Mirror Reflections of a Black Hole
MMirror Reflections of a Black Hole
Michael R.R. Good ∗ Department of Physics, Nazarbayev University, Astana, Kazakhstan
Paul R. Anderson † Department of Physics, Wake Forest University,Winston-Salem, North Carolina, USA
Charles R. Evans ‡ Department of Physics and Astronomy,University of North Carolina at Chapel Hill, North Carolina, USA (Dated: March 12, 2018)
Abstract
An exact correspondence between a black hole and an accelerating mirror is demonstrated.It is shown that for a massless minimally coupled scalar field the same Bogolubov coefficientsconnecting the in and out states occur for a (1+1)D flat spacetime with a particular perfectlyreflecting accelerating boundary trajectory and a (1+1)D curved spacetime in which a null shellcollapses to form a black hole. Generalization of the latter to the (3+1)D case is discussed. Thespectral dynamics is computed in both (1+1)-dimensional spacetimes along with the energy flux inthe spacetime with a mirror. It is shown that the approach to equilibrium is monotonic, asymmetricin terms of the rate, and there is a specific time which characterizes the system when it is the mostout-of-equilibrium. PACS numbers: 03.70.+k, 04.62.+v ∗ [email protected] † [email protected] ‡ [email protected] a r X i v : . [ g r- q c ] S e p . INTRODUCTION The connection between the particle production which occurs at late times after a blackhole forms from collapse [1] and the late time particle production from a mirror in flat spacethat accelerates without bound, asymptotically approaching a null geodesic, was establishedby Davies and Fulling [2, 3]. An interesting question is whether there are mirror trajectoriesfor which their entire history of particle creation, from initial non-thermal phase to late timethermal emission, corresponds to the entire history of particle creation from a spacetimein which a black hole forms from collapse. We have found a specific example in (1+1)dimensions where there is such an exact correspondence.The model for gravitational collapse that we consider consists of a collapsing shell witha null trajectory. The spacetime inside the shell is flat while the geometry outside theshell is the usual Schwarzschild geometry. This model was considered in [4] where the exactBogolubov coefficients connecting the in and out vacuum states were computed for a masslessminimally coupled scalar field. The trajectory for the mirror is a simple modification of onethat was discovered in Ref. [5]. The mirror, which is in flat space, begins at past timelikeinfinity, i − , and accelerates in a monotonic fashion, asymptotically approaching v = v H with v ≡ t + r .One of the advantages of our model is that the Bogolubov coefficients between the in and out states can be computed analytically. It is the equivalence between these coefficients in theblack hole and accelerating mirror cases that establishes the exact connection. Interestingly,in the mirror case there are so far a limited number of specific trajectories for which theBogolubov coefficients have been computed analytically [3, 6–9]. In most of these cases, asin the present case, the actual amount of particle production must be computed numerically.In [10–12] we pointed out this mirror - black hole connection and briefly explored thetime dependence of the particle production and the time dependence of the stress-energytensor in the accelerating mirror case. Here we give the details of the computations ofthe Bogolubov coefficients in both the black hole and accelerating mirror cases. For theblack hole we add a discussion of the computation in (3+1)D. We also give a significantlymore detailed description of the time dependence of the particle production process, whichincludes an estimate, consistent with the uncertainty relation, of the time evolution of thespectrum of the produced particles. The time dependence of the particle production process2as investigated for other mirror trajectories in [8].In Sec. II we compute the Bogolubov coefficients for our mirror trajectory and for the caseof a null shell that collapses to form a black hole in (1+1) and (3+1) dimensions. In the lattercase we ignore the effective potential in the mode equation. In Sec. III the time dependenceof the particle production process and the frequency spectrum of the produced particles areinvestigated. Sec. IV contains a brief discussion of the time dependence of the stress-energyof the quantum field in the accelerating mirror case. Our results are summarized in Sec. V.Throughout we use units such that (cid:126) = c = G = k B = 1 and our conventions are those ofRef. [13]. II. BOGOLUBOV COEFFICIENTS
In this section we compute the particle production that occurs for a massless minimallycoupled scalar field in three different situations: a (1+1)D flat spacetime with an accel-erating mirror moving along a particular trajectory; a (1+1)D spacetime in which a nullshell collapses to form a black hole; and a (3+1)D spherically symmetric spacetime in whicha null shell collapses to form a black hole. We begin with the simplest case which is theaccelerating mirror.
A. (1+1)D flat spacetime with a mirror
The line element for flat space in (1+1)D is simply ds = − dt + dr = − du dv . (2.1)where alternative, null coordinates are u = t − r , v = t + r . (2.2)We denote the trajectory of the mirror by r = z ( t ) (see Fig. 1). Note that we shall only beconcerned with the part of the spacetime that is to the right of the mirror.The wave equation for the massless minimally coupled scalar field is (cid:3) φ = 0 . (2.3)3he field can be expanded in terms of complete sets of mode functions, each of which satisfiesthe equation ∂ u ∂ v f = 0 . (2.4)The general solution is f = a ( u ) + b ( v ) , (2.5)for arbitrary functions a and b .The modes are normalized using the scalar product( φ , φ ) = − i (cid:90) Σ d Σ (cid:112) | g Σ | n a φ ↔ ∂ a φ ∗ , = − i (cid:90) Σ d Σ (cid:112) | g Σ | n a [ φ ∂ a φ ∗ − ( ∂ a φ ) φ ∗ ] , (2.6)with Σ a Cauchy surface and n a the unit normal to that surface. One Cauchy surface weshall use is I − R . In this case the scalar product is( φ , φ ) = − i (cid:90) ∞−∞ dv φ ↔ ∂ v φ ∗ . (2.7)The other consists of the union of I + R with I + L,> , the part of I + L that is to the right of themirror. The scalar product is then( φ , φ ) = − i (cid:90) ∞−∞ du φ ↔ ∂ u φ ∗ − i (cid:90) ∞ v H dv φ ↔ ∂ v φ ∗ . (2.8)The in modes are normalized on I − R and form a complete set for the region to the rightof the mirror. The other set of modes of interest are those which are normalized on I + R and which vanish on I + L,> . We label these as out modes. Another set of modes, labeled left modes, end on I + L,> . Taken together the out modes and left modes form a complete set. Allmodes in either set that impinge upon the mirror must vanish at its surface. The in and out modes thus have the forms f in ω = 1 √ πω (cid:2) e − iωv − e − iωp ( u ) (cid:3) , (2.9a) f out ω = 1 √ πω (cid:2) e − iωh ( v ) θ ( v H − v ) − e − iωu (cid:3) , (2.9b)where the ray tracing functions p ( u ) and h ( v ) are defined so that at the location of themirror p ( u ) = v and h ( v ) = u . See Ref. [8] for details. Note that in [8] the function we call h ( v ) is denoted by f ( v ).
4o find the number of particles produced we first expand the field in terms of both setsof modes φ = (cid:90) ∞ dω [ a in ω f in ω + a in † ω f in ∗ ω ] , (2.10a)= (cid:90) ∞ dω [ a out ω f out ω + a out † ω f out ∗ ω + a left ω f left ω + a left † ω f left ∗ ω ] . (2.10b)We also write f out ω = (cid:90) ∞ dω (cid:48) [ α ωω (cid:48) f in ω (cid:48) + β ωω (cid:48) f in ∗ ω (cid:48) ] . (2.11)Then using the fact that the modes are orthonormal with respect to the scalar product onefinds that α ωω (cid:48) = ( f out ω , f in ω (cid:48) ) , (2.12a) β ωω (cid:48) = − ( f out ω , f in ∗ ω (cid:48) ) , (2.12b) a out ω = ( φ, f out ω ) = (cid:90) ∞ dω (cid:48) (cid:104) a in ω (cid:48) α ∗ ωω (cid:48) − a in † ω (cid:48) β ∗ ωω (cid:48) (cid:105) . (2.12c)Then, if the field is in the in state, the average number of particles found at I + withfrequency ω is (cid:104) in | N out ω | in (cid:105) = (cid:90) ∞ dω (cid:48) | β ωω (cid:48) | . (2.13)We now introduce a specific mirror trajectory that begins at past timelike infinity, i − ,and is asymptotic to the ray v = v H . A Penrose diagram for it is given in Fig. 1. Thetrajectory, which is a slight modification of what was called the Omex trajectory in Ref. [5],is z ( t ) = v H − t − W (cid:0) e κ ( v H − t ) (cid:1) κ , (2.14)with κ and v H constants, and with W the Lambert W (or Product Log) function, which hasthe properties z = W ( z ) e W ( z ) = W ( ze z ) . (2.15)Then writing v = v m ( t ) = t + z ( t ) , (2.16)with v m ( t ) being the value of v for the mirror’s location at time t , we find˜ t m ( v ) = v − κ log[ κ ( v H − v )] , (2.17)5ith ˜ t m ( v ) the time when the mirror intersects the null ray labeled by v . This equation caneasily be verified by substituting (2.14) into (2.16) and using (2.17) along with the secondrelation in (2.15). Then since h ( v ) = u at the surface of the mirror, h ( v ) = ˜ t m ( v ) − z [˜ t m ( v )] = v − κ log[ κ ( v H − v )] . (2.18) I L + I R + I L - I R - i i i + i - FIG. 1. Penrose diagram for a flat (1+1)D spacetime containing an accelerating mirror with thetrajectory (2.14) in the case that κ = 1 and v H = 0. The trajectory is timelike, begins at i − andasymptotically approaches v = v H = 0. The relation p ( u ) = v which is valid at the surface of the mirror is the inverse of therelation h ( v ) = u . We find that p ( u ) = v H − κ W (cid:0) e − κ ( u − v H ) (cid:1) . (2.19)This can be verified by computing h ( p ( u )) and using the first relation in (2.15). Combiningthe equations p ( u ) = v m = t m ( u ) + z [ t m ( u )] and t m ( u ) = u + z [ t m ( u )] one finds t m ( u ) = 12 (cid:20) v H + u − κ W (cid:0) e − κ ( u − v H ) (cid:1)(cid:21) . (2.20)Here t m ( u ) is the time when the mirror intersects the null ray labeled by u .To evaluate the formulas for the Bogolubov coefficients in (2.12a) and (2.12b) we choosethe surface I − R for which the general form of the scalar product is given in (2.7). Combining6hese equations along with (2.9b) and (2.18) and noting that u = −∞ on I − R , we find aftersome algebra that α ωω (cid:48) = 14 π (cid:90) v H −∞ dv e − i ( ω − ω (cid:48) ) v [ κ ( v H − v )] iω/κ (cid:40)(cid:114) ω (cid:48) ω + (cid:114) ωω (cid:48) (cid:20) κ ( v H − v ) (cid:21)(cid:41) , (2.21a) β ωω (cid:48) = 14 π (cid:90) v H −∞ dv e − i ( ω + ω (cid:48) ) v [ κ ( v H − v )] iω/κ (cid:40)(cid:114) ω (cid:48) ω − (cid:114) ωω (cid:48) (cid:20) κ ( v H − v ) (cid:21)(cid:41) . (2.21b)Changing the integration variable to x = v H − v allows for the evaluation of the integrals interms of gamma functions. After more algebra we find α ωω (cid:48) = − e − i ( ω − ω (cid:48) ) v H πκ √ ωω (cid:48) ω − ω (cid:48) (cid:20) − iκ ( ω − ω (cid:48) ) (cid:21) − iω/κ Γ (cid:18) iωκ (cid:19) , (2.22a) β ωω (cid:48) = − e − i ( ω + ω (cid:48) ) v H πκ √ ωω (cid:48) ω + ω (cid:48) (cid:20) − iκ ( ω + ω (cid:48) ) (cid:21) − iω/κ Γ (cid:18) iωκ (cid:19) . (2.22b) B. (1+1)D spacetime with a collapsing null shell v = v u in = v H u out = ∞ I − L I + L I − R I + R FIG. 2. Penrose diagram for a 2D black hole that forms from the collapse of a null shell along thetrajectory v = v . The Cauchy surface used to compute the Bogolubov coefficients is the dotted(blue) surface formed from I + L , part of I − R , and the v = v null ray. Note that the horizon is thefuture light cone of the point ( u in = v H ≡ v − M , v = v ). For a (1+1)D spacetime with a collapsing null shell the line element inside the shell is7till given by (2.1), while outside the shell it is ds = − (cid:18) − Mr (cid:19) dt s + (cid:18) − Mr (cid:19) − dr . (2.23)The Penrose diagram is given in Fig. 2. We define the usual radial null coordinates insidethe shell to be those in Eq. (2.2). Outside both the shell and the horizon, the correspondingcoordinates are u s ≡ t s − r ∗ , (2.24a) v ≡ t s + r ∗ , (2.24b) r ∗ ≡ r + 2 M log (cid:18) r − M M (cid:19) . (2.24c)Following [4, 14] we match the coordinate systems along the part of the trajectory of theshell which is outside the horizon in such a way that both v and r are continuous acrossthe surface and v = v on the surface. This is why we have no subscripts for these twocoordinates. The coordinates t and u are not continuous across the surface. To find therelation between u s and u we note that at the surface and outside the event horizon r = 12 ( v − u ) , (2.25a) r ∗ = 12 ( v − u s ) = r + 2 M log (cid:18) r − M M (cid:19) . (2.25b)Substituting (2.25a) into the right hand side of (2.25b) and solving for u s gives u s = u − M log (cid:18) v H − u M (cid:19) , (2.26)with v H ≡ v − M . (2.27)Note that the event horizon ( u s = ∞ ) is at u = v H .We next show that it is possible to invert (2.26) using the Lambert W function. First itis easy to show that (2.26) can be written in the formexp (cid:18) v H − u s M (cid:19) = (cid:18) v H − u M (cid:19) exp (cid:18) v H − u M (cid:19) . (2.28)Then computing the Lambert W function of both sides the equation and using the secondrelation in (2.15) we find that u = v H − M W (cid:20) exp (cid:18) v H − u s M (cid:19)(cid:21) . (2.29)8he field φ and its mode functions f are solutions to Eq. (2.3). In the flat space regionbelow the null shell the general solution is (2.5). In the Schwarzschild region above the shell,Eq. (2.3) takes the form ∂ u s ∂ v f = 0 . (2.30)The general solution is f = c ( u s ) + d ( v ) , (2.31)with c and d being arbitrary functions. Thus in the flat space region solutions can be anyfunction of u or any function of v while in the Schwarzschild region they can be any functionof u s or any function of v . Given the relations (2.26) and (2.29) it is clear that any solutionin the Schwarzschild region is also a solution in the flat region and vice versa. Once againthe modes are normalized using the scalar product (2.6). There is a complete set of in modesthat are normalized on I − and are given by the expressions f in ω,R = e − iωv √ πω , (2.32a) f in ω,L = e − iωu √ πω . (2.32b)A different complete set of modes consists of subsets that have three different late timebehaviors. Some of the modes end on I + L , others go through the future horizon and endup at the singularity, and the rest end on I + R . As with the accelerating mirror, we areinterested in those that end up on I + R , which we label as out modes and which are given by f out ω = e − iωu s √ πω . (2.33)The other modes we label with the superscripts left and sing .As in the accelerating mirror case (2.13), the average number of particles found at I + R for a given value of ω if the field is in the in state is (cid:104) in | N out ω | in (cid:105) = (cid:104) in | a out † ω a out ω | in (cid:105) . (2.34)The expansions of φ in terms of these complete sets of modes are φ = (cid:90) ∞ dω [ a in ω,R f in ω,R + a in † ω,R f in ∗ ω,R + a in ω,L f in ω,L + a in † ω,L f in ∗ ω,L ] , (2.35a)= (cid:90) ∞ dω [ a out ω f out ω + a out † ω f out ∗ ω + a left ω f left ω + a left † ω f left ∗ ω + a sing ω f sing ω + a sing † ω f sing ∗ ω ] . (2.35b)9n this case the scalar product ( f in ω (cid:48) ,R , f out ω ) = 0, because the out modes vanish on I − R .Hence a out ω = ( φ, f out ω ) = (cid:90) ∞ dω (cid:48) (cid:104) a in ω (cid:48) ,L ( f in ω (cid:48) ,L , f out ω ) + a in † ω (cid:48) ,L ( f in ∗ ω (cid:48) ,L , f out ω ) (cid:105) . (2.36)If we write f out ω = (cid:90) ∞ dω (cid:48) (cid:2) α ωω (cid:48) f in ω (cid:48) ,L + β ωω (cid:48) f in ∗ ω (cid:48) ,L (cid:3) , (2.37)then a out ω = (cid:90) ∞ dω (cid:48) (cid:104) a in ω (cid:48) ,L α ∗ ωω (cid:48) − a in † ω (cid:48) ,L β ∗ ωω (cid:48) (cid:105) , (2.38)and the Bogolubov coefficients can be obtained from α ωω (cid:48) = ( f out ω , f in ω (cid:48) ,L ) , (2.39a) β ωω (cid:48) = − ( f out ω , f in ∗ ω (cid:48) ,L ) , (2.39b)while once again the average number of particles is (cid:104) in | N out ω | in (cid:105) = (cid:90) ∞ dω (cid:48) | β ωω (cid:48) | . (2.40)The Cauchy surface we use to compute the Bogolubov coefficients is shown as dotted(and blue) in Fig. 2. It consists of v = v plus the part of I − R with v > v and all of I + L .However, the modes f out ω,R are nonzero only on the part of the Cauchy surface with v = v that is outside the event horizon ( u s < ∞ , u < v H ). Using (2.32b), (2.33), (2.39a) and(2.39b) one finds α ωω (cid:48) = 14 π (cid:90) v H −∞ du e − i ( ω − ω (cid:48) ) u [ κ ( v H − u )] iω/κ × (cid:34)(cid:114) ω (cid:48) ω + (cid:114) ωω (cid:48) (cid:18) κ ( v H − u ) (cid:19)(cid:35) , (2.41a) β ωω (cid:48) = 14 π (cid:90) v H −∞ du e − i ( ω + ω (cid:48) ) u [ κ ( v H − u )] iω/κ × (cid:34)(cid:114) ω (cid:48) ω − (cid:114) ωω (cid:48) (cid:18) κ ( v H − u ) (cid:19)(cid:35) , (2.41b)where κ = 1 / (4 M ) is the surface gravity of the black hole. These equations are identicalto Eqs. (2.21) for the mirror trajectory considered in Sec. II A if we make the substitution u → v and identify the acceleration parameter, κ , in the mirror case, with the surface gravity10 in the black hole case. Thus the values for α ωω (cid:48) and β ωω (cid:48) are identical with those in (2.22)and we have found an exact correspondence between the particle production which occursin (1+1)D for a mirror with trajectory (2.14) and a black hole that forms from the collapseof a null shell along the surface v = v . C. (3+1)D spacetime with a collapsing null shell
For a (3+1)D spacetime with a collapsing null shell the line element inside the shell isthat of flat space ds = − dt + dr + r d Ω , (2.42)and outside the shell is the Schwarzschild metric ds = − (cid:18) − Mr (cid:19) dt s + (cid:18) − Mr (cid:19) − dr + r d Ω . (2.43)The Penrose diagram is given in Fig. 3. v = v v = v H H + I − I + FIG. 3. Penrose diagram for a (3+1)D black hole that forms from the collapse of a null shell alongthe trajectory v = v . The horizon, H + , is the dotted (red) curve. The Cauchy surface used tocompute the Bogolubov coefficients is the short-dashed (blue) curve. The radial null coordinates have the same definitions as in the (1+1)D case with thoseinside the shell given by (2.2) and those outside the shell given by (2.24). The matching of11he coordinates across the shell is also the same as in the (1+1)D case with the results givenby (2.26) and (2.29).The massless minimally coupled scalar field satisfies Eq. (2.3). The field can be expandedin terms of complete sets of modes where the mode functions are written in the general form f = Y (cid:96)m ( θ, φ ) √ πω ψ ( t, r ) r . (2.44)Inside the shell we have the flat-space radial wave equation − ∂ ψ∂t + ∂ ψ∂r − V eff ( r ) ψ = 0 , (2.45)while outside the shell we have the scalar Regge-Wheeler equation − ∂ ψ∂t s + ∂ ψ∂r ∗ − V eff ( r ) ψ = 0 . (2.46)The effective potential is V eff = (cid:18) − Mr (cid:19) (cid:20) Mr + (cid:96) ( (cid:96) + 1) r (cid:21) , (2.47)which can be seen to work in both cases if inside the shell we set M = 0.The modes are normalized using the full three dimensional version of the scalar product,Eq. (2.6). In the cases we consider the Cauchy surface consists of either a single nullhypersurface or a union of null hypersurfaces, and the integrals are of the forms (cid:90) du (cid:90) d Ω r ↔ ∂ u , (cid:90) dv (cid:90) d Ω r ↔ ∂ v . (2.48)We consider two complete sets of mode functions. Those for the in state are normalizedon past null infinity, I − , and vanish at r = 0 inside the shell. Thus inside the shell they arethe same as the mode functions in flat space in the Minkowski vacuum. On I − they are ψ in ω(cid:96) = e − iωv , (2.49)with 0 ≤ ω < ∞ . They are of course more complicated away from I − , although there areanalytic solutions for them inside the shell. The simplest solution inside the shell is for themode with (cid:96) = 0: ψ in ω = e − iωv − e − iωu . (2.50)12he other complete set of solutions we will consider is a union of two subsets. One subset,of most interest, is normalized on future null infinity, I + . We label them as out modes. On I + they are ψ out ω(cid:96) = e − iωu s , (2.51)where again 0 ≤ ω < ∞ . These modes vanish at the future horizon H + . The other setconsists of modes which vanish at I + and are nonzero on H + . We give them the label H + and will not be concerned with their normalization here. It is easy to show using the scalarproduct and a Cauchy surface for the region outside the horizon, which consists of H + and I + , that these two sets of modes are orthogonal.The expansions for φ in terms of the two complete sets of modes are φ = (cid:90) ∞ dω (cid:88) (cid:96),m (cid:104) a in ω(cid:96)m f in ω(cid:96)m + a in † ω(cid:96)m f in ∗ ω(cid:96)m (cid:105) , (2.52a) φ = (cid:90) ∞ dω (cid:88) (cid:96),m (cid:104) a out ω(cid:96)m f out ω(cid:96)m + a out † ω(cid:96)m f out ∗ ω(cid:96)m + a H + ω(cid:96)m f H + ω(cid:96)m + a H + † ω(cid:96)m f H + ∗ ω(cid:96)m (cid:105) . (2.52b)In this case, the goal is to determine the average number of particles in the out state, as afunction of ω , (cid:96) , and m , if the field is in the in state. This is given by (cid:104) in | N out ω(cid:96)m | in (cid:105) = (cid:104) in | a out † ω(cid:96)m a out ω(cid:96)m | in (cid:105) . (2.53)Using the orthonormality of the mode functions we find from (2.52) that a out ω(cid:96)m = ( φ, f out ω(cid:96)m ) = (cid:88) (cid:96) (cid:48) ,m (cid:48) (cid:90) ∞ dω (cid:48) (cid:104) a in ω (cid:48) (cid:96) (cid:48) m (cid:48) ( f in ω (cid:48) (cid:96) (cid:48) m (cid:48) , f out ω(cid:96)m ) + a in † ω (cid:48) (cid:96) (cid:48) m (cid:48) ( f in ∗ ω (cid:48) (cid:96) (cid:48) m (cid:48) , f out ω(cid:96)m ) (cid:105) . (2.54)If we take the transformation between sets of mode functions to be f out ω(cid:96)m = (cid:88) (cid:96) (cid:48) m (cid:48) (cid:90) ∞ dω (cid:48) (cid:2) α ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) f in ω (cid:48) (cid:96) (cid:48) m (cid:48) + β ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) f in ∗ ω (cid:48) (cid:96) (cid:48) m (cid:48) (cid:3) , (2.55)then the operators are connected by a out ω(cid:96)m = (cid:88) (cid:96) (cid:48) ,m (cid:48) (cid:90) ∞ dω (cid:48) (cid:104) a in ω (cid:48) (cid:96) (cid:48) m (cid:48) α ∗ ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) − a in † ω (cid:48) (cid:96) (cid:48) m (cid:48) β ∗ ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) (cid:105) , (2.56)and the expectation value will be (cid:104) in | N out ω(cid:96)m | in (cid:105) = (cid:88) (cid:96) (cid:48) ,m (cid:48) (cid:90) ∞ dω (cid:48) | β ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) | , (2.57)13ith the Bogolubov coefficients found via α ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) = ( f out ω(cid:96)m , f in ω (cid:48) (cid:96) (cid:48) m (cid:48) ) , (2.58a) β ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) = − ( f out ω(cid:96)m , f in ∗ ω (cid:48) (cid:96) (cid:48) m (cid:48) ) . (2.58b)On any hypersurface where integrals of the form (2.48) are to be computed, the followingorthonormality conditions are useful (cid:90) d Ω Y (cid:96)m ( θ, φ ) Y ∗ (cid:96) (cid:48) m (cid:48) ( θ, φ ) = δ (cid:96),(cid:96) (cid:48) δ m,m (cid:48) , (cid:90) d Ω Y (cid:96)m ( θ, φ ) Y (cid:96) (cid:48) m (cid:48) ( θ, φ ) = ( − m δ (cid:96),(cid:96) (cid:48) δ m, − m (cid:48) . (2.59)It is then possible to show that the Bogolubov coefficients are partially diagonal in the sensethat α ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) ∝ δ (cid:96),(cid:96) (cid:48) δ m,m (cid:48) β ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) ∝ ( − m δ (cid:96),(cid:96) (cid:48) δ m, − m (cid:48) , (2.60)and that the average number of particles is (cid:104) in | N out ω(cid:96)m | in (cid:105) = (cid:90) ∞ dω (cid:48) | β ω(cid:96)mω (cid:48) (cid:96) ( − m ) | . (2.61)To compute the Bogolubov coefficients using Eqs. (2.58) it is necessary to choose a Cauchysurface for the spacetime. The choice we make is driven by the fact that we have exactsolutions for the mode functions f in ω(cid:96)m in the region inside the shell and also everywhere on I − since that is where these modes are normalized. To get their form in the region outsidethe shell it would be necessary either to use a Bogolubov transformation or to solve thepartial differential equation (2.46) numerically. The mode functions f out ω(cid:96)m are normalized on I + so we have analytic expressions for them there. They can be computed in the regionoutside the null shell by separating the functions ψ ω(cid:96) into ψ ω(cid:96) ( t, r ) = e − iωt χ ω(cid:96) ( r ) , (2.62)and numerically solving the resulting radial equation for χ ω(cid:96) , which is d χ ω(cid:96) dr ∗ + ( ω − V eff ) χ ω(cid:96) = 0 . (2.63)However, to extend these solutions to the region inside the null shell to make contact with f in ω(cid:96)m requires either using a Bogolubov transformation such as Eq. (2.55) or solving the14artial differential equation (2.45) numerically. Here we use a Bogolubov transformationand choose the Cauchy surface shown in Fig. 3, which consists of the null surface v = v along with the portion of I − with v < v < ∞ .In a subsequent paper we intend to numerically solve the mode equation (2.63) when theeffective potential is included. In this paper, however, we set V eff = 0 and ignore potentialbarrier effects in order to see what other effects (3+1)D has. Accordingly, inside the shell,the in modes are given by Eq. (2.50) for all values of (cid:96) and m . Similarly, outside the shellthe out modes are given by ψ out ω(cid:96) = e − iωu s , (2.64)which are taken to vanish as u s → −∞ along I − for v > v . Thus α ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) = − i δ (cid:96),(cid:96) (cid:48) δ m,m (cid:48) π √ ωω (cid:48) (cid:90) v H −∞ due − iωu s ↔ ∂ u ( e iω (cid:48) v − e iω (cid:48) u ) , (2.65a) β ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) = i ( − m δ (cid:96),(cid:96) (cid:48) δ m, − m (cid:48) π √ ωω (cid:48) (cid:90) v H −∞ due − iωu s ↔ ∂ u ( e − iω (cid:48) v − e − iω (cid:48) u ) . (2.65b)Note that the terms in the integrands with factors of e ± iω (cid:48) v are total derivatives and canbe integrated trivially. Because e ± iωu s effectively vanishes at u s = ±∞ , these terms vanishalso. The result is that α ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) = − δ (cid:96),(cid:96) (cid:48) δ m,m (cid:48) π (cid:90) v H −∞ du e − i ( ω − ω (cid:48) ) u [ κ ( v H − u )] iω/κ × (cid:34)(cid:114) ω (cid:48) ω + (cid:114) ωω (cid:48) (cid:18) κ ( v H − u ) (cid:19)(cid:35) , (2.66a) β ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) = ( − m +1 δ (cid:96),(cid:96) (cid:48) δ m, − m (cid:48) π (cid:90) v H −∞ du e − i ( ω + ω (cid:48) ) u [ κ ( v H − u )] iω/κ × (cid:34)(cid:114) ω (cid:48) ω − (cid:114) ωω (cid:48) (cid:18) κ ( v H − u ) (cid:19)(cid:35) . (2.66b)The expression for α ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) differs from the (1+1)D case in (2.41a) by the factor of − δ (cid:96),(cid:96) (cid:48) δ m,m (cid:48) and the expression for β ω(cid:96)mω (cid:48) (cid:96) (cid:48) m (cid:48) differs from the (1+1)D case in (2.41b) bythe factor of ( − m +1 δ (cid:96),(cid:96) (cid:48) δ m, − m (cid:48) .As mentioned in the Introduction, Massar and Parentani [4] have computed the Bogol-ubov coefficients for the case of a null shell collapsing to form a black hole. Their computationwas for the s-wave sector in the (3+1)D case when the effective potential is ignored. Thusit was the same as the case done in this subsection. By restricting to the s-wave sector,they effectively considered the (1+1)D case as well. However, because they began with the153+1)D case, their mode functions vanish at r = 0 inside the shell. In our separate (1+1)Dmodel we make no such assumption and instead have modes arising from I − L . Despite thatdifference both models yield the same amount of particle production. (Note that there is amissing normalization factor of 8 M in Eq. (10) of [4].) III. TIME AND FREQUENCY RESOLVED SPECTRA
To investigate the time dependence of the particle production rate we construct localizedwave packets of a form originally used by Hawking [1] and which were used by us in Ref. [8]to examine a set of accelerating mirror models. When this constructive process is applied tomode functions of definite frequency, the resulting packets form a complete orthonormal setthat subdivides (and provides a degree of localization) within both the time and frequencydomains. Following [14], a given mode packet is defined as f out jn ≡ √ (cid:15) (cid:90) ( j +1) (cid:15)j(cid:15) dω e πiωn/(cid:15) f out ω . (3.1)A packet with index j covers the range of frequencies j(cid:15) ≤ ω ≤ ( j + 1) (cid:15) . Since the definitefrequency out modes approach I + R with the behavior f out ω ∼ e − iωu s , a packet with index n covers the approximate time range (2 πn − π ) /(cid:15) (cid:46) u s (cid:46) (2 πn + π ) /(cid:15) . We can write β jn,ω (cid:48) ≡ − ( f out jn , f in ∗ ω (cid:48) ) . (3.2)Using Eq. (3.1) and interchanging the order of integration gives β jn,ω (cid:48) = 1 √ (cid:15) (cid:90) ( j +1) (cid:15)j(cid:15) dω e πiωn/(cid:15) β ωω (cid:48) . (3.3)Then the quantity (cid:104) in | N out jn | in (cid:105) ≡ (cid:90) ∞ dω (cid:48) | β jn,ω (cid:48) | , (3.4)can be thought of as giving the average number of particles detected by a particle detectorthat was sensitive to the frequency range j(cid:15) ≤ ω ≤ ( j + 1) (cid:15) and was turned on during thetime period (2 πn − π ) /(cid:15) (cid:46) u s (cid:46) (2 πn + π ) /(cid:15) . Note that the value of (cid:104) N jn (cid:105) is the same forboth the mirror and the (1+1)D spacetime with a collapsing null shell since the values of β ωω (cid:48) are the same in those cases.A similar expression works for the (3+1)D spacetime with a collapsing null shell for givenvalues of (cid:96) and m . If, as in the previous section, we neglect V eff , then the value of β for given16 and ω (cid:48) is the same for all (cid:96) and m . Thus summing over (cid:96) and m results in an infinitenumber of particles for each value of j and n . If the mode equation is solved by including V eff , then the number of particles for each value of j and n will be finite [14] (a case we willdiscuss elsewhere).If Eq. (2.22b) is substituted into Eq. (3.3) then in the late time, large n limit one cansee that the dominant contribution to the integral comes from values of ω (cid:48) for which thearguments of the oscillating exponentials cancel or nearly cancel and which therefore satisfythe condition ω (cid:48) (cid:29) ω . In this limit | β ωω (cid:48) | ∼ πκω (cid:48) e πω/κ − , (3.5)and one sees that there is a thermal distribution of particles with temperature T = κ/ π .Thus the radiation will asymptotically approach a thermal distribution at the black holetemperature. Such a late time thermal distribution was found for black hole radiation in [1]and for mirrors with a particular class of asymptotically null trajectories in [3] .To compare the exact results with a thermal spectrum, it is useful to write the thermalspectrum in terms of packets. This has been done in [8] for a mirror trajectory studied byCarlitz and Willey [6] in which the particle production is always in a thermal distribution.The trajectory is [8] z ( t ) = − t − κ W ( e − κt ) , (3.6)and the relevant Bogolubov coefficient is β ωω (cid:48) = 14 π √ ωω (cid:48) (cid:34) − ωκ e − πω/ κ (cid:18) ω (cid:48) κ (cid:19) − iω/κ Γ (cid:18) iωκ (cid:19)(cid:35) . (3.7)Substituting Eq. (3.7) into Eq. (3.3) and then into Eq. (3.4) yields (cid:104) N jn (cid:105) = 1 (cid:15) (cid:90) ( j +1) (cid:15)j(cid:15) dωe πω/κ − κ π(cid:15) log (cid:34) e π ( j +1) (cid:15)κ − e πj(cid:15)κ − (cid:35) − . (3.8)Note that the packets depend on the frequency parameters (cid:15) and j but not on the time param-eter n as would be expected if the particles are always produced in a constant-temperaturethermal distribution. Note also that the infrared divergence in Eq. (3.7) results in a diver-gence in the j = 0 bin in Eq. (3.8). Since all real particles detectors have infrared cutoffs,for simplicity we simply ignore the j = 0 bin when making comparisons with our results forthe trajectory (2.14). 17n interesting balance in time and frequency resolution occurs for (cid:15) = κ π log (cid:32) √ (cid:33) = T csch − (2) . (3.9)With this packet width one can show using Eq. (3.8) that a thermal distribution has ∞ (cid:88) j =1 (cid:104) N j (cid:105) = (cid:104) N j =1 (cid:105) + ∞ (cid:88) j =2 (cid:104) N j (cid:105) = 1 + 1 = 2 . (3.10)It is possible, for both the particle production from a mirror following the Carlitz-Willeytrajectory (3.6) and that from a mirror following our accelerating mirror trajectory (2.14),to scale out the dependence of (cid:104) N jn (cid:105) on κ by working with the following dimensionlessquantities: x ≡ ωκ , (3.11a)¯ (cid:15) ≡ (cid:15)κ , (3.11b)¯ v H ≡ κv H . (3.11c)Using Eq. (2.22b), we find for the trajectory (2.14) that (cid:104) N jn (cid:105) = 14 π ¯ (cid:15) (cid:90) ∞ dx (cid:48) (cid:90) ( j +1)¯ (cid:15)j ¯ (cid:15) dx (cid:90) ( j +1)¯ (cid:15)j ¯ (cid:15) dx e i (2 πn/ ¯ (cid:15) − ¯ v H )( x − x ) e − π ( x + x ) / × x (cid:48) √ x x ( x + x (cid:48) ) − ix − ( x + x (cid:48) ) ix − Γ( ix ) Γ( − ix ) . (3.12)For the other mirror trajectories studied in [8], which were all inertial at late times, it wasfound that choosing a small enough value for (cid:15) and thus a small enough range for each valueof j in terms of ω gives fine-grained frequency resolution but coarse-grained time resolution.Similarly choosing a large enough value of (cid:15) results in a fine-grained time resolution butcoarse-grained frequency resolution. It is not possible, of course, to get arbitrary fine-grained simultaneous time and frequency resolution. However, for the asymptotically inertialtrajectories studied in [8] attempts to obtain any significant degree of simultaneous timeand frequency resolution were not successful. As shown below, for the asymptotically nulltrajectory Eq. (2.14) we have had some success in locating an optimal compromise in timeand frequency resolution. The argument of the logarithm is of course the Golden Ratio. It’s significance here is simply that itresults in the sum (3.10).
18e begin by illustrating the time dependence of the particle production rate by choosingthe relatively large number ¯ (cid:15) = 1. Because any realistic particle detector will have aninfrared frequency cutoff, we shall impose one by only considering bins with j ≥
1. For thisvalue of ¯ (cid:15) and for the Bogolubov coefficient (2.22b), we find that most of the particles are inthe bin with j = 1. The time evolution of the average number of particles detected in thisbin is given in Fig. 4 for the case v H = 0. It can be seen from this figure that the particleproduction rate monotonically increases to its thermal value.What can also be seen from Fig. 4 is the very small value that (cid:104) N jn (cid:105) has. This means thatthe actual amount of particle production that one would expect in a specific instance wouldbe very low. This is related to the fact that, even at late times, the flux of radiation due toblack hole evaporation is very sparse [16]. Similar results were found for the asymptoticallyinertial mirror trajectories in [8]. FIG. 4. Average number of particles produced in the j = 1 frequency bin as a function of thetime parameter n for ¯ (cid:15) = 1. The open boxes correspond to the thermal distribution (3.8). To investigate the frequency spectrum we can make use of the specific packet width inEq. (3.9), which is small enough to provide some frequency resolution. First however, in19ig. 5 we show the time dependence of the particle number for the j = 1 bin. It is clear thatthe time resolution is not as good as for the case ¯ (cid:15) = 1 in Fig. 4. The frequency resolutionis shown for three different times in Fig. 6. It is seen that we have reasonably fine-grainedfrequency resolution for the time parameters n = − , ,
1, while the amount of particleproduction in a given time interval is larger in a low frequency bin than a high frequencybin.
FIG. 5. Average number of particles produced in the j = 1 frequency bin as a function of thetime parameter n for the packet width in Eq. (3.9). The open boxes correspond to the thermaldistribution (3.8). The increase in particle production is monotonic with no significant feature in the particlespectrum and production rate near the time of black hole formation in contrast to the initialburst of particles seen for the mirror trajectory in [9]. The approach to a thermal distributionis expected since the mirror trajectory is asymptotically null and in the collapsing null shellcase the backreaction of the black hole radiation on the spacetime geometry is ignored. In20
IG. 6. Plotted are the frequency spectra for the average number of particles produced with thepacket width in Eq. (3.9). From top to bottom the plots are for the values of the time parameter n = − , , IG. 7. Energy flux of a quantized massless minimally coupled scalar field for the acceleratingmirror spacetime. At late times the flux approaches its asymptotic value in Eq. (4.3). contrast, for the asymptotically inertial trajectories studied in [8, 9], one finds a peak in theamount of particle production followed by a steady decline.
IV. STRESS-ENERGY TENSOR
Here we compute the stress-energy tensor for the accelerating mirror spacetime. Thegeneral form of the energy flux for any mirror trajectory as a function of time u is [2] F ( u ) ≡ (cid:104) T uu (cid:105) = 124 π (cid:18) p (cid:48)(cid:48) p (cid:48) − p (cid:48)(cid:48)(cid:48) p (cid:48) (cid:19) , (4.1)where the primes are derivatives with respect to u . The energy flux for the trajectory (2.14)is F ( u ) = κ π (cid:2) W (cid:0) e − κ ( u − v H ) (cid:1) + 1 (cid:3) [ W ( e − κ ( u − v H ) ) + 1] . (4.2)It is shown in Fig. 7. Note that, unlike the case of mirror trajectories which are asymptot- This can also be expressed in terms of the rapidity η ( u ) ≡ tanh − [ ˙ z ( t m ( u ))] = ln p (cid:48) ( u ). The result is12 πF ( u ) = [ η (cid:48) ( u )] − η (cid:48)(cid:48) ( u ). F = κ π , (4.3)which is the value at all times for the case of a mirror following the Carlitz-Willey trajec-tory (3.6).An interesting question is whether there is some way to characterize the non-thermalepoch beyond the observation that the approach to a thermal state is monotonic for boththe particle production and the stress-energy tensor. One way to do so is to look at howquickly a given quantity changes. The rate, F (cid:48) ( u ), at which the energy flux changes is F (cid:48) ( u ) = κ π [ W (cid:0) e − κ ( u − v H ) (cid:1) ] [ W ( e − κ ( u − v H ) ) + 1] . (4.4)The particular time, u max , at which the rate F (cid:48) ( u ) reaches its maximum value, is importantbecause that is the time at which the system is furthest away from both its late-time thermalemission and its early-time zero emission. It is κ ( u max − v H ) = ln 2 − ≈ . . (4.5)It is interesting to note that this is the same time at which | z (cid:48)(cid:48) ( u ) | and | p (cid:48)(cid:48) ( u ) | reach theirmaximum values. This time is also comparable to the time at which the change in theparticle production rate is a maximum. This can be seen from Fig. 4 to be at n ≈
0, whichcorresponds to u ≈
0. Recall that the time corresponding to n is approximately u = 2 πn/(cid:15) .For u > u max the rate of change of the flux falls off rapidly so there is an asymmetry in thegrowth of the flux. This can be seen from the fact that at u = u max the flux is 16 / ≈ V. CONCLUSIONS
We have displayed an exact correspondence between the particle production in (1+1)Dthat occurs for a mirror in flat space with the trajectory (2.14) and the particle productionthat occurs when a black hole forms from gravitational collapse of a null shell. There is alsoa correspondence in the case of a null shell collapsing to form a black hole in (3+1)D if theeffective potential in the mode equation is ignored.23e have used wave packets of the form (3.1) to investigate the time dependence of theparticle production rate in the (1+1)D cases. We found that the particle production rateincreases monotonically with time. We have also computed the stress-energy tensor (cid:104) T ab (cid:105) for the scalar field in the case of the accelerating mirror. The rate of change of the particleproduction mimics the rate of change of energy production in time. With a relativity slow-increase and fast-decrease, the rate of change of energy-particle flux peaks at a maximumtime that corresponds to the most non-thermal, out-of-equilibrium time of the system. Thefact that the rate-loss is greater than the rate-gain, points to an asymmetry in the approachto equilibrium. The energy flux is approximately 60% of its maximum equilibrium value atthe time when the system is the most out of equilibrium.The monotonic increase in particle production underscores the relatively calm approachto equilibrium. There are no characteristic imprints to identify the energy flux in the particleemission. However, the peak non-thermal time can be identified and the rate of change ofenergy flux is mirrored in the rate of change of particle production: clear signatures of theparticle-energy coupling during the non-equilibrium phase. ACKNOWLEDGMENTS
MRRG thanks Yen Chin Ong, Don Page and William Unruh for clarifying some ideas.PRA would like to thank Renaud Parentani, and PRA and CRE would like to thank Alessan-dro Fabbri and Amos Ori for helpful conversations. This work was supported in part by theNational Science Foundation under Grant Nos. PHY-0856050, PHY-1308325, and PHY-1505875 to Wake Forest University, PHY-1506182 to the University of North Carolina,Chapel Hill. CRE acknowledges support from the Bahnson Fund at the University of NorthCarolina, Chapel Hill. Some of the plots were generated on the WFU DEAC cluster; wethank the WFU Provost’s Office and Information Systems Department for their generoussupport. [1] S.Hawking,
Commun. Math. Phys. (1975) 199.[2] S. A. Fulling and P. C. W. Davies, Proc. Roy. Soc. Lond. A (1976) 393.[3] P. C. W. Davies and S. A. Fulling,
Proc. Roy. Soc. Lond. A (1977) 237.
4] S. Massar and R. Parentani,
Phys. Rev. D , 7444 (1996).[5] M. R. R. Good, Ph. D. thesis, UMI Dissertations ISBN: 9781124942285. University of NorthCarolina, (2011).[6] R. D. Carlitz and R. S. Willey, Phys. Rev. D , 2327 (1987).[7] W. R. Walker and P. C. W. Davies, J. Phys. A , L477 (1982).[8] M. R. R. Good, P. R. Anderson, and C. R. Evans, Phys. Rev. D , 025023 (2013).[arXiv:1303.6756 [gr-qc]].[9] M. R. R. Good and Y. C. Ong, JHEP , 145 (2015) [arXiv:1506.08072 [gr-qc]].[10] P. R. Anderson, M. R. R. Good, and C. R. Evans, to appear in the Proceedings of the 14thMarcel Grossmann Meeting, arXiv:1507.03489.[11] M. R. R. Good, P. R. Anderson, and C. R. Evans, to appear in the Proceedings of the 14thMarcel Grossmann Meeting, arXiv:1507.05048.[12] M. R. R. Good, LeCosPA Symposium Proceedings, Taipei, (2015). arXiv:1602.00683 [gr-qc].[13] C. W. Misner, K. S. Thorne, and J. A. Wheeler,
Gravitation (Freeman, San Francisco, 1973).[14] A. Fabbri and J. Navarro-Salas,
Modeling black hole evaporation (Imperial College Press,London, UK, 2005).[15] W. R. Walker,
Phys. Rev. D , 767 (1985).[16] See e.g. F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, arXiv:1506.03975, and referencescontained therein., 767 (1985).[16] See e.g. F. Gray, S. Schuster, A. Van-Brunt, and M. Visser, arXiv:1506.03975, and referencescontained therein.