On the Volume Inside Old Black Holes

On the volume inside old black holes

Marios Christodoulou∗ and Tommaso De Lorenzo†

Aix Marseille Univ, Universit´ede Toulon, CNRS, CPT, Marseille, France

November 4, 2016

Abstract. Black holes that have nearly evaporated are often thought of as small objects, due to their tiny exterior area. However, the horizon bounds large spacelike hypersurfaces. A compelling geometric perspective on the evolution of the interior geometry was recently shown to be provided by a generally covariant definition of the volume inside a black hole using maximal surfaces. In this article, we expand on previous results and show that finding the maximal surfaces in an arbitrary spherically symmetric spacetime is equivalent to a 1+1 geodesic problem. We then study the effect of Hawking radiation on the volume by computing the volume of maximal surfaces inside the apparent horizon of an evaporating black hole as a function of time at infinity: while the area is shrinking, the volume of these surfaces grows monotonically with advanced time, up to when the horizon has reached Planckian dimensions. The physical relevance of these results for the information paradox and the remnant scenarios are discussed.

Introduction large volume start to develop. This characteristic is naturally captured by the manifestly coordinate Since the mid-1970s, the information-loss paradox independent definition of volume employing maxi- [1] has been at the center of a heated debate. The mal surfaces recently proposed by Carlo Rovelli to- fate of the large amount of information fallen in- gether with one of the authors in [10], where it was side the hole is the main topic of several resolution applied to the interior of static black holes 1. proposals in the literature (for a –non-exhaustive– For asymptotically flat geometries, this volume review see [2] and references therein). can be parametrized with the advanced Eddington- In the setting in which the semi-classical approx- Finkelstein time v and is denoted as V (v). In the imation behind Hawking’s computation remains interior of a static spherically symmetric black hole valid up to the very late stages of the evaporation, of mass m0 formed by collapsing matter, the vol- and quantum gravitational effects play an impor- ume grows monotonically with v and is given at tant role only in the strong curvature regime by late times v m0 by “smoothing-out” the singularity [3], a natural pos- 2 sible outcome is the formation of a remnant: a fi- V (v) C m v (1) ≈ 0 nal minuscule object that stores all the informa- where C = 3√3 π for the uncharged case 2. tion needed to purify the external mixed state [4,5] In this article, we expand upon the results in [10] (see [6] for a recent review). and show that the conclusions in that work extend arXiv:1604.07222v2 [gr-qc] 3 Nov 2016 The tiny mass and external size of such objects to the case of an evaporating black hole. The vol- are central to objections against both the existence ume of maximal surfaces bounded by the shrink- of remnants (infinite pair production–see [7] and ref- ing apparent horizon monotonically increases up to erences therein–) and their impossibility of storing when its area has reached Planckian dimensions. inside the large amount of information. The naive Specifically, we show that, at any time, there ex- intuition of “smallness”, however, can be very mis- ists a spacelike maximal surface with proper vol- leading since a remnant contains spatial hypersurfaces of very large volume, see for instance [8,9]. 1Other definitions for the volume have been proposed else- where [11–17]. 2 Once a horizon forms, surfaces of increasingly The Reissner-Nordströmspacetime, in which case C de- pends on the charge Q, was studied in the Appendix of [10] ∗[email protected] and similar results hold also for AdS black holes [18]. The †[email protected] Kerr case is considered in [19].

1 ume approximately given by (1) (where m0 is now functional: the initial mass), that connects the sphere of the Z 3p apparent horizon at that time to the center of the V [Σ] = dy det hab Σ collapsing object before the formation of the singu- Z 3 4 A B1/2 larity . The final remnant hides inside its external = 4π dλ r gABx˙ x˙ Planckian area a volume of order (m /m )5 l 3. γ 0 P P Z A B1/2 = 4π dλ gÃBx˙ x˙ , (4) We first review and clarify some aspects of the γ discussion given in [10] and generalize the results where Σ are spherically symmetric surfaces bounded presented there so that they may be used in an ar- by a given sphere ∂Σ. bitrary spherically symmetric spacetime. In Sec- Thus, the extremization of V [Σ] is equivalent to tionI and the Appendix, we prove the technical re- the 2D geodesic problem for the auxiliary metric sult that finding the spherically symmetric maximal 4 gÃB = r gAB. That is, γ is a solution of surfaces is equivalent to solving a two dimensional A ˜ B A ˜ B geodesic problem. In SectionII we review the defini- x˙ Ax˙ = eλ Aeλ = 0 (5) tion of volume and discuss the analogy between the ∇ ∇ where ˜ is the covariant derivative ing ÃB and λ Minkowski and the Schwarzschild case in order to ∇ illustrate its geometric meaning. In SectionIII we has been chosen to be an affine parameter on γ with examine the evaporating case and calculate the vol- respect tog ÃB. ume enclosed in the horizon as a function of time at infinity. We close with a discussion on the physical The stationary points of V [Σ] solve the “Plateau’s relevance of our result with respect to the debate on problem” or “isoperimetric problem” for ∂Σ. In a the fate of information in evaporating black holes. Euclidean context these are local minima, while in the Lorentzian context they are local maxima. It is αβ simple to show that if the trace K = Kαβg of the I. Maximal surfaces as a 1 + 1 geodesic extrinsic curvature of a hypersurface vanishes, the problem variation of the volume functional is automatically zero (see for instance [21]). For this reason, in the A general spherically symmetric spacetime can be Lorentzian context, surfaces with K = 0 are called described by a line element maximal surfaces. It is the authors understanding that a general 2 α β A B 2 2 ds = gαβdx dx = gABdx dx + r dΩ (2) proof of the opposite statement, namely that for arbitrary spacetimes extremizing V [Σ] for a given ∂Σ with dΩ2 = sin2θ dφ2 + dθ2. We use the notation yields K = 0 surfaces, is missing. Several precise xα = x0, r, θ, φ and xA = x0, r . proofs exist in the mathematical relativity literature { } { } Spherically symmetric hypersurfaces Σ can be (see for instance the seminal papers [22, 23]), that parametrically defined via a coordinate λ: typically rely on energy conditions or other restric- tions on the metric or on the surfaces. “Physicist” 2 A B 2 2 2 dsΣ = (gABx˙ x˙ ) dλ + r dΩ (3) demonstrations can be found in the 3 + 1 literature [21, 24]. A A A d A For completeness, we prove in the Appendix that, where x = x (λ) andx ˙ λ x (λ). We have ≡ d 2 Σ γ S2, with γ : λ xA(λ) being a curve for an arbitrary metric gAB, any surface Σ γ S , ∼ × → ∼ × in the x0- r plane. We denote as ya = λ, φ, θ with γ being a solution of (5), has K = 0. From α β { } well known theorems about the geodesic equation, and hab = ea eb gαβ the coordinates and the induced α ∂xα this also guarantees the local existence of maximal metric on Σ respectively, where ea = ∂ya provides a basis of tangent vectors on Σ. surfaces, see also [25]. We look for the stationary points of the volume

3An argument for the persistence of the large volume in The physical relevance of maximal surfaces has the evaporating case was discussed in [20]. long been recognised in diverse disciplines ranging

2 from problems in mathematical physics [26] to ar- proper volume is what we call the proper volume in 4 3 chitecture and the beautiful tensile structures of special relativity; that is, VΣv = 3 π(2m) . Frei Otto [27]. In general relativity, their useful- In Schwarzschild geometry, the maximal surfaces ness for numerically solving Einstein’s equations is starting from Sv approach the surface r = 3/2m reflected in the popular “maximal slicing” 4 (see (because of this behavior, r = 3/2m will be called for instance [24] and references therein), which in a “limiting surface”), and become null either when sense generalizes the slicing of a Newtonian space- they reach the singularity or when they asymptoti- time by constant (absolute) time surfaces. cally approach the horizon, except one that asymp- Common notions of volume implicitly use maxi- totically becomes r = 3m/2 5. The proper volume mal surfaces. These include the everyday meaning of this surface is infinite. of volume, the special relativistic proper volume and This is a characteristic difference between the two the volume of the Universe, where the latter habit- geometries which underlines the common under- ually refers to the proper volume of the t = const. standing that “space and time exchange roles inside surfaces of the Friedmann-Robertson-Walker met- the hole”. Inside the sphere containing flat space, ric: spherically symmetric maximal surfaces. there are radial timelike curves of infinite length, while all radial spacelike curves have proper length II. Review of the volume definition at most equal to the radius of the sphere. Inside a black hole this is reversed: there are radial space- The volume definition given in [10] can be stated as like curves of infinite length, while radial timelike follows: the volume inside a sphere S is defined as curves have proper time at most equal to πm. the proper volume of the maximal spherically sym- In the physical case of non-eternal black holes metric surface Σ bounded by S, which has the largest formed by collapse, the surface Σv does not have volume amongst all such Σ. Note that this is a ge- infinite volume since it does not extend infinitely ometric statement and as such it is manifestly gen- along r = 3/2m. In fact, it connects the sphere erally covariant. at the horizon Sv with the center of the collaps- In order to illustrate its geometric meaning, we ing object before the formation of the singular- examine in the rest of this section the analogy be- ity, see Fig.2. The surface in its interior will be tween the maximal surfaces of Minkowski spacetime given by solving (5) for the interior metric. For and those of the Schwarzschild solution. The discus- a collapse modeled by a null massive shell or àla sion is summarized in Figures1 and2. Oppenheimer-Snyder [32], the contribution to V (v) R dr will be of the order of that of the flat sphere m3. Using the advanced time v = t + f(r) , the ge- ∼ ometry of the two spacetimes is described by At late times v >> m, this contribution is negligi- ble with respect to the one given by the main part ds2 = f(r)dv2 + 2dvdr + r2dΩ2 , (6) lying on r = 3/2m, and the volume is given by (1). − This characteristic monotonic behaviour is per- with f(r) = 1 and f(r) = 1 2m/r respectively. haps best understood by extending the definition − Consider the sphere Sv defined as the intersection to the case of an eternal black hole. In this case we of r = 2m and the ingoing radial null ray of con- consider the volume difference ∆V (v, v0) between stant v. It bounds a family of maximal surfaces, the two spheres Sv and Sv0 labeled by different times at solutions of (5) for different initial speeds. infinity, in analogy to considering the proper time In Minkowski, these are the simultaneity surfaces between any two points on a timelike curve that of inertial observers, which are straight lines in the otherwise extends to arbitrary values of its affine t-r plane. The one with the biggest volume, Σv, parameter. is that which defines the inertial frame of Sv. Its In Minkowski, this difference is zero: the proper

4The family of surfaces discussed in the next section in- 5The existence of the limiting surface r = 3/2m was ﬁrst cludes the surfaces used for maximal slicing, but keep in mind pointed out in [28]. It is crucial for the singularity avoidance that we do not restrict ourselves to surfaces satisfying the property of the maximal slicing, which is in fact comprised “singularity avoidance” or the “nowhere-null” condition. In by the Σv extended to inﬁnity. Similar elongated surfaces are fact, half of each family of K = 0 surfaces we will study end studied in numerical relativity [29,30] and have been dubbed at the singularity and become null there. “trumpet geometries” [31].

3 t t 3m r 2 S ′ Σv v

Σv′ Sv′

ΔV(v,v')

Σv′ Sv

Σv Sv

r2m r2m r r

Figure 1. Left: Maximal surfaces (blue lines) inside a two-sphere in flat Minkowski spacetime. The largest is the t = const. (bold black lines) defining its inertial frame. Right: Maximal surfaces (blue lines) inside a two-sphere on the horizon of a static black hole. Apart from the transient part connecting it to the horizon, the largest surface (bold black lines) lies on the limiting surface r = 3/2m. The volume difference between the spheres Sv and Sv0 is finite and given by (7). volume of the sphere of fixed radius remains con- f(r, v) = 1 2m(v)/r. For our purposes it is suffi- − stant. In Schwarzschild, by the translation invari- cient to model the formation of the hole by the col- ance inside the horizon, ∆V (v, v0) is given by the lapse of an ingoing null shell at the retarded time volume of the part of Σv0 that lies on the limit- v = 0, and the loss of mass due to evaporation by in- ing surface r = 3m/2 and does not overlap with tegrating the thermal power emission law [34]. The Σv. Thus, this difference is finite, monotonically resulting mass function is increasing and given by 3 1/3 m(v) = Θ(v) m0 3B v , (8) 2 − ∆V (v, v0) = 3√3 π m (v0 v) . (7) − where Θ(v) is the step function, B 10 3 a param- ∼ − Notice that the result for a black hole formed by eter that corrects for back reaction [35] and m0 the collapse, eq. (1), is nothing but the approximate mass of the shell. The spacetime has a shrinking version of the above equation with v = 0. timelike apparent horizon given by rH (v) = 2m(v). The analysis presented in this section can be By numerically solving (5), we can draw the fam- nicely extended to the case of an evaporating black ily of maximal surfaces for the spheres at the ap- hole to which we now turn our attention. parent horizon for different v. The situation, de- picted in Figure3, is in direct analogy with the non- III. The volume of an evaporating black evaporating case. There is again a limiting surface, hole persisting up to very late stages of the evaporation. Thus, as in the static case, the volume of the biggest The spacetime of an evaporating spherically sym- maximal surface Σv inside Sv is the one connecting metric black hole can be described by the Vaidya the latter to the center of the collapsing shell. metric [33], given by replacing f(r) in (6) with We may get an estimate for the volume as a func-

4 t i+

I + Sv

i0 = 0 r

r=2m(v) I − event horizon r = 3/2 m collapsing object r i − Figure 2. Penrose diagram illustrating the surface defin- Figure 3. Eddington-Finkelstein diagram of the two fami- ing the volume (black curve) in the case of a black hole lies of extremal volume surfaces (blue lines) inside an evap- formed by collapse. The details of the surface in the interior orating black hole formed by a collapsing object. The sur- of the collapsing object (dotted curve) will depend on the face defining the volume is in bold black. Note the close specific metric use to describe the latter. For Oppenheimer- analogy with the static case, compare with Fig.1. Snyder and null massive shell collapses, this contribution to the volume is of the order m3. large spacelike maximal surfaces appears to coin- cide with the regime in which the mass has become tion of time and the initial mass as follows: we com- Planckian. These estimates agree with the numer- pute the volume of a surface r = α m(v) and find ical investigation of the actual surfaces, see Fig.4. the α for which this is maximized: We conclude that the volume increases monotoni- 3 45 B 1 cally, following the approximate behavior given in α = 2 + O 4 . (9) (10), up to when its external area becomes Planck- 2 − 8 m0 m0 ian. At this very late time, the internal volume is Indeed, the limiting surface is very well approxi- 5 of order m0 in Planck units. mated by r = α m(v) even for low masses, see Fig.4.

Expanding the volume of r = α m(v) to leading Intuitively, the picture is the following: from the order in 1/m0 we get: perspective of the maximal surfaces, collapse and horizon at any subsequent exterior time are simul- 9 B V (v) 3√3 π m2 v 1 . (10) taneous, see Fig.4. The exterior elapsed time cor- 0 2 m2 ≈ − 0 responds inside the hole to the stretching of space, Thus, for large masses, we have again recovered as given by (1). (1). A direct calculation shows that the surface r(v) = A few numbers α m(v) ceases to be spacelike when the mass func- Before closing this section, let us put the above in tion takes the value perspective: when a solar mass (1030 kg) black hole ! 3/2 becomes Planckian (it needs 1055 times the actual √ 225 B m 3 B 2 mP < mP /10 . (11) age of the Universe), it will contain volumes equiv- ≈ − 8 m0 alent to 105 times our observable Universe, hidden 70 2 This provides an estimate for the regime of valid- behind a Planckian area (10− m ). ity of eq. (10). Interestingly, the non-existence of Perhaps more pertinent is to consider small pri-

5 t hole may end its lifetime much earlier than near- complete evaporation by tunneling to a white-hole geometry. This is possible thanks to quantum gravitational effects that, due to the long times involved, can become important in low curvature regions outside the horizon [40–42] 7. Consider then the setting in which the semi- r=αm(v) classical approximation behind Hawking’s computation remains valid up to the very end of the evaporation. The hole will completely evaporate and the information will unavoidably be lost, as orig- r=2m(v) inally suggested by Hawking [1]. While it seems intuitively reasonable for what appears to be a tiny object to decay away and disappear, it is compelling to ask what became of the macroscopic region in- r side. Conversely, consider the additional hypothesis Figure 4. The surfaces defining the volume enclosed in that quantum gravitational effects play an im- spheres at the apparent horizon of an evaporating black portant role in the strong curvature regime by hole at different times (blue lines). The limiting surface “smoothing-out” the singularity [3]. When the mass lies close to r = α m(v), with α given by (9) (dashed line). becomes Planckian, the semi-classical approxima- Here m = 10 in Planck units. tion underlying Hawking’s computation fails and the evaporation stops (see for instance [45]). The mordial black holes with mass less than 1012 kg. hole does not completely disappear and one can Their initial horizon radius and volume are of the consider the possibility of having a minuscule ob- 15 order of the proton charge radius (10− m) and vol- ject that stores all the information needed to purify 45 3 ume (10− m ) respectively. They would be in the the external mixed state: a remnant [4–6]. final stages of evaporation now, hiding volumes of Standard objections against the remnant scenario 3 3 about one liter (10− m ). such as the infinite pair production [7] and their impossibility in storing inside a large amount of infor- IV. Remnants and the information mation, rely on considering the remnant as a small paradox object. Our result shows that the remnant is in- stead better understood as the small throat of an As was briefly discussed in the introduction, the immense internal region, with a volume of the order 5 results presented above can be relevant in the dis- of m0. General Relativity naturally gives a “bag of cussion about the loss-of-information paradox, par- gold” type description of the interior of a remnant, ticularly in the context of scenarios that assume without the need of ad-hoc spacetimes that involve the semiclassical analysis of quantum field theory some “gluing” of geometries [46, 47]. Notice that on curved spacetimes to be valid in regions of low the result of the previous section is insensitive to curvature and until near-complete radiation of the the details of the would-be-singularity region since initial mass 6. Such scenarios disregard the possi- the limiting surface is in a relatively low-curvature bility of having information being carried out of the region. hole by the late Hawking photons [37, 38], avoiding In [2,8,9] the authors suggest that a large avail- the recent firewall and complementarity debate [39]. able internal space could store a sufficient amount of Another alternative that has recently aroused in- very long wavelength modes that carry all the infor- terest and is not considered here, is that a black mation needed to purify the external mixed state,

6Another potential application of this result is in black 7An alternative scenario in which this process happens hole thermodynamics in view of recent results on the Von must faster by assuming faster-than-light propagation of a Neumann entropy associated to volumes [36]. shock-wave from the bounce region is considered in [43, 44].

6 albeit the available energy being of the order of a i+ few Planck masses. The surfaces studied here are good candidates on which this idea could be tested 8. The details of the mechanism by which information would be stored have not, to our knowledge, I + been made precise; demonstrating this possibility is beyond the aim of this work and, in what follows, we assume this to be possible. We can identify two characteristically distinct

possibilities for the evolution of the large interior re- = 0 i0 gion. The bulk of these large surfaces is causally dis- r connected from their bounding sphere on the horizon [19]. They can remain causally disconnected from the rest of the spacetime, which may lead to a baby universe scenario [49, 50]. v = 0 On the contrary, quantum gravitational effects apparent horizon I − can modify the (effective) metric and bring these re- r = α m(v) gions back to causal contact with the exterior, while inflating phase deflating their volume, allowing for the emission of deflating phase i the purifying information to infinity (the informa- − tion could also be coded in correlations with the Figure 5. Speculative evolution of maximal surfaces in fundamental pre-geometric structures of quantum the case of a long-lived remnant scenario. The volume ac- gravity, as proposed in [9]). This scenario, where quired during the evaporation process (continuous surfaces) deflates back to flat space (dotted surfaces). This is ex- the inflating phase is followed by a slow deflating 4 pected to happen in a time of order m0, during which all phase of the remnant, is sketched in Fig.5. the information stored can be released. We expect this deflating process to be slow, in accordance with bounds on the purification time faces in engineering and to the inspiring work of [51, 52] and the lifetime of long-lived emitting rem- 4 Frei Otto. nants, estimated to be of order m0. The latter scenario can be made precise by constructing an ef- M.C. acknowledges support from the Educational fective metric describing this process through the Grants Scheme of the A.G.Leventis Foundation for evolution of maximal surfaces in the sense of Fig.5. the academic years 2013-14, 2014-15 and 2015-16 It then suffices to numerically solve equation (5) in as well as from the Samy Maroun Center for Space, order to study the evolution. Time and the Quantum.

Acknowledgments Appendix

The authors thank Carlo Rovelli, Alejandro Perez In the notation of SectionI, the mean extrinsic cur- and Thibaut Josset for the many discussions on this vature is defined by subject, Abhay Ashtekar for private communica- α ab α β K = αn = h e e αnβ (12) tions that clarified aspects of the ideas in [8] and ∇ a b ∇ Thomas Baumgart for exchanges on maximal sur- where is the covariant derivative in gαβ and nα faces and their use in numerical relativity. ∇ is the normal to Σ. The Levi-Civita connections of We would like to thank Marina Konstantatou for gAB andg ÃB are related by: pointing us towards applications of maximal sur- 2 8 B ˜B B B Br In [48] it is argued that these surfaces do not store enough Γ AC = Γ AC δCr δA + δAr δC g gAC information for purification. However, in that work the − r − Hawking temperature is assumed constant. The computed (13) information is therefore the one stored in a static black hole, For the calculation that follows, keep in mind the α α α α φφ φφ θθ θθ and it is not pertinent to this discussion. following: eφ = δφ , eθ = δθ , h = g , h = g ,

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