Outgoing gravitational shock-wave at the inner horizon: The late-time limit of black hole interiors
We investigate the interiors of 3+1 dimensional asymptotically flat charged and rotating black holes as described by observers who fall into the black holes at late times, long after any perturbations of the exterior region have decayed. In the strict limit of late infall times, the initial experiences of such observers are precisely described by the region of the limiting stationary geometry to the past of its inner horizon. However, we argue that late infall-time observers encounter a null shockwave at the location of the would-be outgoing inner horizon. In particular, for spherically symmetric black hole spacetimes we demonstrate that freely-falling observers experience a metric discontinuity across this shock, that is, a gravitational shock-wave. Furthermore, the magnitude of this shock is at least of order unity. A similar phenomenon of metric discontinuity appears to take place at the inner horizon of a generically-perturbed spinning black hole. We compare the properties of this null shockwave singularity with those of the null weak singularity that forms at the Cauchy horizon.
- I Introduction
- II Preliminaries: Spherical charged black hole
III Simple examples of late-time perturbations:
- III.1 Simplest example: monotonic, outgoing, test scalar perturbations
- III.2 More realistic linear perturbations
- III.3 Interpretation in terms of late-time Eddington frames
- IV spherically symmetric non-linear perturbations
- V Effective spacetimes for late-infall observers
- VI Discussion
- A Bound on in slightly-perturbed RN
In Einstein-Hilbert gravity coupled to various matter fields, the exterior geometry of a 3+1 dimensional asymptotically flat black hole (BH) spacetime typically approaches a stationary solution at late times. Non-stationary perturbations decay both by falling across the horizon and dispersing to infinity, as described by the ringdown of quasi-normal modes followed by power-law tails. Our purpose here is to explore a corresponding late-time limit of the associated black hole interiors. We will argue that as far as the observations of late-infalling physical observers are considered, the result is well-described by a simple effective geometry which contains the part of the corresponding stationary BH solution to the past of the inner horizon. However, the regular inner horizon is replaced by singular components of two different types: (i) The ingoing section of the inner horizon—the Cauchy horizon (CH)—is replaced by a null, weak, curvature singularity, and (ii) the outgoing section of the inner horizon is replaced by an outgoing shock-wave singularity. The presence of a null, weak, curvature singularity at the CH is a well-known phenomenon since the pioneering works of Hiscock on the Reissner-Nordstrom-Vaidya solution Hiscock () and of Poisson and Israel on the mass-inflation model PI () (see also Ori91 ()). It is the second singular component—the outgoing shock-wave singularity—which will be our main concern in this paper. Our study is motivated in part by the picture of extreme black holes at late times suggested in extremes () and explored further in garf (). This picture agrees with the extreme limit of our results below.
The starting point for our analysis is the large body of literature studying perturbations of the Reissner-Nordström (RN) and Kerr interiors. With the assumption of spherical symmetry, these works establish that perturbations transform at least the initial part of the ingoing inner horizon (the CH) of RN into a null curvature singularity often called the mass-inflation singularity. This singularity is weak in the sense that the metric remains continuous at the singularity, though it is not differentiable. In particular, the area-radius of the spheres is well-defined at this singularity, taking the value near the point marked on the conformal diagram shown in figure 1 (left) and shrinking toward the future as described by the Raychaudhuri equation, eventually reaching . In the spherical case, at least when the matter content includes a minimally-coupled massless scalar field, it was numerically established GG (); BS (); Burko () that when shrinks to zero the weak null singularity meets a strong spacelike singularity333 These numerical simulations showed that a transition from a null to a spacelike singularity occurs in a region which (when mapped to a collapsing-shell spacetime) would correspond to being outside the shell (i.e. where the electric field is non-zero). However, when the initial scalar perturbations are sufficiently weak, the full focussing of the CH to will only occur inside the shell. We strongly expect the formation of a spacelike singularity in this case as well, and have drawn this scenario in figure 1., along which . This situation is depicted in the left panel of figure 1. Note that curvature scalars diverge at both the above spacelike and null singularities. While the establishment of a spacelike singularity is less certain for other forms of matter (e.g. perfect fluids), it is nevertheless expected that all future-directed timelike and null curves inside the black hole will be incomplete due to reaching a curvature singularity of some form.
Investigations of nonlinearly-perturbed spinning BHs reveal a similar scenario. Perturbative analyses Ori92 (); EMGkerr (); OriOsc () again indicate the formation of a null weak scalar-curvature singularity at the CH (though this time the singularity is generically oscillatory OriOsc (), as opposed to the monotonic mass-inflation singularity in the spherical case). The presence of such a null singularity is also supported by an asymptotic local analysis of the Einstein equations Brady (), as well as by exact analytical constructions of locally-generic classes of null weak singularity Generic (). Yet, to the best of our knowledge, no numerical verification of this scenario has yet been carried out in the spinning case.
The results above provide a good (if not yet fully complete) understanding of the internal structure of generic (charged, rotating, non-stationary) asymptotically flat, isolated (i.e. non-accreting accreting ()) black holes in classical general relativity 444See Thorlacius (); OriSemiclassical (); Y1 (); Y2 () for attempts to incorporate semi-classical effects.. However, the detailed experiences of any given observer inside such a black hole will in general depend on the process by which the black hole was formed and on the particular perturbations generated. In contrast, we argue below that the spacetime effectively simplifies from the point of view of observers who enter the black hole at late times, which we call late infall-time (or just late-infall) observers. The simplified spacetime may be described by the simple, stationary BH solution up to the inner horizon—and an effective outgoing shock wave at the outgoing portion of the latter (plus a null weak singularity at the CH). This structure is depicted in the right panel of figure 1.
Due to the key role it plays in our analysis, it is useful to describe the notion of late-infall observers in more detail. We start with a simple demonstration (though not necessarily the most precise or most general one) of this concept. Recall that perturbations outside the BH decay at late time, where of course late means relative to the formation of the black hole and the onset of any significant new perturbations. Therefore, for observers who fall into the BH at sufficiently late time, the exterior geometry will be well approximated by the stationary (and axially-symmetric) BH metric. This allows one to associate specific values of energy () and angular momentum () to the late-time geodesics. More importantly, owing to the approximate time-translation of the external geometry, from any “seed” infalling geodesic , we may construct a one-parameter family of similar geodesics, obtained from by time translation to the future. (We emphasize that in the present construction the members of are exactly geodesics, all related to by the approximate time translation. 555 The geodesics in may be constructed as follows. Let be a set of coordinates for the BH exterior, such that at late time the associated metric functions are approximately independent of . We pick a certain point on the “seed” geodesic outside the BH, and time-translate it to the future by a certain amount . At the new point, which we denote , we set the four-velocity components to be numerically the same as those of at . The geodesic which emanates from with those initial four-velocity components now becomes a member of the set .) Note in particular that all geodesics in share (approximately) the same values of and . Now, each member of is characterized by the parameter , namely the value of the advanced time (Eddington’s advanced null coordinate) at which the geodesic crosses the event horizon. For any given seed geodesic , the late-infall observers are those members of characterized by sufficiently large values of .
Our main objective in this paper is to characterize the experience of such late–infall observers who move toward the outgoing section of the inner horizon in a generically-perturbed charged (or spinning) BH. We shall see that such observers experience abrupt changes in the amplitude of various perturbing fields – as well as the metric itself – while crossing the (would-be) inner horizon. These changes occur within a short proper-time interval whose magnitude decreases exponentially with the infall time . For an observer with fixed resolution and sufficiently large this proper-time interval is so tiny that he experiences the perturbation as an effective shock wave.
The above-mentioned concept of late-infall observers may be generalized, and reformulated in a somewhat more precise manner (though we will not attempt a fully precise definition here). Consider a continuous one-parameter family of inextendible causal curves labeled by the advanced time at which they cross the event horizon, with taking values in some range so that the family includes curves that enter the black hole at arbitrarily late times. Note that since perturbations outside the black hole decay, the advanced time can indeed be used as a coordinate along the horizon at sufficiently late times. We require that, in the limit , the part of these curves to the past of the event horizon approaches a stationary family of such curves in some stationary spacetime; i.e. for which curves with different values of are related by the corresponding time-translation. We further require that, for large , the parts of our curves to the future of the event horizon all have the same proper accelerations when expressed using a reference frame parallel-propagated along the world line in terms of the proper time along the worldline after crossing the event horizon. One may think of this as the assumption that all observers in a given family are equipped with identically pre-programmed rocket ships. Of course we insist that these reference frames at the event horizon are also related by an approximate time translation at large . For our rather qualitative purposes below it will not be necessary to specify the precise rate at which these limiting behaviors are approached, though some such specification will certainly be needed to derive more precise results.
After a brief review of charged spherical black holes in section II, we study the experiences of late-time observers in stationary spacetimes subject to linear perturbations in section III. Such observers experience no perturbation at all until they would expect to encounter an inner horizon. However, they effectively encounter a shockwave at the outgoing inner horizon. While we include a discussion of linearly perturbed Kerr black holes, our treatment mainly focuses on the simpler spherically symmetric case.
Non-linear perturbations are addressed in sections IV and V. Here we consider only spherical black holes. Section IV addresses the model of a charged BH perturbed by a self-gravitating scalar field. We show that the experiences of freely-falling late-time observers again agree with those in unperturbed Reissner-Nordström up to the point where they would expect to encounter the (outgoing section of the) inner horizon. However, instead of finding a smooth null surface at that point, they effectively encounter a gravitational shockwave at which the metric is discontinuous. Section V then gives a heuristic argument that the experiences of more general late-time observers are similar, and in particular that they are described by the effective spacetime shown in the right panel of figure 1. The final discussion in section VI describes possible generalizations to rotating black holes and to black holes in any dimensions and the implications for finite-time observers who fall into astrophysical black holes. We also discuss several aspects of the gravitational shock-wave phenomenon which takes place at the inner horizon.
Ii Preliminaries: Spherical charged black hole
The RN solution is the unique spherically-symmetric electrovac geometry. In Schwarzschild coordinates it takes the form
where , and is the unit two-sphere. Throughout this paper we shall consider the non-extreme black-hole (BH) case, namely . In this case vanishes at two values, . The larger root corresponds to the event horizon, and the smaller one to the inner horizon. Figure 2 (left) depicts a part of the Penrose diagram of the eternal, analytically-extended, RN geometry. Note that the inner horizon has two separate portions – the two intersecting null lines denoted “”.
Later we shall also consider the spherically-symmetric spacetime of a charged collapsing thin shell. The geometry outside the shell is described by the Reissner-Nordstrom (RN) metric (1). Inside the shell the geometry is flat, i.e. Minkowski. Figure 2 (right) displays this hybrid spacetime. In this non-eternal BH spacetime the ingoing portion of the inner horizon is a Cauchy horizon (CH). The diagram only displays the globally-hyperbolic piece of the spacetime—namely, the region up to the Cauchy horizon (the extension beyond the CH will not concern us in this paper).
Let us examine the free-fall orbits of observers who jump into the BH. Throughout this paper we assume, as usual (and without loss of generality) that the motion is confined to the equatorial plane. These geodesic orbits are characterized by two constants of motion: the “Energy” , and the angular momentum . They satisfy the radial equation 666At a certain value of , , flips its sign (a phenomenon known as “gravitational bounce”). In this paper, however, we focus on the behavior of orbits up to their first approach to , hence .
Of particular importance will be the behavior of these worldlines in the neighborhood of , where the last result becomes
In the analysis below it will be useful to express the RN metric in double-null coordinates. A particularly useful form is given by the Eddington coordinates
where is the tortoise coordinate defined by
The line element then becomes
where is to be regarded as a function of and , determined (implicitly) by setting in Eq. (5). Notice that in the region which will mostly concern us here—the domain — is past-directed (this choice of sign simplifies many of the expressions below).
Note that at , implying that either or must diverge there. It follows that at the ingoing section of the inner horizon (as is regular), whereas at the outgoing section (and is regular).
Since vanishes at the inner horizon, in its neighborhood we may approximate , where (note that ). It then follows777Eq. (5) defines up to an integration constant. We use this freedom and choose the convenient pre-factor in the right-hand side of Eq. (7), which fixes this arbitrary constant. from Eq. (5) that near the inner horizon
That is, diverges logarithmically (in ) at the inner horizon.
The metric (6) is singular at the inner horizon (where vanishes). To remove this coordinate singularity we define the inner-horizon’s Kruskal-like coordinates , . With this choice of signs, both and are future-directed. Note that () vanishes at the ingoing (outgoing) section of the inner horizon. The line element now reads . We shall not need here the specific form of the metric functions and . We shall just note that both functions are regular. Furthermore, turns out to be a smooth function of which (unlike ) is nonvanishing at . As a consequence, the Kruskal metric is regular at the inner horizon.
Iii Simple examples of late-time perturbations: Linearized fields
In this section we consider several types of linear perturbations, and explore how these perturbations are experienced by late-time infalling observers.
We shall start our analysis by addressing the simplest type of perturbation, namely a purely-outgoing, spherically-symmetric, test scalar field. Then we shall proceed to consider more realistic types of linear perturbations, deferring non-linear perturbations to the following sections.
iii.1 Simplest example: monotonic, outgoing, test scalar perturbations
Consider a free, massless, minimally-coupled, test scalar field on the RN background. The general behavior of a scalar field of this type inside the BH will be discussed in the next subsection. Here we focus on a spherically-symmetric scalar field, restricting our attention to the very neighborhood of the Cauchy horizon. As it turns out, in this region the field becomes purely outgoing, namely 888This simple form follows from the behavior of perturbation fields inside BHs, which we discuss in the next subsection: In the very neighborhood of the inner horizon, the outgoing and ingoing modes effectively decouple [see Eq.(17)]. Furthermore, the -dependent component of decays as at large , leaving us with only the -dependent component. . In our first example we shall further assume, for simplicity, that vanishes until a certain value ; then it grows monotonically up to , where it reaches its final value (and remains constant afterward). While this type of function is certainly oversimplified, it transparently demonstrates the mechanism responsible for the shock-wave formation. 999Essentially the same analysis will apply in the more general (and more realistic) case, in which the outgoing field starts right after the event horizon, and may also continue its growth beyond the (outgoing portion of the) inner horizon. Also the function needs not be monotonic (see discussion below). Here we picked a monotonic function only for the sake of conceptual simplicity. Both null lines are assumed to reside in the internal range . We shall now explore the behavior of along the worldline of the infalling observer, as a function of proper time —focussing our attention on late-time observers.
A free-falling observer in a RN spacetime crosses the inner horizon through its outgoing section (where is finite). Let us denote the values of as the observer crosses the event and inner horizons by and , respectively. Since we are interested in late-time observers we shall assume . 101010 Within the shell-collapse model we further assume that is sufficiently large that the orbit is confined to the shell’s exterior throughout the range . Also, in the case we further demand , such that as is approached. Along a timelike worldline increases monotonically, hence in the range between the two horizons .
Obviously, and depend on the infalling geodesic. However, for fixed parameters and , the difference will be independent of the infall time, owing to the time-translation symmetry of the RN metric. is typically of order (we assume here that is of order unity, i.e. not too large or too close to zero, and is or smaller; and similarly the BH is not too close to Schwarzschild or to extremality.)
Next we evaluate the proper times at the two events where the worldline intersects the null lines . For convenience we set at the worldline’s intersection with the inner horizon. From the above assumptions (in particular ) it immediately follows that — and hence — throughout the portion of the worldline. We can therefore use Eqs. (3) and (7) in this domain to obtain
Substituting we find for and
where respectively denote the value of at the worldline’s intersections with .
Since , we readily find
hence the difference is bounded by
The last inequality tells us at once that the late-time infalling observers will see a scalar-field profile which rises from zero (at ) to its maximal value (at ) within an arbitrarily short proper-time interval, . For a sufficiently large , this proper-time interval will presumably be unresolved by an observer with fixed sophistication. The large- observers will thus experience the scalar perturbation as a sharp shock-wave of finite amplitude. Note that in terms of the proper time of these observers, the shock wave is detected effectively at [the limit of Eq. (10)]—namely, just at the crossing time of the (outgoing portion of the) inner horizon.
It should be emphasized that the two outgoing null lines and need not be close to the (outgoing section of the) inner horizon in any sense. To further clarify this point, consider the intersection of these two outgoing rays with an ingoing null geodesic located just after the collapse. (To be more specific, within the shell-collapse scenario we may choose such that it passes through the intersection point of the collapsing shell with .) Let us denote the values at the two intersection points of with and by and respectively. Our point here is that and need not be close to ; rather, they can take values anywhere in the range . Still, since , at a sufficiently large both these lines will necessarily attain values, and hence values arbitrarily close to .
Consider now the quantity . For fixed parameters and , is a well-defined function of (independent of ). In the late-infall limit, when the orbit reaches . The quantity at similarly approaches its inner-horizon value, which (for ) turns out to be a definite finite number, . In addition, from Eq. (10) it follows that in the late-infall limit, and thus that . Obviously exactly the same argument applies to as well. Using and in Eq. (12), we thus obtain
(Note that the pre-factor in squared brackets is independent of the infall time .)
The typical behavior of as a function of the observer’s proper-time is depicted in Fig. 3 for several values of infall time .
Summarizing, the late-time infalling observers will experience the scalar perturbation as a sharp shock wave of finite amplitude (), with effectively vanishing rising time ()—located just at the outgoing section of the inner horizon.
So far we have considered the case of a (near-CH) outgoing field which varies monotonically from at until where it saturates at . The extension of the argument to a more general function (not necessarily monotonic; and not necessarily one with well-defined final saturation value) is straightforward: It is sufficient to assume that in the near-CH region, varies over a certain range , from at to some at . Then, the above analysis yields that for a late-infall observer the finite jump in the value of will take place within an extremely small proper-time interval, which decreases as . For an observer of given sophistication, this exponentially small proper time will become unresolvable at sufficiently large . Hence, a late-infall observer will again experience a kind of effective shock-wave phenomenon: a finite change in the field, within an effectively-vanishing proper-time interval. 111111A “classic” shock wave typically contains the following three components: (i) a steady pre-shock value, (ii) a (different) steady post-shock value, and (iii) a sharp transition between the two, through a transition region of arbitrarily-small width. Here we use a somewhat generalized notion of shock wave, because we do have the component (iii), and effectively also (i), but we don’t necessarily have (ii).
iii.2 More realistic linear perturbations
We turn now to consider more realistic perturbations of a spherical charged BH. We still assume that in the region of interest the perturbations are small and can therefore be treated linearly. Again, we would like to explore how these perturbations will be experienced by late-time infalling observers.
iii.2.1 Non-spherical test scalar field
The linearized perturbations—both outside and inside the BH—may conveniently be toy-modeled by a minimally-coupled massless scalar field , satisfying Price (). The perturbation field may be decomposed in spherical harmonics in the usual way,
The individual perturbation modes all satisfy a hyperbolic partial differential equation of the form
with , and with an -dependent effective potential given by
One then finds that outside the BH all modes decay to zero at late time Price (). This decay typically proceeds in two stages: First is the stage of “quasi-normal ringing” (i.e. exponentially-damped oscillations). Subsequently, after the ringing has damped, the late-time perturbations are dominated by inverse-power tails. All modes decay (along worldlines of constant ) as , with (or for initially-static multipoles) Bicak (); Bicak2 ()—the same inverse-power form as in the Schawarzschild case Price (). Throughout the rest of the paper we shall focus our attention on the inverse-power tails (rather than the quasi-normal ringing), because it is this component which eventually dominates at late time.
Investigations of the dynamics of linearized perturbations inside the BH reveal a behavior which parallels the external dynamics in many respects, though there also are some important differences. The infalling power-law tails lead to a similar inverse-power decay inside the BH. Thus, along lines of constant (between the event and inner horizons), the perturbations still decay as Gursel (). Note, however, that this time is a spacelike coordinate, so what we face here is a spatial rather than temporal decay. (For a discussion of this issue see Ref. OriGRG ().)
Being proportional to , the effective potential vanishes at (exponentially in , as it also does at ). As a consequence, in the neighborhood of the inner horizon the perturbations take the simple form of a superposition of outgoing and ingoing modes, that is,
Hereafter we omit the sub-indices from the -coefficients for brevity.
In the asymptotic region , both functions and admit simple inverse-power forms:
We consider now a late-infall observer, and examine how this observer will record the scalar perturbations, focusing on a particular multipole . We restrict our attention to the orbit’s section between two hypersurfaces and (both reside in the domain between the event and inner horizons, and satisfy ). We focus on the quantity , where as before, respectively denote the proper times when the observer crosses the two hypersurfaces .
For a sufficiently late infall time, the observer will reach only when the orbit is already in the near-CH region, where Eq. (17) applies. Therefore,
where . Furthermore, since at large (and ), it follows that the second term in squared brackets vanishes at the late-infall limit. Thus, for a sufficiently large the expression for simplifies to
For generic choice of the RHS is non-vanishing. (This is most explicitly verified in the case where are both in the early domain, , where . 121212Yet, a more appreciable should be achieved when is not (and is not too close to ). ) Yet, the proper-time difference is still given by Eq. (13), namely, . Thus, for sufficiently late infall, the observer will watch a finite jump in the field component , during an effectively vanishing (i.e. physically non-resolvable) proper-time interval.
iii.2.2 Linear electromagnetic and gravitational perturbations
In the RN spacetime, owing to the presence of electric field, the (electrovac) gravitational and electromagnetic perturbations are mutually coupled already at the linear level. Still, one can write decoupled field equations for a pair of combined electromagnetic/gravitational variables (i.e. two specific linear combinations of the electromagnetic and gravitational variables) chandra (). In particular, based on a formalism developed by Moncrief Moncrief (), Gursel et al. Gursel2 () constructed such a pair of electromagnetic/gravitational field variables (for the various angular modes ). These variables satisfy the decoupled equations
with the effective potential
where . In turn, the gravitational and electromagnetic perturbations may be recovered from the fields (by certain linear combinations of the latter fields and their derivatives) chandra ().
The potentials are again and therefore vanish at the two horizons, hence near the CH each of the fields is decomposed into free ingoing and outgoing components, i.e. , as in Eq. (17). Furthermore, the leading-order behavior of these two components in the early-CH domain was found Gursel2 () to be of the same form as in the scalar case, i.e. Eq. (18). As before, our objective is the proper-time variation of the perturbation fields, as recorded by a late-infall observer. The analysis of the previous subsections apply here with no modifications, implying that the variables undergo finite variations within proper-time intervals . The gravitational and electromagnetic perturbations (constructed from the variables and their derivatives) are likely to yield a similar structure of an effective shock wave.
iii.2.3 Linear perturbations in Kerr spacetime
In this subsection we very briefly address the case of a linearly-perturbed Kerr BH. The latter’s internal structure is known to be similar in many respects to that of a RN BH. In particular there are two horizons, an event horizon located at and an inner horizon at , where . Not surprisingly, we find that the effective shock-wave phenomenon takes place in the Kerr case as well.
We focus here on the gravitational perturbation, which is apparently the perturbation field of greatest physical relevance here. (Note that the Kerr background—unlike the electrovac RN background—admits pure vacuum gravitational perturbations.)
The behavior of late-time gravitational perturbations inside a Kerr BH has been analyzed using two different formulations: (i) Analysis of metric perturbations (MP) Ori92 (), (ii) Analysis EMGkerr (); OriOsc () of the evolution of the Teukolsky variables Teukolsky () and . Both analyses examined the late-time gravitational perturbations, employing the so-called late-time expansion OriGRG (); OriKERRco (); EMGkerr (). 131313Ref. Ori92 () also considered nonlinear metric perturbations, that is, higher-order terms in the nonlinear perturbation expansion (which turned out, however, to be negligible compared to the linear metric perturbations). Ref. EMGkerr () also considered linear electromagnetic perturbations. In this section, however, we only consider linear gravitational perturbations. They both focussed on the near-CH behavior, and led to similar (and mutually-consistent) results.
For the sake of the present analysis the key result may be summarized as follows: Near the CH (that is, ), the linear MP decouple into a superposition of outgoing and ingoing components, namely 141414In the Kerr case we still define and , with now defined through (where and are the Boyer-Lindquist time and radial coordinates).
Furthermore, decays at as an inverse power of , hence for a late-infall observer near the CH.
In the Kerr background (unlike the RN case), infalling timelike geodesics may intersect the inner horizon either at its ingoing or outgoing section. Here we shall focus on those orbits intersecting the outgoing section (these include, for example, all infalling geodesics with positive and ). The effective shock-wave phenomenon will only occur for this class of orbits.
Consider now an observer which falls into the Kerr BH (and heads towards the outgoing section of the inner horizon), at the late-infall limit. As before, we focus on the observer’s history while moving between and . Just like in the RN case, both proper-times scale at the late-infall limit as — and so does their difference . Here is a certain positive constant, the inner-horizon’s surface gravity. Again, we find a finite jump in the metric, within an effectively-vanishing proper time — namely, an effective gravitational shock wave.
On the other hand, geodesics with will intersect the ingoing section of . In the late-infall limit, these observers will hit the CH at its past boundary.
iii.3 Interpretation in terms of late-time Eddington frames
We shall provide here a simple interpretation of the effective shock-wave phenomenon derived above. For concreteness and simplicitly we present the explicit argument only for the RN case, but it applies to the Kerr case as well.
The line element (6) preserves its form under a coordinate transformation of the form
where is any constant (this invariance reflects the time-translation symmetry of RN). We shall refer to different sets of Eddington coordinates —corresponding to different choices of —as different Eddington frames (this terminology is borrowed from the analogous notion of Lorentz frames in Minkowski spacetime). Note that all tensors constructed from the metric are unaffected by this transformation. In addition, preserves its functional form, .
Consider two infalling observers, which move along two identical worldlines related to each other by a time translation. These observers cross the event horizon (EH) at Eddington times and , respectively (with ). Owing to this difference in , the two observers will not share the same function (or ). To bridge this difference, we equip the second observer with its own Eddington frame , setting in Eq. (23). Since now , it is not difficult to show that of the second observer will be the same function as of the first observer—and the same relation will apply between and .
Consider now some linear perturbation field on the background spacetime (6). Like all other tensorial quantities, is invariant under shifts in the Eddington frame. We assume that near the CH decouples to ingoing and outgoing components (like all linear fields considered above); and we shall be concerned here with the field’s outgoing component, which we denote .
Pick two values , with the only requirement that , where . We shall now examine how the second observer will experience this variation in from to , as a function of its own proper time. To this end we re-formulate the problem in terms of rather than . The change from to thus occurs while the second observer moves from to , where
Let us now fix , and yet consider the late-infall limit for the second observer: . Evidently, in this limit both and are pushed toward . The corresponding (second-observer) proper times will thus be pushed to the same (finite) limiting value , that is, the moment of inner-horizon crossing. In particular, the proper-time difference will vanish in this late-infall limit.
We conclude that at the late-infall limit, the finite variation in the perturbation field (which takes place between a certain pair of values ) occurs within a vanishing proper-time interval—at a moment which (at the limit) coincides with that of inner-horizon crossing. Thus we recover the effective shockwave phenomenon for late-infall observers.
Iv spherically symmetric non-linear perturbations
In this section we shift our focus from linear perturbations on a RN (or Kerr) background, to nonlinearly-perturbed BHs. The main new ingredient is that now the infall orbit is disturbed by the MP, which in turn may influence the observer’s experience of the perturbation. For simplicity, we shall restrict attention here to a spherically-symmetric model of a nonlinearly-perturbed charged BH. We shall first present the model and describe the perturbed BH geometry, and then analyze the experience of late-infall observers in such a spacetime—demonstrating that the effective shock-wave phenomenon occurs in nonlinearly-perturbed BHs as well.
iv.1 Self-gravitating scalar field perturbations of a charged BH
Let us consider a spherical charged BH perturbed by a spherically-symmetric self-gravitating scalar field. This model was investigated by several authors, primarily numerically GG (); BS (); Burko () but also analytically ORIup (); BOint () (assisted by insights gained from earlier analytical investigations of the mass-inflation model PI (); Ori91 ()). The model consists of a massless, minimally-coupled, scalar field , satisfying the covariant wave equation on the (self consistently-perturbed) metric of a spherically-symmetric charged BH. The scalar-field energy-momentum tensor
acts as a source term in the Einstein equations (in addition to the electromagnetic contribution to energy-momentum), yielding a system of nonlinear field equations for the metric functions [e.g. and in Eq. (25) below]. As for initial conditions, we consider here initial configurations wherein is initially compactly supported outside the BH, as in Burko (). (Alternatively, one may prescribe the initial data for , including its presumed inverse-power tails, directly on the EH, as done in BS ().) Evolving the initial data one then finds—not surprisingly—that at late times perturbations die out, and the BH settles down asymptotically to a member of the RN family, with charge and a certain final mass . The scalar perturbations decay as inverse-power tails. In particular, along the EH, (typically with ) at late times Price (); Bicak (); Burko (). These radiation tails fall into the BH and perturb its internal geometry.
The perturbed metric in the BH interior is conveniently expressed in double-null coordinates. In particular, in Eddington-like coordinates 151515 The Eddington-like coordinate may naturally be defined in the perturbed spacetime by using characteristic initial-value formulation, and setting and along the outgoing initial ray (beyond the end of the compact initial support of ) to be the same functions as in the unperturbed RN spacetime with the Eddington metric (6) (setting therein). The key property of is that it diverges at future null infinity, with on approaching the latter. An analogous procedure may in principle be applied to define , yielding an ingoing null coordinate which diverges at the EH. we write the line element in the form
The infalling scalar-field tail triggers the formation of a curvature singularity at the CH. This is a direct consequence of the infinite blue-shift that takes place at the inner horizon Penrose (), which leads to (almost) exponential divergence of the gradient of —and of curvature. It turns out, however, that this is actually a weak Tipler (); OriWeak () curvature singularity, located exactly at the CH (). The metric tensor (in appropriate coordinates) approaches a finite, non-singular, limit as . For example, we may use the Kruskal-like coordinates
with the line element
(with ; note that and are both future-directed, and correspondingly ). The CH is located at . Both and have finite, nonvanishing values at the CH. Yet, diverges at , implying the presence of a null curvature singularity there (though a weak one). The scalar field behaves in a manner similar to : It is finite at , yet diverges at that limit.
Perturbation theory predicts ORIup () (and numerical simulations Burko () confirm) that in the early portion (i.e. ) of the CH, the metric functions deviate only slightly from their respective values in the unperturbed RN solution. The domain is amendable to perturbative treatment. Correspondingly we express and as
where the suffix “RN” denotes the corresponding function in the unperturbed RN spacetime. The perturbations vanish in the limit . (This limit corresponds to the past boundary of the CH, but also to along spatial lines of constant .) In the domain , the scalar field is dominated by its linear perturbation term, and the MP by the second-order perturbation, as described below (see Appendix).
As was mentioned above, (like and ) is finite along the CH. It initially starts at with , but subsequently shrinks steadily with time (), due to the focussing induced by the outflux of scalar-field energy-momentum across the CH. At some stage shrinks to zero—at which point the null weak CH singularity terminates, and connects to a strong, spacelike, singularity GG (); BS (); Burko ().
iv.2 Late-infall orbits
We turn now to investigate the experience of late-infall observers in this spacetime of a nonlinearly-perturbed spherical charged BH. We consider (equatorial) infalling geodesics which are not necessarily radial. The angular-momentum parameter is conserved in these geodesics, though is no longer conserved.
For concreteness we shall focus here on the behavior of the metric function , which the infalling observer probes as a function of his proper time . Physically, a rapid change in will imply (for a finite-size observer) a rapid deformation in the tangential directions . Again, we choose two hypersurfaces, denoted (with ), requiring that both hypersurfaces intersect the CH (before shrinks to zero). While the observer progresses from to , changes by the amount . Hereafter a quantity with a sub-index “1” or “2” will denote the value of this quantity as the worldline crosses the hypersurface or , respectively. Since along the CH is steadily shrinking, one finds that .
Let us now evaluate the proper-time interval , using
Note that (like ) is bounded throughout the domain , so we can easily bound by
where denotes the maximal value of in the worldline’s section between and . The integral on the RHS is nothing but the proper-time length of a timelike curve connecting the two points and , in a fiducial two-dimensional spacetime with the flat metric . It is bounded above by the (timelike) geodesic connecting these edge points, whose length is , where and . Clearly, , therefore
Consider now the late-infall limit, which is the limit of large . In this limit . Therefore approaches , which is the maximal value of along the section of the CH. We obtain our bound on (for late-infall observers) in its final form:
where is a parameter which depends on and but not on the orbit’s infall time.
We conclude that late-infall observers will measure a non-vanishing variation in the metric function , within a short proper-time difference which shrinks exponentially in the infall time —which is again an effective shock-wave phenomenon.
iv.3 Do the late-infall orbits cross the CH?
Our analysis so far did not make use of the perturbative nature of the metric field (at the early portion of the CH). We merely assumed that the CH singularity is null and weak—and more specifically, that admits a finite limiting value along the CH. Correspondingly, there was no need to restrict and to the perturbative domain (): We only required that the surfaces and intersect the CH, rather than the spacelike singularity. However, there still was one hidden assumption: We implicitly assumed that the late-infall observers will make it all the way from to without intersecting the CH (that is, with finite ). Once an observer intersects the CH, we cannot make any concrete statement about his subsequent experience, because the CH is by definition the boundary of the domain of unique prediction (for e.g. the metric functions). 161616Furthermore, owing to the divergence of curvature at the CH singularity, it is unclear whether a classical extension beyond the CH will be physically meaningful.
We therefore still need to complete this missing piece of the analysis. We shall show that as long as is located in the weakly-perturbed domain of the CH (that is, is large compared to ), the late-infall orbits indeed arrive at with finite .
The control on the growth of will be achieved by monitoring the evolution of the geodesic’s “energy” parameter
Note that (unlike ) is no longer conserved, because the perturbations destroy the exact -translation invariance of the RN background. Yet, following the evolution of will enable us to control , and thereby the evolution of along the orbit.
iv.3.1 Equation of motion for
The lower-index covariant geodesic equation, applied to the Eddington-like metric (25), reads
To get rid of the term in the RHS, we use the normalization condition , which for the metric (25) yields
The quantity then becomes
Noting that the last term in squared brackets is equal to , we re-express in terms of the Kruskal-like metric function :
Note that in the background RN metric, both and are functions of , hence is conserved. Evolution of will thus only result from the MP and .
So far all equations were exact. To proceed beyond this point, we restrict attention to the early portion of the CH, where perturbations are presumably small, and carry out the analysis at the leading order in the MP. 171717 That is, we expand the various prescribed background functions (i.e. the functions of obtained from the metric functions etc.) to first order in the MP. However, we do not linearize the worldline-related quantities, like , , , , etc. Also, since we are dealing with the orbit’s evolution very close to the CH, we may replace the background’s functions and by their corresponding inner-horizon values and . We obtain
where and are constants,
Recall that (like ) is a finite, non-vanishing constant.
iv.3.2 Analyzing the evolution of
We re-write Eq. (33) as
in which we view as a (yet unknown) function of the parameter along the geodesic. One might choose to approximate this function by the corresponding function for geodesics in the unperturbed RN geometry. We shall not proceed here in this way, because we do not want to assume a-priori that the accumulating perturbations in the orbital parameters must be small. Instead, we shall proceed by expressing in terms of . To this end we use the contravariant version of Eq. (30), namely
This, together with Eq. (31), constitutes a closed algebraic system for the two unknowns . One can of course write down the exact solution of this algebraic system. However, it will be more instructive to employ here the approximate solution, associated with the smallness of : We are dealing here with the near- region, where 181818Note that (valid in the perturbative domain considered here) also implies that (the perturbation in ) satisfies ; that is, . . The algebraic system thus yields the simple approximate solution (to leading order in the small parameter ) 191919The algebraic system also admits a second solution, in which and are interchanged. However, since our late-infall observers enter the region with large , it is the solution (35) which actually takes place.
Substituting this expression for in Eq. (34) we find
Setting (where, recall, ), we obtain
where we have absorbed the above (in ) in and . Note that the RHS in this equation is a prescribed function of and (with no reference whatsoever to four-velocity).
At this point it will be useful to refer to the concrete form of the MP and . We focus here on the early portion of the CH (i.e. ), where the MP are small and decay as inverse powers of and/or . We denote by the modification in acquired in the near-CH region, up to some , due to the presence of MP. Based on the inverse-power form of the MP, in the Appendix we derive the bound
where is a certain parameter (independent of ). In particular, we find that for late-infall orbits .
iv.3.3 Analyzing the evolution of
We proceed now to analyze the evolution of , showing that it remains finite throughout (for any . From Eq. (35) we have
Now for late-infall orbits we already found that and hence we may regard as constant (essentially the entrance value of ). Also we may set , and take the near-CH form of , namely . We obtain
where is some positive constant. Re-writing this as , we obtain
[Notice that the last two equations are the same as those describing late-infall geodesics in the unperturbed RN geometry.] This expression is bounded above (by the in the RHS). Thus, remains finite throughout the range —meaning that the orbit cannot cross the CH (located at ) in that domain.
iv.4 Concluding Remarks
We found that the late-infall observers cannot cross the CH in the regime . The analysis in section IV.2 then shows that the proper-time for these observers to move from to , for any , decreases exponentially in the infall time . Since varies during this range by a finite amount , we inevitably face here the phenomenon of effective gravitational shock-wave: A discontinuity in the metric tensor, which propagates along a null hypersurface (the outgoing section of the inner horizon in our case). Physically, this means that any extended object will undergo a sudden deformation, by a certain amount, within an effectively-vanishing proper time.
The amplitude of such a gravitational shock-wave may naturally be characterized by the (dimensionless) magnitude of the object’s deformation. Specifically in the spherical model studied here, the shock’s amplitude may be taken to be the dimensionless quantity .
Two additional remarks are in order here:
Since the discussion in section IV.3 assumed in order to treat perturbations perturbatively, one might mistakenly conclude that the amplitude of the gravitational shock wave can be weak (i.e. ). But this is not the case. It is clear from the above analysis that late-infall geodesics cannot fall across the CH until after the perturbations grow to be of order 1. We therefore see from section IV.2 that late freely-falling observers must face a strong gravitational shock wave whose amplitude is at least of order 1.
In order to allow a simple discussion of the quantity , which is conserved along geodesics in exact RN, we have so far assumed freely-falling worldlines. But let us now consider an accelerated late infall-time observer. We choose the acceleration as a function of proper time to be bounded and such that, in the unperturbed RN geometry, the worldline would reach the outgoing inner horizon before crossing the ingoing inner horizon. Note that in the limit of large all of these accelerations occur before reaching . As a result, in the unperturbed spacetime becomes some constant for all . Corresponding late infall-time observers in the perturbed spacetime may thus be analyzed just as for the freely-falling observers discussed above but with the entrance value of replaced by . We conclude that any accelerated late infall-time observer who “would have reached the outgoing inner horizon in unperturbed RN” also experiences a shockwave in the perturbed spacetime.
V Effective spacetimes for late-infall observers
We have seen above that, in the limit of late infall times, observers who enter a perturbed Reissner-Nordström black hole experience an effectively unperturbed Reissner-Nordström geometry up to the point where they would expect to encounter an inner horizon at . At this point, observers who would have reached the outgoing inner horizon in exact RN then encounter a shockwave across which the metric changes discontinuously.
Describing the detailed nature of this discontinuity requires further investigation. While we will not attempt a precise treatment here, it is natural to expect that the above shockwave in fact contains a curvature singularity, as we know that our observers will eventually reach such a singularity and we expect that, since they are already “nearly null” in the region described in section IV, all proper times along their worldlines will be compressed to zero in the large limit202020 Note that this argument is equally valid if, for some kinds of matter, the spacetimes contain no spacelike singularity. In such cases the ingoing worldline will reach the weak null singularity, which is also a curvature singularity..
In addition, it is clear that some accelerated observers will reach the ingoing weak null singularity shown in figure 1 (left). For observers who enter the black hole very late, in the region where perturbations are very small, this may be accomplished using roughly the same set of accelerations (as defined in their own reference frame) as would be required to reach the ingoing part of the inner horizon in an unperturbed RN black hole. Furthermore, in the limit of late infall times, such observers arrive at the portion of the weak null singularity close to in figure 1 where the singularity is extremely weak, so that again such observers will measure no noticeable departure from unperturbed RN until is exceedingly close to