−2GM ln( r

2GM−1) (14)

v=t+r+2GM ln( r

2GM−1). (15)

Note that the term r∗ =r+2GM ln(_2GMr −1)is obtained by solving equa-tion 13 explicitly for t, yielding t=r∗+C, where C is some constant.

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With a little algebra, metric 2 with Q=0 can be written as

−(1−2GM

r )dv

2₊_dvdr₊_drdv₊_r2_dΩ2_. ₍₁₆₎ Since this metric is not diagonal we can still compute the determinant at r=2GM as being g = −r4_sin2_θ ₆₌ _{0, although g}

vv vanishes. The metric is therefore invertible at r=2GM, and hence this radius is no singularity anymore.

The similar result for the Reissner-Nordstr ¨om metric can be obtained by replacing −(1− 2GM

r ) by 1− 2GMr + GQ2

r2 in 16. Obviously, the steps for

the Schwarzschild metric in radiation co ¨ordinates can be repeated for this metric. And just like in the Schwarzschild case, it turns out that the inner and the outer horizon are just co ¨ordinate singularities.

The same steps as above are now repeated for ingoing and outgoing null curves. For ingoing null curves, v is constant, so dv_dr =0. v is constant since radial null curves obey equation 13, and this equation can be (formally) integrated, and the solution can be denoted as t=r∗. Now v can be defined as v=t+r∗. Hence we see that v is constant for null curves, just like in the Schwarzschild case.

The case where the null curves are outgoing, is different however. Since v is not constant, but u instead is constant, the metric yields

dv dr =

2 1−2GM_r +GQ_r22

. (17)

Physically, since any timelike path has to be ‘in-between’ the infalling and the outgoing null curves (thus spanning the so called light cone), this for-mula tells us that outgoing null curves in-between the inner and the outer horizon still have a decreasing radius. Also, inside the inner horizon, the null curves behave ‘normal’ again (i.e. the outgoing nullcurves have an in-creasing radius) Note that by appropriately choosing co ¨ordinates, we can analytically extend the solution. It can be shown that the observer can re-turn through the inner and the outer horizon, and end up in a different universe in this co ¨ordinates. For more details see [10].

The reason why the behavior of the metric inside the inner horizon is trou-blesome is that a possible observer doesn’t necessarilly need to hit the line r=0. He can also choose to park his spaceship around there and observe the singularity: the singularity at r=0 is a timelike singularity. It is pictured in figure 2(a). This is in contradiction with the strong cosmic conjecture, since the evolution of any initial data in a Reissner-Nordstr ¨om is incom-plete. One way to make things right, would be by saying that observers cannot cross the inner horizon. For then there wouldn’t be any observers who can be in region of influence of the singularity.

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physical for some reason. Of course spherical symmetry is not physical in any way, so small perturbations could be added to study the behavior of the metric. Poisson and Israel calculated (for the first time in non-linear order) the effect of such a perturbation. [33]. It was found that the inner horizon is unstable under perturbations: an infinite energy density develops from a small perturbation. Also the effective mass from the black hole increases. This phenomenon is called mass inflation. It is studied in the next sections. It turns out that this might provide a principle which prevents observers from crossing the inner horizon.

Instead of an external source which provides the perturbation (which is assumed with mass inflation), one could also study the possibility of an in-ternal source of energy. In the next section it is studied if Schwinger pair production around a charged black hole might provide such a mechanism. The idea is that the electric field polarizes the vacu ¨um and creates electrons and positrons.

Since the charged black hole is essentially a point charge, it is easy to calcu-late the radial component of the electric field as a function of r, so that we can calculate the Schwinger pair production rate at the inner horizon.

Er= Q 4πe0r2

We will calculate how much pairs are produced per cubical meter per sec-ond as a function of the radius r, the charge Q of the black hole and the mass M of the black hole. Next this rate is integrated over the volume be-tween r−and r+, to obtain an indication of the total number of pairs which

fall on the inner horizon. For simplicity, it is assumed that only pairs of electrons and positrons are produced. By computing the invariant scalar p FµνFµν ₌ Q

4πe0r2 from the maxwell tensor, we see that we don’t need to

worry about the value of E under a change of co ¨ordinates. So Er will be used as the value of the electric field of the RNBH in the following.

7.1 Pair production

It is not our purpose to go into the details of quantum field theory (QFT), so the full calculation won’t be given here. The formula which is used to calculate the pair production rate in a constant electric field will simply be stated here, and it is assumed that the black hole electric field varies slowly locally. Then one can calculate the electron positron production rate as a function of r. Obviously, close to the singularity this approximation becomes invalid, and one would have to do the full quantum gravitational calculation. The distance between the two particles of one pair will be given here, to obtain an indication where the approximation of a constant electric field becomes invalid. The full calculation for the pair production of spin

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1/2 particles in a constant electric field, and more, can be found in [30]. The probability for pair production in a constant electric field E0 per unit time per unit volume is given by

ρ= 1 L3_T(1−e −2γ₎_. Where γ= L3Te 2_E2 0 8π3 ∞

∑

1 1 n2exp(−πn m2 eE0 ).

If we taylor expand the expression for ρ and keep the first term, we get approximately (restoring the right factors of c and ¯h)

ρ= e 2_E2 0 4π2_¯h2_cexp(−π m2_c3 e¯hE0 ). (18)

Since we have only electrons in our model, the mass m is extremely small. Hence, if a sufficiently big electric field (at the inner horizon) is used (if

−π2m2r2c34e0

eQ¯h 1), in which we will be interested, one can safely use this approximation.[30]

Since pair production is a local effect, it is assumed that we can use this formula for nonconstant electric fields as well, approximately. Later in this section some justification will be given. As it turns out, if a suffi-ciently large electric charge is assumed (suffisuffi-ciently large charge to mass ratio α= √ Q

4πe0GM), the approximation can be used. These claims will later

be made precise.

Obviously, since the electric field goes with _r12 , the pair production will be

the highest on the inner horizon. Therefore if one takes the charge to mass ratio α fixed and large enough (large enough for the above approximation to be reasonable, since for small ratios the inner horizon is very small), the value of ρ at the inner horizon can be calculated as a function of M, the black hole mass (figure 3a). One can do this the other way around, to see the behavior of ρ at the inner horizon if the black hole is slowly overcharged (figure 3b).

Next ρ can be integrated over the region between r−and r+to obtain ρint. By definition of integration on manifolds, ρ can just be integrated by us-ing appropriate charts where the function is well defined. That is, on the domainU where the r and t co ¨ordinates are well defined. On charts, inte-gration is done in theRn, so denoting the chart from the manifold to t, r, θ, φ co ¨ordinates with h, we get a total pair production of

N = Z Uρ p −g= Z h−1_{(U )}ρ(x µ₎p −gd4x (19) = Z t 0 Z 2π 0 Z _π 0 Z r₊ r− r2sin(θ)ρ(r)drdθdφdt=4πt Z r₊ r− r2ρ(r)dr. (20)

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Here g is the determinant of the metric, in our case −r4sin(θ)2. Since N

is proportional to t, if we integrate over a large enough time interval, then any amount of pair production (in the approximation that the pair produc-tion is time independent, which might also break down at late times) is going to lead to a disturbance of the metric. Remember however that we are working with an approximation, so for the class of black holes where this approximation doesn’t apply, it is still possible that the total pair pro-duction is negligible, and further research is needed. For the class of black holes where the approximation holds, we will only look at the pair pro-duction per unit volume in a finite amount of time, to avoid divergence problems.

To get a characteristic time scale for the system, one can calculate the char-acteristic time τf f for a black hole, which is the free fall time of an electron from the outer to the inner horizon radius. In figure 3c and 3d, the pair production in a time τ_{f f} is plotted as a function of α and a function of M. We compute the free fall proper time τf f from the outer to the inner horizon of a charged particle with charge q = ±e and mass m = me. We assume that the particle has no initial angular momentum, and that the energy is approximately equal to the mass for simplicity. (If E and L get bigger, the square root (and hence the tunneling distance) in formula 28 gets bigger, and the tunneling probability is exponentially surpressed as the tunneling distance (energy) gets bigger, so this assumption is correct to first order.) Hence the angular velocity vanishes. We then start with equation 5, where we notice that E2₋_c2_p2 ₌ _c4_m2

e, where me is the electron (or positron) mass. Hence we get (using the concerved quantities 6 and 7):

cτf f = Z r −gµνdx µ dτ dxν dτ dτ = Z r −gtt(dt dτ) 2₋_g rr(dr dτ) 2_dτ ₌ Z r −gtt(dt dr) 2₋_g rrdr = Z r₋ r+ v u u t −(_E˜− _4πeqQ 0mcr) 2₋_V₍_r₎ V(r)∆(r) dr= Z r₋ r+ v u u t−(c 2₊ L˜2 r2) V(r) dr. (21) Where V(r) = (c2+ L˜2 r2)∆(r) − (E˜− qQ 4πe0rmc )2 (22) ∆(r) =1−2GM rc2 + GQ2 4πe0c4r2 . (23)

Here ˜L = L/m and ˜E = E/mc. To obtain the expression (_dτdr)2 _{= −}_V₍_r₎_, which is used here, one can use the fact that pµpµ _{= −}_m2_c2_{, and expand} this expression. It is useful to plot the pair production per unit time τf f,

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ρint =τf fdN_dt . This is plotted in figure 3, c and d.

As can be seen from figures 3 a and b, for an arbitrary M > 0, there is a charge to mass ratio α for which the pair production at the inner horizon is not neglectable. That is if the charge is small enough, for a given black hole mass M, the pair production at the inner horizon is not neglectable anymore. If we taylor expand r−around Q=0, we see that

r−= Q

2 8πe0c2M

+O(Q4).

Filling in this taylor expanded r− in 18, it is seen that if we take the limit

Q → 0, ρ becomes indeed infinite. It seems that the distance of the in-ner horizon to the singular point decreases faster than the electric field increases as a function of Q. What’s wrong? Obviously if Q is too small, and the inner horizon radius becomes comparable to the Planck scale, we can no longer use the approximation that E is constant and we will have to do the full quantum field theory calculation to see what happens with the pair production around singularities. Probably some quantum gravity will also be involved. Hopefully this calculation will show that if no charge is present in the singularity, there also is no pair production, and the Q=0 limit of ρ will go to zero, as we expect it to be zero in the Schwardschild case.

However, this limit doesn’t even make sense: Q cannot be arbitrarily small, since the charge is quantized. So if the black hole has a charge e, the ques-tion is if there will immediately be a pair producques-tion which neutralizes the black hole. Given the large values of 18 in the Q=0 limit, this will probably be the case. Also since an observer can get arbitrarily close to the singu-larity, where the electric field will be very high, the pair production will probably be very high close to the singularity. It is possible that due to this effect any charged black hole will be neutralized quickly.

Looking at figures 3c and d, the pair production between the inner and the outer horizon per unit free fall time gives the same picture. If α is in-creased, the total pair production decreases for a given M. Of course this is explained as before, since the inner horizon radius goes with Q2_{for small} Q. Looking at figure 3(c), it is seen that for a given time interval τf f the pair production is non neglectable if the mass M of the black hole is between approximately 1035 and 1056 planck masses, which roughly agrees to the mass of the black holes we observe in the universe. So it seems to be the case that for α not close to one (not near extremal or extremal), all RNBH with M in some interval have a non neglectable pair production, and for which we have a principle which provides a mass perturbation, indepen-dent of a source outside the black hole (for example a star or a galaxy). To make this more clear, one can compare the energy density of the pro-duced pairs (mec2ρ) with the energy density of the electric field, 1₂e0E2. The energy density of the electric field at the inner horizon is plotted in figure

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(a) The solid line is the pair production rate per unit time per unit volume at the inner horizon plotted as a function of M, the black hole mass. The dashed line is the electric field energy density at the inner horizon divided by the electron mass (mec2), as a function of M in planck

masses, both for fixed α=0.00063.

(b) The solid line is the pair production rate per unit time per unit volume at the inner horizon plotted as a function of α, for fixed M = 1.9·1052 planck masses. The dashed line denotes the electric field energy density at the inner horizon di-vided by the electron mass (mec2), as a

function of α at the same mass as the pair production rate.

(c) The pair production per unit volume in a time interval of the length of the free fall time of an electron between the in-ner horizon and the outer horizon per unit time as a function of M, for fixed

α=0.00062 as a function of the mass M

in planck masses.

(d) The pair production per unit volume in a time interval of the length of the free fall time of an electron between the inner horizon and the outer horizon per unit time as a function of α, for fixed M=1.8∗1052planck masses.

Figure 3: Pair production rates, with the mass in units of the Planck mass mp = q ¯hc G. Here α = Q √

4πe0GM. In figure a and b, the energy density of the

electric field is also plotted (with the dashed lines), in units of the electron rest energy. In figures c and d, the pair production rate is multiplied by the freefall time τf f, which gives a characteristic time scale for a given black hole.

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3 (with the solid line) in units of the electron rest energy, together with the pair production rates (with the dashed lines). As one can see, if α is not too big, and M in the mentioned interval, the pair production dominates the electric field energy density. For the values of alpha for which the pair production dominates, the schwinger effect is not negligible.

However, several approximations have been made. Remember that α couldn’t be too small, since then the inner horizon radius would be of the order of magnitude of the separation between the electron and the positron created. If this is the case, we cannot longer use the approximation in which the elec-tric field varies slowly locally. In the next few alineas we will see what this distance is exactly. So the values of α for which the approximation holds are needed, to see if there exists a class of black holes for which the above conclusion is valid.

To justify the approximation of a constant electric field made above, an in-terpretation of pair production is needed. Consider the electrons with neg-ative energy in the Dirac sea. Since a charge is present in the RNBH, there is also a potential present around the black hole. This potential causes energy level crossing around the black hole, of the energy of the electrons in the dirac sea and the possible positive energy electrons. This effect is shown in figure 4. If no potential is present, the energy levels of the electrons in the Dirac sea start at E= −mec2. So at E < −mc2, if a potential is present, it is possible for electrons in the Dirac sea to tunnel through the potential bar-rier to a ‘positive’ energy level.[32] This of course leaves a hole in the Dirac sea, which is interpreted as a positron. In this way a particle anti-particle pair is created. With the WKB approximation it is possible to derive the tunneling frequency of particles. We will assume that the created particles are at rest, since this situation requires the least energy.

The calculation is taken from [31]. The situation is illustrated in figure 4. The tunneling takes place from r−to r+(not to be confused with the inner

and outer horizon). As one can see, the electric potential bends the energy levels of the electrons with negative energy. This gives rise to so called ‘en-ergy level crossing’. In other words, there exist en‘en-ergy ε and r+, r−, such

that at r+, resp. r−εcrosses ε±resp, the positive and negative energy

spec-tra. If no potential is present, the positive energy levels correspond to all the particles (since by assumption there are only electrons and positrons) with energy ε > mec2. In the same way the particles with negative energy would have ε< −mec2without potential, in the relativistic description. If a potential is added, these energy levels are bended like for example in figure 4.

Let’s calculate the value of these bended energy levels for a coulomb po-tential, so that the exact value or r− and r+ can be calculated for a given

field strength at the inner horizon of the black hole. To obtain an expres-sion for r±, the radial momentum is calculated, as a function of the energy εand r. Where ε is calculated with the Klein-Gordon equation. Later ε will

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Figure 4: In gray, the Dirac sea, from where the tunneling takes place at an energy ε. The tunneling takes place from r− to r+, which are obtained by

setting the radial momentum to zero. Here the energy is in units of mec2, the energy levels are plotted as a function of ˆr= r/λc, for l=0, pr = 0 and

Q

4πe0 = c¯h1.27. Here λc is the Compton wavelength. The picture is taken

from [31].

be taken as small as possible, and the field strength at the inner horizon is used, since this will give the biggest tunneling frequency. Then the radii at which the momentum vanishes are calculated for a given energy.

The central equation of this problem is the Klein-Gordon equation

[(ε+ Q

4πe0r

)2+c2¯h2∇2−m2_ec4]ψε(x) =0.

By separation of variables with ψε(x) = Rε,l(r)Ylm(ˆx), the equation can be rewritten for the radial part as

{ε2+ εQ r4πe0 +c2¯h2∂2_r− c2¯h2l(l+1) − (_4πeQ 0) 2 r2 −m 2 ec4}rRε,l(r) =0. (24) However to apply the WKB approximation to this formula to compute r±

and the tunneling frequency (which we will not be doing), it turns out that l(l+1) has to be replaced by(l+1₂)2.[34] This is called the Langer-correction, and comes from the fact that the radial wave equation is not exactly a ‘Schr ¨odinger’ like equation. So the Langer-correction gives a small cor-rection to the potential.

With this Langer-correction, the energy levels now become, using the for-mula

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ε±(pr, l; r) = ± s (cpr)2+ (c¯h)2 (l+1₂)2 r2 +m2ec4+V(r). (26) Where V(r) = _4πeQ

0r. Notice that the angular momentum equals(¯h)

2(l+12)2 r2 , and pr= 1 c s ε2+2ε/4πe0r− (c2¯h2(l+1/2)2− ( Q 4πe0 2 )/r2₋_m2 ec4, (27) by rewriting 26, and using that ε± = ε. The energy levels are drawn in

figure 4. Notice that if_4πeQ

0 >c¯h(l+1/2), then if r→0, ε+→ −∞.

If we set pr =0, we can calculate

r±= ε_4πeQ 0 ± q (_4πeQ 0ε) 2_{+ (}_c2_¯h2₍_l₊_1/2₎2_{− (} Q 4πe0) 2₎₍_ε2₋_m2 ec4) m2 ec4−ε2 . (28) Here ε< −mec2.

Obviously, the biggest contribution to the tunneling rate will come from the states for which l=0 and pr = 0, since the tunneling distance is small in this case. Hence from now on, l=0 and pr = 0, since the goal was to obtain an estimate of the tunneling distance. Then, for the energy levels to behave nicely, one needs Q >2πe0c¯h. Since we already assumed that Q is reasonably big to taylor expand the pair production formula and keep only the first term, we can safely assume this.

The ratio of the electron-positron distance of pairs produced at the inner horizon and the inner horizon radius can now be calculated as function of the black hole mass and charge. For this, the inner horizon radius is filled in in 27, to obtain ε= − Q 4πe0rinn + s ( c¯h 2rinn )2₊_m2 ec4, (29) r_inn = GM c2 (1− p 1−α2). (30)

This energy is used in 29 to calculate the tunneling distance for l=0 and pr =0 ∆ =r−−r+= q (c¯h)2_ε2₋_¯h2_m2 ec6+4m2ec4(4πeQ0) 2 ε2−m2_ec4 . (31)

The inner horizon radius rinn depends on M and Q, the black hole mass and charge. In figure 5(a)∆ is plotted as a function of M for fixed charge to mass ratio α. In figure 5(b)_r∆

inn is plotted as a function of α for fixed M.

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(a) The ratio _r∆_inn as a function of M, for

α=9.8·10−33. M is in Planck units.

(b) The ratio _r∆_inn as a function of α, for M=10.0·1016mp.

Figure 5: The ratio _r∆

inn of the tunneling distance to the inner horizon

ra-dius both as a function of the black hole mass M and the black hole charge parameter α.

let us first look at the regime where α is small in 5(a). For small α (hence small Q and small rinn), the inner horizon goes with approximately Q2, as we saw earlier. So for small Q, ε is going to scale with approximately _Q12

(neglecting the potential term and the mass term). Approximately then, the numerator of 31 goes with _Q12. The dominator of 31 goes with _Q14. But

remember that we divide by rinn, which goes with Q2. So like we see in figure 5(a), for small alpha we expect _r∆

inn to be constant.

Now for α close to one, the inner horizon radius is very large, if M is not too small. This regime is called the macroscopic regime, and can be stated as the regime where

c¯h rinn c2me Q 4πe0rinn (32) c¯h GM(1−√1−α2) me α √ 4πe0G(1− √ 1−α2) . (33)

In that case ε=mec2−_4πeQ₀_r_inn approximately, so the dominator of formula 31 goes with Q2

r2 inn

(neglecting the meterms). The numerator of 31 goes with Q, neglecting the other terms in the square root. Then for large rinn, delta goes with mer2inn

Q , so the ratio r∆inn goes with

merinn

Q , which is by assumption much smaller then 1 in this regime. This agrees nicely with figure 5(a) and 5(b).

The conclusion of this calculation seems to be that if we take α ‘not too small’, as we are assuming, and M not too small, so that the black hole is in the macroscopic regime, then the calculation of the pair production rate at the inner horizon can be trusted. Remember that the conclusion of the

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cal-culation of the pair production rate was that for large α and large mass M the pair production rate is neglectable and for any α there exists a mass Mα small enough such that the black hole with mass Mα has neglectable pair production. So if α and M are taken to be big enough, there seems to be a class of black holes which have neglectable pair production rate (remember that the production rate at the inner horizon decreases both as a function of M and as a function of α). So there is another principle needed which alters the metric for this class of black holes. Let us consider mass inflatioin for this class of black holes.

Note however that for the class of black holes with a mass of 1035 to 1056 mp, the mass of the black holes we observe in the universe, the pair produc-tion is non neglactable, and we have found a mass perturbaproduc-tion principle which would destroy the cauchy horizon.

7.2 Mass inflation

According to mass inflation, a small perturbation outside the black hole will cause an infinite energy density at the inner horizon. Hence a cur-vature singularity will develop on the inner horizon, which prevents ob-servers from passing through the Cauchy horizon of the Reissner-Nordstr ¨om space-time. It will be assumed that the perturbation consists of radiation, for simplicity. The calculation is taken from [37]. We work in units where G and c are 1.

Intuitively, the argument for an infinite energy density is displayed in fig-ure 6. Consider a small radiation source present outside the outer radius, say a distant galaxy, following the path AC in figure 6. Suppose that the galaxy existed since the very beginning of the universe (when galaxies could form), and will exist all eternity. As it goes to spatial infinity (the point C) , the radiation piles up before the inner horizon CE. So any ob-server who crosses the inner horizon CE will observe the entire future his-tory of the universe in one flash.[36] Intuitively, this would give an infinite energy density at the inner horizon. Let’s check if this is indeed the case.

Our goal is to calculate the Ricci scalar at the Cauchy horizon of the RNBH. To do this, the metric needs to be determined at the Inner horizon (which is the Cauchy horizon). However the situation is more complicated since radiation is sent to the black hole. The result is that the mass can’t be taken constant, and the proper metric to use is the charged Vaidya metric. This metric is very similar to the Reissner-Nordstr ¨om metric. However this metric doesn’t describe the whole region needed, so there are two forms

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Figure 6: Electromagnetic radiation falling in a RNBH, from a source fol-lowing path AC outside the black hole. The picture is taken from [35] needed. ds2 = −f(r, v)dv2+2drdv+r2dΩ2 (34) ds2 = −e2σdUdV+r2dΩ (35) f =1−2m(v) r + e2 r2 (36) e2σ = −2(∂Ur)(∂Vr) f (37)

Here v is the ingoing null co ördinate, r is the radial co ördinate and U is the outgoing null co ördinate. Unfortunately, there doesn’t exist an explicit coordinate transformation from double null co ördinates (equation 15) to so called radiation co ördinates. To obtain the expression 15, one has to solve the Einstein equations for a spherically symmetric metric of the form (15), where e2σand r are arbitrary functions of U and V. [39] In the Schwarzschild case however, there does exist an explicit co ördinate transformation. It is just u and v from formula 15, only rescaled with a power of e: V =ev/4GM and U = −e−u/4GM_.

Notice that v is time dependent. The lines with v is constant describe the ingoing radial geodesics. Also the mass function m is time dependent. This time dependence is needed since there is infalling energy present. This en-ergy is falling in in the form of radiation. This layer of outgoing radiation

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Figure 7: The conformal diagram of a RNBH with the value of the null and radiation co ¨ordinates indicated. The picture is taken from [36].

(a null surface for which U is constant, say at U=A) will be called S. The null layer is parameterized by the affine parameter V = λ. The value of

the radial co ¨ordinate r at S will be defined as R(λ):= r(V = λ, U = A). S

crosses the inner apparent horizon at V=0. Note that S divides the region between the inner and the outer horizon (the latter is called the event hori-zon for obvious reasons) into two parts. Quantities defined for the region U< A (resp. > A) will get a subindex 1 (resp. 2) in the following analysis. As follows from the equations 15, v = ∞ at the inner horizon (this holds

true in the Reissner-Nordstr ¨om case).

The goal is to calculate the mass (energy density) along S in the limit where

λ → 0(V → 0). This mass function∆m(λ)of S is calculated by taking the

difference between the mass functions m2and m1of region 2(U > A)and region 1. Remember that we are working in the charged Vaidya metric, where the mass is not constant.

From equation it follows that on S R0/v0 = 1

2(1+Q

2_/R2_{) −}_m/R. ₍₃₈₎ The prime denotes the derivative with respect to λ. One can also calculate the geodesic equations for the Reissner-Nordstr ¨om black hole in radiation co ¨ordinates, but this will not be done here. From these geodesic equations

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it follows that along the line U= A,

v00 =v02(Q2/R3−m/R2). (39)

One can define the function z=R/v’, and compute the derivative z0 = R0/v0+

Rv00/v02. By rewriting equations 38 and 39, we get z0 = 1

2(1−Q

2_/R2₎_. ₍₄₀₎

Hence one gets the following expressions, which follow from resp. equa-tion 38, the definiequa-tion of z and equaequa-tion 40:

m(λ) = (R/2)(1+Q2/R2) −zR0 (41) v(λ) = Z λ (R/z)dλ (42) z(λ) =Z+ 1 2 Z λ 0 (1−Q2/R2)dλ. (43) In the equation for v, the constant of integration is set to zero, since adding a constant to a co ¨ordinate doesn’t make any physical difference. Using the expression for m, and using that R1(λ) = R2(λ) and Q is constant

(it is all situated in the singularity at the origin r=0), we get ∆m(λ) = (Z1−Z2)R0(λ). Here Z1and Z2are integration constants.

Define m0to be the mass of the black hole after it has absorbed all the radia-tion. The mass function in region one can now be written as m1(v1) =m0−

δm(v1), where δm is a perturbation. This perturbation might be caused by radiation originating from the collapse of the star, and approximately goes with v1₁−pat late times. Here p depends on the type of radiation and p≥12 [36]. As it turns out, the exact form of this formula won’t be needed. For λ close to zero, we can approximate the expression for z in 43 by ap-proximating the integral, and writing R=r0 = m0−

q m2₀−Q2_{, the inner} horizon radius: z(λ) =Z+1 2(1−Q 2_/r2 0)λ= Z−k0r0λ. (44)

Where k0= _2r1₀(Q2/r₀2−1) >0. Since v1diverges at λ=0, v0₁also diverges at λ= 0. Hence by equation 43 Z1 =0, so that z goes to zero for λ goes to zero. By equation 44 then, v1(λ) = −(1/k0)ln(|λ|).

Remember that, since we are interested in finding the behavior of the mass function∆m near the inner horizon, the function R needs to be determined. For this, define δR = R−r0. Since r0 is a constant, the derivative of δ is the same as that of R. At the inner horizon, v0 = ∞. So equation 38 can be

On the Cosmic Censorship Conjecture