Geometry near the inner horizon of a rotating, accreting black hole
aa r X i v : . [ g r- q c ] F e b Geometry near the inner horizon of a rotating, accreting black hole
Tyler McMaken ∗ and Andrew J. S. Hamilton † JILA and Dept. Physics, U. Colorado Boulder, CO 80309, USA (Dated: February 23, 2021)Here we present a novel classical model to describe the near-inner horizon geometry of a rotat-ing, accreting black hole. The model assumes spacetime is homogeneous and is sourced by radialstreams of a collisionless, null fluid, and it predicts that the standard Poisson-Israel mass inflationphenomenon will be interrupted by a Kasner-like collapse toward a spacelike singularity. Such amodel is shown to be valid at the inner horizon of astrophysically realistic black holes through com-parison to the conformally-separable model, which provides a natural connection of the Kerr metricto a self-similar, accreting spacetime. We then analyze the behavior of null geodesics in our model,connecting them to the Kerr metric in order to answer the practical question of what an infallingobserver approaching the inner horizon might see.
I. INTRODUCTION
Physicists have long wondered what happens inside as-trophysically realistic black holes. The exterior geome-try of a back hole has been well-established to be de-scribed completely by the Kerr-Newman metric, sinceany perturbations during a generic collapse will be ra-diated away and leave the black hole with only threeuniquely-identifying parameters: a charge Q , angularmomentum J , and mass M . But when the Kerr-Newmansolution is extended to the interior of a black hole’s eventhorizon, some puzzling, nonphysical structures emerge.Within the simplest model of black holes (theSchwarzschild solution), no major peculiarities or non-physical structures arise except the divergence of thespacetime curvature at r =0 at a spacelike singularity.Prior to the 1960s, many opposed the idea that a realis-tic gravitational collapse would lead to a singularity, sincemost known models of collapse were highly idealized andunstable to perturbations [1]. However, in 1965, Penrosepublished a theorem demonstrating the inevitability ofsingularities within event horizons of black holes [2], andsoon after, Belinskii, Khalatnikov, and Lifschitz found arealistic model for such a collapse to a spacelike singular-ity (the so-called BKL collapse), which is highly complexand oscillatory [3].In spite of these efforts, the fact remains that most, ifnot all, black holes are not spherically symmetric and in-stead carry at least some angular momentum. The struc-ture of the Kerr-Newman interior differs drastically fromthat of a Schwarzschild black hole—instead of a space-like singularity at the center, the Kerr-Newman solutionhas a timelike singularity along with a second horizonwithin the event horizon. This inner horizon coincideswith the singularity when J = 0 and Q = 0, but fornonzero spin or charge, the spacetime between the innerhorizon and the singularity forms a region in which pre-dictability breaks down—general relativity is powerless ∗ [email protected] † [email protected] to predict unambiguously what would happen if an ob-server passes through the inner horizon, because such anobserver would be able to view the singularity.Aside from the breakdown of predictability, the addedinterior structure of a rotating or charged black hole isproblematic for another reason. As first pointed out byPenrose in the context of a Reissner-Nordstr¨om (charged)black hole, the inner horizon is a surface of infiniteblueshift, so that an infalling observer at an inner horizonwould see the entire history or future of the Universe flashbefore their eyes as the energy of any incoming radiationbecomes classically unbounded [4]. Penrose conjecturedthat the added effects of this diverging radiation wouldchange the underlying spacetime curvature of the vac-uum solution [5]. This conjecture was finally confirmed afew decades later, when Poisson and Israel performed afull nonlinear perturbation analysis in a seminal 1990 pa-per [6, 7]. Poisson and Israel concluded that the crossingof ingoing and outgoing shells of null dust at the innerhorizon would lead to a divergence of the spacetime cur-vature. Poisson and Israel dubbed this divergence “massinflation,” because an observer near the inner horizonwould measure an exponentially large quasi-local internalmass parameter (though the global mass as measured atinfinity would remain finite). The observer near the innerhorizon would then see an asymptotically Schwarzschild-like geometry with an enormous Schwarzschild mass M ,and the journey to the inner horizon would encompassall but the last Planck time of the black hole’s classicalhistory.Classically, the Poisson-Israel toy model of mass infla-tion leads to the formation of a null weak curvature sin-gularity, in which the curvature locally diverges but thetidal distortion of extended objects travelling along time-like geodesics remains finite, allowing for the continuationof spacetime beyond the Cauchy surface [8, 9]. Dafer-mos extended this result for the less-simplified Einstein-Maxwell-real scalar field equations [10], and Ori andothers found that null curvature singularities providea generic class of solutions to the Einstein equations,adding it to the list of known possible singularities thathad previously only included the BKL singularity [11–13]. The BKL and null curvature singularities are quitedifferent in nature—though they both may be oscillatoryin nature, the BKL singularity is strong and spacelike,whereas the null curvature singularity is weak and light-like [14].Despite the enticement of the null weak curvature sin-gularity, both in its mathematical accessibility and in itspotential to allow for a gateway to another Universe, itsuffers one fatal flaw. One of the key assumptions for allthe models that predict a null weak singularity is thatthe collapsed black hole is in complete isolation. Underthis assumption, which still dominates much of the re-search program for mass inflationary phenomena to thisday [15], the only source of ingoing perturbations is thePrice tail, a stream of gravitational waves emitted andbackscattered during the collapse. The Price tail decayswith an inverse power law in advanced time, and calcula-tions for the formation of a null weak singularity assumethat no additional radiation perturbs the metric abovethat power law [16]. However, astrophysical black holescontinue to accrete material long after the initial gravi-tational collapse, and even the cosmic microwave back-ground radiation would dominate over the longest-livedPrice tail modes of a stellar-mass black hole after only 1second [17].Motivated by this shortcoming, Burko found numeri-cally that a null weak singularity only forms for a suffi-ciently steep radiation power law drop-off, and that if itdoes not drop off quickly enough, a spacelike singularitywill form at the intersection of the ingoing and outgoinginner horizons and grow until it has completely sealed offthe Kerr tunnel [18, 19].Hamilton subsequently developed a self-similar modelfor the inner horizon spacetime that generalizes the massinflation phenomenon to include arbitrary ingoing andoutgoing collisionless streams of radiation at arbitrarytimes, first for spherical-symmetric spacetimes [20] andsoon after for the more realistic case of rotating blackholes [21–23]. The rotating case, which assumes con-formal separability, is reviewed in more detail in SectionIII A. The key conclusion from this model is that the con-tinued streams eventually slow the inflation of the cur-vature, causing the spacetime to collapse radially. Theresulting global geometry, used throughout this paper, isshown in the Penrose diagram of Fig. 1.Though the conformally-separable model of Hamiltonis valid through the inflation and the beginning of thesubsequent collapse of the spacetime, it eventually failsonce the rotational motion of the streams becomes com-parable to their radial motion. After this point, numeri-cal calculations indicate that the collapse follows a BKL-like behavior [17]. However, recent semiclassical calcu-lations suggest that quantum backreaction effects mayalter or even invert the collapse [24]. Thus, one maywish to calculate the renormalized stress-energy tensorin the conformally-separable Kerr spacetime. Such a cal-culation has not yet been attempted because of the com-plexity of Hamilton’s model; however, here we derive anew model that considerably simplifies the conformally- t FIG. 1. (Color online). Penrose diagram for the late-time evo-lution of a collapsed star with a Kerr exterior (white) matchedto an inflationary Kasner regime (shaded blue). The innerhorizons (dashed lines) of the Kerr metric are superseded bythe BKL singularity of the inflationary Kasner model (squig-gled line). The gray arrows labelled t indicate the directionof increasing Boyer-Lindquist time. separable Kerr model while still retaining its essentialfeatures of inflation and collapse. This model, which wecall the inflationary Kasner model, will be shown to pro-vide a reasonable continuation of the Kerr metric nearthe inner horizon through the first two Kasner epochs ofits BKL collapse, and it will hopefully allow for futurequantum calculations in this region.In Section II, the inflationary Kasner metric is derivedas a general solution to Einstein’s equations for a homo-geneous spacetime sourced by a null, perfect fluid. Inparticular, we find that for a line element of the form ds = − α ( T ) dT + a ( T ) dx + a ( T ) dy + a ( T ) dz , (1)the general solution for an energy-momentum tensorwhose only non-negligible components are T = T takes the form a ∝ T − p/ exp (cid:0) T p (cid:1) ,a ∝ T p ,a ∝ T p , (2)for some arbitrary constant p . This metric is dubbedthe inflationary Kasner metric because it is a naturalnon-vacuum extension of the Kasner metric, a vacuumsolution used to model collapse to a BKL singularity.During a BKL collapse, the spacetime undergoes a seriesof “BKL bounces,” between which the evolution is de-scribed by the Kasner metric’s power law behavior. TheBKL model is described in more detail in Sec. II C, inwhich we show how the two epochs of the inflationaryKasner solution can be reduced in certain limits to pre-viously obtained results.Then, in Sec. III, we show how the inflationary Kasnermodel can be applied to the inner geometry of astro-physical black holes. To do so, we connect our model tothe Kerr metric in a regime where both are valid, em-ploying Hamilton’s conformally-separable Kerr model tofacilitate the matching and to determine the degree towhich the assumptions of the inflationary Kasner modelare valid near the inner horizon. Sec. III A is devotedto reviewing the conformally-separable Kerr model andcomparing it to the inflationary Kasner model, where wefind that the inflationary exponent ξ of the conformally-separable model is related to the inflationary Kasner time T via the relation T p ∝ e − ξ . Then, in Sec. III B, we an-alyze the behavior of null geodesics in our model, ray-tracing them from an observer in the inflationary Kasnerspacetime backwards until they connect to null geodesicsin the Kerr spacetime. Such a matching allows us inSec. IV to answer the practical question of what an ob-server falling toward the inner horizon of an astrophysi-cal, classical black hole might see. II. INFLATIONARY KASNER METRICA. Preliminaries
Throughout this paper, we use the metric signature − + ++ and geometric units where c = G = 1.In our analysis we use an orthonormal tetrad formal-ism, in which quantities are defined in the tetrad basis { e ˆ m } to yield physically measured components in the lo-cal, Cartesian frame of an observer. In such a formalism,coordinate-frame quantities can be converted into tetrad-frame quantities through the vierbein e ˆ mµ , which can beread off directly from a line element via ds = g µν dx µ dx ν = η ˆ m ˆ n e ˆ mµ e ˆ nν dx µ dx ν . (3)Indices for abstract, Einstein-summed tetrad-framequantities are denoted by lowercase Latin letters withhats, while indices for abstract, Einstein-summedcoordinate-frame quantities are denoted by lowercaseGreek letters. Indices for specific components of tetrad-frame quantities are denoted by Arabic numerals, whilethose of coordinate-frame quantities are given by theirstandard Latin or Greek letters. Thus a tetrad-framefour-vector can be expressed as k ˆ m = { k , k , k , k } , acoordinate-frame one as k µ = { k t , k r , k θ , k φ } , and theconversion between the two is given by k ˆ m = e ˆ mµ k µ . (4)For a more complete review of the tetrad formalism,see M¨uller’s “Catalogue of Spacetimes” [25] or Chan- drasekhar’s The Mathematical Theory of Black Holes [26].
B. Derivation of the line element
The purpose of this subsection is to derive the lineelement for the inflationary Kasner metric, which in itsfinal form reads ds = a (cid:0) − dT + dx (cid:1) + a (cid:0) dy + dz (cid:1) , (5)with a = e ( T − T ) / √ πΦ T (cid:18) TT (cid:19) − / ,a = 1 √ πΦ T (cid:18) TT (cid:19) / , (6)where the time coordinate T progresses backward fromthe positive constant T to 0, and the positive constant Φ represents the mass-energy density of the streams offluid seen by an observer at T = T . In general, the mass-energy density will be found to depend on T through therelation Φ ( T ) = Φ p T /T e T − T . (7)The form of the inflationary Kasner line element ofEq. (5) relies on two main assumptions. First, assumethe metric is spatially homogeneous, so that the met-ric coefficients are functions only of the time coordinate T . Such a requirement exists in a more stringent form forthe Kasner metric described in the next section, in whichthe metric coefficients are power laws in T during a BKLcollapse. Second, assume the solution to Einstein’s equa-tions is sourced by a collisionless, null, perfect fluid inthe radial direction. In a tetrad frame, such a sourcecorresponds to the energy-momentum tensor T ˆ m ˆ n = Φ Φ , (8)where Φ is the mass-energy density of the null streams.For a realistic accreting black hole, even if the accretedmaterial is not null and purely radial, near the inner hori-zon, all streams of matter and radiation are expected tofocus along the principal null directions ultrarelativisti-cally, so that the energy-momentum tensor to a goodapproximation takes the form above.Assume the line element (and therefore the vierbein)can be written in a diagonal basis. Thus, the tetrad1-forms may be written as e µ dx µ = a a a T dT, (9a) e µ dx µ = a dx, (9b) e µ dx µ = a dy, (9c) e µ dx µ = a dz, (9d)where the scale factors a i are functions only of the timecoordinate T but are otherwise left arbitrary. The choiceof the present form of e T will help to simplify later cal-culations; in general, e T may be any function of T aftera suitable transformation of the T coordinate.Instead of working in a coordinate basis and using themetric components to find the Christoffel connection co-efficients Γ µνρ , here we work entirely in a tetrad basiswithout reference to the coordinate frame, so we mustfirst find the analogous tetrad connection coefficients. Inthe tetrad basis, the connection 1-forms ω ˆ m ˆ n (which areantisymmetric in their tetrad-frame indices) can be de-fined by the torsion-free condition de ˆ m + ω ˆ m ˆ n ∧ e ˆ n = 0 , (10)or in component form, ω ˆ m ˆ nρ = e ˆ mµ ∇ ρ e µ ˆ n . (11)Converting all indices of the connection 1-form com-ponents to a tetrad basis then yields the Ricci rotationcoefficients ω ˆ m ˆ n ˆ r , antisymmetric in their first two indices: ω ˆ m ˆ n ˆ r = η ˆ ℓ ˆ m e ρ ˆ r ω ˆ ℓ ˆ nρ = e µ ˆ m e ρ ˆ r ∇ ρ e ˆ nµ . (12)For the tetrad 1-forms of Eqs. (9), the six nonvan-ishing Ricci rotation coefficients are as follows, where i ∈ { , , } and a dot above a variable indicates differ-entiation with respect to the time coordinate T : ω ii = − ω i i = Ta a a ˙ a i a i . (13)The tetrad-frame Riemann curvature tensor compo-nents R ˆ k ˆ ℓ ˆ m ˆ n = e κ ˆ k e λ ˆ ℓ e µ ˆ m ( ∇ κ ∇ λ − ∇ λ ∇ κ ) e ˆ nµ can thenbe calculated, yielding 18 nonzero components: for i, j ∈ { , , } and i = j , R i i = R i i = − R ii = − R i i = T a a a (cid:18) ˙ a i a i d ln ( a a a /T ) dT − ¨ a i a i (cid:19) , (14a) R ijij = − R ijji = T a a a ˙ a i ˙ a j a i a j . (14b)Then, the tetrad-frame Ricci tensor R ˆ k ˆ m = η ˆ ℓ ˆ n R ˆ k ˆ ℓ ˆ m ˆ n ,Ricci scalar R = η ˆ k ˆ m R ˆ k ˆ m , and tetrad-frame Einstein ten-sor G ˆ k ˆ m = R ˆ k ˆ m − η ˆ k ˆ m R follow naturally. The result-ing four nonzero Einstein components, where i ∈ { , , } with cyclic addition, are G = T a a a (cid:18) ˙ a ˙ a a a + ˙ a ˙ a a a + ˙ a ˙ a a a (cid:19) , (15a) G ii = G − Ta a a ddT (cid:18)(cid:18) ˙ a i +1 a i +1 + ˙ a i +2 a i +2 (cid:19) T (cid:19) . (15b)Under the assumption that the tetrad-frame energy-momentum tensor has the form of Eq. (8), Einstein’sequations give a system of four nontrivial equations: G = 8 πΦ, (16a) G = 8 πΦ, (16b) G = 0 , (16c) G = 0 . (16d)The most natural solution to Eqs. (15) and (16) canbe obtained by setting a = a , which reduces Eqs. (16c)and (16d) to the same equation. Physically, this corre-sponds to the assumption that the y - z plane, orthogonalto the streams of matter, remains isotropic, a reasonableassumption close to the horizon, given that any streamswill focus ultrarelativistically in the x -direction. The re-maining three equations simplify to8 πΦ = T H a a (2 H + H ) , (17a)8 πΦ = T H a a H + H − H H − T ! , (17b)0 = 2 H H + H − H + H T − ˙ H − ˙ H , (17c)where we have introduced the quantities H i ≡ ˙ a i a i = ⇒ ˙ H i = ¨ a i a i − H i (18)for i ∈ { , } . Combining Eqs. (17a) and (17b) to elimi-nate Φ yields ˙ H H = − T = ⇒ H = pT = ⇒ a = C T p , (19)where p and C are arbitrary integration constants. Sub-stituting this solution into Eq. (17c) yields˙ H − (2 p − T H − p T = 0= ⇒ H = − p T + qT p − = ⇒ a = C T − p/ exp (cid:18) q p T p (cid:19) , (20)where q and C are arbitrary integration constantsand the first-order differential equation in H is mosteasily solved with the help of the integration factorexp (cid:0) − R p − T dT (cid:1) = T − p .Thus, the tetrad 1-forms for the inflationary Kasnermetric are e µ dx µ = C C T p/ − exp (cid:18) q p T p (cid:19) dT, (21a) e µ dx µ = C T − p/ exp (cid:18) q p T p (cid:19) dx, (21b) e µ dx µ = C T p dy, (21c) e µ dx µ = C T p dz. (21d)Without loss of generality, replace the constants C , C , and p through a set of redefinitions and coordinatetransformations with the constants T and Φ , so thatthe vierbein becomes e µ dx µ = e ( T − T ) / √ πΦ T (cid:18) TT (cid:19) − / dT, (22a) e µ dx µ = e ( T − T ) / √ πΦ T (cid:18) TT (cid:19) − / dx, (22b) e µ dx µ = 1 √ πΦ T (cid:18) TT (cid:19) / dy, (22c) e µ dx µ = 1 √ πΦ T (cid:18) TT (cid:19) / dz. (22d)These tetrad 1-forms lead directly to the line elementof Eqs. (5) and (6) through the relation in Eq. (3).This form is chosen to illuminate the physical mean-ing of the constants (as described in the next section)and to keep the exponential dependence in T as simpleas possible. The radial component of the tetrad-frameenergy-momentum tensor, T = T = Φ , then reducesto Eq. (7). C. Interpretation
The evolution of the inflationary Kasner geometry isvisualized in Fig. 2. The evolution begins at T = T ,when the mass-energy density Φ is at its small initialvalue of Φ . At this point, an observer might see such ageometry if, for example, they fall inside a rotating, ac-creting black hole and approach its inner horizon. Oncethe observer has come close enough to the inner horizon,the inflation epoch will begin, characterized by the rapidexponentiation of the observed stream energy density. Asthe observer’s proper time progresses forward and T pro-gresses backward from T , the inflation will slow until a turns around (at the vertical gray line in Fig. 2), signal-ing the start of the collapse epoch. The inflation-collapsetransition occurs when T is of order unity, or more pre-cisely, when H changes sign from positive to negativeat T = 1 / T ). Duringthe collapse epoch, Φ continues to increase as the space-time collapses in the y - and z -directions and the observer - - - T / T inflation collapse Φ / Φ a a FIG. 2. (Color online). Evolution of the inflationary Kas-ner geometry and radial energy-momentum from the initialKerr vacuum at time T = T through inflation and collapse.The plotted quantities are the normalized tetrad-frame mass-energy density Φ/Φ (black) from Eq. (7) and the scale factors a (orange) and a (purple) from Eq. (6). The parameterschosen here are T = 9 .
09 and Φ = 0 . approaches the inflationary Kasner singularity at T = 0where the inner horizon would have been.We have chosen to call the metric of Eq. (5) the in-flationary Kasner metric because of its similarity to ahomogeneous vacuum solution first found by the mathe-matician Edward Kasner in 1921 [27]. For three spatialdimensions, the Kasner metric has the line element ds = − dT + a dx + a dy + a dz , (23)where the scale factors a i evolve purely as power lawswith T , a i = T p i , (24)for exponents p i that were found in the vacuum solutionto satisfy the following conditions: X i p i = X i p i = 1 . (25)From these Kasner conditions it can be shown that oneof the exponents must always be negative or zero whilethe other two are nonnegative. More specifically, if theKasner exponents are labelled in increasing order, theywill satisfy the condition − ≤ p ≤ ≤ p ≤ ≤ p ≤ . (26)This mathematical picture can be physically inter-preted as an evolution in which one spacetime axis ex-pands while the other two collapse (assuming the timecoordinate T is positive and decreases with increasingproper time; otherwise the Kasner solution would de-scribe a globally expanding spacetime).The significance and applicability of the Kasner met-ric for black hole interiors was explored in the 1970s byBelinskii, Khalatnikov, and Lifschitz, who described acollapse consisting of a series of “Kasner epochs” duringwhich the geometry is approximated by a Kasner met-ric with constant Kasner exponents p i [3]. Accordingto the BKL model, the three spatial components of themetric evolve in such a way so that the metric determi-nant decreases monotonically to zero in a finite time, butone spatial component always increases while the othertwo decrease (cf. Eq. (26)). Once one of the decreas-ing components has collapsed to a small enough value,the geometry then undergoes a “BKL bounce,” in whichone of the two collapsing components begins to grow, thepreviously expanding component begins to collapse, andthe angles of orientation for the collapsing and expandingaxes change.In 2017, the evolution of the inner horizon of a ro-tating, accreting black hole was explored numerically byHamilton, who found that the spacetime approximatelyundergoes a BKL collapse as predicted a half a centuryearlier [17]. Here, we find that the inflationary Kasnersolution is an analytic model of such a collapse, with twoKasner epochs as described below.The first epoch in the inflationary Kasner solution, la-belled “inflation” on Fig. 2, begins at T = T . Duringinflation, the exponential terms in the line element ofEq. (5) dominate the evolution of the geometry, so thatthe scale factor for the x -axis collapses while those of the y - and z -axes remain approximately static. The behav-ior can thus be approximated as that of Minkowski spacewith accelerated radar-like coordinates in the x -direction[28]. The inflation epoch resembles a Kasner epoch withexponents ( p , p , p ) = (1 , , , (27)corresponding to a spacetime collapsing only in the radialdirection (indeed, the growing streams focus along theprincipal null directions during inflation). The inflationcontinues as the locally-measured energy-momentum Φ grows at an absurdly fast rate with a scale factor of order ∼ e /Φ (as confirmed in the next section, in which it isfound that the Kasner time T scales as 1 /u ∼ /Φ ). Foran astronomically realistic black hole, in which the initialmass-energy density of accreted matter or radiation isgenerally quite small after the initial collapse, Φ couldreach 10 and beyond. It is perhaps fitting that Kasnerhimself (with his nine-year-old nephew) was the coiner ofthe term “googol” [29].Once the Kasner time T has grown small enough, theexponential terms in the Eq. (5) freeze out, leaving thepower laws in T to dominate the geometry’s evolution. The result is the collapse epoch, beginning at around T = 1 /
2, in which the scale factor for the x -axis turnsaround and begins to grow, the scale factors in the y -and z -directions continue to collapse, and the streams’energy-momenta continues to grow, albeit at a slowerrate in log( T ). This corresponds to a Kasner epoch withexponents ( p , p , p ) = (cid:18) − , , (cid:19) , (28)which can be found by a coordinate transformation of T from the (cid:0) − , , (cid:1) form of Eq. (5) in order to satisfythe Kasner conditions of Eq. (25). This epoch approxi-mates a Schwarzschild geometry asymptotically close tothe Schwarzschild singularity. To see why this is the case,note that in the limit as r →
0, the Schwarzschild line el-ement takes the form ds ≈ Mr dt − r M dr + r do , (29)where do = dθ + sin θ dφ is the 2-sphere line element.With the coordinate transformations r → T / and t → x (note that r is timelike and t spacelike in thisregime), the line element becomes ds ≈ − dT + T − / dx + T / do , (30)where the constants have been absorbed into the coor-dinates for simplicity. This is precisely the (cid:0) − , , (cid:1) Kasner epoch when the θ - φ plane is transformed into the y - z plane.Thus, the inflationary Kasner metric provides a sim-ple model that encompasses all the relevant features ofthe evolution of the geometry near the inner horizonof a rotating, accreting black hole as it undergoes aBKL-like collapse. That collapse consists of two Kasnerepochs, an inflationary epoch characterized by Kasnerexponents (1 , ,
0) that matches the behavior of the tradi-tional Poisson-Israel mass-inflation regime, and a subse-quent collapse epoch characterized by Kasner exponents (cid:0) − , , (cid:1) as the geometry approaches a spacelike singu-larity at T = 0.In the next section, we confirm the applicability of thismodel to astrophysical inner horizons by comparing it toa more complex model, the conformally-separable Kerrmodel, with the eventual goal of finding the necessaryboundary conditions to attach the inflationary Kasnermetric to the Kerr metric far enough above the innerhorizon. III. MATCHING NEAR THE INNER HORIZON
We have yet to verify explicitly that the assumptionsof the inflationary Kasner metric hold true near the innerhorizon of an astrophysical black hole. In order to do so,we employ Hamilton’s conformally-separable Kerr met-ric, which has already been shown to provide a reason-able classical model of the inner workings of an accretingblack hole [21–23].In Sec. III A, we review the conformally-separablemodel, finding that it exactly matches the behavior of theinflationary Kasner model for asymptotically small accre-tion rates. Specifically, we find in Eqs. (40) a set of trans-formations between the parameters and coordinates ofthe inflationary Kasner and conformally-separable Kerrmodels. These relations confirm the validity and applica-bility of the inflationary Kasner model to an astrophysi-cal inner horizon.Then, in Sec. III B, we use the transformations ofEqs. (40) to match the inflationary Kasner solution tothe Kerr metric. Such a matching allows us to ray-tracenull geodesics across both regimes, from the Kerr back-ground to an inflationary Kasner observer.
A. Conformally-separable Kerr solution
Just as the inflationary Kasner metric provides a non-vacuum generalization of the vacuum solution of the Kas-ner metric to include the effects of accretion, so too doesthe conformally-separable Kerr metric provide a gener-alization of the vacuum Kerr solution to include the ef-fects of accretion. Here we present the main results ofthe conformally-separable model; a more complete reviewcan be found in Refs. [21–23] (or, in the Boyer-Lindquistform used here, in Ref. [17]).Consider a rotating, accreting black hole with externalmass M . For an ideal, rotating Kerr black hole, threeassumptions hold true: the black hole is axisymmet-ric, the spacetime is stationary, and its Hamilton-Jacobiequations are separable [30]. The conformally-separablemodel presented below was developed in an attempt tofind the most general metric that still satisfies these con-ditions. To allow for the inclusion of accreting matter orradiation, however, the conditions required slight modi-fication. In particular, instead of strict stationarity, theassumption of conformal stationarity adopted here im-plies that the spacetime expands in a self-similar fashionwith time at an asymptotically small rate (this rate isthe accretion rate v that is taken to be asymptoticallyclose to zero in Eq. (39)) [22]. Such a condition may notapply at the onset of a gravitational collapse when theaccretion is supplied by the bulk of the collapsing mat-ter, but that collapse occurs within a small proper time,and at late times, a black hole will only grow at a rateon the order of its light crossing time divided by the ageof the Universe, a very small number. However, it willstill accrete, so the assumptions of isolation and Pricetail decay from models with a null weak singularity atthe inner horizon will not apply.It should be noted that strictly speaking, there is noinner horizon in the conformally-separable model (norin the inflationary Kasner model), since mass inflation near that region of spacetime will give way to collapse.When we refer to the inner horizon, we thus mean theregion of spacetime within the black hole asymptoti-cally close to the dimensionless Boyer-Lindquist radius r − ≡ − √ − a , in which crossing streams focus alongthe principal null directions and cause inflation and col-lapse. Also, strictly speaking, the conformally-separablemodel does not hold for extremal black holes, for which∆ ′ defined in Eq. (36) is zero. However, this shouldnot be too worrisome, since astronomically realistic blackholes are expected to have spins no higher than theThorne limit [31].Under the assumptions of conformal stationarity, axialsymmetry, and conformal separability, the conformally-separable line element takes the form ds = ρ s e vt − ξ ) (cid:18) dr ( r + a ) e ξ ∆ r + sin θ ∆ θ dθ + − e ξ ∆ r ( dt − ω θ dφ ) + ∆ θ ( dφ − ω r dt ) (1 − ω r ω θ ) (cid:19) (31)([17]), where x µ = { r, t, θ, φ } are dimensionless Boyer-Lindquist coordinates (the radial coordinate is writtenfirst to emphasize that r is timelike within the outer hori-zon). The function ∆ r is the horizon function, whose ze-ros define the location of the geometry’s horizons, and ∆ θ is the polar function, whose zeros define the location ofthe north and south poles. Additionally, ω r is the angularvelocity of the principal frame through the coordinates,and ω θ is the specific angular momentum of principal nullcongruence photons. The r and θ subscripts denote func-tions of only r and θ , respectively, and ρ s is the separablepart of the conformal factor. Eq. (31) reduces to the fa-miliar Kerr line element when the following definitionsare made:∆ r = r − r + a ( r + a ) , ∆ θ = sin θ, (32a) ω r = ar + a , ω θ = a sin θ, (32b) ρ s = M p r + a cos θ, (32c) ξ = v = 0 , (32d) { r, t, θ, φ } = (cid:26) r BL M , t BL M , θ BL , φ BL (cid:27) , (32e)where M is the black hole’s external mass, a ≡ J/M is the black hole’s dimensionless spin parameter, and { r BL , t BL , θ BL , φ BL } are the standard (dimensionful)Boyer-Lindquist coordinates.If the vacuum Kerr form of Eq. (31) is generalized toinclude the effects of accretion, the solution to Einstein’sequations sourced by ingoing and outgoing collisionlessnull streams implies that three of the above definitionsin Eqs. (32) are amended:(1) The dimensionless factor v becomes an arbitraryfree parameter, which can be interpreted (with the propergauge choice) as the black hole’s net accretion rate ˙ M ,or equivalently, the difference in the flux of outgoing andingoing streams near the inner horizon. This factor canbe treated as very small and reduces to zero for equalstreams of ingoing and outgoing radiation.(2) The inflationary exponent ξ , which measures thedegree to which the geometry has undergone self-similarcompression, changes with the radius and accretion pa-rameters, behaving like a step function near the innerhorizon as inflation is ignited.(3) The horizon function ∆ r strays from its Kerr valuenear the inner horizon, “freezing out” at a small, negativevalue during collapse instead of reaching zero at r = r − .In the conformally-separable solution, ξ and ∆ r aregoverned by the highly nonlinear pair of relations inEq. 88 of Ref. [22] (where x = a cot − (cid:0) ra (cid:1) , y = − cos θ ,and ∆ x = e ξ ∆ r ). To simplify their behavior, it sufficesto assume their Kerr values (Eqs. (32a) and (32d)) for allportions of spacetime except just above the inner horizon.In the regime near the inner horizon, ξ rapidly increasesfrom zero as r remains frozen at its inner horizon valueof r − , and the equations governing the evolution of ξ and∆ r simplify toe ξ = (cid:18) ( U r + v )( U r − v )( u + v )( u − v ) (cid:19) / , (33)∆ r = ∆ (cid:18) ( U r + v )( u − v )( U r − v )( u + v ) (cid:19) ∆ ′ / (4 v ) , (34)where ∆ is a constant of integration equal to the lin-ear extrapolation of ∆ r evaluated away from the in-ner horizon when ξ = 0, U r = u , and ∆ r still equals itsKerr value. The dimensionless parameter u , the counter-streaming velocity, represents the average of the initialaccretion rates from the two streams. The accretion pa-rameters satisfy 0 < v < u ≪
1, and the outgoing andingoing accretion rates are proportional to u ± v . Thefunction U r is defined by U r ≡ dξdr ( r + a ) e ξ ∆ r . (35)In addition, the constant ∆ ′ in Eq. (34) is proportionalto the radial derivative of the Kerr horizon function eval-uated at the inner horizon:∆ ′ ≡ − d ∆ r dr (cid:12)(cid:12)(cid:12)(cid:12) r − ( r − + a )= 2 (cid:0) r − − r − + a r − + a (cid:1) ( r − + a ) . (36)The conformally-separable Kerr model predicts thatthe geometry of the inner horizon will be divided intothree distinct epochs, as shown in Fig. 3. The parame-ter ξ represents the timelike coordinate separating theseepochs, just as T does for the inflationary Kasner model(in fact, it will be shown later, Eqs. (40), that the iden-tification T ∝ e − ξ generally holds).Initially, the geometry resembles the Kerr vacuumwhen ξ is negligibly small and r is just above its inner - - - ξ T / T Kerr inflation collapse T Φ U r - Δ r FIG. 3. (Color online). Evolution of quantities in theconformally-separable Kerr model. Plotted are the tetrad-frame energy-momentum component T from Eq. (38)(black), the corresponding inflationary Kasner energy-momentum Φ (the v → T ) (gray dashed), the parameter U r from Eq. (33) (red),and the magnitude of the horizon function ∆ r from Eq. (34)(blue). The parameters of this model have been chosen toavoid numerical overflow while still allowing the solution tocapture the full behavior; in particular, u = 0 . v = 0 . a = 0 .
96. The difference in the appearance of Φ here vis-`a-vis Fig. 2 is solely due to a difference in the scaling of theaxes. horizon value. Then, as r approaches r − , the mass infla-tion epoch begins as the locally-measured radial energy-momentum T of the streams rapidly inflates (along withthe internal mass parameter and the Weyl curvature).The horizon function dominates the geometry’s evolutionduring this epoch as ∆ r deflates toward 0 − . Through-out the inflation and collapse epochs, r remains approx-imately frozen at its inner horizon value of r − .Finally, inflation is slowed once ξ grows large enoughand begins to dominate, causing a self-similar collapseof the geometry. During the collapse epoch, the curva-ture and T once again begin to diverge, while the hori-zon function freezes out at an exponentially small value.The collapse epoch then continues until Eqs. (33) and(34) are no longer valid because of the increasing angu-lar motion of the streams. However, it is possible thatthe conformally-separable solution will break down re-gardless after this point, once the curvature exceeds thePlanck scale.The connections between the conformally-separableKerr model and the inflationary Kasner model becomeevident when considering the energy-momentum tensorseen in a tetrad frame. A natural tetrad frame to chooseis the one encoded by the line element in Eq. (31), with1-forms e µ dx µ = ρ s e vt − ξ/ ( r + a ) √− ∆ r dr, (37a) e µ dx µ = ρ s e vt + ξ/ √− ∆ r − ω r ω θ ( dt − ω θ dφ ) , (37b) e µ dx µ = ρ s e vt − ξ sin θ √ ∆ θ dθ, (37c) e µ dx µ = ρ s e vt − ξ √ ∆ θ − ω r ω θ ( dφ − ω r dt ) . (37d)In the Kerr limit, the tetrad frame in Eqs. (37) reducesto the interior Carter frame, in which observers at restsee the principal null directions as purely radial (in the x -direction) as the frame follows them freely falling androtating inward. The interior Carter frame differs fromthe standard (exterior) Carter frame only in the swappingof e µ ↔ e µ and √− ∆ r ↔ √ +∆ r , since below the outerhorizon, r becomes timelike and ∆ r becomes negative.In this tetrad frame, Einstein’s equations yield thefollowing non-negligible components of the energy-momentum tensor seen by a Carter observer: T = T = U r ∆ ′ − v πρ s e vt + ξ ( − ∆ r ) (38)(cf. Eqs. 125-128 in Ref. [22]). These components, whichrapidly diverge during inflation and collapse (see Fig. 3),represent the net combination of the energy-momenta ofingoing and outgoing collisionless streams observed in theradial direction. Their behavior is dominated by the van-ishing of ∆ r during inflation and by the conformal piecee − ξ once the horizon function freezes out during collapse.In terms of the counter-streaming velocity u , a Taylor ex-pansion for small v yields T = T ≈ e ξ u ∆ ′ πρ s ( − ∆ ) e ∆ ′ u ( − e − ξ ) + O ( v ) . (39)The radial energy-momentum thus grows as ∼ e /u , sothat, perhaps counterintuitively, the smaller the value of u , the more rapid the inflation. For astronomically real-istic black holes, the above expansion is generally valid,since v scales as the black hole light crossing time t BH divided by the accretion (mass-doubling) time t acc , andfor most of the lifetime of the black hole, t acc ≫ t BH [20].When comparing the energy-momentum tensor of theconformally-separable Kerr metric in Eq. (39) with theenergy-momentum tensor of the inflationary Kasner met-ric in Eq. (7), the two are equivalent in the limit v →
0, when the following definitions are made: T = T e − ξ , (40a) T = ∆ ′ u , (40b) Φ = u ∆ ′ πρ − ( − ∆ ) , (40c)where ρ − is the value of the separable conformal factor ρ s at the inner horizon.Thus, the inflationary Kasner solution provides asimple yet precise approximation of the conformally-separable Kerr spacetime seen in the tetrad rest frameof a Carter observer, through the matching of Eqs. (40).The conformally-separable solution, in turn, provides anapproximation of the geometry of a rotating, accretingblack hole, which reduces to the inflationary Kasner so-lution near the inner horizon in the limit of an asymp-totically small accretion rate v . T and T remain the only non-negligible componentsof T ˆ m ˆ n through inflation and collapse, and the collapseepoch of the conformally-separable solution is defined toend when other components of T ˆ m ˆ n (namely, T and T ), which initially diverge at a much slower rate, be-come comparable in magnitude to the radial components.This occurs at ξ = ∆ ′ / (6 u ) − ln √− ∆ (or equivalently,at T /T = − ∆ e − ∆ ′ / (3 u ) ≈ − in Fig. 2), and beyondthis point, the approximations of Eqs. (33), (34), and(38) are no longer valid. The classical solution can becontinued numerically for higher ξ , yielding a series ofeven more complex Kasner epochs and BKL bounces, al-though an extension of the classical solution may fail ifquantum effects become important once the curvaturepasses the Planck scale [17]. B. Null geodesic behavior
What will an observer in the inflationary Kasner space-time see? To answer this question, consider a Carter ob-server (at rest in the tetrad of Eqs. (37)) falling into arotating black hole from rest at infinity and approachingthe inner horizon.The Kerr metric provides an excellent approximationof a rotating black hole’s geometry far above the innerhorizon, so the Kerr null geodesic equations will providethe trajectory of a photon in this regime. However, oncethe observer approaches the inner horizon, streams ofingoing and outgoing matter will focus along the radialdirections in the Carter tetrad and will begin to inflate,causing the geometry to be better approximated by theinflationary Kasner metric. Thus, here we find the equa-tions for null geodesics in the inflationary Kasner space-time and then connect them to null geodesics in the Kerrspacetime in a regime near the inner horizon where bothare valid.To find null geodesic trajectories in the inflationaryKasner spacetime, note that because the metric is ho-0mogeneous, there are three conserved quantities corre-sponding to each of the spatial coordinates x , y , and z .These momenta are simply the covariant forms of thespatial components of a photon’s coordinate-frame four-momentum, k i = g iµ dx µ dλ , where i ∈ { x, y, z } . (41)When Eqs. (41) are combined with the condition k µ k µ = 0, the four components of the four-momentumcan be expressed in terms of the coordinate time T andconserved quantities k x , k y , and k z . In the tetrad frameof Eq. (5), these components take the form( k IK ) = − s k x a + k y + k z a , (42a)( k IK ) = k x a , (42b)( k IK ) = k y a , (42c)( k IK ) = k z a , (42d)where the subscript IK denotes quantities valid in theinflationary Kasner regime. The negative sign for ( k IK ) is chosen so that the affine parameter increases as T de-creases from T just above the inner horizon to 0 at theinflationary Kasner singularity.In the coordinate frame, Eqs. (42) lead to the followingequations of motion that can be integrated: dTdλ = − E a s(cid:18) a obs1 a (cid:19) cos χ + (cid:18) a obs2 a (cid:19) sin χ, (43a) dxdλ = − E a a obs1 a cos χ, (43b) dydλ = − E a a obs2 a sin χ cos ψ, (43c) dzdλ = − E a a obs2 a sin χ sin ψ, (43d)where a obs i is the value of a i at the observer’s position,and the constants of motion k i have been replaced by theobserver’s viewing angles χ ∈ [0 , π ] and ψ ∈ [0 , π ) (andthe normalization factor E ), which indicate the positionof the photon in the observer’s field of view. More detailsabout the definitions of these angles and their relationsto other quantities used throughout the paper are givenin the Appendix. The important point to note here isthat χ = 0 ◦ corresponds to an ingoing photon reaching anobserver looking in the principal null direction away fromthe black hole, and χ = 180 ◦ corresponds to an observerlooking directly toward the black hole in the principalnull direction.The evolution of null geodesics seen by an observerat T obs looking in different directions is shown in Fig. 4 during both the inflation and collapse epochs. In theseplots, the positive x -direction is aligned with the prin-cipal null direction away from the black hole. Since theinflationary Kasner metric is isotropic in the y - z plane,the dependence on the viewing angle ψ is trivial—thegeodesics of Fig. 4 can be revolved around the x -axis in3D space to obtain solutions with different values of ψ .The inflation epoch is characterized by the focusingof null geodesics along the principal null directions. Anobserver in the inflation epoch (upper panel of Fig. 4)will thus see both ingoing and outgoing null geodesicsthat have begun to align along the x -axis, so that anincreasingly large portion of the observer’s sky is takenup by a narrowing band of the inflationary Kasner back-ground orthogonal to the principal null axis. The sameinflation power law behavior from the upper panel ofFig. 4 is also seen in the lower panel, in which the pho-tons undergo both inflation and collapse. These pho-tons all start at T = T , corresponding to x = ± T forall but the χ = 90 ◦ geodesic (which begins at approxi-mately p y + z ∝ ( T /T obs ) / , far outside the range ofthis plot). The photons in this plot begin by proceedinginward toward the origin, curving toward the x -axis asthey undergo inflation. Then, once the photons reach thecollapse epoch, they turn sharply, orthogonal to the x -axis, until they reach the observer at the origin. As theobserver continues farther into the collapse epoch, theturns sharpen even more, and the locations of the turnsspread out farther in the y − z plane as most of the back-ground radiation from T reaching the observer becomessqueezed into a band around χ = 90 ◦ . Once the observerhas reached the singularity at T = 0 in the Carter tetradframe, the entire inflationary Kasner background in theobserver’s field of view will be squashed into the ring at χ = 90 ◦ , and photons arriving at any other position inthe sky must have originated from a vanishingly smallpatch of the background along one of the principal nulldirections.From the behavior of the null geodesics in Fig. 4, onemust be careful not to jump too quickly to any con-clusions about what an observer near the inner horizonwould see, especially since, as we shall see, most of thephotons arriving at an observer deep in the collapse epochtend to align almost exactly with part of the boundary ofthe black hole’s shadow. To be certain about each pho-ton’s complete path, we must continue the ray-tracingbackwards beyond T to r ≫ r − , where only the Kerrsolution is valid.In the Kerr spacetime, any geodesic is characterized bythree conserved quantities: the energy E , angular mo-mentum L , and Carter constant K , defined by E ≡ − k t , L ≡ k φ , K ≡ k θ + ( k φ + ω θ k t ) ∆ θ , (44)where k t , k φ , and k θ are the covariant components ofa photon’s Kerr coordinate-frame four-momentum. Justas with the inflationary Kasner metric, these conservedquantities lead in a straightforward way to the following1 - x y + z T o(cid:0)(cid:1) = T χ= ° χ= (cid:7)(cid:8)(cid:9) ° χ= ° χ= ° χ= ° - - y + z T (cid:12)(cid:13)(cid:14) = - T χ= (cid:15)(cid:16)(cid:17) °χ= (cid:18)(cid:19)(cid:20) ° χ= ° χ= (cid:21)(cid:22) °χ= ° FIG. 4. Null geodesics seen by an observer at the origin in the inflationary Kasner spacetime. Geodesics are parametrized bythe observer’s viewing angle χ at an equal spacing of 15 ◦ , from an observer looking directly outward ( χ = 0 ◦ ) to one lookingdirectly inward toward the black hole’s center ( χ = 180 ◦ ). All geodesics end at the origin at T = T obs and are ray-tracedbackwards via Eqs. (43) to T = T . The upper plot shows an observer at T obs = 0 . T near the end of the inflation epoch,and the lower plot shows an observer at T obs = 10 − T near the end of the collapse epoch. The parameters chosen here are T = 9 .
09 and Φ = 0 . four-momentum components in the Carter tetrad frame,defined by ( k K ) ˆ m ≡ e ˆ mµ ( dx µ /dλ ):( k K ) = ± ρ s s K − ( ω r L − E ) ∆ r , (45a)( k K ) = ω r L − Eρ s √− ∆ r , (45b)( k K ) = ± ρ s s K − ( L − ω θ E ) ∆ θ , (45c)( k K ) = L − ω θ Eρ s √ ∆ θ , (45d)where the subscript K indicates quantities valid in theKerr regime.In the coordinate frame, Eqs. (45) lead to the following equations of motion: drdλ ′ = ± p R ( r ) , (46a) dtdλ ′ = E − ω r L ∆ r + L − ω θ E ∆ θ ω θ , (46b) dθdλ ′ = ± p Θ( θ ) , (46c) dφdλ ′ = E − ω r L ∆ r ω r + L − ω θ E ∆ θ , (46d)written in terms of the Mino time dλ ′ ≡ dλ/ρ s and theeffective potentials R ( r ) ≡ (cid:0) ( ω r L − E ) − K ∆ r (cid:1) ( r + a ) , (47a)Θ( θ ) ≡ (cid:0) K ∆ θ − ( L − ω θ E ) (cid:1) csc θ. (47b)Below the outer horizon, ( k K ) must be negative, sincethe radial coordinate is timelike and decreases as theaffine parameter increases, so that all geodesics are nec-essarily infalling. Thus, only the lower sign for Eq. (45a)and Eq. (46a) is valid below the outer horizon. How-ever, ( k K ) may be positive or negative in this regime2depending on the relative magnitudes of L and E , and ageodesic with positive (negative) ( k K ) is said to be out-going (ingoing). Additionally, a positive (negative) signfor Eq. (45c) and Eq. (46c) corresponds to a geodesicwhose polar angle θ increases (decreases) as the affineparameter increases.The Kerr and inflationary Kasner metrics are bothvalid in a small domain just above the inner horizon, andwe choose to match their null geodesics at the Boyer-Lindquist radius r and corresponding Kasner time T .The exact value of these parameters is not too important;the results of matching the null geodesics are robust fora range of values as long as T is close enough to T that∆ r is well-approximated by the Kerr horizon function butfar enough into the inflation epoch that r has frozen outand the streams have begun to focus along the principalnull directions, so that the inflationary Kasner solutionis valid. Practically, for the parameters used in the plotsthroughout this paper, we choose to match geodesics at r = 0 .
73 (with the inner horizon at r − = 0 . T ≈ . T .The assumption ( k IK ) ˆ m | T = T = ( k K ) ˆ m | r = r , matchingEqs. (42) and (45), leads to a direct mapping betweenthe orbital parameters ( k x , k y , k z ) and ( E, L, K ): E = ρ s − ω r ω θ (cid:18) k z ω r √ ∆ θ a − k x √− ∆ r a (cid:19) , (48a) L = ρ s − ω r ω θ (cid:18) k z √ ∆ θ a − k x ω θ √− ∆ r a (cid:19) , (48b) K = ρ s a (cid:0) k y + k z (cid:1) , (48c)where the functions ∆ r , ∆ θ , ω r , ω θ , ρ s , a , and a are allevaluated at the point of matching just above the innerhorizon, where T = T , r = r , and θ = θ . Additionally,in order to obtain the complete Kerr solution, the propersigns must be specified. With reference to Eqs. (46), onemust require: sgn (cid:16) ± p R ( r ) (cid:17) = − , (49a)sgn (cid:16) ± p Θ( θ ) (cid:17) = sgn ( k y ) . (49b)With this matching, it is then possible to continue theinflationary Kasner geodesics of Fig. 4 to their points oforigin in the Kerr spacetime. Here we consider two do-mains for the points of origin of Kerr photons: the firstsource is the fixed background of stars, galaxies, and ra-diation travelling inward from infinity, and the secondsource is the collapsing surface of the star that formedthe black hole, emitting radiation outward. By the timephotons from the latter source reach the observer, theywill be so redshifted and dimmed that the star’s surfacewill be practically imperceptible, so any part of the ob-server’s sky consisting solely of photons from this sourcewill form the black hole’s shadow. As an example, theschematic Penrose diagram of Fig. 1 shows the paths of ingoing and outgoing photons from both of these sourcesreaching an observer near the inner horizon at point O.It may seem counterintuitive that outward-directedphotons from the collapsing star’s surface near r ≈ r + could reach an observer near r − . The paths of thesephotons fall under two general cases: if the photons wereemitted during the collapse just before the formation ofthe event horizon (at r > r + ), they may reach a turningpoint below the photon sphere and travel inward untilreaching the observer. Alternatively, if they were emit-ted below the event horizon (at r < r + ), they can remainoutgoing as their Boyer-Lindquist radius decreases, untilthey are detected by an observer looking inward.Some examples of photon paths reaching observersnear the inner horizon are shown in Fig. 5. To avoidthe effects of any coordinate singularities at the horizons,the paths are plotted using Doran coordinates, which arerelated to the Boyer-Lindquist coordinates by the trans-formations r D = r BL , (50a) dt D = dt BL + p M r ( r + a ) r + a − M r dr BL , (50b) θ D = θ BL , (50c) dφ D = dφ BL + a p M r/ ( r + a ) r + a − M r dr BL (50d)[32]. We limit our analysis to two equatorial observers,one in the inflation epoch ( T obs = 0 . T ) and one deepinto the collapse epoch ( T obs = 10 − T ); a more completeanalysis of which photons arrive from which sources fordifferent observer latitudes and radii is given in Sec. IV.The two left panels of Fig. 5 show null geodesics in theequatorial plane, reaching an observer at ( x, y ) ≈ (1 . , ψ = 270 ◦ —oncethe inflationary Kasner geodesics have been traced backfrom the observer to the point of matching at T = T ,here they are continued backward in the Kerr metric totheir point of origin at infinity (blue) or the outer hori-zon (red). As χ increases, the geodesics become more andmore skewed until they asymptotically wrap around thephoton sphere given by the dashed curve. The χ = 180 ◦ geodesic is omitted from the top left panel for simplicity;its form is identical to the χ = 180 ◦ geodesic in the lowerleft panel.The behavior of the geodesics in the left two panels ofFig. 5 matches that of Fig. 4. In particular, when the ob-server has progressed deep into the collapse epoch (when T obs ≪ T ), most light tends to focus along the princi-pal null directions, so that most of the observer’s fieldof view contains light originating from a small patch ofthe background (when χ ≤ ◦ ) or illusory horizon (when χ ≥ ◦ ). In the collapse epoch, therefore, the observersees most of the background sky squashed into a thinband close to χ = 90 ◦ .The right panels of Fig. 5 show geodesics for a fixedvalue of χ instead of ψ . Here, the observer is looking up3 - - - - x y T (cid:23)(cid:24)(cid:25) = (cid:26)(cid:27)(cid:28)(cid:29) T , ψ = ° χ = ° χ = ! ° χ = " ° χ = ° χ = $%& ° χ = ’() ° - - - - x * T +-. = - / T ψ = ° χ≤ :; ° χ = ° χ≥ <=> ° FIG. 5. (Color online). Null geodesics ray-traced backwards from an equatorial inflationary Kasner observer at T obs = 0 . T (top two panels) and T obs = 10 − T (bottom two panels) to their Kerr origins. The left panels show a slice of the equatorialplane with Doran azimuthal coordinates, viewed from over the pole, and the right panels show a polar slice in co-rotatingcoordinates. In all panels, the two thin solid black curves shows the locations of the inner and outer horizons, and the dashedcurves show the location of the null circular prograde equatorial (left) and polar (right) orbits. All geodesics are labelledby the viewing angle of the inflationary Kasner observer, equally spaced at intervals of 15 ◦ , and they originate either fromthe background at infinity (dark blue) or from the surface of the collapsing star (dark red). The parameters chosen here are u = 0 . r = 0 . θ = 90 ◦ , and a = 0 . and down instead of only looking within the equatorialplane. With this polar view, some geodesics ( ψ = 195 ◦ to ψ = 270 ◦ ) originate from infinity, but the others ( ψ = 90 ◦ to ψ = 180 ◦ ) originate at some arbitrary location belowthe outer horizon, where the collapsing star’s surface ex-isted at some point in the past. Though it may not beapparent from this view, these geodesics become increas- ingly skewed in the direction of the black hole’s rotationas ψ decreases, with the equatorial geodesic with ψ = 90 ◦ occupying a single point in the polar view. Additionally,note that the geodesics in this right panel can be reflectedacross the z = 0 line to obtain the geodesics for ψ < ◦ and ψ > ◦ .The polar null geodesics in the right two panels of4Fig. 5 remain unchanged for an observer travelling frominflation to collapse, a consequence of the fact that theinflationary Kasner metric is isotropic in the y - z plane,so that the dependence on ψ in this case is trivial. Thus,an infalling equatorial Carter observer will see more andmore of the sky flattening out and piling up toward theedges of the black hole’s shadow, though the view at dif-ferent altitudes will remain relatively unaffected by theinflationary Kasner metric. IV. THE CARTER OBSERVER’S EXPERIENCE
As a brief caveat, it should be noted that the observer’sfield of view and the angles ( χ, ψ ) defined in this paperare completely dependent on the choice of tetrad frame.The interior Carter tetrad is adopted in this paper be-cause of its simplicity and natural alignment with theprincipal null directions, but it is only a valid inertial restframe for a free-falling observer below the outer horizon.In particular, an observer of mass m at rest in the Carterframe must have orbital parameters E = 0, L = 0, and K = ( ma cos θ ) (where E , L , and K are defined analo-gously to Eq. (44) but for a timelike geodesic). Neverthe-less, a free-falling observer can decelerate to E = 0 oncethey have passed through the outer horizon in order tostay at rest in the Carter frame and reproduce the resultsfound here.With that caveat out of the way, consider the completefield of view of a Carter observer during their descent intoa black hole. The relevant object of analysis here is theblack hole’s shadow, the portion of the observer’s sky voidof any background photons. The perceived boundary be-tween the black hole’s shadow and the sky is determinedby the location of the photon sphere, where photons cir-culate on a null, circular orbit for an indefinitely longamount of time before peeling off and reaching the ob-server. The orbital parameters of these photons (and thecorresponding viewing angles) are given by the solutionsto the equations R ( r ) = 0 , dR ( r ) dr = 0 , (51)parametrized by the allowed prograde ( − ) and retrograde(+) photon orbital radii, whose extremes are given by r c = 2 M (cid:18) (cid:18)
23 cos − ( ± a ) (cid:19)(cid:19) (52)[33]. Though we have been working with dimensionlessBoyer-Lindquist coordinates, in this section we restorefactors of M to connect our equations to physical quan-tities.The black hole’s shadow is shown in Fig. 6 for an equa-torial observer at rest in the Carter frame at various radiiand inflationary Kasner times. The observer’s sky is dis-played with a Mollweide projection, where the center cor-responds to the observer’s view directly ahead toward the black hole at χ = 180 ◦ , the leftmost and rightmost pointscorrespond to the view directly behind the observer at χ = 0 ◦ , and the top and bottom points correspond tothe view directly above ( χ = 90 ◦ , ψ = 270 ◦ ) and below( χ = 90 ◦ , ψ = 90 ◦ ) the observer, respectively. More de-tails about the projection are given in the Appendix.The black hole’s spin axis is pointed to the right, so thatthe flow of spacetime is towards the observer above theshadow and away from the observer below the shadow.The progression of images in Fig. 6 from top left to bot-tom right shows the view of the black hole as a Carterobserver gets progressively closer to the inner horizon.Far from the black hole, the characteristic asymmetricalsilhouette is seen in the top left image, with the back-ground sky slightly blueshifted and the collapsing star’ssurface extremely redshifted (the color in these images iscalculated from the energy component k of the photon’sfour-momentum, normalized to k at its point of origin).Then, as the observer approaches the outer horizon at r + = 1 . M in the (non-inertial) exterior Carter frame,the shadow takes up more and more of the observer’sview until the entire background sky is reduced to a sin-gle point behind the observer at the outer horizon. Then,as the Carter frame continues inward, the background skybehind the observer begins to grow again, until it takesup a little less than half the field of view once the observerreaches near the inner horizon (here r − = 0 . M ).As detailed in the Appendix, the field of view in Fig. 6changes orientation between the exterior to the interiorof the black hole. For r > . M , the black hole is infront of the observer and the sky is behind the observer,but for r < . M , we choose the black hole to be belowthe observer and the sky to be above, just as it is for thefamiliar case of an observer on the surface of Earth.How does the inflationary Kasner solution modify theobserver’s view as they approach the inner horizon? Thebottom two rows of Fig. 6 show a Carter observer’s viewin the inflation and collapse epochs. As inflation pro-gresses, the black hole’s shadow takes up approximatelyhalf of the equatorial observer’s field of view, and the skybecomes more and more blueshifted. Then, as the ob-server continues into the collapse epoch, the black hole’sshadow changes orientation until it appears as an infi-nite plane below the observer, taking up half of the fieldof view (in comparison, at a Schwarzschild singularity,an observer in free-fall also sees the shadow take up ex-actly half the field of view). Most of the sky above be-comes squashed into a narrow band around χ = 90 ◦ (thehorizontal midline in these images) as the observer ap-proaches the inflationary Kasner singularity, as shown inthe previous section. The validity of these images canbe at least partially verified by comparing the points oforigin of the geodesics of Fig. 5 in conjunction with theirlocation in the images of Fig. 6 for T obs = 0 . T and T obs = 10 − T . To find the location of each ( χ, ψ ) pointon the images of Fig. 6, refer to Fig. 9 in the Appendix.What about observers outside of the equatorial plane?The final shape of the black hole’s shadow depends on5 ω obs / ω em FIG. 6. (Color online). Mollweide projection of the full field of view of an infalling Carter observer in the equatorial plane atvarious radii and inflationary Kasner times recorded above each image. The black hole silhouette (black curve) separates the(generally) blueshifted photons sourced from r → ∞ from the extremely redshifted photons sourced from r ≈ r + . The colorrepresents the degree of redshift/blueshift. Note the change in the observer’s orientation between the exterior ( r > . M ) andinterior ( r < . M ) regions. The parameters used here are u = 0 . r = 0 . θ = 90 ◦ , T ≈ . T and a = 0 . T ?@A = - B T FIG. 7. (Color online). Black hole silhouettes for an infla-tionary Kasner observer near the end of the inflation (upperpanel) and collapse (lower panel) epochs, at a Boyer-Lindquistlatitude ranging from the equator at θ obs = 90 ◦ (blue) to thepole at θ obs = 0 ◦ (red). The projection is the same as that ofFigs. 6 and 9b, and the parameters chosen here are u = 0 . r = 0 .
73, and a = 0 . the Boyer-Lindquist latitude of the observer, as shown inFig. 7. Near the end of the inflation epoch (top image), inthe black hole’s equatorial plane ( θ obs = 90 ◦ ), the blackhole takes up a little more than half of the observer’s fieldof view. But at higher latitudes ( θ obs < ◦ ), the shadowtakes up more and more of the field of view, so that thesky in front of the observer appears as a thinner and thin-ner band connecting the principal null directions. Then,above some critical latitude, all photons must be ingoing,so that the shadow takes up the entire field of view at theend of the inflation epoch. An observer approaching theinner horizon at these latitudes close to the pole will seethe sky constrict to a single point directly behind them.However, deep into the collapse epoch (bottom image),regardless of whether the observer is above or below thecritical latitude, the black hole’s shadow will always takeup half the field of view below the observer, shifted 90 ◦ from its location during the inflation epoch.How much time passes for an observer experiencing theinflation and collapse of a black hole’s inner horizon ge-ometry? In the simplest case, for an equatorial observerof mass m at rest in the interior Carter tetrad frame, theproper time that passes from the point of matching at T = T to the singularity at T = 0 is given by τ = − M Z T e ( T − T ) / √ πΦ T (cid:18) TT (cid:19) − / dT. (53)For the parameters used in Fig. 2, the proper time expe-rienced by the observer is approximately τ ≈ (cid:18) MM ⊙ (cid:19) − seconds , (54)where M/M ⊙ is the mass of the black hole in units ofsolar masses. This proper time only changes by an orderof magnitude or two at most across the physically validdomains of a , θ , r , and u . In particular, the integral inEq. (53) approaches a constant value in the limit of anasymptotically small initial counter-streaming velocity u .However, in the same limit, the total time spent just inthe collapse epoch ( T < /
2) becomes exponentially tiny(for the parameters used in Fig. 2 the time spent in thecollapse epoch is already less than 1% of the time spentin the inflation epoch).As a final note, the inflationary Kasner proper timecalculated above is about an order of magnitude smallerthan the proper time experienced by an equivalent ob-server in the Kerr spacetime travelling from the point ofmatching ( r = r ) to the inner horizon ( r = r − ). V. CONCLUSIONS
The general classical outcome of the effect of accretedmatter and radiation on a rotating black hole is the in-flation and subsequent collapse of the spacetime near theinner horizon into a spacelike, BKL-like singularity. Herewe have developed a simplified model that connects thiscollapsing geometry near the inner horizon to the Kerrexterior. The model, which we have called the inflation-ary Kasner model, is derived under the assumption thatstreams of matter near the inner horizon focus alongthe principal null directions at ultrarelativistic speeds,so that the Einstein tensor in the Carter frame approxi-mately corresponds to that of a null, perfect fluid stream-ing at equal rates along the x -direction. Such an assump-tion leads to a Kasner-like form with two epochs, onecorresponding to a purely radial collapse with Kasner ex-ponents (1 , , − , , ). The end result of the model is the terminationof geodesics at a spacelike singularity at T = 0; notably,the inner horizon and all the additional structure beyondit never get the chance to form.We have verified the applicability of the inflation-ary Kasner metric to the near-inner horizon geome-try of rotating, accreting black holes through compar-ison to a previously-derived solution, the conformally-separable Kerr metric. This solution comes equippedwith a natural connection to the Kerr metric, along witha continuous evolution through the inflation and col-lapse epochs (and beyond, as has been shown compu-tationally, through several BKL bounces [17]). In the7limit of asymptotically small accretion rates ( v → ◦ from its position at the end ofthe classical Poisson-Israel mass inflation epoch, and un-like that latter case, the view is independent of the ob-server’s latitude. Once the collapse epoch has proceededlong enough, the curvature will have diverged to such alarge extent that the classical solution will surely breakdown. A calculation of the quantum back reaction willthus be necessary if one wishes to explore the spacetimeevolution past this point in order to determine the fi-nal outcome of the collapse. The inflationary Kasnermetric will hopefully provide a simpler basis for quan-tum calculations than more complicated models like theconformally-separable solution. Appendix
In this appendix we define and elaborate on the us-age of the observer’s viewing angles χ and ψ employedthroughout the paper. To determine the path of a nullgeodesic in a 3+1D spacetime uniquely, one needs tospecify at most two constants of motion. For the infla-tionary Kasner metric, the spatial four-momentum com-ponents k x , k y , and k z uniquely label a null geodesic,but there is an extra degree of freedom associated withan arbitrary normalization factor for the four momen-tum’s magnitude. Thus, we transform to a new set ofconstants that represents the celestial coordinates for anobserver in the inflationary Kasner tetrad frame. In par-ticular, the viewing angle χ ∈ [0 , π ] is defined to be theangle between the x -axis and − k IK (negative since theobserver is seeing the photon reach the origin of theirframe of reference), and ψ ∈ [0 , π ) is the angle betweenthe x -axis and the projection of − k IK onto the x - x plane. These viewing angles are shown in Fig. 8.The definitions of the observer’s viewing angles andtheir relation to the spatial covariant momenta viaEqs. (42) are given bytan χ ≡ q ( − k ) + ( − k ) − k = q k y + k z − k x a obs1 a obs2 , (A.1a)tan ψ ≡ − k − k = − k z − k y , (A.1b) or equivalently, k x = −E a obs1 cos χ, (A.2a) k y = −E a obs2 sin χ cos ψ, (A.2b) k z = −E a obs2 sin χ sin ψ, (A.2c)where a obs1 and a obs2 are the values of the scales factorsfrom Eq. (6) at time T = T obs , and E is some positivenormalization factor (the additional degree of freedommentioned earlier).Physically, the x -axis of the tetrad frame is parallelto the principal null directions of the black hole, and the x -axis points in the ˆ θ direction. When χ = 0 ◦ , the ob-server is looking along the positive x -axis, away fromthe black hole, at ingoing photons. When ψ = 0 ◦ and χ = 90 ◦ , the observer is looking straight down along thepositive x -axis, in the ˆ θ direction. Geodesics with con-stant Boyer-Lindquist latitude are then given by ψ = 90 ◦ and ψ = 270 ◦ .When matching the tetrad-frame four-momenta of theinflationary Kasner solution with the Kerr solution ata Boyer-Lindquist radius of r = r , one can find therelation between the observer’s viewing angles and theKerr orbital parameters defined by Eqs. (44). InvertingEqs. (48) and combining with Eqs. (A.1) yieldstan χ = p K ( − ∆ r ) E − ω r L a obs1 a obs2 , (A.3a)sin ψ = ω θ E − L p K ∆ θ . (A.3b)This relation holds for viewing angles defined for anobserver in the interior Carter frame, within the eventhorizon. Outside the event horizon, the exterior Carterobserver also possesses a set of viewing angles ( χ, ψ ), stilldefined by Fig. 8. However, those angles’ relations tothe Kerr orbital parameters will differ from the interior (cid:1) (cid:2) (cid:3) (cid:3) (cid:3) (cid:4) (cid:5) FIG. 8. Definition of the viewing angles χ and ψ with respectto the tetrad frame axes x , x , and x , for a photon withtetrad-frame four-momentum k ˆ m . (a)(b)FIG. 9. (Color online). Coordinate grid of the viewing angles χ and ψ on a Mollweide projection of the full field of view of anexterior (a) and interior (b) Carter observer. Lines of constant ψ are equally spaced at 15 ◦ intervals from ψ = 0 ◦ (red) to ψ = 360 ◦ (blue), and lines of constant χ are equally spacedat 15 ◦ intervals from χ = 0 ◦ (yellow) to χ = 180 ◦ (cyan). case, since the interior Carter frame differs from the ex-terior frame of Eqs. (37) in the swapping of e µ ↔ e µ and √− ∆ r ↔ √ +∆ r . In the Kerr exterior, the viewingangles are related to the Kerr parameters bysin χ = p K ∆ r E − ω r L , (A.4a)sin ψ = ω θ E − L p K ∆ θ . (A.4b) The interior versus exterior region also differs in howwe treat the Mollweide projections of the observer’s sky,with reference to Figs. 6 and 7. In the exterior regionoutside the event horizon, it is natural to choose the − x direction ( χ = 180 ◦ ) for the center of the projection, sinceit corresponds to the direction toward the center of theblack hole. However, in the interior region, it is morenatural to choose the − x direction ( χ = 90 ◦ , ψ = 270 ◦ )for the center of the projection, so that the black hole’sshadow occupies the lower half of the field of view andthe sky occupies the upper half, so as to coincide withour general notion of “uprightness” as we perceive of iton Earth.In terms of the Mollweide projection’s latitude ϕ ∈ [ − π , π ] and longitude λ ∈ [ − π, π ) for Figs. 6 and 7,defined by ϕ = sin − (cid:18) π (cid:16) y p − y + sin − y (cid:17)(cid:19) , (A.5a) λ = π x p − y , (A.5b)where x ∈ [ − ,
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