Remnant-free Moving Mirror Model for Black Hole Radiation Field
MMIT-CTP/5144
Remnant-free Moving Mirror Model for Black Hole Radiation Field
Michael R.R. Good , , Eric V. Linder , , and Frank Wilczek , , , Physics Department, Nazarbayev University, Astana, Kazakhstan. Energetic Cosmos Laboratory,Nazarbayev University, Astana, Kazakhstan. Berkeley Center for Cosmological Physics & Berkeley Lab,University of California, Berkeley, CA, USA. MIT, Cambridge, MA, USA. T. D. Lee Institute and Wilczek Quantum Center,Shanghai Jiao Tong University, Shanghai, China. Arizona State University, Tempe, AZ, USA. Stockholm University, Stockholm, Sweden. (Dated: September 4, 2019)We analyze the flow of energy and entropy emitted by a class of moving mirror trajectories whichprovide models for the radiation fields produced by black hole evaporation. The mirror radiationfields provide natural, concrete examples of processes that follow thermal distributions for longperiods, accompanied by transients which are brief and carry little net energy, yet they ultimatelyrepresent pure quantum states. A burst of negative energy flux is a generic feature of these fields,but it need not be prominent.
In the context of quantum field theory, moving mirrormodels consider the effect of imposing boundary condi-tions on a moving surface [1–3]. There are prospects torealize interesting versions of these models experimen-tally [4–6] (and see below). Moving mirror models arethought to provide useful idealizations of black hole evap-oration [7], though it remains unclear how far the analogycan be taken. Here we analyze a class of moving mirrortrajectories leading to radiation fields which have severalproperties that are remarkable in themselves, and desir-able in a model of black hole evaporation: • The emitted radiation follows a thermal (Planck)spectrum for arbitrarily long times. This simulatesthe Hawking radiation process which, according toa semiclassical analysis, is thought to dominate theevaporation of an isolated black hole over most ofits lifetime. • The radiation field is limited in space and time. • Apart from the radiation, the quantum fields ap-proach their normal ground states (i.e. “vacuum”)at early and late times. In this sense, there is noremnant [8–12]. • Despite the thermal appearance of the bulk of theradiation field, the final state, including the radia-tion field, is a pure quantum state.This last point is especially interesting, for it embodiesthe “information loss” paradox and, within this class ofmodels, resolves it. This bundle of properties suggeststhat with use of these trajectories the moving mirror ide-alization of evaporating black hole radiation may be re-markably appropriate.Through a sum rule connecting the flows of energyand entanglement entropy, we show that these propertiesimply a period of negative energy flux. The required flux need not be large, however, if it occurs at near-maximalentropy.
Model.
In moving mirror models we impose Dirich-let boundary conditions along the worldline of a “mir-ror” on quantum fields in 1+1 dimension. For concrete-ness we will focus on the case of a single massless scalarfield, though most of our results generalize to generalconformal field theories. For our detailed analysis, wewill focus on emissions accompanying a particularly sim-ple and symmetric 2-parameter family of trajectories. Asthe analysis will make clear, the radiation fields emittedby moving mirrors following a wide variety of trajectoriesthat share some broad qualitative features will share thegood properties of this symmetric family.Our model trajectories are given by t ( x ) = − x − sinh(2 κx ) g . (1)where κ and g are free parameters. They are inspired bythe “black mirror” trajectory [13–15] t ( x ) = − x − e κx κ . (2)The black mirror evolves to an asymptotic light-like tra-jectory [16, 17] of an eternal black hole horizon [18, 19].It emits an exactly Planckian spectrum of radiation withtemperature T = κ π . Our trajectories are P T symmetricversions of the black mirror, in which we also introducean additional parameter.Fig. (1) displays trajectories defined by Eq. (1) within aPenrose conformal diagram. Note that for large t the mir-ror becomes static, ˙ x → − κt : that is, t ( x ) approaches avertical asymptote as x → ±∞ . Thus this trajectory dis-plays asymptotically static boundaries [17, 20–22]. Near t = 0, on the other hand, the mirror bends toward anull trajectory, where its velocity approaches the speed a r X i v : . [ g r- q c ] A ug of light. The mirror trajectory simulates the effect of dy-namical geometry in generating radiation fields. Whenstatic, i.e. at early and late times, it plays the role of thecenter of radial coordinates, but at intermediate times itsimulates the mathematics of the black hole horizon inHawking’s calculation.Consistent with the “no remnant” interpretation, thestatic initial and final states represent empty space. Moreprecisely, we have the velocity dxdt ≡ V ( x ) = − gg + 2 κ cosh(2 κx ) . (3)It is zero in the limit x → ±∞ , and has its maximumabsolute value at x = 0, where V max = − g/ ( g + 2 κ ). For g (cid:29) κ , V max → −
1. At late times the velocity scalesas inverse time. Small and innocuous modifications ofthe trajectory at late (and early) times could bring themirror strictly to rest finally (and initially). To do thisone can employ smooth but non-analytic functions, asare used to construct smooth partitions of unity [23]. I L + I R + I L - I R - i i i + i - FIG. 1. The trajectory, Eq. (1), plotted in a conformal dia-gram. Here κ = 1 and g = 10 n . Red, Blue, Green, Black are n = 0 , , ,
3, respectively.
Radiation Field.
The expectation value of the stresstensor can be evaluated analytically, in terms of the mir-ror trajectory t ( x ). It is straightforward to evaluate theenergy flux F ( x ) at the mirror, as derived from the stresstensor [1], at right null infinity.As can been seen in Fig. (2), when g/κ (cid:29) F thermal = κ π , (4)which is the energy flux value associated with thermalemission in the analog black mirror. To leading order in κ/g (cid:28) F ( x ) /F thermal = 1 − (2 κx ) + O ( κ/g ) . (5)A striking feature and perhaps surprising feature is theburst of negative energy flux around x = 0. It saturatesto twice the height of the plateau, and finite width asthe plateaus expand. We will demonstrate below, usingthe connection between energy flow and geometric en-tropy, that negative energy flux is a necessary feature ofthe radiation fields associated with remnant-free movingmirror models, following from unitarity.The total energy flux observed at right null-infinity isfinite. For large g (cid:29) κ we have E = κ π ln gκ . (6)The moving mirror model contains no variable that corre-sponds directly to the black hole mass, but one can definean effective mass parameter by imposing the semiclassi-cal (Hawking) relationship between the radiation flux andmass. For our trajectories the effective mass, so defined,remains constant on the long plateaus. Thus they do notreflect the expected increase of temperature (associatedwith decrease of black hole mass). It should be possibleto construct approximately “self-consistent” trajectorieswhich incorporate that feature, but we will not attemptthat here. Here we only note that we are free to choose g in such a way that the energy E corresponds to themass of the black hole which emits radiation at tempera-ture T . This entails ln( g/κ ) ∼ M , with M measured inPlanck units. Thus g/κ (cid:29) - - x - - - - F FIG. 2. The energy flux from the symmetric mirror is plottedwith κ = √ π = 12 .
279 so that thermal equilibrium is seento be F ( x ) = 1. We show g = 10 n where Red, Blue, Green,Yellow, Orange, Black is n = 1 , , , , ,
6, respectively. Notethe color scheme differs from Fig. (1) but is consistent withall other figures.
The central quantities for calculating particle produc-tion are the beta Bogolyubov coefficients, which can becalculated analytically. The particle spectrum per modeper mode is | β | = ωω (cid:48) e − πωκ π κ ω p (cid:12)(cid:12)(cid:12)(cid:12) K iωκ (cid:18) ω p g (cid:19)(cid:12)(cid:12)(cid:12)(cid:12) , (7)where K n ( z ) is a modified Bessel function of the secondkind and ω p ≡ ω (cid:48) + ω , the sum of the in and outgoingmode frequencies. The spectrum, N ω , or particle countper mode, detected at right null infinity surface I + R isfound by integrating Eq. (7), N ω = (cid:90) ∞ | β | dω (cid:48) . (8)Fig. (3) illustrates the results for different g values. Forlarge g we approach a Planck spectrum, but not uni-formly. There is always a zero at strictly 0 frequency,and the total number of radiated particles is always fi-nite. The total energy can be retrieved using the particlesas a sum over quanta, E = (cid:90) ∞ (cid:90) ∞ ω · | β | dω dω (cid:48) , (9)This consistent global energy result via Eq. (7) is thesame energy as Eq. (6), which was derived earlier by in-tegrating the local stress tensor.For large g/κ (cid:29) | β | ≡ N ωω (cid:48) thenhas the simple form, to lowest order and assuming a trueCauchy principal value, N ωω (cid:48) = e − ωπ ω (cid:48) csch( πω )2 π ( ω (cid:48) + ω ) ≈ πω (cid:48) e πω − , (10)in units of κ , where the last step simplifies the prefactorby considering the large frequency regime, ω (cid:48) (cid:29) ω . Theappearance of the Planck distribution for large g is con-sistent with the constant plateau energy flux of Eq. (4)for g (cid:29) κ as seen in Fig. (2) and the flattening plateaufor constant particle emission over time in Fig. (5). Itparallels Hawking’s calculation of radiation from a fixedblack hole background.For any finite value of r ≡ g/κ the total number of par-ticles created is finite, though it increases without limitas r grows. With r = 10 we find N ≈ particles.The absence of an “infrared catastrophe” in the numberof soft quanta reflects the absence of a physical remnant. Entropies.
Both thermodynamic entropy and entan-glement entropy play an important role in particle cre-ation models [24]. For our purposes, the most enlight-ening measure of entanglement (see also harvesting [25])is the renormalized entanglement entropy of the state atfuture null infinity to the left (or right) of a given valueof null time u . Heuristically, this represents a flow ofentanglement entropy through u . In the moving mirrormodel, it has the simple form − S = η , where η is therapidity of the mirror as it crosses u . In our model wehave a simple expression for S expressed as a function of x , the corresponding position of the mirror: S ( x ) = 112 ln (cid:18) gκ cosh(2 κx ) (cid:19) . (11) ω N ω FIG. 3. The spectrum, N ω , Eq. (8), or particle count permode detected at right null infinity, of the asymptoticallystatic thermal mirror is plotted vs frequency ω , in units of κ . This asymptotically static solution shows no infrared di-vergence in the number of soft particles. The entropy vanishes for the asymptotic spatial posi-tions, as it should since the evolution is unitary and en-tails no information loss. Fig. (4) illustrates this entropyflux. - - x S ( x ) FIG. 4. The asymptotically static mirror with finite energyand finite particle count has entropy flux, Eq. (11), plottedwith different g values (same color scheme as Fig. 2). The energy flux F is related analytically to the entropy S according to (e.g. [22, 26, 27]) F ( u ) = 12 π (6 S (cid:48) + S (cid:48)(cid:48) ) (12)= 12 π e − S ddu (cid:18) e S ddu S (cid:19) . (13)From Eq. (13) and the assumption that S becomes con-stant for u → ±∞ we derive the sum rule, ∞ (cid:90) −∞ du e S ( u ) F ( u ) = 0 . (14)Thus, on general principles, the flux F ( u ) will have anegative region for the radiation field of a remnant-freepure state. Note that for purposes of fulfilling the sumrule the negative flux has greatest leverage when it occursat the maximal S ( u ), as we see realized in Figs. (2) and(4). The flux of statistical mechanical entropy associatedwith the thermally-distributed flux on the plateau tracks S (cid:48) for x > − S (cid:48) for x < Particle Counts in Time.
Time evolution can be re-solved with the use of wave packets [7] β (cid:15)jn that pick outfrequencies near j(cid:15) at retarded time u = 2 πn(cid:15) − withwidth 2 π(cid:15) − . This localization of the global beta coef-ficients corresponds to the sensitivity response of a par-ticle detector at a given time, frequency, and bandwidth[17, 28]. With large g (cid:29) κ , and good time resolution(i.e., large (cid:15) ; the particles pile up in the single j = 0 bin)one observes a flattening plateau, Fig. (5). Remarkably,the plateau extends through the n = 0 time bin, withno obvious scar from accompanying the negative energyflux. - - - - - - - - - - - - n N jn Particle Flux in Time
FIG. 5. The discrete spectrum, N jn , time evolved. Here thesystem is set with κ = 1, and g = 10 . The detector is setwith j = 0, n = ( − , (cid:15) = 4. Notice the flattened plateaucentered around n = 0. Summary and Discussion.
The moving mirror modelsdiscussed here produce radiation fields that look ther-mal for long periods of time, and limited transients, yetrepresent pure states. This, together with their geomet-ric character, suggest that they may provide instructivemodels for quantum evaporation of black holes, whichplausibly – yet paradoxically – have those properties.A striking feature, here simply shown to be generic, ofremnant-free moving mirror models is the occurrence ofnegative energy flux. Although states with locally nega-tive values of energy density are known to occur in severalcontexts, including the intensely studied Casimir effect,their occurrence is somewhat unusual and there seemsto be no general understanding of their properties. In our context, the derivation of the sum rule Eq. (14) con-nects negative energy flux to the purity of the quantumradiation field, though it does not provide a mechanisticexplanation of the connection.Within the specific models analyzed here, the inde-pendent signatures of the negative energy flux appear tobe subtle. Specifically, as a fraction of the overall pro-cess it is strictly limited in time and energy (for g (cid:29) κ , E NEF = − ( κ/ π )[ √ − tanh − (cid:112) / ≈ − . κ ), andit does not appear prominently in the response of quasi-realistic particle detectors, Fig. (5), unlike many positiveenergy flux signatures [29].In order to satisfy the sum rule Eq. (14) with an incon-spicuous negative energy flux, it is important that theflux occur where the entanglement entropy S is large. Ifthis possibility is to be relevant to physical black holeevaporation, it should therefore act early in the blackhole’s history, or at regular intervals throughout. To ourknowledge no existing semiclassical treatment of blackhole evaporation, including ones which attempt to incor-porate back-reaction, contains such effects. It is conceiv-able that better approximations, either within generalrelativity itself or in the larger framework of string the-ory, could display them. It is tempting to speculate, inview of the connection to entropy, that entropic forcescome into play, modifying the space-time geometry. Aswe have seen, the required effects may not need to belarge.Independent of their possible connection to black holes,the unusual features predicted to occur in radiation fieldsproduced by moving mirror models are interesting inthemselves, especially as they confront the tension be-tween quantum purity and apparent thermality. Theavailability of arrays containing very large numbers ofmirrors whose orientation can be programmed flexiblymight offer another road (in addition to [4–6]) to realiz-ing such models. ACKNOWLEDGMENTS
MG is funded by the ORAU FY2018-SGP-1-STMMFaculty Development Competitive Research Grant No.090118FD5350 at Nazarbayev University, and the state-targeted program “Center of Excellence for Fundamentaland Applied Physics” (BR05236454) by the Ministry ofEducation and Science of the Republic of Kazakhstan.EL is supported in part by the Energetic Cosmos Lab-oratory and by the U.S. Department of Energy, Officeof Science, Office of High Energy Physics, under AwardDE-SC-0007867 and contract no. DE-AC02-05CH11231.FW’s work is supported by the U.S. Department of En-ergy under grant Contract Number DE-SC-0012567, bythe European Research Council under grant 742104, andby the Swedish Research Council under Contract No.335-2014-7424. [1] S. A. Fulling and P. C. W. Davies, Proc. Roy. Soc. Lond.A (1976) 393.[2] F. Wilczek, In
Black holes, membranes, wormholesand superstrings ed. S. Kalara and D. Nanopoulos,pp. 1-21, January 1992, (World Scientific, Singapore).[arXiv:9302096 [hep-th]].[3] A. Fabbri and J. Navarro-Salas,
Modeling black hole evap-oration , London, UK: Imp. Coll. Pr. (2005).[4] P. Chen and G. Mourou, Phys. Rev. Lett. , no. 4,045001 (2017).[5] C. M. Wilson, G. Johansson, A. Pourkabirian, et al. Na-ture (London), 479, 376, (2011).[6] H. Wang, M.P. Blencowe, C.M. Wilson, A.J. Rimberg,Phys. Rev. A 99, 053833 (2019).[7] S. W. Hawking, Commun. Math. Phys.
199 (1975).[8] M. R. R. Good, K. Yelshibekov and Y. C. Ong, JHEP , 013 (2017), [arXiv:1611.00809 [gr-qc]].[9] P. Chen, Y. C. Ong and D. h. Yeom, Phys. Rept. , 1(2015), [arXiv:1412.8366 [gr-qc]].[10] M. R. R. Good, Y. C. Ong, A. Myrzakul, K. Yelshibekov,Gen. Rel. Grav. , 92 (2019).[11] M. R. R. Good, Universe , 122 (2018).[12] A. Myrzakul and M. R. R. Good, MG15 Proceedings,World Scientific, (2019).[13] M. R. R. Good, P. R. Anderson and C. R. Evans, Phys.Rev. D , 065010 (2016), [arXiv:1605.06635 [gr-qc]].[14] P. R. Anderson, M. R. R. Good and C. R. Evans, MG14,[arXiv:1507.03489 [gr-qc]].[15] M. R. R. Good, P. R. Anderson and C. R. Evans, MG14,[arXiv:1507.05048 [gr-qc]]. [16] M. R. R. Good, Int. J. Mod. Phys. A , 1350008 (2013),[arXiv:1205.0881 [gr-qc]].[17] M. R. R. Good, P. R. Anderson and C. R. Evans, Phys.Rev. D , 025023 (2013), [arXiv:1303.6756 [gr-qc]].[18] R. D. Carlitz and R. S. Willey, Phys. Rev. D , 2327(1987).[19] M. R. R. Good and E. V. Linder, Phys. Rev. D ,065006 (2018), [arXiv:1711.09922 [gr-qc]].[20] W. R. Walker and P. C. W. Davies, J. of Phys. A, 15, 9,L477, (1982).[21] M. R. R. Good and E. V. Linder, Phys. Rev. D ,125010 (2017), [arXiv:1707.03670 [gr-qc]].[22] M. R. R. Good and E. V. Linder, Phys. Rev. D ,025009 (2019), [arXiv:1807.08632 [gr-qc]].[23] L. Tu, An Introduction to Manifolds , Ch 13. Universitext(2nd ed.) (2011).[24] C. Holzhey, F. Larsen and F. Wilczek, Nucl. Phys. B , 443 (1994), [arXiv:9403108 [hep-th]].[25] W. Cong, E. Tjoa and R. B. Mann, JHEP , 021(2019), [arXiv:1810.07359 [quant-ph]].[26] P. Chen and D. h. Yeom, Phys. Rev. D , 025016 (2017),[arXiv:1704.08613 [hep-th]].[27] E. Bianchi and M. Smerlak, Phys. Rev. D , 041904(2014), [arXiv:1404.0602 [gr-qc]].[28] P. R. Anderson, R. Clark, A. Fabbri and M. R. R. Good,Phys. Rev. D , 012345 (2019), [arXiv:1906.01735 [gr-qc]].[29] M. R. R. Good and Y. C. Ong, JHEP1507