Infrared Effects in the Late Stages of Black Hole Evaporation
PPrepared for submission to JHEP
Infrared Effects in the Late Stages of Black
Hole Evaporation ´Eanna ´E. Flanagan
Department of Physics, Cornell University, Ithaca, NY 14853Cornell Laboratory for Accelerator-based Sciences and Education (CLASSE), Cornell Uni-versity, Ithaca, NY 14853
E-mail: [email protected]
Abstract:
As a black hole evaporates, each outgoing Hawking quantum carriesaway some of the black holes asymptotic charges associated with the extended Bondi-Metzner-Sachs group. These include the Poincar´e charges of energy, linear momentum,intrinsic angular momentum, and orbital angular momentum or center-of-mass charge,as well as extensions of these quantities associated with supertranslations and super-Lorentz transformations, namely supermomentum, superspin and super center-of-masscharges (also known as soft hair). Since each emitted quantum has fluctuations thatare of order unity, fluctuations in the black hole’s charges grow over the course ofthe evaporation. We estimate the scale of these fluctuations using a simple model.The results are, in Planck units: (i) The black hole position has a uncertainty of ∼ M i at late times, where M i is the initial mass (previously found by Page). (ii) Theblack hole mass M has an uncertainty of order the mass M itself at the epoch when M ∼ M / i , well before the Planck scale is reached. Correspondingly, the time at whichthe evaporation ends has an uncertainty of order ∼ M i . (iii) The supermomentum andsuperspin charges are not independent but are determined from the Poincar´e chargesand the super center-of-mass charges. (iv) The supertranslation that characterizes thesuper center-of-mass charges has fluctuations at multipole orders l of order unity thatthat are of order unity in Planck units. At large l , there is a power law spectrumof fluctuations that extends up to l ∼ M i /M , beyond which the fluctuations fall offexponentially, with corresponding total rms shear tensor fluctuations ∼ M i M − / . a r X i v : . [ h e p - t h ] F e b ontents A Derivation of late time predictions of stochastic process 29B Independent charges in stationary regions of future null infinity 32C Choice of basis of algebra of charges 34D Derivation of changes in charges in stationary-to-stationary transi-tions 36 – 1 –
Introduction and summary
Hawking’s black hole information loss paradox is one of the most enduring myster-ies in theoretical physics: how does information escape from a black hole during itsevaporation? [1–3]. Great progress has been made on this issue in the past few years,using explicit Euclidean path integral methods. It is now possible to explicitly computethe Page curve that describes the time evolution of the entanglement entropy of theemitted Hawking radiation and the black hole, and to show that it is consistent withunitarity [4–9]. In addition, the amount of time taken for the information in a diarythrown into a black hole to return in the Hawking radiation can be reliably computed[5]. Nevertheless, some of the processes and computational prescriptions that arise inthe Euclidean domain remain mysterious in the Lorentzian domain, so there is stillmuch to be understood.A central role in this subject is played by the semiclassical approximation, wherethe gravitational field is treated classically (aside from linear perturbations that can betreated as free gravitons) and the matter fields are treated quantum mechanically. Thisapproximation excludes macroscopically large quantum fluctuations in the geometry.It is the only approximation in which we can compute the full state of the outgoingHawking radiation. In addition, the new computational prescriptions for computing theentanglement entropy of the exact state of the Hawking radiation [4–9] are expressed interms of a single semiclassical geometry, as are other similar powerful theoretical toolsand results (the Ryu-Taganacki formula [10, 11], the quantum focussing conjecture [12]and the covariant entropy bound [13]).On the other hand, it has been known since the work of Page in the 1980s [14] thatthe semiclassical approximation actually fails drastically during the course of blackhole evaporation. This failure arises as follows: each emitted quantum carries of amomentum ∼ M − in a random direction, where M is the mass of the black hole inPlanck units, and the corresponding change in the velocity of the black hole is of order ∼ M − . This change in velocity causes a net displacement in the center-of-mass ofthe black hole of order ∼ M after an evaporation time ∼ M . During the evaporationprocess we have n ∼ M such kicks that accumulate as a random walk, giving a totalnet uncertainty in the black hole location of order ∼ √ nM ∼ M , much larger thanthe size of the black hole. Thus we have superpositions of macroscopically distinctgeometries.Several authors have argued for the importance of the center-of-mass fluctuations inunderstanding the unitarity of black hole evaporation [15–17]. They note that unitarityis required only for the evolution in the complete Hilbert space, not in the subspace thatcorresponds to a single semiclassical geometry, and that the relative phases of different– 2 –emiclassical geometries in a quantum superposition contain information. However,there are counterarguments [18, 19] which suggest that the breakdown of the semiclas-sical approximation is fairly innocuous. First, there are situations where black holesevaporate in anti-de Sitter space where the center of center-of-mass spreading is sup-pressed but where there is still an information loss paradox [19]. Second, the dimensionof the Hilbert space associated with the center-of-mass motion is negligibly small com-pared to the relevant scale of the exponential of the Bekenstein-Hawking entropy, sinceit scales as a power law in the mass M of the black hole .In Ref. [20] we show that the center-of-mass fluctuations give rise to large correc-tions to the angular distribution of the Hawking radiation. We also argue there thatthose corrections remove one of the primary objections to the proposal that soft hairon black holes plays a key role in how unitarity of the evaporation is achieved [21–25],by increasing the number of soft hair modes that can interact with outgoing Hawkingquanta.The purpose of this paper is to study the macroscopic fluctuations in the geometryof evaporating black holes, in more detail than hitherto. A more detailed understandingmay be useful for eventually extending some of the theoretical tools discussed above tosituations where infrared quantum fluctuations are large. It may also shed light on therole of soft hair. Finally, some of the results derived here were used in the computationsof Ref. [20].The theoretical framework we use to study these fluctuations is as follows. In theclassical theory, the geometry of a stationary black hole is determined by the conservedcharges on future null infinity, including the soft hair charges associated with extensionsof the Bondi-Metzner-Sachs (BMS) group [21, 22]. We assume that the this propertyremains true in the quantum theory, and evolve the charges using a simple model,described in Secs. 2.1 and 3.6 below. We extend similar previous studies [14, 15] of thefluctuations in a number of directions: • We extend the computations to late times when the black hole mass M is smallcompared to its initial mass M i , by making use of the approximation that thefluctuations in charges are small compared to their expected values [Eqs. (2.5)below]. This approximation is valid until M ∼ √ M i . The variance in the center ofmass location does not evolve significantly as the black hole shrinks from M ∼ M i to M (cid:28) M i , but remains ∼ M i . We can impose an infrared cutoff by assuming that the black hole moves on a torus of size theevaporation timescale ∼ M , and impose a maximum kinetic energy of motion of order ∼ /M [Eqs.(2.2b) and (2.5b) below, neglecting logarithmic factors]. This gives a Hilbert space dimension ∼ M . – 3 – We compute the evolution of the fluctuations in the mass of the black hole. This isof order unity in Planck units after an evaporation times, but grows at late timesaccording to ∆ M ∼ M i /M [Eq. (2.5c) below]. It follows that ∆ M ∼ M when M ∼ M / i . At this epoch, there is an order unity amplitude for the evaporationto be completed ( M = 0), but there is also an order unity amplitude for the blackhole mass to be macroscopic, M ∼ M / i . • We extend previous studies to include the charges associated with an extensionof the BMS group [26–29]. These charges are reviewed in Sec. 3.1 below, andare higher- l analogs of the center-of-mass, momentum and spin that are encodedin the asymptotic metric near future null infinity. We show that for evaporatingblack holes only some of these charges are independent. The independent chargescan be taken to be the so called super center-of-mass charges, or soft hair. Thesecharges can be parameterized in terms of the supertranslation required to setthem to zero, a function Φ on the two-sphere with dimensions of length with only l ≥ l of order unity, the fluctuations are of order unity in Planck units[Sec. 3.8]. There is also a contribution to the fluctuations associated with quantathat have only partially arrived at future null infinity by the time the chargesare measured, which we compute in Sec. 4. This contribution gives a power lawspectrum of fluctuations extending up to l ∼ M i /M which is subdominant atearly times, but becomes dominant when M becomes small compared to M / i [Sec. 4.1].The organization of this paper is as follows. The predictions of our model for theevolution of the Poincar´e charges are given in Sec. 2. Section 3 extends this modelto include soft hair charges. In Sec. 4 we consider some additional contributions tothe fluctuations of the soft hair charges that are associated with quanta that haveonly partially arrived at future null infinity by the time the charges are measured.Preliminary versions of some of the results were presented at the conferences [30, 31]. In this section we define a stochastic process that gives a crude model of the evaporationof a black hole, including fluctuations in its Poincar´e conserved charges. We believe– 4 –hat it captures the dominant effects of the fluctuations. A variant of this modelwas first introduced by Page [14] and explored in more detail by Nomura, Varela andWeinberg [15]. Page estimated the fluctuations in position of the black hole after anevaporation time. Below we extend his computations to late times and also estimatethe mass fluctuations of the black hole. A more sophisticated model which includes allthe BMS and extended BMS charges will be given in Sec. 3 below. The results fromthat more sophisticated model for the Poincar´e charges agree qualitatively with thoseof the simple Page model discussed here.We start by describing the motivation for the model. Consider the evaporationof a black hole of mass M (cid:29)
1, in Planck units with 8 πG = (cid:126) = c = 1. Roughlyone Hawking quanta per time ∆ t ∼ M is emitted; this is explicit in an orthonormalwavepacket mode basis. Each quantum carries an energy ∆ E ∼ M − and spatialmomentum ∆ p ∼ M − , which change the energy and momentum of the black hole bycorresponding amounts. The fractional fluctuations in ∆ E and ∆ p are of order unity,since they are carried by a single quantum. Also the spatial momentum can be in arandom direction.The classical stochastic process is defined by M n +1 = M n − (cid:15) n M n , (2.1a) t n +1 = t n + M n , (2.1b) p n +1 = p n − (cid:15) n δ n M n , (2.1c) x n +1 = x n + p n . (2.1d)Here n = 1 , , . . . labels the steps, one for each emitted quantum. The variable M n is the mass of the black hole at step n . The quantity (cid:15) n is a random variable whichtakes on the values 0 and 1 with probability each of 1 /
2. The mass evolution equation(2.1a) describes the mass of the black hole being reduced, with a probability to occur oforder unity, by each emitted quantum. The emission process takes a time of order thecurrent mass of the black hole, as encoded in the time evolution equation (2.1b); t n isthe time of step n . The spatial momentum of the black hole after step n is p n , and itsevolution is governed by Eq. (2.1c). The magnitude of the emitted spatial momentumis (cid:15) n /M n , the same as the energy, since the Hawking quantum is massless. Howeverthe momentum can be in either direction (we are using a one dimensional model ofthe black hole motion). The directionality is encoded in the random variable δ n , whichtakes on the values − . The change in the position x n of the black hole during step n is the momentum p n divided by the mass M n times the The interpretation of individual steps in the model (2.1) as individual quanta should not be taken – 5 –ime interval M n , which yields the evolution equation (2.1d). All of the variables (cid:15) n , δ n for n = 1 , , . . . are uncorrelated. The initial conditions at n = 1 are taken to be M = M i , the initial black hole mass, and p = x = t = 0.The evaporation model (2.1) clearly incorporates a number of simplifications andapproximations. However we believe that the key predictions of the model are robustand are insensitive to these simplifications. Some of the approximations are: • It incorporates only one spatial dimension, and treats only the Poincar´e conservedcharges of the black hole, neglecting the additional charges associated with theBMS algebra and its extensions. These restrictions will be lifted to some extentin Sec. 3 below. • It treats all the fluctuations classically rather than quantum mechanically. Thisrestriction will be lifted to some extent in Sec. 2.5 below. • It models a continuous process as a discrete process. However we believe thatthis idealization does not affect the scale of the late time fluctuations predictedby the model. • It neglects any initial fluctuations in the black hole’s conserved charges. This isacceptable since at late times the fluctuations will be dominated by the cumulativeeffects of the emitted Hawking quanta, for any reasonable estimate of initialfluctuations . • Clearly, the the model could be generalized and made more precise by insertingdimensionless parameters of order unity into each of the equations (2.1); thiswould not change the qualitative predictions. too literally. In particular, if we quantize a scalar field near future null infinity on a spherical harmonicbasis, then the operator (3.12) below that describes linear momentum radiated consists entirely of crossterms between different modes, rather than individual modes carrying linear momentum. However,consider the following slight generalization of the model: at each timestep there are two independentrandom variables (cid:15) n and (cid:15) (cid:48) n that take on the values 0 and 1 with equal probability, with (cid:15) n = 1representing the emission of a l = 0 scalar quantum and (cid:15) (cid:48) n = 1 representing an l = 1 quantum. ThenEqs. (2.1) effectively hold with the (cid:15) n in Eq. (2.1a) replaced by (cid:15) n + (cid:15) (cid:48) n and with the (cid:15) n in Eq. (2.1c)replaced by (cid:15) n (cid:15) (cid:48) n . The qualitative predictions of the model are unchanged by this refinement. For example, for a particle of mass M i , the standard quantum limit on the uncertainty in positionafter a time t is ∆ x (cid:38) (cid:112) t/M i [32]. This uncertainty is of order ∼ M i after an evaporation time t ∼ M i , much smaller than the late time uncertainty ∆ x ∼ M i due to the Hawking quanta, cf. Eq.(2.5a). – 6 – The motion of the black hole is treated non-relativistically. This is consistentsince the motion is still non-relativistic at the time the model breaks down, cf.Eq. (2.5b) below.The key feature of the model is that independent, uncorrelated fluctuations areintroduced into the black holes conserved charges at each timestep or each emissionevent. Those fluctuations ultimately originate in incoming modes of quantum fields atpast null infinity, which are orthonormal and so uncorrelated for the incoming vacuumstate.
At times small compared to an evaporation time, t (cid:28) M i , simple random walk argu-ments can be used to estimate the fluctuations in the black hole charges, as first doneby Page [14] and explored in more detail by Nomura, Varela and Weinberg [15]. In thissection we review these early time predictions of the model.We define τ to be the time since black hole formation in units of the evaporationtime, τ = t/M i . The fluctuations in the various quantities are∆ x = M i τ / √ (cid:2) O ( τ ) + O ( τ − M − i ) (cid:3) , (2.2a)∆ p = (cid:112) τ / O ( τ )] , (2.2b)∆ M = √ τ / O ( τ )] . (2.2c)Thus, after an evaporation time, the fluctuations in the mass and momentum are oforder unity in Planck units, whereas the fluctuations in position are large, of order ∼ M i .To derive these estimates we use the approximation M n = M i on the right handsides of Eqs. (2.1); this is valid up to fractional corrections of order ( M i − M n ) /M i ∼ τ [Eq. (A.4) below]. Solving the momentum evolution equation (2.1c), squaring andtaking an expectation value and using (cid:104) (cid:15) n (cid:15) m (cid:105) = 14 (1 + δ nm ) , (cid:104) δ n δ m (cid:105) = δ nm , (2.3)gives (cid:104) p n (cid:105) = ( n − / (2 M i ). Combining this with t n = ( n − M i from Eq. (2.1b)yields the momentum fluctuation estimate (2.2b). Similarly the mass evolution equa-tion (2.1a) in this approximation yields M n = M i − (cid:80) nj =1 (cid:15) j /M i , and an analogousargument yields the mass fluctuation estimate (2.2c). Finally solving the positionevolution equation (2.1d) yields x n +1 = (cid:80) nj =1 (cid:80) jk =1 (cid:15) k δ k /M i . Squaring and taking the– 7 –xpectation value gives (cid:104) x n (cid:105) = n / (6 M i ) , up to fractional corrections ∼ /n , whichyields the estimate (2.2a). We now extend the computations of the previous subsection from the regime τ (cid:28) τ ∼
1, by approximating the black hole mass fluctuations to be small compared to theexpected value of the mass. We will show that this approximation remains valid until M ∼ √ M i .The results for the fluctuations at a time when the expected black hole mass is M are derived in Appendix A and are∆ x = (cid:26)
14 ( M i − M )( M i − M ) + M ln (cid:18) M i M (cid:19) (cid:27) / × (cid:34) O (cid:32) (cid:112) M i − M M (cid:33)(cid:35) , (2.4a)∆ p = (cid:115) ln (cid:18) M i M (cid:19) (cid:20) O (cid:18) M (cid:19) + O (cid:18) M i − M M (cid:19)(cid:21) , (2.4b)∆ M = (cid:26) M i − M i M + 12 M i M − M M (cid:27) / (cid:34) O (cid:32) (cid:112) M i − M M (cid:33)(cid:35) . (2.4c)These results are consistent with Eqs. (2.2) above in their common domain of validity τ (cid:28)
1, using the relation M = M i (1 − τ / M (cid:28) M i ,to ∆ x = 12 M i (cid:20) O (cid:18) MM i (cid:19) + O (cid:18) M i M (cid:19)(cid:21) , (2.5a)∆ p = (cid:115) ln (cid:18) M i M (cid:19) (cid:20) O (cid:18) M i M (cid:19)(cid:21) , (2.5b)∆ M = M i √ M (cid:20) O (cid:18) MM i (cid:19) + O (cid:18) M i M (cid:19)(cid:21) . (2.5c)The key qualitatively new feature in this regime is the enhanced fluctuations of theblack hole mass, which become of order the mass itself, ∆ M ∼ M , at M ∼ M / i . As Using similar methods one can show that (cid:104) xp (cid:105) = (cid:112) / x ∆ p at leading order, so the positionand momentum fluctuations are somewhat correlated as one might expect. Here we mean the mass fluctuations at fixed n given by Eq. (A.7) below, not the much largerfluctuations at fixed time t given by Eq. (2.4c). – 8 –iscussed in the introduction, this implies that there is an order unity amplitude forthe evaporation to be completed at the same time as there is an order unity amplitudefor the black hole mass to be macroscopic, M ∼ M / i .There is an elementary argument for the large, late time mass fluctuations, whichis as follows [33]. Consider a black hole with initial mass M i . After an evaporationtime ∼ M i , when the mass is M i /
2, the spread in mass is of order unity, from therandom walk estimates of Sec. 2.2. Consider now two different histories, one with mass M i / M i / t later are [( M i / − t/ / and [( M i / − t/ / , and when themass of the first history is zero, the mass of the second history is ∼ M / i . Thus theoverall fluctuations in mass at this time must be at least ∼ M / i . It is also straightforward to numerically simulate the stochastic model (2.1) of theblack hole evaporation. Representative results are shown in Fig. 1, for an initial massof M i = 10 in Planck units, and showing 20 independent trials. The results confirmthe analytic predictions of a spread of ∆ t ∼ M i in the endpoint of evaporation, and aspread ∆ M ∼ M ∼ M / i in mass at the endpoint. Although the analytic calculationsbreak down once ∆ M ∼ M , the numerical results indicate that the fluctuations do notdramatically change after this occurs. So far, the motion of the black hole has been treated classically. However, generalizingto a more detailed quantum mechanical treatment does not qualitatively change theresults, as we now outline. The black hole center-of-mass motion can be describedby its Wigner function W ( p , x ), with variance-covariance matrix Σ AB = (cid:104)(cid:104) ζ A ζ B (cid:105)(cid:105) −(cid:104)(cid:104) ζ A (cid:105)(cid:105)(cid:104)(cid:104) ζ B (cid:105)(cid:105) with ζ A = ( p , x ) and (cid:104)(cid:104) f (cid:105)(cid:105) ≡ (cid:82) d p (cid:82) d xf W for any function f . Asbefore, we idealize the evolution as a series of n steps, each of which has two parts.First, the black hole emits a quantum, under which by momentum conservation Σ transforms as Σ → Σ + (cid:18) σ γ δ ij
00 0 (cid:19) , (2.6)where σ γ ∼ M − is the uncertainty in momentum of the emitted Hawking quantum.Second, the black hole evolves freely for a time ∆ t ∼ M , under which Σ → R · Σ · R T ,– 9 – nitial Mass: 10000Number of trials: 20 Figure 1 : Black hole evaporation histories showing mass as a function oftime in Planck units, obtained by numerically integrating Eqs. (2.1). Thesimulation confirms the late time prediction ∆ M ∼ M ∼ M / i at the end-point of evaporation. where R = (cid:18) δ ij δ ij ∆ t/M δ ij (cid:19) . (2.7)After n ∼ M steps, the effect of the initial value of Σ is negligible, and the predictedscalings of position uncertainty and momentum uncertainty agree with Eqs. (2.2) above. In this section we generalize the Newtonian model of the previous section to includeall of the Bondi-Metzner-Sachs (BMS) charges of the black hole and in addition thecharges associated with the extended BMS algebra [34–38].As discussed in the introduction, in this paper we focus on charges of the black holemeasured at future null infinity. One could instead consider the symmetries and charges– 10 –efined on the black hole horizon, which has a different symmetry algebra [21, 22, 39–47]. These charges are related to charges at future null infinity by global conservationlaws, as detailed in Ref. [21], assuming that one can find the appropriate identificationbetween horizon symmetry generators and asymptotic symmetry generators (currentlyknown in some special cases). However our approach here follows the perspective of adistant asymptotic observer.
We start by reviewing the nature of the BMS and extended BMS charges of asymptot-ically flat spacetimes. For more details on this topic see the review by Strominger [26]and the expositions [29, 48]. The BMS group is the group of asymptotic Killing vectorsthat act on spacetimes which are asymptotically flat at future null infinity [49–51].Associated with each asymptotic Killing vector (cid:126)ξ or generator of the group, and witheach cut of future null infinity, there is a conserved charge Q ( (cid:126)ξ ) [52, 53].We follow the notation of Flanagan and Nichols (FN) [38]; the notation (FN,2.1)will mean Eq. (2.1) of FN. In retarded Bondi coordinates ( u, r, θ A ) = ( u, r, θ , θ ), themetric of an asymptotically flat spacetime near future null infinity can be written as[34, 37, 54–58]. ds = − (cid:20) − mr + O (cid:18) r (cid:19)(cid:21) du − (cid:20) O (cid:18) r (cid:19)(cid:21) dudr + r (cid:20) h AB + 1 r C AB + O (cid:18) r (cid:19)(cid:21) ( dθ A − U A du )( dθ B − U B du ) , (3.1)where U A = − r D B C AB + 1 r (cid:20) − N A + 116 D A ( C BC C BC ) + 12 C AB D C C BC (cid:21) + O ( r − ) , (3.2) A, B = 1 ,
2, and all the functions that appear in the metric are are functions of u and θ A .The metric h AB is the unit round metric on the two-sphere and is used to raise and lowercapital Roman indices, and D A is the associated covariant derivative. There are threeimportant, leading-order functions in the metric’s expansion coefficients [34, 37, 49, 54–58]: the Bondi mass aspect m ( u, θ A ), the angular-momentum aspect N A ( u, θ A ), andthe symmetric tensor C AB ( u, θ A ) whose derivative N AB = ∂ u C AB (3.3)is the Bondi news tensor. Evolution equations for these metric functions in terms ofretarded time are given by (FN,2.4), (FN,2.11a) and (FN,2.11b). The leading order– 11 –omponents of the stress energy tensor are given by (FN,2.6) and involve functionsˆ T uu ( u, θ A ) and ˆ T uA ( u, θ A ).The BMS algebra is the algebra of infinitesimal diffeomorphisms on null infinitythat map from one Bondi frame ( u, θ A ) to another. A general BMS generator can bewritten as [Eq. (FN,2.13)] (cid:126)ξ = (cid:20) α ( θ A ) + 12 uD A Y A ( θ B ) (cid:21) ∂ u + Y A ( θ B ) ∂ A , (3.4)where Y A is a globally smooth conformal Killing vector on the 2-sphere (the set ofwhich is isomorphic to the Lorentz algebra), and α is an arbitrary smooth functionthat parameterizes the supertranslation transformations.Two different extensions of the BMS algebra have been proposed. Barnich andTroessaert [34, 54] suggested an extension that includes all local conformal Killingfields Y A on a 2-sphere, allowing isolated singular points. This replaces the Lorentzalgebra of vector fields with an infinite dimensional Virasoro algebra; see Ref. [26]for more details. Campiglia and Laddha [27, 28] suggested extending the Lorentztransformations to include all smooth infinitesimal diffeomorphisms Y A on a 2-sphere.We focus here on the second extension, whose status can be summarized as follows.It arises as the symmetry group of an extended phase space of general relativity atfuture null infinity, in which fewer of the diffeomorphism degrees of freedom are fixedthan is usual [27, 28, 59]. While the presymplectic current of general relativity divergesat null infinity in the extended phase space, by exploiting a redefinition freedom[53]in the presymplectic current it can be made finite. Comp`ere, Fiorucci and Ruzziconiuse this method to construct a finite presymplectic current and derive charges Q ( (cid:126)ξ )associated with each symmetry generator (cid:126)ξ in the extended algebra [29]. However,their construction uses a specific coordinate system, and so is not obviously local andcovariant (which would be necessary for uniqueness). Indeed it can be shown thereis no redefinition of the presymplectic current that is local and covariant throughoutthe spacetime and which makes the presymplectic current finite at null infinity [59].Nevertheless, the charges defined by Ref. [29] can be shown indirectly to be covariantand unique [60]. They have also been shown to be consistent with the leading andsubleading soft graviton theorems [29].The charges associated with a symmetry of the form (3.4), with Y A an arbitrarysmooth vector field on the two sphere, on a cut u = constant of a stationary region of The key idea justifying the extension is that one should only fix diffeomorphism degrees of freedomthat correspond to degeneracy directions of the presymplectic form, and the standard construction ofthe BMS algebra fixes some degrees of freedom that are not degeneracy directions. – 12 –uture null infinity can be written as [29] Q ( (cid:126)ξ ) = 18 π (cid:90) d Ω (cid:104) αm + Y A ˆ N A (cid:105) , (3.5)where ˆ N A = N A − uD A m − D A ( C BC C BC ) − C AB D C C BC − u D B D A D C C BC + u D B D B D C C CA . (3.6)In a given Bondi frame , the charges we consider are (Sec. III of FN): • The
Bondi four momentum P α which is encoded in l = 0 , m ( u, θ A ) and conjugate to normal translations. • The supermomentum charges which are encoded in the l ≥ m ( u, θ A )and are conjugate to supertranslations. They encode a separate energy conserva-tion law at each angle [55]. • The angular momentum J αβ which is encoded in the l = 1 piece of ˆ N A ( u, θ A ) andis conjugate to the Lorentz generators Y A (conformal Killing vectors). As usualthis can be split into intrinsic angular momentum, and orbital angular momentumor center-of-mass charge (center-of-mass minus velocity times time). • The superspin charges which are encoded in the magnetic parity piece of the l ≥ N A ( u, θ A ), and are conjugate to l ≥ Y A in the extended algebra. They encode a separate conservation lawfor intrinsic angular momentum at each angle [58]. • The super center-of-mass charges which are encoded in the electric parity pieceof the l ≥ N A , and are conjugate to l ≥ Y A in the extended algebra. They encode a separate conservation lawfor orbital angular momentum or center-of-mass charge at each angle [38, 61]. Inthe context of black holes they are also called soft hair [21, 22, 37]. The formula (3.6) corrects Eq. (3.5) of FN, which is valid only for BMS symmetries and for evenparity extended symmetries, since the correction terms (fifth and sixth terms) in Eq. (3.6) are parityodd. We will apply this formula however only in the even parity case for which the BMS formulawould suffice. Note that the various charges discussed here mix together under transformations of Bondi frame,see, for example, Appendix B of FN. In this paper we adopt the convention of using the initial Bondiframe, associated with the stationary state to which the black hole settles down after it is first formed. – 13 –e can parameterize these charges in terms of a set of symmetric, tracefree tensors J ij , J ijk , . . . as follows. For any symmetric, tracefree Cartesian tensor Y i ...i l , weconsider the symmetry generator (3.4) with α = 0 and with Y A = D A ( Y L n L ),where L is the multi-index i . . . i l , n L = n i . . . n i l , and n i is the unit vector(sin θ cos ϕ, sin θ sin ϕ, cos θ ). We define the symmetric tracefree tensor J L bydemanding that the corresponding charge (3.5) is Y L J L . The tensor J L is relatedto the l th multipole electric parity piece of ˆ N A byˆ N el A = g l D A ( J L n L ) (3.7)with g l = 2(2 l + 1)!! / ( l ( l + 1) l !), from Eq. (3.5) and using the identity (C5) ofRef. [48]. We will idealize the evaporation of a black hole a sequence of transitions betweenstationary states: after each Hawking quantum is emitted, the black hole settles downto a stationary state, then the next quantum is emitted, and so on.In regions of future null infinity that are stationary, the various charges discussedabove are not all independent. This follows from the fact that there exists a canonicalBondi frame associated with the stationary region in which the metric functions takea simple form that encode the mass and intrinsic spin (see, for example, Sec. II.D ofFN): m ( θ A ) = m = constant , (3.8a) C AB ( θ A ) = 0 , (3.8b) N A ( θ A ) = magnetic parity , l = 1 . (3.8c)Now a general Bondi frame will be related to the canonical frame (3.8) by a nonlinearBMS transformation of the form (FN,2.12), parameterized by a Lorentz transformationand a supertranslation, and therefore the metric functions m , C AB and N A in the generalframe encode just one infinite family of charges, and not three (see Appendix B). Wewill focus here on the independent charges, which we take to be the momentum P α ,angular momentum J αβ , and super center-of-mass charges; the other charges can bedetermined from these.We also show in Appendix B that the super center-of-mass charges are determinedto a good approximation by the shear tensor C AB , so we focus on this quantity insubsequent sections rather than on N A . Specifically, we decompose C AB in terms of anelectric parity potential Φ and a magnetic parity potential Ψ via C AB = D A D B Φ − h AB D Φ / (cid:15) C ( A D B ) D C Ψ , (3.9)– 14 –here we take the l = 0 , (cid:88) l ≥ (cid:88) i ...i l Q i ...i l n i . . . n i l = (cid:88) l ≥ (cid:88) L Q L n L . (3.10)The relation between the tensors Q L and the super center-of-mass charges J L is givenin Eqs. (B.9) and (B.10) of Appendix B. In our model of black hole evaporation, each emission of a Hawking quantum is idealizedas a stationary to stationary transition as viewed at future null infinity. Specifically,this means that the spacetime at some early retarded time u is vacuum near futurenull infinity, and is also approximately stationary there. There is subsequently a burstof gravitational waves and/or matter energy flux to infinity, and the spacetime is againvacuum and approximately stationary near future null infinity at some later retardedtime u . In this section we will give formulae for the changes in the BMS and extendedBMS charges in such transitions, in terms of fluxes to null infinity of mass-energy orgravitational-wave energy. These formulae will be one foundation of our model of blackhole evaporation of Sec. 3.4 below, and are derived in Appendix D.The changes in the linear momentum P α and angular momentum J αβ have thesame form as in special relativity, but with the stress-energy fluxes supplemented bygravitational wave terms. The total energy radiated per unit solid angle in either matteror gravitational waves is ∆ E = (cid:90) u u du (cid:104) ˆ T uu + T uu (cid:105) , (3.11)where ˆ T uu = lim r →∞ r T uu , T uu = N AB N AB / (32 π ) and N AB is the news tensor (3.3).The Bondi 4-momentum is given by P α = ( E, P ) = (cid:82) d Ω(1 , n ) m/ (4 π ), from Eqs.(FN,3.5), (FN,3.7) and (FN,3.9), and the change in Bondi 4-momentum is∆ P α = (∆ E, ∆ P ) = − (cid:90) d Ω(1 , n )∆ E . (3.12)Similarly we define ∆ E A = (cid:90) u u du (cid:104) ˆ T uA + T uA (cid:105) , (3.13)where ˆ T uA = lim r →∞ r T uA and T uA is a kind of gravitational wave angular momentumflux given in terms of C AB and N AB by Eq. (FN,3.23). The quantity ∆ E A can be– 15 –nterpreted as angular momentum radiated per unit solid angle in either matter orgravitational waves. We also define the quantity (cid:102) ∆ E = (cid:90) u u du u (cid:104) ˆ T uu + T uu (cid:105) . (3.14)The components of angular momentum are given in terms of ˆ N A by J ij = 14 π (cid:90) d Ω e A [ i n j ] ˆ N A , (3.15a) J i = 18 π (cid:90) d Ω e Ai ˆ N A , (3.15b)from Eqs. (3.5), (FN,3.5), (FN,3.8) and (FN,3.9), where e iA = D A n i . The changes inthese quantities are (see Appendix D)∆ J ij = − (cid:90) d Ω e A [ i n j ] ∆ E A , (3.16a)∆ J i = − (cid:90) d Ω (cid:20) e Ai ∆ E A − n i (cid:102) ∆ E (cid:21) . (3.16b)Finally we turn to the super center-of-mass charges. As discussed in Sec. 3.2 above,the center-of-mass charges are encoded in the electric parity potential Φ for the sheartensor C AB defined in Eq. (3.9). The change ∆Φ in this potential (which encodes thegravitational wave memory) is given by D ∆Φ = 4 π P ∆ E + 6 n i n j m (cid:20) ∆ P i ∆ P j −
13 ∆ P δ ij (cid:21) . (3.17)Here m is the rest mass of the initial Bondi 4-momentum, ∆ P is the momentum change(3.12), P is the projection operator that sets to zero the l = 0 , D is the angular differential operator D = D / D / D = D A D A . The formula (3.17) is valid in initial rest frames, i.e., Bondi framesin which the spatial components of the initial Bondi 4-momentum vanish. The formula (3.17) is valid to quadratic order in the radiated momentum − ∆ P , see Ref. [62] foran exact version. – 16 – .4 Evolution model We now describe the evolution model for the black hole evaporation process, whichis based on the same philosophy as the simple Newtonian model of Sec. 2 above. Asbefore the evaporation is treated as a series of discrete steps, and each step is a classicalstochastic event. The only generalization is that all of the BMS charges are includedinstead of just the Poincar´e charges.After the black hole is first formed, with initial mass M i , it rapidly settles down to astationary state, on a timescale ∼ M i . We will call the canonical Bondi frame associatedwith this initial stationary state the initial Bondi frame . After each Hawking quantumis emitted, we assume that the black hole settles down again to a new stationarystate, with a new associated Bondi frame which we call the instantaneous Bondi frame ,before the next quantum is emitted. The changes in the BMS charges, that is, thecharges carried off by the Hawking quantum, will have a simple universal form in theinstantaneous Bondi frame, but their form in the initial Bondi frame will be morecomplicated and will depend on the values of all the BMS charges.We now describe the model in more detail. We denote by x α = ( u, r, θ A ) theinitial Bondi frame. At the n th step, the black hole BMS charges in this frame are4-momentum P αn , angular momentum J αβn , and the tensor C nAB ( θ A ) which encodes thesuper center-of-mass charges. Our goal is to derive a formula for the charges at step n + 1 in the initial Bondi frame, in terms of the corresponding values at the n th step,and also the changes in the charges in the instantaneous Bondi frame.We compute the BMS transformation from the initial frame to the n th instan-taneous frame in two stages. First, we make a supertranslation parameterized by afunction β on the 2-sphere (see Appendix B of FN), to a Bondi frame x ˆ α = (ˆ u, ˆ r, θ ˆ A ).We can write this function as β = t − t i n i + β , where n i = (sin θ cos ϕ, sin θ sin ϕ, cos θ ), t µ is a 4-vector associated with the normal translation piece of the transformation, and β is purely l ≥
2. The charges in the hatted Bondi frame are [Eqs. (FN,B7) and(FN,B8)] P ˆ αn = P αn , (3.19a) J ˆ α ˆ βn = J αβn − t αn P βn + t βn P αn + δJ αβ [ β ] , (3.19b) C n ˆ A ˆ B = C nAB − D A D B β + h AB D β . (3.19c)Here δJ αβ [ β ] is given by Eqs. (FN,B7) with β replaced by β . Next, we choose thesupertranslation β to make C n ˆ A ˆ B = 0 , (3.20)– 17 –hich determines β uniquely as shown in Sec. II.D of FN. We also choose the trans-lation to make P n ˆ α J ˆ α ˆ βn = 0 , (3.21)which makes the hatted frame be a center-of-mass frame. A translation which achievesthis is t βn = 1 M n P n α ( J αβn + δJ αβ [ β ]) , (3.22)where M n = − (cid:126)P n .Next, we perform a boost Λ ¯ α ˆ α from the hatted Bondi frame to the instantaneousBondi frame x ¯ α = (¯ u, ¯ r, θ ¯ A ). From Eqs. (FN,B3) and (FN,B6) the charges transformas P ¯ αn = Λ ¯ α ˆ α P ˆ αn , (3.23a) J ¯ α ¯ βn = Λ ¯ α ˆ α Λ ¯ β ˆ β J ˆ α ˆ βn , (3.23b) C n ¯ A ¯ B = ω ϕ ϕ ∗ C n ˆ A ˆ B = 0 . (3.23c)Here ϕ : S → S is the conformal isometry of the two sphere associated with the boost,as described after Eq. (FN,B3), and ω ϕ is given by Eq. (B.2). The boost is determinedin the usual way by the requirement that the new frame be a rest frame, ie P ¯ in = 0 . (3.24)Finally, in the instantaneous Bondi frame, the changes in the charges due to theemission of a Hawking quantum are P ¯ αn +1 = P ¯ αn + ∆ P ¯ αn , (3.25a) J ¯ α ¯ βn +1 = J ¯ α ¯ βn + ∆ J ¯ α ¯ βn , (3.25b) C n +1¯ A ¯ B = C n ¯ A ¯ B + ∆ C n ¯ A ¯ B . (3.25c)The prescription we use for the changes ∆ P ¯ αn , ∆ J ¯ α ¯ βn and ∆ C n ¯ A ¯ B is discussed in Sec. 3.6below.We now transform the new charges back to initial Bondi frame, and express theresults in terms of the initial Bondi frame components of the old charges. The final– 18 –esult is P αn +1 = P αn + Λ α ¯ α ∆ P ¯ αn , (3.26a) J αβn +1 = J αβn + Λ α ¯ α Λ β ¯ β ∆ J ¯ α ¯ βn + 2 M n P n γ (cid:0) J γ [ αn + δJ γ [ α (cid:1) Λ β ]¯ β ∆ P ¯ βn , (3.26b) C n +1 AB = C nAB + ω ϕ − ϕ − ∗ ∆ C n ¯ A ¯ B . (3.26c)Here the right hand sides are functions of the changes in the charges in the instanta-neous Bondi frame, and of the charges at step n : the Lorentz transformation Λ α ¯ α andassociated map ϕ are determined as a function of P αn by Eq. (3.24), while the quantity δJ αβ is given as a function of C nAB by Eqs. (3.19c), (3.20) and Eq. (FN,B7) with β replaced by β .From the structure of these evolution equations we see that the evolution (3.26a) ofthe 4-momentum is uncoupled from that of the angular momentum and super center-of-mass. In particular, this implies that the results of the simple model of Sec. 2above for the 4-momentum evolution should still be valid. We also see that the supercenter-of-mass evolution (3.26c) is uncoupled from the angular momentum, and can becomputed once the 4-momentum evolution is known. Finally, the angular momentumevolution (3.26b) depends on both the 4-momentum evolution and the super center-of-mass evolution. The velocity of the black hole is of order ∼ /M at late times, up to a logarithmicfactor, from Eq. (2.5b) above. This is small compared to unity until the Planck scale M ∼
1; in particular it is small compared to unity when M ∼ M / i , the epoch when∆ M ∼ M . Therefore in the evolution equations (3.26) it is a good approximation totreat the velocity as small. Expanding to linear order in the velocity v n and splitting We have replaced ˆ α with α in the indices of the Lorentz transformation. This is a slight notationalinconsistency but there is no real inconsistency because the Bondi frames x ˆ α and x α differ only by asupertranslation. – 19 –nto space and time components and writing (cid:126)P n = ( E n , P n ) = M n (1 , v n ) yields M n +1 = M n + ∆ E n + v in ∆ P in , (3.27a) P in +1 = P in + ∆ P in + v in ∆ E n , (3.27b) J in +1 = (cid:18) E n M n (cid:19) J in + ∆ J in + ∆ E n M n J ijn v jn + 1 M n ( v jn ∆ P jn ) J in − M n ( v jn J jn )∆ P in −
85 ∆ E n Q nij v jn , (3.27c) J ijn +1 = J ijn + ∆ J ijn − M n J in ∆ P j ] n − J in v j ] n − E n M n J in v j ] n + 2 M n v kn J k [ in ∆ P j ] n + 165 v kn Q n k [ i ∆ P j ] n , (3.27d) C n +1 AB = C nAB + ∆ C nAB + L (cid:126)Y ∆ C nAB −
12 ∆ C nAB D C Y C , (3.27e)where Y A = − v in e Ai . Here we have parameterized the electric quadrupole piece of C nAB in terms of a tensor Q nij using the definitions (3.9) and (3.10). Equations (3.27) showexplicitly the leading coupling of the super center-of-mass to the angular momentumevolution, which occurs through the quadrupole Q nij .The velocity terms in these equations are suppressed relative to the other terms bya factor ∼ M , so we can take the zero velocity limit. This yields M n +1 = M n + ∆ E n , (3.28a) P in +1 = P in + ∆ P in , (3.28b) J in +1 = (cid:18) E n M n (cid:19) J in + ∆ J in , (3.28c) J ijn +1 = J ijn + ∆ J ijn − M n J in ∆ P j ] n , (3.28d) C n +1 AB = C nAB + ∆ C nAB . (3.28e)The angular momentum evolution is more transparent if we switch to a differentset of variables, namely the center of mass X in = ( P in u n − J in ) /M n and the intrinsicangular momentum S ijn = J ijn − X [ in P j ] n . Here u n is the value of the retarded timecoordinate u along future null infinity I + at step n . We also note from Eq. (3.16b)that for u large compared to ∆ u n = u n +1 − u n ∼ M , the change in the space-timecomponents of the angular momentum can be written as∆ J in = ∆ J in + u n ∆ P in , (3.29)where ∆ J in is given by the first term on the right hand side of Eq. (3.16b). Using thesenew variables to rewrite Eqs. (3.28c) and (3.28d) and making use of Eqs. (3.27a) and– 20 –3.27b), we obtain our final result for the evolution prescription in the slow motionapproximation: M n +1 = M n + ∆ E n , (3.30a) P in +1 = P in + ∆ P in , (3.30b) X in +1 = X in + ∆ u n M n P in − ∆ J in M n , (3.30c) S ijn +1 = S ijn + ∆ J ijn + 2 ∆ E n M n X [ in P j ] n + 2 M n ∆ J in P j ] n , (3.30d) C n +1 AB = C nAB + ∆ C nAB . (3.30e) To complete the model we need to give a prescription for the changes ∆ E n , ∆ P n , ∆ J ijn ,∆ J in and ∆ C nAB to the charges in the n th instantaneous Bondi frame. For the firstfour of these charges, we can use simple order of magnitude estimates as we did inthe Newtonian model of Sec. 2 above. However, for the super center-of-mass charges,the relevant physics is less familiar, which is why we derived general formulae for thechanges in the charges in terms of fluxes to infinity in Sec. 3.3 above. We now use thoseformulae as a guide to develop a prescription.For a single Hawking quantum, Eqs. (3.11) and (3.12) are consistent with a radiatedenergy of order ∼ M − and a radiated linear momentum of order ∼ M − in a time∆ u ∼ M if the flux is of order ˆ T uu + T uu ∼ M − . (3.31)Similarly, the angular momentum ∆ J ij carried by a single quantum should be of orderunity . This implies from Eq. (3.16a) that the angular momentum flux should scaleas ˆ T uA + T uA ∼ M − , (3.32)and so we expect the first line in Eq. (3.16b) to scale as ∆ J in ∼
1. A simple modelfor the changes in the Poincar´e charges, consistent with these estimates and along the The linear momentum is of order ∼ M − and the effective displacement from the center of massof the black hole is at most of order ∼ M . These scalings are also consistent with a scalar field model of the outgoing Hawking flux: Forthe outgoing solution Φ = f ( u, θ A ) /r + O ( r − ) we have ˆ T uu = f ,u and ˆ T uA = f ,u f ,A , with f ∼ ∂ u ∼ M − , ∂ A ∼ – 21 –ines of the Newtonian model of Sec. 2, is given by∆ E n = − (cid:15) n M n , (3.33a)∆ P in = (cid:15) n M n Ξ in , (3.33b)∆ J ijn = (cid:15) n (cid:15) ijk χ kn , (3.33c)∆ J in = (cid:15) n Θ in . (3.33d)Here as before (cid:15) n = 0 or 1 is a random variable, and Ξ n , χ n and Θ n are independentlyand randomly distributed on the unit sphere.For the super-center-of-mass charges, we parameterize ∆ C nAB in terms of a potential∆Φ n as in Eq. (3.9), which in turn is given by Eq. (3.17). We can neglect the secondterm on the right hand side, since it is smaller than the first term by a factor ∼ M .The remaining term is the l ≥ ∼ M − . We therefore adopt the simple model ∆Φ n = (cid:15) n M n ϕ n (3.34)where ϕ n is the random process on the twosphere given by ϕ n = (cid:88) l ≥ ϕ lmn Y lm (3.35)with (cid:104) ϕ lmn (cid:105) = 0 and (cid:104) ϕ lmn ϕ ∗ l (cid:48) m (cid:48) n (cid:48) (cid:105) = c l δ ll (cid:48) δ mm (cid:48) δ nn (cid:48) . (3.36)Here c l are dimensionless constants which are of order unity for l of order unity. Wewill take the coefficients c l to fall off exponentially with l as l → ∞ , since this is thebehavior of the transmission coefficients that enter into the amplitudes for emittedHawking quanta . Here the Kronecker delta δ nn (cid:48) enforces the fact that successiveemitted quanta are uncorrelated, while the other two Kronecker delta factors ensureisotropy. In a more complete treatment, the quantum field would be split up into independent modes, andthe stress energy components would be given by expressions quadratic in the field. The nonlinearitieswould then induce correlations between the l = 0 , l ≥ There should also be a factor of ∼ l − in c l due to the presence of the operator D in Eq. (3.17).We neglect this factor since it is unimportant compared to the exponential factor. – 22 – .7 Results for Poincar´e charges The evolution prescription for the Poincar´e charges given by Eqs. (3.30a) – (3.30d)is similar to the simple Newtonian model of Sec. 2 above; here, however, it has beenderived from the full BMS kinematics. There are also some differences from the modelof Sec. 2, aside from the trivial generalization to three dimensions. First, there is theevolution equation (3.30d) for the intrinsic angular momentum, which was not trackedin Sec. 2. We can make order of magnitude estimates of the terms in Eq. (3.30d),using X in ∼ M , P in ∼
1, ∆ P in ∼ ∆ E n ∼ M − , and ∆ J ijn ∼ ∆ J in ∼
1. The first twoterms on the right hand side are of order ∼ ∼ M − and canbe neglected. The random walk estimate then gives that the fluctuations δS ij in S ij at late times are of order ∼ √ n ∼ M i , where n ∼ M i is the number of steps. Thedimensionless angular momentum parameter is then of order δS ij M ∼ M i M . (3.37)Hence the spin fluctuations are unimportant until M ∼ √ M i , which occurs after theregime M ∼ M / i of interest in this paper. A similar conclusion was reached inAppendix C of Ref. [15].A second difference from the model of Sec. 2 is the angular momentum term (thirdterm) on the right hand side of Eq. (3.30c) for the position evolution, which does notappear in the corresponding Eq. (2.1d). This term is of order ∆ J in /M n ∼ M − , andis statistically independent of the second term which is ∼ Turn now to the super center-of-mass charges. Combining the definition (3.9) of thepotential Φ n , the result (3.30e) for how the charges are updated at each step, andthe model (3.34) for the charges carried away by the n th quantum, we obtain for thepotential Φ n at late times Φ n = (cid:88) l ≥ (cid:88) m ˆ ϕ lmn Y lm (3.38) Here we are assuming zero mean spin; if the black hole starts with some net spin, this will ofcourse bias the evolution and lead to a spin down [63]. However the fluctuations will be still given byEq. (3.37) at leading order. – 23 –here ˆ ϕ lmn = (cid:80) n (cid:48) ≤ n ϕ lmn (cid:48) . Combining this with Eq. (3.34) gives (cid:104) ˆ ϕ lmn ˆ ϕ ∗ l (cid:48) m (cid:48) n (cid:105) = δ ll (cid:48) δ mm (cid:48) c l (cid:88) n (cid:48) ≤ n M n (cid:48) ≈ δ ll (cid:48) δ mm (cid:48) c l ln( M i /M ) , (3.39)where M = M n = (cid:112) M i + 1 − n and we recall that c l = O (1) for l = O (1). Hence,for l of order unity, the fluctuations in each l, m component ˆ ϕ lmn are of order unity inPlanck units (neglecting logarithmic factors), and they fall off exponentially with l forlarge l . Equivalently, the tensors Q L defined by Eq. (3.10) have fluctuations of orderunity for l of order unity.Consider now the super center-of-mass charges J L defined by Eq. (3.7). For l ≥ Q L by a factor of the mass, by Eq. (B.10). Hence we havethat the fluctuations are of order J L ∼ M, (3.40)for l ≥ l of order unity. For l = 2, by contrast, the fluctuations are muchlarger, since in the second term in Eq. (C.8) the orbital angular momentum is of order J i ∼ M at late times and the momentum is of order P i ∼
1, from Eqs. (2.2) and(C.3), giving J ij ∼ M . (3.41)How large are the fluctuations in the spacetime geometry associated with the fluc-tuations (3.40) and (3.41)? The fluctuations (3.40) correspond to displacements oforder unity in Planck units, and so the fluctuations in the geometry are small. Bycontrast, the displacements associated with the fluctuations (3.41) are of order ∼ M ,and correspond to macroscopic modifications to the geometry of order unity. However,these fluctuations are not independent of the fluctuations in the center of mass and mo-mentum. In particular, the charges P α , J αβ , J L , which are in the initial Bondi frame,determine a BMS transformation to the comoving Bondi frame in which the metricfunctions take the simple form (3.8). The l = 2 supertranslation piece of this transfor-mation is determined from Q ij which has the small fluctuations (3.39), not from J ij which has the large fluctuations (3.41) (see Eqs. (B.8)). The macroscopic fluctuationsin the geometry are dominated by the fluctuating boost and fluctuating translation. This is true both for u = 0 and for u ∼ M i , that is, for both versions of the super center-of-masscharges discussed in Appendix C. – 24 – Transient effects
The calculations so far in this paper have neglected some transient effects that areimportant for the super center-of-mass fluctuations at high multipole orders. Specifi-cally, consider the black hole charges evaluated on the cut S of I + given by u = u in the initial Bondi frame. We have assumed that each emission event impacts I + either completely before S , or completely after S , since we have counted only completeemission events and not partial events. In fact, there will be approximately ∼ M i /M emission events or outgoing quanta for which the associated stress energy impacts I + partially before S , and partially after S , whose contributions to the charges have notbeen correctly accounted for. In this section we will estimate the contribution fromthese events.We start by summarizing the results. We denote by Φ tr the “transient” contributionfrom the partially counted quanta to the potential Φ for the shear tensor at u = u .We parameterize the spectrum of angular fluctuations of Φ tr by a quantity d Φ /d ln l defined so that (cid:28)(cid:90) d Ω Φ (cid:29) = (cid:90) d ln l d Φ d ln l , (4.1)where the angular brackets denote expected value. Here l denotes multipole order,which we treat as a continuous variable at large l . Our result for the fluctuations ofΦ tr at large l is d Φ d ln l ∼ σ ∆ l M (cid:18) M l σ (cid:19) exp (cid:20) − l M σ (cid:21) , (4.2)where the symbol ∼ means that we have dropped constant factors of order unity. Here M is the expected mass of the black hole and σ the variance in the center-of-masslocation at retarded time u . Using the late-time result (2.5a) for this variance wecan rewrite the power spectrum as d Φ d ln l ∼ M i l M (cid:18) M l M i (cid:19) exp (cid:20) − l M M i (cid:21) . (4.3)Thus the transient contribution to the spectrum is a power law up to a critical angu-lar scale given by l crit ∼ M i /M , and at higher l it is exponentially suppressed. The This notation is an alternative to that used in Eq. (3.39). The two notations are related byreplacing the right hand side of Eq. (3.39) with [ l (2 l + 1)] − δ ll (cid:48) δ mm (cid:48) d Φ /d ln l . Note that the result (2.5a) is correct up to an unknown constant factor of order unity, arising fromthe idealized model (2.1), and hence the argument of the exponential factor in Eq. (4.3) is similarlysubject to a correction factor of order unity. – 25 –ransient contribution (4.3) is small compared to the previously computed contribu-tion (3.39) for l of order unity, but dominates for 1 (cid:28) l (cid:28) l crit where the previouscontribution is exponentially suppressed.The spectrum (4.3) characterizes the fluctuations in the potential Φ, or equivalentlyof the tensors Q L , at large l . The super center-of-mass charges J L are related to thesetensors by a factor of the black hole mass, as discussed in Sec. 3.8 above.Finally, we note that the shear tensor (3.9) is related to Φ by two angular deriva-tives. Hence the spectrum of fluctuations of C AB is given by the right hand side of Eq.(4.3) multiplied by l , which scales ∝ l − . The total rms fluctuation in C AB is of order C ∼ ln (cid:18) M i M (cid:19) + M i M . (4.4)Here the first term is the contribution (3.39) previously computed, and the second termis the transient contribution (4.3). The first term dominates at early times, while thesecond term begins to dominate when M becomes small compared to M / i . We now turn to the derivation of the spectrum (4.2). Our derivation is based on thesame kind of heuristic model as used in earlier sections of the paper. A more rigorousderivation based on the two point function of the flux operator yields qualitatively thesame result and will be given elsewhere.As previously discussed, each outgoing quantum is characterized by an outgoingflux ∼ M − over a timescale ∼ M . For simplicity we will assume that the dependenceon time is identical for each outgoing quantum. Thus for the n th quantum we assume,consistently with Eqs. (3.33) and (3.34),ˆ T uu ( u, θ A ) + T uu ( u, θ A ) = M − n (cid:15) n F (cid:20) u − u n M n (cid:21) ϕ n ( θ A ) , (4.5)where u n +1 − u n = M n [cf. Eq. (2.1b)] and F is a fixed smooth nonnegative functionwith F ( x ) = 0 for | x | > (cid:82) dx F ( x ) = 1. Also ϕ n is the random process givenby Eq. (3.35) but now with the l = 0 , T uu ( u, θ A ) + T uu ( u, θ A ) = (cid:88) n M − n (cid:15) n F (cid:20) u − u n + n · ∆ M n (cid:21) ϕ n ( θ A ) , (4.6)– 26 –here ∆ = ∆ ( u ) is the center of mass location of the black hole. Finally we evaluate thenet change in the potential Φ for the shear tensor from u = −∞ to u = u by applyingEq. (3.17) in the initial Bondi frame, neglecting the subdominant second term, andusing Eq. (3.11). This gives D Φ( u , θ A ) = 4 π P (cid:88) n M − n (cid:15) n ˆ F (cid:20) u − u n + n · ∆ M n (cid:21) ϕ n ( θ A ) . (4.7)Here we have defined the function ˆ F ( x ) = (cid:82) x −∞ dx (cid:48) F ( x (cid:48) ), which satisfiesˆ F ( x ) = 0 , x < − , (4.8a)ˆ F ( x ) = 1 , x > . (4.8b)There are two types of terms that arise in the sum (4.7). For sufficiently earlyquanta for which the argument of ˆ F is larger than 1, we can drop the ˆ F factor by Eqs.(4.8), and the computation reduces to that of Sec. 3.8. For later quanta for which theargument of ˆ F is less than one in absolute value, the computation is modified. Theseare the terms with | u − u n | (cid:46) ∆ ∼ M , that is, the final ∼ M quanta in the sum.These terms will enhance the fluctuations at high multipole orders l .We now make an order of magnitude estimate the spectrum of fluctuations asa function of angular scale of the expression (4.7). We specialize to high multipoleorders l , for which the sum over quanta will be dominated by the late quanta justdiscussed. For these terms we can drop the factor ϕ n ( θ A ), since its dependence on θ A is exponentially small at large l , and we have ϕ n ∼ l . We also approximate ∆ ( u ) by its final value ∆ ( u ), and M n by its final value at u = u which we denotesimply by M . We assume for simplicity that there is a value ¯ n of n for which u ¯ n = u ,and define j = n − ¯ n , so that to a good approximation we have u n = u + jM . Weassume initially that ∆ is fixed with ∆ = | ∆ | (cid:29) M ; later we will consider the effectof fluctuations in ∆ . We will also approximate ∆ /M by the integer l ∗ that is closestto it, l ∗ = (cid:20) ∆ M (cid:21) , (4.9)and define µ to be the cosine of the angle between n and ∆ . With these definitionsand approximations we have that the relevant terms in Eq. (4.7) are D Φ tr ( u , θ A ) = 4 πM P l ∗ (cid:88) j = − l ∗ (cid:15) ¯ n + j ˆ F ( l ∗ µ − j ) , (4.10)where the subscript “tr” denotes the transient contribution to Φ from the late incom-plete quanta. – 27 –e can divide up the sphere into 2 l ∗ strips of width ∼ /l ∗ , with the k th stripgiven by µ k ≤ µ ≤ µ k +1 for − l ∗ ≤ k < l ∗ , where µ k = k/l ∗ . On the k th strip the termsin the sum (4.10) with j > k + 1 vanish, while the terms with j < k are constant, byEqs. (4.8), which yields D Φ tr ( u , θ A ) = 4 πM P (cid:40) k − (cid:88) j = − l ∗ (cid:15) ¯ n + j + (cid:15) ¯ n + k ˆ F [ l ∗ ( µ − µ k )] + (cid:15) ¯ n + k +1 ˆ F [ l ∗ ( µ − µ k +1 )] (cid:41) . (4.11)We now estimate the total power in the fluctuations by squaring and integrating overthe two sphere, which reduces to a sum over strips of the integral over each strip. In thiscalculation we drop the second and third terms in the brackets in Eq. (4.11), therebymaking a fractional error of order 1 /l ∗ (cid:28)
1. The projection operator P subtracts off l = 0 , (cid:15) ¯ n + j with δ(cid:15) ¯ n + j = (cid:15) ¯ n + j − (cid:104) (cid:15) ¯ n + j (cid:105) . Italso changes the final answer by a factor of two which we will neglect. We obtain (cid:28)(cid:90) d Ω( D Φ tr ) (cid:29) ∼ M l ∗ l ∗− (cid:88) k = − l ∗ (cid:42)(cid:32) k − (cid:88) j = − l ∗ δ(cid:15) ¯ n + j (cid:33) (cid:43) ∼ M l ∗ (cid:88) k ( k + l ∗ ) ∼ l ∗ M , (4.12)where the angular brackets denote expected value and we have used Eq. (2.3).Now the function (cid:80) j δ(cid:15) ¯ n + j describes a random walk, and hence the spectrum d ( D Φ tr ) /d ln l of the fluctuations scales ∝ /l where we are using the notation (4.1),since this is a well-known property of random walks [64]. This powerlaw spectrumcontinues up to the maximum scale l ∼ l ∗ of the individual strips. Combining this withthe normalization (4.12) and the definition (4.1) (with Φ tr replaced by D Φ tr ) yields d ( D Φ tr ) d ln l ∼ l ∗ M l Θ( l ∗ − l )Θ(2 − l ) , (4.13)where Θ is the step function. Using D ∼ l from Eq. (3.18) it follows that d Φ d ln l ∼ l ∗ M l Θ( l ∗ − l )Θ(2 − l ) . (4.14)So far in this discussion we have treated ∆ as fixed. We now take into accountthat ∆ has a distribution that is very nearly Gaussian, by the central limit theorem,since it is a sum of a large number of independent contributions (cf. Sec. 2.2 above): d P d ln ∆ ∼ ∆ σ exp (cid:20) − ∆ σ (cid:21) , (4.15)where σ is the variance. We can now integrate this against the expression (4.14) toget the total spectrum: d Φ d ln l ( l ) = (cid:90) d ln ∆ d Φ d ln l ( l ; ∆) d P d ln ∆ (∆) , (4.16)– 28 –hich using Eq. (4.9) yields the final result (4.2). Acknowledgments
I thank Abhay Ashtekar, Venkatessa Chandrasekharan and Kartik Prabhu for helpfuldiscussions. This research was supported in part by NSF grants PHY-1404105 andPHY-1707800.
A Derivation of late time predictions of stochastic process
In this appendix we derive the late time predictions (2.4) of the Newtonian stochasticmodel (2.1) of black hole evolution of Sec. 2.We start by defining ¯ M n = (cid:104) M n (cid:105) , δM n = M n − ¯ M n , (A.1a)¯ t n = (cid:104) t n (cid:105) , δt n = t n − ¯ t n , (A.1b)where the angular brackets denote an expectation value. Substituting the decomposi-tion (A.1a) into the mass evolution equation (2.1a), expanding in powers of δM n , andseparating the expected value and the remaining part of the equation gives¯ M n +1 = ¯ M n −
12 ¯ M n (cid:20) O (cid:18) δM n ¯ M n (cid:19)(cid:21) , (A.2a) δM n +1 = δM n + (cid:18) δM n M n − δ(cid:15) n ¯ M n (cid:19) (cid:20) O (cid:18) δM n ¯ M n (cid:19)(cid:21) , (A.2b)where δ(cid:15) n = (cid:15) n − (cid:104) (cid:15) n (cid:105) . The first equation gives just the usual semiclassical evolution ofthe expected mass of the black hole, while the second gives the evolution of the massfluctuations.The solution for the expected mass can be obtained by approximating Eq. (A.2a)as a differential equation, and is ¯ M n = (cid:113) M i + 1 − n (cid:20) O (cid:18) M i − MM i M (cid:19)(cid:21) . (A.3)Here on the right hand side we have written M for ¯ M n inside the error estimates,for simplicity. We have also temporarily dropped the error term δM n / ¯ M n from Eq. The error estimate can be obtained by computing the solution to subleading order, which is¯ M n = Q n − ln( Q n /M i ) / (4 Q n ) with Q n = M i + 1 − n . – 29 –A.2a); we will restore this fractional error estimate at the end of the computation,when we have computed (cid:104) δM n (cid:105) . Next from the expected value of the time evolutionequation (2.1b) and the definitions (A.1) we find ¯ t n +1 = (cid:80) nk =1 ¯ M k . Converting this toan integral and using the expression (A.3) gives¯ t n = 23 ( M i − ¯ M n − ) (cid:20) O (cid:18) M i (cid:19) + O (cid:18) M (cid:19)(cid:21) . (A.4)We now turn to computing the fluctuations. From Eq. (A.2b) we obtain δM n +1 = − n (cid:88) j =1 (cid:34) n (cid:89) k = j +1 (cid:18) M k (cid:19)(cid:35) δ(cid:15) j ¯ M j (cid:20) O (cid:18) δM n ¯ M n (cid:19)(cid:21) (A.5)The product inside the square brackets can be evaluated by taking the logarithm,converting the sum to an integral, and using Eq. (A.3), which yields δM n +1 = − q n +1 ¯ M n (cid:20) O ( ¯ M − n ) + O (cid:18) δM n ¯ M n (cid:19)(cid:21) , (A.6)where q n +1 = (cid:80) nk =1 δ(cid:15) j . Squaring and taking the expected value gives (cid:104) δM n +1 (cid:105) = n/ (4 ¯ M n ). Using this expression to evaluate the error estimate in Eq. (A.6) and elimi-nating n in favor of M = ¯ M n using (A.3) finally yields (cid:104) δM n (cid:105) = M i − M M (cid:34) O (cid:32) (cid:112) M i − M M (cid:33)(cid:35) . (A.7)Note that this is the fluctuation in mass at fixed n , to be distinguished from the morephysically relevant fluctuations in mass at fixed time t [cf. Eq. (2.4c) above], which wecompute below.We next compute the fluctuations δt n . From the time evolution equation (2.1b) weobtain δt n +1 = (cid:80) nk =1 δM k , and squaring and taking the expected value using Eq. (A.6)gives (cid:104) δt n +1 (cid:105) = n (cid:88) k,l =1 min( k − , l − M k ¯ M l (cid:34) O (cid:32) √ k ¯ M k (cid:33) + O (cid:32) √ l ¯ M l (cid:33)(cid:35) . (A.8) We use the approximation b (cid:88) k = a f ( k ) = (cid:90) b + a − dkf ( k ) (cid:20) O (cid:18) f (cid:48)(cid:48) f (cid:19)(cid:21) . – 30 –onverting the sums to integrals as before yields (cid:104) δt n +1 (cid:105) = 12 ( M i − M ) ( M i + M/ (cid:34) O (cid:32) (cid:112) M i − M M (cid:33)(cid:35) . (A.9)Now in this simple discrete model of the black hole evolution, the black hole mass M ( t ) at a given time t is obtained by evaluating M n at the value of n for which t n isclosest to t . Hence the fluctuations in mass at fixed time t are given by δM ( t ) = δM n − M (cid:48) ( t ) δt n , (A.10)where the right hand side is evaluated at the value of n = n ( t ) obtained by solvingEqs. (A.3) and (A.4), and the derivative is given by M (cid:48) ( t ) = − / (2 M ). SquaringEq. (A.10), taking the expected value, dropping the cross term which one can show issubdominant, and using the expressions (A.7) and (A.9) finally yields the result (2.4c).Note that the mass fluctuations (2.4c) at fixed time are dominated by the fluctuationsin t n .To compute the momentum fluctuations, we expand Eq. (2.1c) in powers of δM n and solve, obtaining p n +1 = n (cid:88) k =1 (cid:15) k δ k ¯ M k (cid:20) − δM k ¯ M k + O (cid:18) δM k ¯ M k (cid:19)(cid:21) . (A.11)Squaring and taking the expected value gives (cid:104) p n +1 (cid:105) = n (cid:88) k =1
12 ¯ M k (cid:20) O (cid:18) δM k ¯ M k (cid:19)(cid:21) , (A.12)since δM k , (cid:15) k and δ k are statistically independent. Converting the sum to an integraland using Eqs. (A.3) and (A.7) now yields the formula (2.4b).Finally, by combining the position evolution equation (2.1d) with the expression(A.11) for the momentum yields x n +1 = n (cid:88) r,s =1 r − (cid:88) k =1 s − (cid:88) l =1 (cid:15) k δ k (cid:15) l δ l ¯ M k ¯ M l (cid:20) O (cid:18) δM ¯ M (cid:19)(cid:21) . (A.13)Taking the expected value and converting the sums to integrals gives (cid:104) x n +1 (cid:105) = 12 n (cid:88) r,s =1 min( r,s ) − (cid:88) k =1 M k (cid:20) O (cid:18) δM ¯ M (cid:19)(cid:21) = 4 (cid:90) M i M n dM r (cid:90) M i M n dM s (cid:90) M i max( M r ,M s ) dM k M r M s M k (cid:20) O (cid:18) δM ¯ M (cid:19)(cid:21) , (A.14)and evaluating the integral gives the expression (2.4a).– 31 – Independent charges in stationary regions of future nullinfinity
In this appendix we derive the relationships between the various charges of the ex-tended BMS algebra that apply in stationary regions of future null infinity, and showthat the independent charges can be taken to be the 4-momentum P α , the angularmomentum J αβ , and the super center-of-mass charges. Equivalently, we show that thesupermomentum and superspin charges are determined in terms of the other charges,and so can be neglected for our purposes.In the canonical Bondi frame associated with the stationary region, the metricfunctions take the simple form (3.8). We now make a nonlinear BMS transformation toa general Bondi frame, of the form (FN,2.12), following Appendix B of FN. Quantitiesin the new frame will be denoted with overbars. The transformation is parameterizedin terms of a conformal isometry ϕ : S → S of the 2-sphere into itself, and a function β on the two sphere [denoted by α in Eq. (FN,2.12)]. The Bondi mass aspect in thisgeneral frame is given from Eqs. (FN,B5) and (3.8) as¯ m ( θ A ) = m ω − ϕ , (B.1)where ω ϕ ( θ A ) is defined by ϕ ∗ h AB = ω − ϕ h AB and ϕ ∗ is the pullback. The quantity ω ϕ is determined by the boost part of the Lorentz transformation ϕ and is given explicitlyby ω ϕ = cosh ψ − n · m sinh ψ, (B.2)where n = (sin θ cos φ, sin θ sin φ, cos θ ), ψ is the rapidity parameter of the boost and m is a unit vector giving the direction of the velocity of the general frame with respectto the canonical frame. The 4-momentum in the general frame is now given from Eqs.(3.5), (FN,3.7) and (FN,3.9) as¯ P α = ( ¯ P , ¯P ) = 14 π (cid:90) d Ω(1 , n ) m ω − ϕ = m (cosh ψ, sinh ψ m ) . (B.3)We therefore see that that the Bondi mass aspect (B.1) and all the supermomentumcharges are determined in terms of the Bondi 4-momentum (B.3).Turn now to the superspin and super center-of-mass charges, which are encodedin the function ˆ N A defined by Eq. (3.6). The transformation law for this function instationary regions of I + can be obtained by combining Eqs. (FN,B1), (FN,B2) and(3.5) together with footnote 25 of FN and is¯ˆ N A = ω − ϕ ϕ ∗ ˆ N A + 3 ω − ϕ ( ϕ ∗ m ) D A β + βD A ( ω − ϕ ϕ ∗ m ) . (B.4)– 32 –ere the overbar denotes the value of this function in the general Bondi frame. Thetransformation law (B.4) can be simplified by defining the new quantity S A = ˆ N A + 32 mD A Φ + 12 Φ D A m, (B.5)where the potential Φ for the electric parity piece of the shear tensor C AB is definedin Eq. (3.9). Now in the canonical BMS frame, S A will coincide with ˆ N A and willbe purely l = 1 and of magnetic parity, encoding the intrinsic spin of the spacetime.Combining Eqs. (B.4), (B.5), (FN,B5), (FN,B6) and (FN,B8) yields that in the generalBMS frame this function will be¯ S A = ω − ϕ ϕ ∗ S A + 3 ¯ mD A β + β D A ¯ m, (B.6)where β consists of the l = 0 , β . Combining this with a barred versionof Eq. (B.5) gives¯ˆ N A = −
32 ¯ mD A ¯Φ −
12 ¯Φ D A ¯ m + ω − ϕ ϕ ∗ S A + 3 ¯ mD A β + β D A ¯ m. (B.7)The last three terms in this equation are determined from the intrinsic spin, 4-momentumand from the Poincar´e transformation ( ϕ, β ) relating the two frames, so they are de-termined by the linear and angular momentum ¯ P α and ¯ J αβ . It follows that ¯ˆ N A andthe superspin and super center-of-mass charges are determined from ¯ J αβ and ¯ P α andthe electric parity potential ¯Φ, in a general Bondi frame in a stationary region .We next derive the explicit form of the super center-of-mass charges, expanding tosecond order in the velocity of the boost. Since the metric is stationary, shifting thesupertranslation β by a constant times ω ϕ does not affect the metric in the generalBondi frame, from Eqs. (FN,2.12). Therefore without loss of generality we take the l = 0 component of β to vanish, and we parameterize the translation β as β = β i n i .From Eq. (B.7) we can now read off the orbital angular momentum (3.15b) and the l = 2 super center-of-mass charge (C.4) in the general frame, using Eqs. (B.1) and(B.2) to expand to expand in powers of the velocity v = m tanh ψ of the boost:¯ J i = − J ij v j + m (cid:18) v (cid:19) β i − m ¯ Q ij v j , (B.8a)¯ J ij = − m ¯ Q ij (cid:2) O ( v ) (cid:3) + 85 v k v k + 125 m β . (B.8b)Here we have used the definition (3.10), J ij is the intrinsic angular momentum inthe canonical Bondi frame, and the angular brackets < . . . > denote the symmetric This result was previously derived within a limited approximation in Sec. III.E of FN. – 33 –racefree projection. Combining Eqs. (B.8) to eliminate β i , dropping the intrinsicangular momentum terms and using Eq. (B.3) gives¯ J ij = − m ¯ Q ij (cid:2) O ( v ) (cid:3) + 125 m ¯ J (cid:2) O ( v ) (cid:3) . (B.9)A similar calculation for l ≥ J L = − m g l ¯ Q L + O ( v ) . (B.10) C Choice of basis of algebra of charges
In this appendix we discuss two different versions of the super center-of-mass charges.To explain these versions, it is useful to distinguish between two different kinds of timeevolution of the charges. The first is just the kind discussed in Sec. 3.3, associated withevaluating the charges as surface integrals on cuts of I + of the form u = constant,and varying u .A second kind of time evolution is associated with the choice of basis in the algebraof asymptotic symmetries. Consider for example the orbital angular momentum J i that is associated via Eqs. (3.4) and (3.5) with the boost symmetry generator (cid:126)ξ = D A n i ∂ A − un i ∂ u . This choice of boost symmetry is associated with a particular choiceof origin of the retarded time coordinate u (which in the body of the paper we took tobe the time of formation of the black hole, see Sec. 3.4). However, by conjugating thesymmetry generator with a time translation u → u − u where u is a constant, we canobtain the new boost symmetry generator (cid:126)ξ = D A n i ∂ A − ( u − u ) n i ∂ u . (C.1)We denote the corresponding charge by J i ( u, u ), where the first argument reflects thedependence on the cut of I + and the second argument the choice of generator. Thedependence on u is given by J i ( u, u ) = J i ( u,
0) + u P i ( u ); (C.2)changing u amounts to a change of basis in the algebra of symmetry generators orequivalently in the algebra of charges. The center of mass at retarded time u , given by X i ( u ) = 1 P ( u ) J i ( u, u ) = 1 P ( u ) [ J i ( u,
0) + uP i ( u )] , (C.3)encodes both types of time dependence. It is this quantity and not the charge J i ( u, u , from Eqs. (3.1), (3.2),(3.5) and (3.6). – 34 –here is an exactly analogous story for the super center-of-mass charges [48]. Wespecialize for simplicity to the quadrupole l = 2 case. Consider the superboost sym-metry generator given by (3.4) for α = 0, Y A = D A ( n i n j ), (cid:126)ξ = D A ( n i n j − δ ij / ∂ A − u ( n i n j − δ ij / ∂ u . (C.4)We denote the corresponding charge (3.5) by J ij ( u ), a symmetric traceless tensor [cf. thediscussion around Eq. (3.7) above]. As before by conjugating with a time translation wecan obtain a new superboost symmetry generator, given by Eq. (C.4) with u replacedby u − u , and we denote the corresponding charge by J ij ( u, u ). The dependence on u is given by, from Eqs. (3.5) and (C.4), J ij ( u, u ) = J ij ( u,
0) + 3 u P ij ( u ) , (C.5)where P ij = 14 π (cid:90) d Ω m (cid:18) n i n i − δ ij (cid:19) (C.6)is the l = 2 supermomentum charge. As before we can define a super center-of-massquantity that incorporates both types of time evolution, and which is the quantity thatappears most directly in the metric, via X ij ( u ) = 1 P ( u ) J ij ( u, u ) = 1 P ( u ) [ J ij ( u,
0) + 3 u P ij ( u )] . (C.7)We will call the charge (C.5) the super center-of-mass charge, and the quantity (C.7)the comoving super center-of-mass, following Comp`ere [48].In the special case of stationary regions of I + the charges J ij ( u, u ) and J i ( u, u )are related by the formula (B.9) derived in Appendix B: J ij ( u, u ) = − m Q ij ( u ) (cid:2) O ( v ) (cid:3) + 125 m J ( u ) (cid:2) O ( v ) (cid:3) . (C.8)Here m is the rest mass associated with the Bondi 4-momentum, v is the velocityof the Bondi frame with respect to the canonical Bondi frame, Q ij is defined in Eq.(3.10), and the angular brackets denote symmetric tracefree projection. This formula isconsistent with the transformation laws (C.2) and (C.5) because we have in stationaryregions P ij = 45 m (cid:18) P i P j − P δ ij (cid:19) (cid:20) O (cid:18) P m (cid:19)(cid:21) , (C.9)from Eqs. (C.6), (B.1) and (B.2). Now evaluating Eq. (C.8) at u = u , dividing by P ( u ) and using the definitions (C.3) and (C.7) gives the relation between the comovingsuper center-of-mass and normal center of mass X ij ( u ) = − Q ij ( u ) (cid:2) O ( v ) (cid:3) + 125 m X ( u ) (cid:2) O ( v ) (cid:3) . (C.10)– 35 – Derivation of changes in charges in stationary-to-stationarytransitions
In this appendix we derive the formulae (3.12), (3.16) and (3.17) for the changes inBMS and extended BMS charges in stationary to stationary transitions in terms offluxes to null infinity of mass-energy or gravitational-wave energy. A similar analysisin a different notation can be found in Ref. [48].The change (3.12) in Bondi 4-momentum is obtained by multiplying Eq. (FN,4.3)by (1 , n ), integrating over solid angles, and noting that the l = 0 , Q Y the charge(3.5) specialized to α = 0 . To derive the change in this charge we first define thesubleading memory observables (cid:103) ∆Φ = (cid:90) u u du u∂ u Φ , (D.1a) (cid:103) ∆Ψ = (cid:90) u u du u∂ u Ψ . (D.1b)These observables parameterize the relative displacement, produced by a burst ofgravitational waves, of two test masses that are initially co-located with an initialrelative velocity; see, e.g, Sec. II.A of Ref. [65]. Now differentiating Eq. (3.6) withrespect to u and combining with Eqs. (FN,2.7) and (FN,2.11) gives ∂ u ˆ N A = − π ˆ T uA + 4 πuD A ˆ T uu + u D A ( N BC N BC ) − N AB D C C BC + 38 C AB D C N BC + 18 D B C AC N BC − D B N AC C BC − u D B D A D C N BC + u D B D B D C N AC − u D A D B D C N BC − π∂ u ˆ T rA . (D.2)We now multiply by Y A and integrate over solid angles and over u . The left hand sidethen becomes the change ∆ Q Y in the charge, from Eq. (3.5). On the right hand side,the last term gives a vanishing contribution since we assume the stress energy tensorvanishes at u = u and u = u . The second and third terms can be simplified usingthe definition (3.14) of u -weighted energy flux (cid:102) ∆ E , while the first, fourth, fifth, sixthand seventh terms can be similarly simplified using the definition (3.13) of angularmomentum flux ∆ E A , making use of Eq. (FN,3.23). The eighth, ninth and tenth terms The electric parity piece (cid:103) ∆Φ is called center-of-mass memory [61], while the magnetic parity piece (cid:103) ∆Ψ is called spin memory [58]. – 36 –an be written in terms of the subleading memory observables (D.1) using Eqs. (3.3)and (3.9), and using R ABCD = h AC h BD − h AD h BC . The final result is ∆ Q Y = − (cid:90) d Ω (cid:20) ∆ E A Y A + 12 (cid:102) ∆ E D A Y A (cid:21) − π (cid:90) d Ω (cid:104)(cid:103) ∆Φ D ( D A Y A ) + (cid:103) ∆Ψ D ( (cid:15) AB D A Y B ) (cid:105) , (D.3)where the differential operator D is given by Eq. (3.18). The changes (3.16) in angularmomentum components can now be obtained by taking Y A = 2 e A [ i n j ] and Y A = e Ai ,using D A e Ai = D n i = − n i and the fact that the operator D annihilates the l = 0 , C AB , which encodes the super center-of-mass charges. We multiply Eq.(FN,4.3) by the projection operator P that sets to zero the l = 0 , PD = D to obtain D ∆Φ = 4 π P ∆ E + P ∆ m. (D.4)To evaluate ∆ m we now specialize to an initial rest Bondi frame in which the spatialcomponents of the Bondi 4-momentum vanish, so that the initial Bondi mass aspect isa constant m , from Eqs. (B.1) and (B.3). The final Bondi mass aspect will be of theform (B.1) with m replaced by m + δm for some δm , for some boost parameters ψ and m . Eliminating the boost parameters in terms of the radiated spatial momentumusing Eqs. (B.2) and (B.3) and expanding to second order in this momentum now givesthe formula (3.17). References [1] S.W. Hawking,
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