Global Causal Structure of a Transient Black Object
aa r X i v : . [ g r- q c ] O c t Global Causal Structure of a Transient Black Object
Tehani Finch ∗ and James Lindesay † Computational Physics LaboratoryHoward University, Washington, D.C. 20059
Abstract
A singularity-free and spherically symmetric transient black object whose cen-ter remains always timelike, yet directly manifests a trapped region, has beenconstructed and numerically implemented. The exterior geometry is shown tobe similar to that of a long-lived transient black hole, with a few subtle dif-ferences. The large-scale global structure of the geometry is examined throughthe construction of a conformal diagram, which exhibits no event horizon andbears resemblance to that of a Minkowski spacetime. Since there is no singular-ity within the geometry, the evolution of the exchange of information betweentimelike observers, including those that fall through the trapped region, can bedirectly explored. The dynamics of generic “standard” communications, as wellas entangled communications, is exhibited through both t vs. r and conformalspacetime diagrams.
Quantum aspects of gravitating systems continue to be actively explored inthe physics literature. Cosmologies with trapping surfaces are of particular in-terest, since the geometry presents regions of interplay between quantum andgravitational phenomena. In particular, black holes manifest horizons, which arelightlike surfaces bounding regions of information exchange. For black holes, thevery coupling of geometrodynamics to quantum processes likely implies that de-scriptions of the spacetime should qualitatively differ from classical static systems.For instance, any temperature associated with a static black hole generates radi-ations that modify the black hole. Furthermore, the classical, static horizon ofSchwarzschild geometry is a t = ∞ surface, implying that those exterior to thehorizon can never observe infalling objects reach it. However, infalling energieslikewise modify the dynamic surfaces through which they fall. Therefore, ge-ometries with explicit time dependence are needed to model dynamic spacetimesconsistent with actual systems. ∗ e-mail address, tkfi[email protected] † e-mail address, [email protected] o gain further insights into the surfaces generated in dynamic cosmologies,dynamic black holes embedded in asymptotically Minkowski spacetimes have beendeveloped and explored [1, 2]. The metric forms utilize non-orthogonal coordi-nates inspired by river models of stationary spacetimes [3], with a temporal de-pendency parameterized by the river time rather than the Schwarzschild time.The geometries are free of singularities away from r = 0. Fixed temporal coordi-nate curves remain spacelike surfaces throughout the dynamic geometries, corre-sponding to the times measured by certain inertial observers. The motivation fordeveloping these dynamic descriptions is that such a temporal foliation simplifiesone’s descriptions of quantum coherent processes on the geometry. Those inertialobservers satisfying u ctobs = 1 have been referred to as being geometrically sta-tionary [4], and in Robertson-Walker cosmology such observers are usually called co-moving observers. These qualitative differences from Schwarzschild time makeexaminations of quantum behaviors and information dynamics on such geometriesmore straightforward.The results presented will develop as follows. In Section 2, the geometry ofan example spherically symmetric transient black object is developed. Its stress-energy densities will depend upon time and radial coordinates, and will satisfy allenergy conditions everywhere during accretion, and in the exterior during evap-oration. Consistency conditions on the geometry are demonstrated, and lightliketrajectories within and near the trapped region are explored. Section 3 developsthe conformal diagram of the dynamic spacetime, demonstrating global causalproperties of the singularity-free and horizon-free geometry. To complete the dis-cussion, Section 4 then examines the recovery of information temporarily trappedwithin the transient black object. The temporal and kinematic dynamics of theoutgoing communications of an infalling emitter are displayed via spacetime di-agrams and plots of emission rates. The evolution of an entangled photon pair,one of which traverses the trapped region, is also explicitly demonstrated. Because of their relative mathematical simplicity, static black holes have beenof considerable interest in the study of quantum mechanics in gravity, as well astheir influence upon the formative dynamics of galaxies. An uncharged, sphericallysymmetric black hole has a spacelike “center” r = 0 that implies the existenceof a horizon. This horizon delineates the outermost boundary of a region ofspacetime within which all future-trending trajectories ultimately hit the “center.”However, it is possible to construct a geometry that behaves very similarly to atransient black hole in the exterior, but has an innermost boundary to the trappedregion, for which all causal trajectories have decreasing radial coordinate [5, 6].The innermost region then serves as a “depository” that temporarily stores anyinformation that falls into the region of trapped trajectories. The development of2uch a transient black object will be the subject of this section. The radially dynamic spacetime metric for a spherically symmetric, temporallytransient black object will be developed from the general metric ds = − − R M ( ct, r ) r ! ( dct ) +2 s R M ( ct, r ) r dct dr + dr + r dθ + r sin θ dφ , (2.1)whose properties and virtues have been discussed in [1, 7]. In this equation, afinite radial mass scale R M ( ct, r ) ≡ G N M ( ct, r ) /c is a length scale of the mass-energy content of the black object, G = r ∂∂r R M ( ct, r ) = − πG N c T . Themetric takes the form of Minkowski spacetime both asymptotically ( r >> R M ) aswell as when the radial mass scale vanishes.Radial trajectories for test particles of mass m in this geometry have 4-velocitycomponents that satisfy u r = − s R M r u ct ± q ( u ct ) − Θ m , Θ m ≡ ( m = 00 m = 0 , (2.2)where the + sign signifies “outgoing” trajectories, and the - sign signifies “ingoing”trajectories. For massive systems whose proper time (up to an additive constant)is given by t , the temporal component of their 4-velocities satisfy u ct = 1, andthe trajectory is neither ingoing nor outgoing. Freely falling trajectories sharingthis temporal coordinate represent what have been referred to as geometricallystationary trajectories [4]. For this physical setting, observer trajectories with4-velocity components satisfying u ctobs = 1 , u robs = − q R M r obs can be shown to satisfygeodesic equations for massive gravitating systems which share proper time withthe asymptotic observer, dt = dτ . These are the co-moving observers of thisgeometry.The radial coordinate provides the length scale for local angular proper dis-tances dℓ θ = r dθ and transverse areas d σ = r sinθ dθ dφ . One should also notethat any geometrically stationary observer ( u ct = 1) in this geometry will mea-sure a proper radial distance interval at a fixed time value (i.e. a synchronousproper length measurement shared by other geometrically stationary observers)given by ds = dr . This implies that r can also be interpreted as the properdistance between a geometrically stationary observer with coordinates ( ct, r ) andthat geometrically stationary observer that is encountering the center r = 0 atthe same value of t . This fact motivates the use of the term “center” for r = 0 [8].Such an interpretation does not hold for fiducial (fixed r ) observers, who mustundergo accelerations in order to maintain their radial coordinate.3s can be seen from (2.1) and (2.2), at the trapping surfaces R T S , instanta-neously given by solutions to 1 = s R M ( ct, R T S ) R T S , (2.3)outgoing light will be momentarily stationary. In addition, curves of fixed radialcoordinate labeled by r are spacelike in the region between these surfaces. Thus,if solutions to this equation exist, the surfaces R T S bound regions that exclude thepossibility of having fiducial observers, since even light cannot be stationary withinthese regions. Such regions are referred to as trapped regions of the spacetime. Iftrapped regions exist, the geometry contains a “black object.” If there is a horizon,the geometry contains a “black hole.”Several points of interest directly follow from these equations and the form ofthe metric: • The radial coordinate is a proper distance as well as a measure of transverseareas for a class of observers; • Outgoing photons at the trapping surface are momentarily stationary inthe radial coordinate. For dynamic geometries, outgoing photons that arecrossed by the outermost trapping surface have u rγ + = 0 at that instant.Thus, there can be no observers with stationary or increasing radial co-ordinate in regions for which u rγ + ≤
0. The radial scales R T S represent static limits in this geometry. These surfaces are sometimes referred to as“apparent horizons”.A calculation of curvature components for the metric (2.1) exhibits no inherentsingularities away from r = 0. This means that no observer measures singularcurvatures on any surface or transition time of the geometry. The functional formof the radial mass scale R M ( ct, r ) can also be chosen to preclude any singularbehavior at r = 0 itself. Such geometries are thus singularity free.It is quite straightforward to choose a form for the radial mass scale thatgenerates a region in the spacetime for which the center r = 0 is spacelike, yetnon-singular, generating a singularity-free black hole. However, for such a ge-ometry, energy densities in the vicinity of the center are necessarily exotic, sincetheir constituents cannot be timelike. Alternatively, one can directly develop dy-namic geometries for which the center remains timelike perpetually, yet containtransient bounded trapped regions. The exterior of such a geometry behavessimilarly to a transient black hole. It should be straightforward to examine uni-tarity and information dynamics everywhere since the spacetime is horizonlessand singularity-free. Such a geometry, referred to henceforth as a transient blackobject , will be examined in what follows.4t should be noted that in certain circumstances, black objects within whichthe center remains perpetually timelike nevertheless develop horizons, thus alsobecoming black holes. For instance, if the black object does not evaporate away,a region develops within the object for which outgoing lightlike trajectories reach t = ∞ , but not exterior lightlike future infinity, thus implying a horizon. Timeliketrajectories interior to the trapped region cannot escape, but exterior observerscan fall in and become trapped. However, one cannot have a transient black holewithout developing a spacelike center r = 0. Therefore, the transient black objectconsidered in this paper is distinct from a transient black hole in that the trappedinformation can be recovered in its “original” form in the future, in principle. One of the motivations for this model was the fulfillment of energy conditionsover the broadest possible region of spacetime. Classical gravitating systems areexpected to satisfy various energy conditions everywhere. However, quantumsystems exhibit spacelike coherent behaviors, which can violate these conditions.Such violations are necessary for the evaporation of black holes [9].The null energy condition (NEC) and weak energy condition (WEC) state thatthe quantity I null/weak ≡ − u µnull/weak T µβ u βnull/weak (2.4)should be non-positive ( i.e. I null/weak ≤ T µβ refers to the energy-momentum tensor. The dominant energy condition (DEC) refers to the 4-momentum p µsource of the matter distribution as seen by the observer with 4-velocity ~u obs , givenby p µsource ≡ − T µβ u βobs . The DEC requires that this 4-momentum satisfy I DECobserver ≡ ~p source · ~p source ≤ ≤ ∂∂ct R M ( ct, r ) ≤ s R M ( ct, r ) r ∂∂r R M ( ct, r ) . (2.6)A solution to this non-linear relation satisfying the energy conditions, R ECM , isgiven by R ECM ( ct, r ) = 49 r ( ct B − ct ) . (2.7)5hen substituted into Einstein’s equation, this form generates a pressureless col-lapse of matter whose edge will be referred to as r exterior ( ct ). During accretion,the exterior surface satisfies the equation R M ( ct, r = r exterior ) = R So ≡ G N Mc , (2.8)where M denotes the total mass in the cosmology. R So represents the Schwarzschildradius for a static geometry with mass M . Thus, during the period of accretiononly, R M ( ct, r ) = ( R ECM ( ct, r ) interior, R ECM ( ct, r ) < R So R So exterior, static vacuum. (2.9)The exterior surface r exterior ( ct ) can be shown to collapse at a sub-luminal rate.The accretion will be assumed to continue until the exterior surface reaches abounce scale L bounce defined by micro-physics that will be described in the nextsubsection. A region in spacetime within which any outgoing lightlike trajectory will havedecreasing radial coordinate initiates once R M ( ct dark , r = R So ) = R So , i.e. thematter has collapsed within its Schwarzschild radius. This defines the time of theonset of a trapped region as t dark . At that time, evaporation due to quantumeffects is assumed to begin. However, the collapse will continue indefinitely unlesssome microscopic effect prevents complete degeneracy or the formation of a singu-larity. It will be assumed that such micro-physics contains a fundamental lengthscale limiting gravitational degeneracy. This finite, arbitrarily small, scale will bechosen here to have a fixed value L bounce , beyond which microscopic pressures andquantum non-locality prevent further collapse. The functional behavior of R M interior to L bounce must then be of a form that prevents the formation of a sin-gularity, i.e. such that Lim r → R M ( ct,r ) r < ∞ . If this limit attains a value greaterthan unity, a horizon will form, creating a non-singular black hole . Otherwise,the center r = 0 will remain timelike everywhere, and if a trapped region forms,the geometry will manifest a generic black object . In the exterior, both types ofdark geometries are quite similar prior to complete evaporation. Since there isno spacelike center, a sturdy system on a timelike trajectory can in principle bedetected after evaporation of the black object.The dynamics of information and the global causal structure of a dynamicblack object is the subject of this investigation. During black object evaporation,the exterior region is delineated by the outermost trapping surface R S ( ct ), whichis defined in terms of the interior mass of the the black object that has yet toevaporate R S ( ct ) ≡ G N M ( ct ) c . (2.10)6he dynamics describing the evaporation of this radial surface scale will be mo-tivated using Hawking-like thermal radiation rates expected from a quasi-staticgeometry. The emission of such radiation from trapping surfaces has been dis-cussed in the literature [10]. The rate of interior mass change is expected to be ofthe form ˙ M c = characteristic energyemitted quantum × number of quanta emittedunit time , where the dot indicates a derivative with respect to ct . The energy of the emittedquantum is expected to be defined by the spatial extent of the radial surface scale R S ( ct ), which likewise defines the temperature of a quasi-static geometry. Forgenerality, an arbitrarily small additional microscopic scale δ R will be includedto prevent an indefinitely large energy, giving a form for this term of the order ¯ hcR S ( ct )+ δ R . The rate of emission of the quanta for fiducial observers in the staticgeometry is expected to be of the order of one quantum per unit Rindler time[11], which is likewise inversely proportional to the radial surface scale. Again, forgenerality, an arbitrarily small microscopic scale δ T will be added, resulting in arate of emission κR S ( ct )+ δ T . where κ is a dimensionless number of order one. Thus,multiplying by G N c , the dynamics of the surface scale will be taken to satisfy˙ R S ( ct ) = − L P lanck R S ( ct ) + δ R κR S ( ct ) + δ T . (2.11)After the interior mass has decreased to a point where the radial surface scaleis equal to the bounce scale L bounce , the geometry no longer contains a trappedregion, and the black object vanishes. Subsequently, all outgoing lightlike trajec-tories will have increasing radial coordinates. If there is no further quantum decay,a remnant of this mass will remain. However, for the present investigation, thecore region will be assumed to continue quantum decay through massless quanta.The gravitationally stabilized quantum decay will be presumed to generate quantathat satisfy quantum measurement constraints as well as geometric consistency,as will be discussed shortly. During this decay, the exterior surface scale delin-eating the interior and exterior regions will take the fixed value L bounce . ‡ Theradial surface scale and the the exterior surface scale are demonstrated in Figure1. The particular form chosen for the decay is given by R S ( ct ) = A ( ct − ct final ) ,where ct ≤ ct final and the constant A is chosen to smoothly match the final rateof thermal evaporation with the initial rate of decay. The quadratic form allowsa smooth transition to Minkowski space. ‡ Alternatively, if this scale were to maintain a time-dependent fractional value relative to R S ( ct ), the black object would continue thermal decay until all interior mass has evaporated.The development of such a model is quite straightforward, and its features do not exhibitsignificant modification from the model herein explored. S H ct L and R S H ct L -10 10 20 30 40 50 60123456 r exterior H ct L Figure 1: Plots of R S ( ct ) and ˙ R S ( ct ) (left), and r exterior ( ct ) (right). The blackcurve on the left plot represents the radial surface scale, while the negative graycurve represents its derivative.In the exterior region, changes in the interior mass will be presumed to becommunicated via collections of massless quanta that carry energy and change thegeometry. The quanta collectively carry sufficient energy to change the interiorradial surface scale by δR S , where δR S <
0. The quanta then propagate througha static affine space with lesser interior radial surface scale R S + δR S . If this isdone consistently, all energy conditions will be satisfied in the exterior.Quantum measurement constraints can be estimated by examining the energy-at-infinity δE carried per emission by the outgoing quanta. The fact that δR S =˙ R S δct , implies that δE = − c G N δR S = ¯ hcR S ( ct )+ δ R . Incorporating the uncertaintyprinciple δE δt ≥ ¯ h implies that the rate of decrease in the radial surface scale(for δ R ≃ δ T ) has a lower limit given by δR S δct emissions > L P lanck . However,there is also an upper limit upon the rate of evaporation/decay, due to geomet-ric consistency and causality. Specifically the rate of thermal evaporation mustbe dynamically consistent so that energy carrying quanta do not leave spacetimeflat enough that subsequent quanta eventually catch up. Indeed, a rapid enoughevaporation would violate geometric consistency. The chosen decay form generatessuccessive outgoing massless quanta that asymptotically have separations demon-strated in Figure 2. In the figure, geometry-changing quanta are emitted fromthe emission surface at small fixed intervals and propagated as outgoing quantaconsistent with the metric (2.1) on null trajectories satisfying ˙ r Q = 1 − q R M /r Q .Therefore, the specific model developed incorporates the following points: • The geometry-changing quanta must emit exterior to the background radialsurface scale R bgS through which they propagate, if they are to transportmass away; • The geometry-changing quanta will thus propagate through a backgroundgeometry with surface scale R S + δR S carrying energy δE = − c G N δR S ; • The emissions communicate the transport of energy as a lightlike outgoing8 emit D r Γ¥ Evaporant Spread ³ Figure 2: Asymptotic spread of sequentially emitted energy-carrying quanta.energy that includes gravitational binding; • Since the exterior radial mass scale incorporates the geometry-changingmassless quanta, all energy conditions are expected to be satisfied in theexterior.Each massless geometry-changing quantum “sees” a static geometry; it merelypropagates through the geometry left behind by its immediate predecessor. Ex-terior outgoing quanta emitted at ( ct o , r o ), located at r γ ( ct ), and propagatingthrough a background geometry parameterized by R bgS satisfy ct − ct o = r γ ( ct ) − r o + 2 q R bgS ( q r γ ( ct ) − √ r o ) + 2 R bgS log q r γ ( ct ) − q R bgS √ r o − q R bgS . (2.12)The geometry-changing quanta generated by evaporation or decay will thus propa-gate through a local background geometry characterized by by R bgS ( ct o ) = R S ( ct o )+ δR S ( ct o ). In order to calculate the radial mass scale at an arbitrary exterior point( ct, r ), the retarded event of emission ( ct o , r o ) ≡ ( ct ret , r o ( ct ret )) must be deter-mined. During evaporation, the exterior emission scale will be chosen to satisfy r o ( ct ret ) = R S ( ct ret ) + δ stretch (where δ stretch can be arbitrarily small because thequanta propagate on a background of scale R S + δR S as opposed to R S itself).The radial mass scale at a general exterior point will therefore be given by R M ( ct, r ) ≡ R S ( ct ret ( ct, r )) , r > r exterior ( ct ) = R S ( ct ) . (2.13)This form then ensures a causal propagation of the evaporation of mass from thecore region. During evaporation the core region is chosen to maintain its previous r -dependence from the accretion period, such that R M ( ct, r ) ≡ R S ( ct ) R S ( ct dark ) R ECM ( ct dark , r ) , r ≤ L bounce (2.14)9here the pressureless form satisfying energy condtions R ECM is given by Eqn.(2.7). The region between the exterior and the bounce scale L bounce will be as-sumed to maintain the spatially coherent form R S ( ct ). The chosen forms smoothlymatch the behaviors during the transition from accretion to evaporation.As previously mentioned, this form for the radial mass scale has been chosen soas to satisfy energy conditions in the broadest region of the spacetime. Plots of theevolution of the radial mass scale, along with local measures of fulfillment of theenergy conditions, are represented in Figure 3. The diagrams in the figure depict H ct,r L H ct,r L H ct,r L H ct,r L Figure 3: Radial mass scale and energy conditions for late accretion (upper left),final accretion with evaporation (upper right), mid-evaporation (lower left), andlate decay (lower right). Both axes are measured in units of R So . The radialmass scale is shown in gray with a black overlay of trapped region. The DECinvariant I DECobserver is shown for radial observer motions in red and for azimuthalobserver motions in green. The NEC invariant I null is shown in purple. Theenergy conditions are satisfied for I DECobserver ≤ I null ≤
0, respectively.snapshots of these quantities during late accretion, the final stage of accretionafter the black object has formed and evaporation has begun, the late stage ofthermal evaporation, and the late stage of a gravitationally suppressed quantumdecay of the remnant.The gray curves in Figure 3 represent the spatial dependence of the radialmass scale at the given time. If there is a trapped region, it is represented bya black segment upon the radial mass scale curve. It is clear from the diagramthat the trapped region has both an inner boundary R − T S and an outer boundary R + T S . Curves are also demonstrated depicting local values for the DEC invariant I DECobserver for radial observer motions (red), I DECobserver for azimuthal observer motions(green), and the NEC invariant I null (purple), using a convention in which theDEC and NEC are satisfied when I DECobserver ≤ I null ≤
0, respectively. All10nergy conditions are everywhere satisfied prior to the beginning of evaporation,as expected. Also, energy conditions are satisfied everywhere in the exterior region r > r o ( ct ), implying that any measurements in the exterior will never detect exoticenergy forms. The snapshots demonstrate violation of the energy conditions onlywithin the region of spatial coherence where production of the quanta occurs. Suchviolations of energy conditions should be expected within the trapped regions,since these are also classically forbidden regions. The exploration of the dynamic features of a system is straightforward on aconformal diagram. Unfortunately, many useful dynamic geometries such as thistransient black object do not afford a direct calculation of a set of conformal coor-dinates. A general technique is therefore necessary to construct Penrose diagramsin complicated geometries.The technique relies only upon constructing lightlike surfaces for the givenmetric, in this case given in Eqn. (2.1). Once those null geodesics have beengenerated, the conformal coordinates can be labeled ( v, u ), based upon the cor-respondence of the lightlike curves on reference hypersurfaces (in this case, theexterior surface and ultimately skri ± ). Conformal spacetime coordinates ( ct ∗ , r ∗ )can be introduced such that v = ct ∗ + r ∗ and u = ct ∗ − r ∗ . For ingoing nullgeodesics, the required equation labeled by conformal coordinate v takes the form˙ r v = − − s R M r v , (2.15)while outgoing null geodesics labeled by u satisfy˙ r u = 1 − s R M r u . (2.16)For this geometry, ingoing lightlike trajectories labeled by v initiating on pastlightlike infinity have access to all regions of spacetime. Outgoing lightlike trajec-tories labeled by u terminating on future lightlike infinity likewise have access toall regions of spacetime, since there is no horizon.Ingoing lightlike trajectories defining the conformal coordinate v for the tran-sient black object are demonstrated in Figure 4. The ingoing photons are chosento terminate at r = 0, temporally separated in units of the Schwarzschild ra-dius of the geometry R So . For the chosen parameters, the trapped region firstdevelops at ct ≃ − . R So , and vanishes at ct ≃ R So . Accretion ends at ct ≃ +0 . R So , and final decay occurs at ct ≃ . R So . Ingoing photon trajec-tories are seen to propagate through essentially flat spacetime until they approachthe trapped region, within which their radial coordinates decrease more rapidly11 .5 5 7.5 10 12.5 15 17.5 20r Γ -10-7.5-5-2.52.557.510ct Ingoing Photons Figure 4: Ingoing lightlike trajectories. Results are displayed in units of R So .than in Minkowski spacetime. In the static exterior of the past, ingoing masslessquanta satisfy ct o − ct = r γ ( ct ) − r o − q R So ( q r γ ( ct ) − √ r o ) + 2 R So log q r γ ( ct ) + √ R So √ r o + √ R So . (2.17)Thus, the outgoing lightlike surface of constant u communicating the beginningof evaporation can serve as the exterior surface of correspondence for assigningthe parameter v during evaporation.Outgoing lightlike trajectories defining the conformal coordinate u for the tran-sient black object are demonstrated in Figure 5. The outgoing photons are emittedfrom r = 0, temporally separated in units of the radial mass of the geometry R So .Photons emitted after the final decay of the black object propagate through flatspacetime. Likewise, photons emitted in the distant past propagate through nearlyMinkowski spacetime. However, photons emitted just prior to the formation ofthe black object initially propagate with increasing radial coordinate, then havedecreasing radial coordinate as the trapped region forms. Photons emitted dur-ing the lifetime of the black object (while there is a trapped region) also initiallypropagate with increasing radial coordinate, but slow as they approach the innersurface delineating the trapped region, being temporarily trapped within the in-nermost core region. It should be noted that all outgoing photons eventually willreach lightlike future infinity in this transient black geometry.12
10 15 20 25 30r Γ -1010203040506070ct Outgoing Photons Figure 5: Outgoing lightlike trajectories (that originate at r = 0). Results aredisplayed in units of R So . 13 Conformal diagram of the transient black ob-ject
Light-cone conformal coordinates ( v = ct ∗ + r ∗ , u = ct ∗ − r ∗ ) parameterized inthe previous section will be made compact using the identification Y → = [ tanh ( ct ∗ + r ∗ scale ) − tanh ( ct ∗ − r ∗ scale )] / ,Y ↑ = [ tanh ( ct ∗ + r ∗ scale ) + tanh ( ct ∗ − r ∗ scale )] / . (3.18)The coordinates ( Y → , Y ↑ ) can be used to construct the Penrose diagram of thisgeometry with a transient black object. The geometry indeed has some finiteperiod with real solutions of Eqn. (2.3) for the outer and inner surfaces of thetrapped regions satisfying R − T S = R + T S , with the chosen parameters for the metric(2.1).The conformal diagram, with significant transitional epochs indicated, is shownin Figure 6. The diagram is bounded from the left by the timelike center r = 0,from the lower right by past lightlike infinity skri − , and from the upper right byfuture lightlike infinity skri + . There are no horizons on the diagram. The staticradial mass (Schwarzschild radius) of the geometry depicted by the (orange) curvelabeled R So sets the scale of the diagram. The center of the conformal coordinates( Y → = 0 , Y ↑ = 0) is chosen to correspond with the coordinate ( ct = 0 , r = R So ).The timelike surface that delineates the exterior region of the geometry is depictedby the (red) dashed curve labeled R X . During accretion, this curve represents theradial coordinate within which all mass in the spacetime is interior. After evapo-ration begins ( ct > ct dark ≈ ct bounce ), R X represents the radial scale within whichmass that has yet to evaporate or decay away is contained. The spacelike surfaceshowing the beginning of evaporation ( ct dark ) is represented by the dashed (green)curve just prior to the solid (green) curve labeled ct bounce in the diagram on theleft. The latter depicts the end of accretion. The geometry-changing quanta emitfrom the surface labeled r o , depicted by the black dashed curve. The dashedspacelike surface labeled ct remnant depicts the end of thermal evaporation, and thebeginning of decay of the remnant. Finally, the solid (green) spacelike curve la-beled ct final depicts the end of decay of the remnant of evaporation. The region ofthe spacetime above the communication of the end of decay is flat. The diagramon the right demonstrates the boundaries of the trapped region.Fixed-coordinate surfaces in the geometry are demonstrated in Figure 7. Oneof the most useful characteristics of this type of metric is that fixed-time coordi-nate surfaces are everywhere spacelike, and are shared by a set of geometricallystationary co-movers. This structure of the geometry is apparent in the figure.However, fixed-radius surfaces are also spacelike within the trapped region. In thediagram on the left, fixed-radius coordinate surfaces are everywhere timelike for r > R So . Also, fixed-radius surfaces interior to the minimum value of the inner14 X r o R So ct remnant ct final ct bounce R X ct remnant R TS ct Dark
Figure 6: Conformal diagrams of the transient black object, demonstrating dy-namic features of interest. The edge r exterior has been abbreviated R X . Thediagram on the right emphasizes the boundaries of the trapped region. Once theblack object forms, the outer boundary R X is always timelike, while the innerboundary R T S is spacelike during evaporation.15 a LH b LH c L R X r o R S Figure 7: Fixed- ct and fixed- r surfaces on a conformal diagram for the transientblack object. Left: Spacelike surfaces of fixed ct are represented by green curvesgraded in units of R So . Curves of fixed radial coordinate r are represented by redcurves graded in units of 0 . R So from the center out to 2 R So , then integral valuesof this unit, then decades of this unit. Right: Close-up view of (a) r = 0 . R So ,(b) r = 0 . R So , and (c) r = 0 . R So . 16oundary of the trapped region r < R − T S are always timelike, including the center r = 0. The boundaries of the trapped region represent surfaces for which outgoinglightlike trajectories are momentarily stationary in the radial coordinate. Thosefixed-radius surfaces that lie within the trapped region demonstrate transitionfrom timelike to spacelike, then back to timelike behaviors exhibited in the left-most region of the diagram. For clarity, this region is expanded in the diagram onthe right of Figure 7. In this diagram, fixed-radius surfaces near the static massscale R So are depicted. The slope of each of the surfaces r = constant is timelikeprior to the formation of the black object, becomes unity as the innermost surfaceof the trapped region R − T S crosses that coordinate prior to the fixed-coordinatesurface becoming spacelike, again becomes unity as the outermost surface of thetrapped region R + T S = R S ( ct ) crosses that coordinate, and is ultimately againtimelike. This behavior has been observed in prior explorations of coordinatesurfaces in transient trapped geometries [2, 8].The behaviors of the outgoing null trajectories near the black object displayedin Figure 5 demonstrate that the outgoing communications from any system justbefore crossing the trapping surface are temporarily held near that surface. Thisis true for infalling systems at any time throughout the existence of the dynamicblack object. However, all of these communications will eventually reach a nearbyexterior observer after a finite time. In contrast, the analogous outgoing lightliketrajectories near a Schwarzschild horizon will release communications that onlyreach the exterior observer after an indefinitely long period of time. The next sec-tion will examine the dynamics of the release of these communications as systemsenter the black object. In order to examine the information dynamics of the transient black object,a standard exterior observer will be chosen to be geometrically stationary, thussharing proper time with that of the asymptotic observer ( t ) parameterized in themetric. The initial conditions of this nearby observer can be chosen such that theobserver avoids ever encountering the trapped region. This co-moving observer,whose trajectory is given by the outermost bold trajectory in Figure 8 will serveas the observational platform examining gravitating systems entering the trappedregion. The trajectory was chosen to correspond to that of a stationary observerat r = R So after the remnant has completed its decay.The observed infalling system will emit “standard” frequency photons at a“standard” rate in its proper coordinate frame of reference. In what follows,the emitter will be a freely falling, geometrically stationary system that reaches r = R So at time ct = 10 R So (which is during evaporation), then falls through the17 Figure 8: Standard photons that are emitted by a geometrically stationary in-falling source and detected by a nearby external observer. A standard spacetimediagram (which displays the boundaries of the trapped region) is represented onthe left, and the conformal diagram is represented on the right.18rapped region. This infalling emitter is depicted by the innermost bold trajectoryin Figure 8. The emitter approaches, but never reaches, the center r = 0. In thefigure, the outgoing photons are emitted at ct = R So , ct = 2 R So , ct = 3 R So , etc .There is an extended period of time during which only highly redshifted photonsare observed from the emitter. This especially occurs between the communicationof the emitter crossing into the trapped region and the communication of the endof the black object. After the emitter passes through the inner boundary of thetrapped region, all outgoing communications are temporarily trapped inside thatboundary, prevented from entering the trapped region. These photons “bunchup” within the inner surface of the trapped region, but are rapidly released afterthe trapped region vanishes, in a manner similar to that illustrated in Figure 5.Eventually, all emitted communications will be received by the observer. In order to examine the redshift of the emitted quanta, the null trajecto-ries (2.16) that are obtained directly from the metric are not sufficient; the nullgeodesic equation du β dλ + Γ βµν u µ u ν = 0 , (4.1)is required (here λ is the affine parameter). The geodesic equation is used tocalculate the 4-velocities for massless quanta. Subsequently, it is seen that thequantity − ~u obs · ~u γ , the observed value of the temporal component of the 4-velocityof a radially outgoing photon, satisfies − ~u obs · ~u γ = (cid:18) u ctobs ∓ q ( u ctobs ) − (cid:19) u ctγ , (4.2)where the ∓ sign refers to outgoing/ingoing observers. This is directly propor-tional to the observed energy of the photon. For a geometrically stationary ob-server, u ctobs = 1, and the observed photon temporal component is seen to be simply u ctγ . The relationship (4.2) can likewise be used to express u ctγ in terms of its value( u ctγ ) proper = − ~u ∗ · ~u γ in the proper frame * of the emitter: u ctγ = ( u ctγ ) proper u ct ∗ ∓ q ( u ct ∗ ) − , (4.3)where the ∓ sign refers to outgoing/ingoing emitters. Since both the observer andthe emitter are geometrically stationary, the geodesic equation directly calculatesthe observed frequency changes of the gravitating photons.The rate of reception of information emitted from the infalling emitter canbe obtained by examining the component u ctγ for the outgoing standard photons.19he standard frequency (in the frame of the source) for photons emitted will bechosen such that u ctγ standard =1. The temporal behavior of u ctγ for standard photonsfrom the infalling emitter as measured by the geometrically stationary observer isshown in Figure 9. The figure demonstrates both the evolution of the frequency
10 20 30 40 50 ct0.20.40.60.81u Γ ct Standard Outgoing Quanta 10 20 30 40 50 ct12345dN €€€€€€€€€€€€€ dt obs (cid:144) dN €€€€€€€€€€€€€€€ d Τ emit Figure 9: Left: Observed u ctγ of standard photons as a function of observationtime. Right: Ratio of rate of observation of detected quanta to the rate of standardemission.of individual photons on the left, as well as the rate of observation of photonemissions on the right. The rate of emission of photons is related to the interval dτ emit between successive photons. Likewise, the interval between the observationof those successive photons by the co-moving observer dt obs relates to the rate ofdetection. Thus, prior to the disappearance of the trapped region, the ratio ofthe rate of observation to the standard rate of emission should equal the ratioof the observed frequency to the standard frequency. As a check for numericalaccuracy, this independent measure of redshift on the right of Figure 9 indeedfunctionally coincides with the geodesic calculation displayed on the left prior tothe communication of the end of the black object. However, after communicationsfrom the emitter emerge following the evaporation of the trapped region, theindividual photons depicted in the left diagram are initially quite redshifted andultimately approach unit ratio from below, while the rate ratio in the diagram onthe right is quite enhanced and ultimately approaches unit ratio from above. Thisshould not be surprising, since such behavior is apparent in Figure 8.The observation of infalling emitters through the trapped region of a tran-sient black object is somewhat different from the observation of infalling emittersthrough the horizon of a transient black hole [8]. For emitters falling into transientblack holes, any photons emitted by infalling systems before the horizon is crossedwill continue to be redshifted until those systems are seen to completely vanishas the black hole itself fully evaporates away. This is true independent of the ac-tual time that the system falls through the horizon, i.e. , all infalling emitters areseen to simultaneously vanish as they traverse the vanishing horizon of a transientblack hole. However, for the transient black object, all emissions rapidly reverttoward standard frequency and rate of emission as the trapped region evaporates.20or a static black hole, emitters are never observed to traverse the horizon (see,for instance, reference [11], page 23).To complete this examination, a measure of the power of emissions from theinfalling system as observed by the external observer will be developed. The powershould be related to the energy of each photon times the rate at which photonsare observed. This power parameter measured for radially outgoing photons atthe radial coordinate of the external observer is displayed in Figure 10. After
10 20 30 40 50 ct2468101214Power Standard Power
Figure 10: Normalized power P standard received by the geometrically stationaryexterior observer.final decay of the remnant, and for very early times, this standard power takesthe value of unity. Just as the trapped region vanishes, there is a considerablespike observed in this measure of power.If the detector used by a single external observer is small enough (or if therewere an isotropic set of infalling emitters all radiating radially) the emissionsare uniform over the observed solid angle. The actual detections are then ex-pected to fall off with the inverse square of the radial distance traveled by thephoton, which for a geometrically stationary emitter and observer is given by r obs ( ct observed ) − r ∗ ( ct emit ). A diagram of this observed measure of “intensity” isdemonstrated in Figure 11. Early measurements of this intensity are smaller thanlater measurements because of the greater distance between the emitter and thesource. This intensity is normalized to take the value unity after the remnant hascompletely decayed. At this time, both the observer and system are motionlesssince geometrically stationary objects have fixed spatial coordinates in Minkowskispacetime. The figure demonstrates a spike in intensity as the trapped photonsare released after the termination of thermal evaporation, which corresponds tothe end of the black object. 21 D r f2 €€€€€€€€€€€€D r P standard Figure 11: “Intensity” measured by the exterior observer using a small detector.
The absence of a spacelike center for the singularity-free black object allowsa direct exploration of the trajectories of gravitating entangled photons. As anexample, consider a massive unstable particle coincident with the infalling geo-metrically stationary system that decays into an entangled pair of photons justas the particle encounters the outer trapping surface R S ( ct decay ). One of the pho-tons will be emitted radially outward, and the other radially inward to conservemicroscopic momentum. One observer (Alice) will be a geometrically stationaryobserver with final location x = R So , while the other observer (Bob) will be a geo-metrically stationary observer located diametrically opposite the original observerwith final location x = − R So .This arrangement is demonstrated in Figure 12. The left diagram of the figureis a spacetime plot using the coordinates ( ct, x ), while the right diagram is aconformal plot derived from ( ct, r ). The infalling unstable particle is representedby the dashed black curve, while the entangled photons which are the products ofthe decay are represented as the outgoing and ingoing null light-blue trajectories,ultimately detected by Alice and Bob. In the spacetime diagram on the left, theinterior mass scale R S ( ct ), which coincides with the outer trapping surface whilethere is a black object present, is also demonstrated as a pair of solid (red) timelikecurves slowly approaching zero.The decay occurs just as the unstable particle encounters the trapping surface,indicated by the initial vertical slope of the outgoing photon trajectory that ulti-mately reaches Alice. The ingoing photon crosses the center, but remains trappeduntil the trapped region vanishes due to the evaporation of the black object, atwhich time it crosses the interior mass scale and ultimately is detected by Bob.In the radial conformal diagram on the right, Alice has azimuthal location φ = 0,while Bob has azimuthal location φ = π . Thus, these observations are actually22 AliceBob
Figure 12: Trajectories of the entangled photon pair emitted by the unstableparticle just as it crosses into the trapped region. The left diagram is a standardspacetime diagram using ( x, ct ), while the right diagram utilizes conformal (radial)coordinates ( Y → , Y ↑ ) and superposes two opposite values of the azimuthal angle φ . 23pacelike separated, which is not depicted on the radially symmetric Penrose plot.In this diagram, the ingoing entangled photon “reflects” from the center as itspolar angle switches from zero to π . It is interesting to note that, in contrastto the situation in flat spacetime, Alice and Bob likely detect the photons dur-ing differing epochs. In addition, the energy ratio ǫ ≡ u ctγ u ctγ ∗ measured by Alice is ǫ ≈ .
01, while that measured by Bob is ǫ ≈ × − . Thus, the spacetime andenergy-momentum entanglement information measured by the different observersis both temporally-shifted and redshifted in energies. A singularity-free transient black object whose center remains always time-like has been developed, and its evolution has been numerically explored. Suchobjects are difficult to distinguish from long-lived transient black holes in the exte-rior, yet are everywhere analytic. The geometry was constructed to satisfy energyconditions everywhere during accretion. In addition, the geometry satisfies energyconditions external to the trapped region, and external to any region expected toinvolve prolific production of quanta. The coordinates utilized are particularlyuseful since they admit a class of geometrically stationary (co-moving) observers.These observers share proper time with that of the asymptotic observer, directlyparameterized by the metric time t . Geometrically stationary observers can serveas convenient platforms for performing measurements within the dynamic geom-etry.The particular model explored involves a geometry with an overall energydistribution of fixed mass M that undergoes a pressureless collapse initiated inthe distant past. The collapse terminates after quantum non-locality effects arepresumed to dominate the dynamics [13]. Once mass is contained within a regionsmaller than its Schwarzschild radius, a trapped region ( i.e a region in which allcausal trajectories move toward decreasing radial coordinate) forms. However,since the center remains timelike and the black object is transient, no horizondevelops, and the object never becomes a black hole. This collapsed black objectthen undergoes thermal decay analogous to that expected from a black hole untilthe trapped region vanishes. The remnant of thermal evaporation is then chosento decay away in a manner consistent with quantum and geometric constraints.A conformal diagram demonstrating the large-scale causal structure of thegeometry has been demonstrated. The modification of the spacetime from thatof Minkowski is minimal, since the center r = 0 remains everywhere timelike. Inthese coordinates, the development of a trapped region merely expands the volumeof the conformal diagram, as expected from previous studies [8, 12]. Fixed-timesurfaces remain everywhere spacelike, providing global foliation for parameterizingthe dynamics. Surfaces of fixed radial coordinate have been demonstrated to be24imelike exterior to the trapped region, and spacelike within the trapped region.Information exchange in the vicinity of the black object has also been explored.Outgoing photons from an infalling emitter were seen to redshift as the emitterapproaches the trapped region. However, in contrast to a transient black hole,direct emissions that are temporarily trapped while the black object persists arelater observed once the trapped region vanishes. This rapid release of informationhas been demonstrated for the model. The redshift of energies and the dynamicsof the rate of communications are shown to behave as expected.In order to explore the dynamics of entangled information, the trajectories ofentangled photons have been examined. The photons were formed as a particledecayed while crossing into the trapped region. The presence of the transient blackobject alters the relative times of observation and energy redshifts of the entangledphotons. The exploration directly demonstrates that the loss of entanglementinformation is only temporary for this geometry, and that there are no obviouscases of violation of the standard laws of physics.The analytic and causal properties of this dynamic black object should con-siderably simplify the exploration of quantum geometrodynamic behaviors on thegeometry, as will be demonstrated elsewhere [14]. Also, one should be able toincorporate such a transient black object within a dynamic de Sitter geometryconsistent with big bang cosmology, as has been done with a transient black hole[8]. In principle, such a transient black object should introduce no exotic behav-iors in the global geometry, while yet introducing an additional trapped regioninto the spacetime. Acknowledgments
The authors warmly acknowledge their association with Beth Brown. TF ispleased to acknowledge past discussions with Ted Jacobson, along with supportfrom Chanda Prescod-Weinstein. JL gratefully acknowledges useful past discus-sions with James Bjorken, Paul Sheldon, and Lenny Susskind.
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