Black hole radiation in the presence of a universal horizon
BBlack hole radiation in the presence of a universal horizon
Florent Michel ∗ and Renaud Parentani † Laboratoire de Physique Th´eorique, CNRS UMR 8627, Bˆatiment 210,Universit´e Paris-Sud 11, 91405 Orsay CEDEX, France
In Hoˇrava and Einstein-æther theories of modified gravity, in spite of the violation of Lorentzinvariance, spherically symmetric stationary black hole solutions possess an inner universal horizonwhich separates field configurations into two disconnected classes. We compute the late time radia-tion emitted by a dispersive field propagating in such backgrounds. We fix the initial conditions onstationary modes by considering a regular collapsing geometry, and imposing that the state insidethe infalling shell is vacuum. We find that the mode pasting across the shell is adiabatic at late time(large inside frequencies). This implies that large black holes emit a thermal flux with a temperaturefixed by the surface gravity of the Killing horizon. In turn, this suggests that the universal horizonshould play no role in the thermodynamical properties of these black holes.
PACS numbers: 04.50.Kd, 04.62.+v, 04.70.Dy
I. INTRODUCTION
The laws of black hole thermodynamics are firmly established in Lorentz invariant theories, and they play a crucialrole in our understanding of black hole physics [1]. In particular, the entropy and the temperature are governed bythe area and surface gravity of the event horizon. In Lorentz violating theories, the status of these laws is unclearbecause essential aspects are no longer present [2–7]. For instance, the thermality of the Hawking flux is inevitablylost in the presence of high frequency dispersion, although it is approximatively recovered for large black holes, i.e.,when the surface gravity κ is much smaller than the UV scale Λ setting the high frequency dispersion [8].The origin of the difficulties can be traced to the fact that the event horizon no longer separates the outgoing fieldconfigurations into two disconnected classes. In fact, when the dispersion is superluminal, it can be crossed by outgoingradiation. However, it was recently discovered that in some theories of modified gravity such as Hoˇrava gravity [9–11]and Einstein-æther [12–15], spherically symmetric black hole solutions possess a second inner horizon. This horizon,named universal , cannot be crossed by outgoing configurations, even for superluminal dispersion relations which allowfor arbitrarily large group velocities. (The difficulty mentioned in [2] is thus evaded.) Following this discovery, ithas been argued that the universal horizon should play a key role in the thermodynamics of such black holes. Eventhough they seem to obey a first law [16, 17], a key question concerns the temperature of the Hawking radiation theyemit. Would it be essentially governed by the (higher) surface gravity of the universal horizon, or would it still befixed by the surface gravity κ of the Killing horizon?Two recent works concluded that the universal horizon emits a steady radiation with properties governed by itssurface gravity. Because of the complicated nature of the field propagation near that horizon, this conclusion wasindirectly obtained, in [18], by making use of a “tunneling method”, and, in [19], by analyzing the characteristics ofthe radiation field. In the present paper, we reexamine this question by performing a direct calculation and reach theopposite conclusion that no radiation is emitted from the universal horizon at late time.We proceed as follows. As in the original derivation of Hawking [20], we identify the boundary conditions on theoutgoing modes in the near vicinity of the universal horizon by considering a simple collapsing shell geometry, andby assuming that the state of the field is vacuum inside. We then compute the mode mixing across the shell betweeninside modes propagating outwards φ in ω , and outside stationary modes with a fixed Killing frequency ψ λ . The late timebehavior is obtained by sending the inside frequency ω → ∞ . In this limit, we show that the scattering coefficientsinvolving modes with opposite norms vanish. This result can be understood from the fact that the modes ψ λ areaccurately described by their WKB approximation in the immediate vicinity of the universal horizon. In other words,the pasting across the shell is adiabatic in the limit ω → ∞ . Hence, for large outgoing radial momenta, the state ofthe field outside the shell is the usual vacuum, as explained in [21].It then remains to propagate these high momentum dispersive modes from the universal horizon till spatial infinity.This propagation has already been studied in detail; see [22] for a recent update. It establishes that large blackholes emit a stationary flux which is (nearly) thermal, and with a temperature approximatively given by the standard ∗ fl[email protected] † [email protected] a r X i v : . [ g r- q c ] J u l relativistic value. In other words, the robustness of the Hawking process, i.e. its insensitivity to high frequencydispersion which was first established in [23], is now extended to black holes with a universal horizon.From this it is tempting to conclude that the laws of black hole thermodynamics should also be robust, andthey should involve the properties of the Killing horizon. This conclusion is reinforced by the fact that the fieldconfigurations propagating on either side of a universal horizon come from two disconnected Cauchy surfaces, and arehighly blueshifted. Hence, it seems that no Hadamard condition of regularity [7] could be satisfied on the universalhorizon. This raises the question of the fate of the universal horizon; see [6]. This difficult question shall not bediscussed in the present work.Appendix A gives the details of the calculation which is summarized in the main text. In Appendix B we compareour model with previously-studied dispersive ones without a universal horizon, and show the role of the accelerationof the preferred frame. Appendix C shows the results of numerical simulations confirming the approximately thermalcharacter of the emission at infinity governed by the surface gravity of the Killing horizon. II. MASSLESS RELATIVISTIC SCALAR FIELD IN A COLLAPSING SHELL GEOMETRY
In this section, we briefly review the computation of the Hawking radiation emitted at late time in a collapsinggeometry [20]. Although these concepts are well known, we present them in a way which prepares the more involvedcalculation of the late time flux when dealing with a dispersive field in the presence of a universal horizon. As explainedin the Introduction, we shall use a direct calculation which consists of pasting the modes across the infalling shell.We closely follow the derivation of [24].For simplicity, we consider an infalling spherically symmetric lightlike thin shell. In this case, it is particularlyappropriate to work with advanced Eddington-Finkelstein (EF) coordinates v, r , where v is the advanced null time.At fixed r , one has dv/dt S = 1, where t S is the usual Schwarzschild time. Hence, outside the shell, the stationaryKilling field K µ ∂ µ is simply ∂ v . On both sides of the shell taken to be v = 4 M , the line element reads ds = (cid:18) − M ( v ) r (cid:19) dv − dvdr − r (cid:0) dθ + sin( θ ) dϕ (cid:1) . (1)where M ( v ) = Θ( v − M ) M . These coordinates cover the entire space-time, shown in the right panel of Fig. 1. Onthe left panel, the infalling and outgoing null radial geodesics are represented in the ( v − r, r ) plane. One clearly seesthat the Killing horizon (where the norm K µ K µ vanishes) divides the outgoing geodesics into two separate classes.We work in Planck units: c = (cid:126) = G = 1.Let Φ be a massless real scalar field with the action S = (cid:90) d x √− g ( ∂ µ Φ) ( ∂ µ Φ) . (2)We define ψ ≡ r Φ and consider radial solutions independent of ( θ, ϕ ). Inside the shell, for v < M , we introduce thenull outgoing (affine) coordinate U = v − r . Outside the shell and for r > M , we introduce the null coordinate u = v − r ∗ K ∈ ( −∞ , ∞ ) , (3)where r ∗ K = r + 2 M ln | r/ M − | is the usual tortoise coordinate, which diverges on the Killing horizon. To coverthe region inside the Killing horizon, one needs another coordinate u L = − ( v − r ∗ K ). The field equation then reads (cid:26) ∂ U ∂ v ψ = 0 , v < M, (cid:0) ∂ u ∂ v + (cid:0) − Mr (cid:1) Mr (cid:1) ψ = 0 , v > M. (4)For simplicity, we neglect the potential engendering the grey body factor and work with the conformally invariantequations ∂ U ∂ v ψ = ∂ u ∂ v ψ = 0. The solutions can be decomposed as ψ ( u, v ) = ψ u ( u ) + ψ v ( v ) , (5)and similarly for v < M with u replaced by U . The infalling v sector and the outgoing u sector completely decouple.Moreover, the v modes ψ v are regular across the horizon and play no role in the Hawking effect. We thus consideronly the u modes, and, to lighten the notations, we no longer write the upper index u on outgoing modes. This sign guarantees that dU/du L is positive. As we shall see in Section III, a similar sign must be taken when studying a dispersivefield on both sides of a universal horizon. rM - - v - rM ℐ + ℐ - FIG. 1. (Left panel) Null radial geodesics in the ( v − r, r ) plane in units of M . The coordinate v − r coincides with theMinkowski time T inside the mass shell, and with the Schwarzschild time t S for r → ∞ . The solid black lines are null radialgeodesics which are reflected on r = 0. The dashed green line represents the trajectory of the null shell v = 4 M . The blue,dot-dashed one shows the Killing horizon at r = 2 M outside the shell. The dotted purple line shows the locus r = M , v > M which will play a crucial role in Section III. The wavy line shows the singularity, located at r = 0 , v > M . (Right panel)Penrose-Carter diagram of the collapsing shell geometry. The vertical line corresponds to r = 0 , v < M . I − corresponds to u, U → −∞ , and I + to v → ∞ . To compute the global solutions, we need the matching conditions across the null shell. In the present case, ψ iscontinuous along v = 4 M . Hence ψ inside ( U ) = ψ outside ( u ( U )), where the relation between null coordinates is u ( U ) = U − M ln (cid:18) − U M (cid:19) , (6)for r > M ( U < r < M ( U > u L ( U ) = − u ( | U | ).To obtain the Hawking flux one needs to relate the in modes φ in ω characterizing the vacuum inside the shell, tothe out modes φ out λ characterizing the asymptotic outgoing quanta with Killing frequency λ . In the internal region, acomplete orthonormal basis of positive-norm modes is provided by the plane waves φ in ω ≡ e − iωU √ πω , (7)where ω ∈ R + is the inside frequency i∂ U . In the external region, the (positive-norm) stationary modes for r > M ,are φ out λ ≡ Θ( r − M ) e − iλu √ πλ , λ ∈ R + . (8)A similar equation defines φ ( L ) λ ( u L ) in the trapped region, for r < M . The modes φ out λ , φ ( L ) λ (cid:48) and their complexconjugate form a complete orthonormal basis. One easily verifies that the conserved scalar product for the u modescan be written as ( ψ , ψ ) = i (cid:90) ∞−∞ du ( ψ ∗ ∂ u ψ − ψ ∂ u ψ ∗ ) . (9)The Bogoliubov coefficients encoding the Hawking flux are then given by the overlaps between the two sets ofmodes: α λ,ω = (cid:0) φ out λ , φ in ω (cid:1) ,β λ,ω = (cid:0) ( φ out λ ) ∗ , φ in ω (cid:1) . (10)Using u ( U ) of Eq. (6), they can be computed explicitly; see [24] for details. The late time behavior is obtained bysending the inside frequency ω → ∞ . In this limit, one recovers the standard thermal result (cid:12)(cid:12)(cid:12)(cid:12) β λ,ω α λ,ω (cid:12)(cid:12)(cid:12)(cid:12) ∼ ω →∞ e − πMλ . (11)To prepare for the forthcoming analysis, it is instructive to compute the Bogoliubov coefficients by the saddle pointmethod [25, 26]. For the α λ,ω coefficient, when ω (cid:29) λ , i.e., at late time, the location of the saddle is given by λ = ω (cid:16) e − κ ( u − u ) + O ( e − κ ( u − u ) ) (cid:17) , (12)where u is a constant which drops out of the late time flux. (In the present model, u vanishes.) From this equationwe recover the time-dependent redshift relating ω , the large frequency emitted from the collapsing star, to λ , thefrequency received at infinity and measured using the proper time of an observer at rest. In particular, we recover thecharacteristic exponential law governed by the surface gravity κ = ∂ u ln U ( u ) = M . Had we considered a collapsingshell following a (regular) infalling timelike curve, Eq. (12) would still have been obtained at late u − u time.This is the kinematical root of the universality of Hawking radiation in relativistic theories. Indeed, when studyingthe β λ,ω coefficient, one finds that the saddle point is now located at λ = − ωe − κu s . p . . When taking into accountthe fact that the integration contour should be deformed in the lower u -complex plane, one finds that u s . p . has animaginary part (cid:61) ( u s . p . ) = − π/κ , whereas its real part in unchanged. This gives a relative factor exp ( − πλ/κ ) withrespect to the α λ,ω coefficient. Upon squaring their ratio, we recover Eq. (11). We also recover here that the Hawkingtemperature κ/ π is fixed by the late time exponential decay rate entering Eq. (12). We finally notice that thestationarity of the flux is nontrivial. It follows from the fact that the ratio of Eq. (11) is independent of ω , and fromthe fact that | β λ,ω | ∝ /ω for ω → ∞ [26]. III. EMISSION FROM A UNIVERSAL HORIZONA. The model
We aim to compute the late time radiation of a dispersive field propagating in a collapsing geometry. In principle,the radiation and the background fields should both obey the field equations of some extended theory of gravity, suchas Hoˇrava-Lifschitz gravity [9] or Einstein-æther theory [12, 13]. Since our aim is to study the radiation rather thanthe collapse, the latter shall be described by a simplified model. At the end of the calculations, we shall argue thatour results do not qualitatively rely on the particular model we use.For reasons of simplicity, we assume that the collapsing object is a null thin shell, and that the external geometryis still Schwarzschild. In this case, the metric is again given by Eq. (1), and the Penrose diagram of Fig. 1 still coversthe whole space-time. To describe the (unit time like) æther field u µ in the external region outside the shell, weadopt the solution of [16] (also used in [19]) with c = 0, r = 2 M , and r u = 0. The Killing horizon is still at r = 2 M , whereas the universal horizon, where u µ K µ = 0, is located at r = M . Inside the shell, we assume that theæther field is at rest. To our knowledge, this configuration has not been shown to be a solution of the field equations.However, as explained in Subsection III E, small deviations from this configuration should not significantly modifyour conclusions.In EF coordinates, on both sides of the shell, the æther field u µ , and its orthogonal spacelike unit field s µ are givenby u µ ∂ µ = ∂ v − M ( v ) r ∂ r ,s µ ∂ µ = ∂ v + (cid:18) − M ( v ) r (cid:19) ∂ r , (13)where M ( v ) = Θ( v − M ) M . We introduce the “preferred” coordinates t, X by imposing that u µ dx µ ∝ dt and s µ ∂ µ = sgn( r − M ) ∂ X . Their precise definition is given in Appendix A 1. In these coordinates, the metric takes thePainlev´e-Gullstrand form: ds = c dt − ( dX − V dt ) , (14)where V = − M ( v ) r ,c = | K µ u µ | = (cid:12)(cid:12)(cid:12)(cid:12) − M ( v ) r (cid:12)(cid:12)(cid:12)(cid:12) . (15)At fixed t , outside the shell, V and c only depend on X . We notice that u µ dx µ = cdt. (16)The factor c ensures that dt is a total differential. Moreover, as explained in Appendix B, c is constant when theæther field is geodesic. Here we work with an accelerated æther, which is a necessary condition to have a universalhorizon. Importantly, c vanishes on the universal horizon. In fact, the novelties of the present situation with respectto the standard case studied in [8] only arise from the vanishing of c , and the associated divergence of the dispersivescale Λ /c .In Fig. 2 we show the lines of constant preferred time, and the direction of the aether field u µ , in the v, r plane.The coordinate t is discontinuous across the shell trajectory, as was the null coordinate u in the former section. Asin the relativistic case, outside the shell we must use two coordinates t and t L , now on either side of the universalhorizon. The inside coordinate T evaluated along the shell, at v = 4 M − , is a monotonically increasing function ofboth t ( v = 4 M + , r ) for r > M and of t L ( v = 4 M + , r ) for r < M . So, the foliation of the entire space-time by theinside coordinate T is globally defined and monotonic.We consider a real massless dispersive field Φ with a superluminal dispersion relation. Its action is given by Eq. (2)supplemented by a term quartic in derivatives: S = (cid:90) d x √− g (cid:20) ( ∂ µ Φ) ( ∂ µ Φ) − ( ∇ µ ( h µν ∇ ν Φ)) ( ∇ ρ ( h ρσ ∇ σ Φ)) (cid:21) , (17)where ∇ µ is the covariant derivative and h µν ≡ g µν − u µ u ν is the projector on the hyperplane orthogonal to u µ . Thedispersive momentum scale is given by Λ. The field equation reads ∇ µ ∇ µ Φ + 1Λ ( ∇ µ h µν ∇ ν ) Φ = 0 . (18)Using a (1 + 1)-dimensional approximation, Eq. (18) reduces to (cid:20) [ ∂ t + ∂ X V ] 1 c [ ∂ t + V ∂ X ] − ∂ X c∂ X + 1Λ ∂ X c∂ X c ∂ X c∂ X (cid:21) ψ = 0 , (19)when working in the preferred coordinates. Since this (self-adjoint) equation is second order in ∂ t , the Hamiltonstructure of the theory is fully preserved. In particular, the conserved scalar product has the standard form( ψ | ψ ) = i (cid:90) dX ( ψ ∗ Π − Π ∗ ψ ) , (20)where Π = u µ ∂ µ ψ = ( ∂ t ψ + V ∂ X ψ ) /c is the momentum conjugated to ψ . For more details; see Appendix A 2.The Hamilton-Jacobi equation associated with Eq. (19) isΩ = ( λ − V ( X ) P ) = c ( X ) (cid:20) P + P Λ (cid:21) . (21)We introduce the Killing frequency λ , the preferred frequency Ω, and the preferred momentum P : λ = − K µ ∂ µ S = − ∂ t S, (22)Ω = − c ( X ) u µ ∂ µ S = λ − V ( X ) P, (23) P = s µ ∂ µ S = ∂ X S. (24)In these equations S should be conceived as the action of a point particle; see [21, 22, 29]. As explained in theseworks, S governs the WKB approximation of the solutions of Eq. (19). Notice that Eq. (22) only applies outside theshell, whereas all the other equations make sense on both sides. In an analogue gravity perspective [27, 28], to reproduce such a situation one needs a medium in which the group velocity of low-frequency waves vanishes at a point. From Eq. (21), we see that the effective dispersive scale Λ /c must be divergent at the point where c →
0. It would be interesting to find media which could approximatively reproduce this behavior.
FIG. 2. In this figure we show the lines of constant preferred time for the collapsing geometry in the plane ( v − r ) /M, r/M .The dashed line represents the trajectory of the null shell v = 4 M , and arrows show the direction of the aether field u µ . Noticethat the external preferred time t diverges on the universal horizon r = M , v > M , whereas the internal time T , which isequal to v − r inside the shell, covers the entire space-time. B. The modes and their characteristics
To compute the late time radiation one should identify the various solutions of Eq. (19), and understand theirbehavior. In the presence of dispersion, one loses the neat separation of null geodesics into the outgoing u ones,and the infalling v ones. In what follows, we call P u ( P v ) the roots of the dispersion relation which have a positive(negative) group velocity in the frame at rest with respect to the “fluid” of velocity V ; see [8]. Similarly, thecorresponding modes will also carry the upper index u or v .
1. The in and out asymptotic modes
In the internal region v < M , the situation is particularly simple. Since the velocity field V vanishes, the preferredfrequency is ω = − ∂ T S , and the dispersion relation Eq. (21) becomes ω = P + P Λ . (25)This relation is shown in the left panel of Fig. 3. At fixed ω , the positive frequency modes with wave vectors P u ( ω ) > P v ( ω ) < in modes φ u, in ω and φ v, in ω . They both have a positive norm, which can easily be set tounity through a normalization factor. The mode φ u, in ω is the dispersive version of the relativistic in-mode of Eq. (7). P ω u P ω v - - P Λ - - ΩΛ P λ v P λ u - P - λ u , ← - P - λ u , → - - P Λ - - - λΛ FIG. 3. (Left panel) Dispersion relation in the internal region, where the preferred frame is at rest, in the Ω , P plane. Thesolid line shows ω = Ω versus P for the positive-norm modes. The dashed line corresponds to negative Ω, i.e., negative-normmodes. The intersections with a line of fixed ω > P uω and P vω . Right: Dispersion relationin the “superluminal” region for M < r < M in the λ, P plane. The two additional roots on the negative Ω u branch areclearly visible. Outside the shell, for v > M , at fixed Killing frequency λ >
0, the situation is more complicated as the numberof real roots depends on r . Outside the Killing horizon, for r > M , one has c > | V | . So, Eq. (21) possesses two realroots P u ( λ ) > P v ( λ ) <
0, which describe outgoing and infalling particles, respectively. The WKB expressionfor the corresponding stationary modes [the solutions of Eq. (19)] is ψ ( i ) λ ( t, X ) ≈ exp (cid:16) − i (cid:16) λt − (cid:82) X P ( i ) ( λ, X (cid:48) ) dX (cid:48) (cid:17)(cid:17) π (cid:113)(cid:12)(cid:12) Ω( λ, P ( i ) ) / ( c ( X ) ∂ λ P ( i ) ) (cid:12)(cid:12) , (26)where P ( i ) ( λ, X ) is a real solution of Eq. (21) at a fixed λ , and Ω( λ, P ( i ) ) the corresponding preferred frequency.These WKB modes generalize the expressions of [22, 30] in that c is no longer a constant. Using Eq. (20), oneeasily verifies that they have a unit norm. One also verifies that the group velocity along the i th characteristic is dX ( i ) /dt = 1 /∂ λ P ( i ) . When considered far away from the black hole, r/ (2 M ) (cid:29)
1, the u -WKB mode is the dispersiveversion of the relativistic out-mode of Eq. (8).From this analysis, we see that there is no ambiguity to define the asymptotic behavior of the in and out modes,solutions of Eq. (19). As before, these two sets encode the black hole radiation through the overlaps of Eq. (10). Tobe able to compute these overlaps, we need to construct the globally defined modes. To this end, we must study boththe behavior of ψ ( i ) λ near the horizon and the third kind of stationary modes which propagate in this region.
2. Near horizon modes
Inside the Killing horizon but outside the universal horizon, for
M < r < M , one has c < | V | . As can be seenfrom the right panel of Fig. 3, one recovers the two roots P u ( λ ) > P v ( λ ) < u root P u ( λ ) has been significantly blueshifted, whereas the infalling root P v ( λ ) hardly changed. Locally, inthe WKB approximation, the corresponding modes are again given by Eq. (26).In addition, below a certain critical frequency λ c that depends on c and V , we have two new real roots we call − P ( u, → ) − λ and − P ( u, ← ) − λ , where the arrow indicates the sign of the group velocity given by 1 /∂ λ P . (The minus signs infront of these roots and λ come from the fact that they have a negative preferred frequency Ω for λ >
0. Hence, for λ = − | λ | , the mirror image roots, P ( u, → ) −| λ | and P ( u, ← ) −| λ | , have a positive Ω.) Since Ω <
0, the WKB modes associatedwith these roots have a negative norm [22]. We call the right-moving one (cid:0) ψ u, →− λ (cid:1) ∗ and the left-moving one (cid:0) ψ u, ←− λ (cid:1) ∗ ,so that the modes without complex conjugation have a positive norm. Both of them carry a negative Killing energy − λ . Using Eqs. Eq. (26) and Eq. (20), one easily verifies that (cid:0) ψ u, →− λ (cid:1) ∗ and (cid:0) ψ u, ←− λ (cid:1) ∗ have a negative unit normwithin the WKB approximation. As we shall see they describe the negative energy partners trapped inside the Killinghorizon before and after their turning point, respectively. To summarize the situation, it is appropriate to representthe characteristics of the three types of modes. We proceed as in [21, 22]. ψ λ u ψ λ v ( ψ - λ u , ← ) * ( ψ - λ u , → ) * rM - v - rM λΛ = ψ λ u ψ λ v ( ψ - λ u , ← ) * ( ψ - λ u , → ) * rM v - rM λΛ = FIG. 4. Characteristics in a Schwarzschild stationary geometry for λ = 10 − Λ (left) and λ = Λ (right). The arrows indicatethe direction of increasing preferred time along each characteristic. Solid lines correspond to positive-norm modes, and dashedones to negative-norm modes. For r > M , each characteristic is named by the corresponding mode. The green dashed linecorresponds to an extra v mode confined in r < M , as discussed in footnote 3. In this footnote, we also explain that theinfalling v mode (described by the blue line) possesses a turning point inside the universal horizon. The mode correspondingto the orange, dashed line is the high momentum WKB mode (cid:0) ψ u, →− λ (cid:1) ∗ before the turning point, and the low momentum mode (cid:0) ψ u, ←− λ (cid:1) ∗ after it.
3. The characteristics
As said above, the characteristics are solutions of the Hamilton-Jacoby equation dXdT = ∂ λ P . Since the frequencyis a constant of motion on each side of the mass shell, they can then be computed straightforwardly. In Fig. 4, theyare shown in the external region v > M for a small value of | λ | / Λ = 0 .
01 (left panel) and a moderate one | λ | / Λ = 1(right panel). The solid lines correspond to positive energy solutions while the dashed ones correspond to negativeenergy solutions.The infalling v -like characteristics corresponding to ψ vλ (in blue) approach the universal horizon from infinity andcross it at a finite value of v . (When sending λ → v = cst. ) As their wavevectors are finite for r → M + , these characteristics will play no role in the sequel. As in the relativistic case, the v modes act as spectators in the Hawking effect. The u -like characteristics with positive energy (in red), corresponding to the WKB modes ψ uλ , emerge from theuniversal horizon from its right ( r > M ) at early times. When t increases, the momentum P uλ is redshifted while r increases. At a finite time, the characteristics cross the Killing horizon, and go to infinity as t → ∞ (almost alongnull outgoing geodesics when λ/ Λ (cid:28) t → −∞ , they also emerge from r = M + . However, when increasing t they have a turning point inside the Killinghorizon, after which they move towards the universal horizon, smoothly cross it, and hit the singularity at r = 0 atfinite values of v and t L . Before the turning point, they are described by the WKB mode (cid:0) ψ u, →− λ (cid:1) ∗ , and after theturning point by (cid:0) ψ u, ←− λ (cid:1) ∗ .It is important to notice that the only novel aspect with respect to the standard dispersive case (treated in fulldetail in [22]) concerns the behavior near the universal horizon. To clarify these new aspects, we represent in Fig. 5the global structure of the characteristics in the collapsing mass shell geometry. Interestingly, v -like characteristics have a turning point inside the universal horizon r < M . (The presence of the turning point may beunderstood from the fact that, close to r = 0, | V | and c go to infinity but | V | /c goes to 1. So, at fixed λ two roots merge at a point r = r tp >
0. The turning point approaches r = 0 in the limit λ → v → −∞ , t L → + ∞ , and are highly blueshifted. In addition, for r < M , there is a new v modewith negative norm for λ >
0. It is indicated by a dashed green line in Fig. 4. It emerges from the singularity and approaches theuniversal horizon while closely following the positive Killing frequency characteristic after its turning point. (In fact this new v modeis directly related to the u modes emerging from the singularity in [2]: inside a universal horizon, u and v modes are swapped becauseof the vanishing of c at r = M .) Since some of the v modes originate from the singularity, and since the blueshift they experience isunbounded for r → M − , the v part of the state will not obey Hadamard regularity conditions. This strongly indicates that the inner side of the universal horizon should be singular. This interesting question goes beyond the scope of the present paper. rM v - rM FIG. 5. Characteristics crossing the infalling shell in the v − r, r plane. The Killing frequencies of the outgoing u modes and theincoming v modes is λ = ± . λ of the out-going u mode is positive (negative). The arrows indicate the future direction associated with the aether field.When tracing backwards the u -like characteristics associated with the Hawking quanta ( λ >
0) and their inside negative energypartners ( λ < v -like superluminal characteristics with a high and positiveKilling frequency λ in . The v mode which emanates from the singularity (the dashed line) returns to it after having bounced atthe center of the shell (not represented), closely following the characteristic of the v mode coming from infinity which hits theshell at the same value of r .
4. The characteristics in the collapsing geometry
In the internal region v < M , the characteristics are straight lines. Coming backwards in time from the outsideregion, the inside trajectories are fixed by the value of the inside frequency which is determined (as in the relativisticcase), by continuity of the field ψ across the mass shell; see Appendix A 3 for details. As a result, the derivative ∂ r ψ must be continuous across v = 4 M . At the level of the characteristics (i.e., in the geometrical optic approximation),this implies that k v , the radial momentum at fixed v , is continuous along the shell. In terms of the inside and outsidepreferred momenta P u ( ω ) and P u ( λ, r ) evaluated at v = 4 M − and v = 4 M + , respectively, the continuity conditiongives (cid:12)(cid:12)(cid:12)(cid:12) rr − M (cid:12)(cid:12)(cid:12)(cid:12) ( λ + P u ( λ, r )) = ω + P u ( ω ) . (27)This equation has two solutions, but only one is well behaved as the other one gives a trajectory along which thepreferred time is not monotonic. A straightforward calculation using the dispersion relation Eq. (21) also shows thatthe sign of Ω is preserved. It should be noted that Eq. (27) is the dispersive version of the relativistic equation | r/ ( r − M ) | λ = ω , which gives back Eq. (12) for r > M , ω (cid:29) λ , and when using u rather than r .It should be also emphasized that all outgoing u -like characteristics originate from inside the shell, as in therelativistic case. This is shown in Fig. 5. Therefore, thanks to the universal horizon, the state of the field inside theshell determines the state of the u modes. In this we avoid the problem discussed in [2], namely that in the absence of a universal horizon, the u modes of a superluminal field originate from the singularity at r = 0. As discussed infootnote 3, these modes still exist, but they are now trapped inside the universal horizon.Finally, we notice that the Killing frequency λ in of the incoming v modes which generate the outgoing u modesexiting the shell at r c ≈ M is very large. More precisely, when dealing with u characteristics with positive Ω(i.e., modes with positive norm), irrespective of the sign of their Killing frequency λ , the Killing frequency λ in ispositive. A straightforward calculation (based on the continuity of k v applied to the v modes) shows that it scales as λ in ≈ M/ ( r c − M ).For completeness, we have also represented in Fig. 5 a couple of infalling v characteristics which enter the shell for0 < r < M . One comes from r = 0 (the dashed line), and one from r = ∞ (the solid line). They both reach thesingularity after having bounced at r = 0 inside the shell. These characteristics, although interesting, play no role inthe Hawking process.0 C. Behavior of the WKB modes near the universal horizon
To be able to compute the late time behavior of the Bolgoliubov coefficients, we need to further understand theproperties of the stationary modes in the immediate vicinity of the universal horizon at r = M . For r > M , the tworoots P vλ and P u, ←− λ remain finite as r → M . As can be seen in Fig. 4, the associated trajectories smoothly cross thehorizon. They thus play no role in the large ω limit. In fact they describe out modes.The two other roots P uλ and P u, →− λ both diverge as r → M . Importantly, they both satisfy P in ± λ = Λ Mr − M ± rM λ + O (cid:18) − Mr (cid:19) , (28)where the + sign applies to P uλ , and the - sign to P u, →− λ . We have added a superscript in to emphasize that thisbehavior is relevant at early time t , just after having crossed the shell. The simple relation between P uλ and P u, →− λ implies that for r → M , the two WKB modes ψ uλ and ψ u, →− λ are also related to each other by flipping the sign of λ .In the forthcoming discussion, to implement these points, we shall replace ψ u, →− λ by ψ u − λ , and add a superscript in tothe WKB modes ψ u ± λ .The appropriate character of this superscript can be understood as follows. Although the divergence in 1 / ( r − M )in Eq. (28) resembles to what is found in the relativistic case, it has a very different nature due to the differentrelationship between r and the preferred coordinate X . This can be seen by looking at the validity of the WKBapproximation for ψ u, in λ close to the universal horizon. Deviations from this approximation come from terms in( ∂ X r ) / ( rP ), ( ∂ X ( r − M )) / (( r − M ) P ), and ( ∂ X P ) /P . Using ∂ X = r − Mr ∂ r , (29)we find that these three terms go to zero as r → M . Therefore, close to the universal horizon, the WKB approximationof Eq. (26) becomes exact for ψ u, in ± λ . In fact, these modes behave as the dispersive in modes near a Killing horizon [21,22]. Namely, they have a positive norm for all values of λ and, moreover, contain only positive values of P u . Werecall that this is the key property which also characterizes the so-called Unruh modes [26, 31] for a relativistic field.These are strong indications that no stationary emission should occur close to the universal horizon, as the pairproduction mechanism rests on deviations from the WKB approximation. This is confirmed in the next subsection. D. Bogoliubov coefficients from the scattering on the shell
We are now in a position to determine the scattering coefficients which govern the propagation across the nullshell. Inside the shell, one has the in mode φ u, in ω . Along the shell, for v = 4 M − , it is a plane wave which behaves as φ u, int ω ∼ exp [ i ( ω + P u ( ω )) r ]. After having crossed the shell, for r/M − (cid:28)
1, it may be expanded in terms of thefour WKB modes (which form a complete basis) φ u, in ω = (cid:90) ∞−∞ dλ (cid:16) γ ω,λ ψ u, in λ + δ ω,λ ( ψ u, in − λ ) ∗ + A ω,λ ψ vλ + B ω,λ ( ψ u, ← λ ) ∗ (cid:17) . (30)We are interested in the coefficients γ ω,λ and δ ω,λ which multiply the two modes with divergent wave vectors andopposite norms. It should be pointed out that the integral over λ runs from −∞ to ∞ . The other two coefficients A ω,λ and B ω,λ multiply the two modes which remain regular across the universal horizon in the ( v, r ) coordinates.They vanish in the limit ω → ∞ .The calculation of γ ω,λ and δ ω,λ is straightforward in the ( v, r ) coordinates; see Appendixes A 4 and A 5. For | λ | (cid:46) Λ, we find that their ratio decays as (cid:12)(cid:12)(cid:12)(cid:12) δ ω,λ γ ω,λ (cid:12)(cid:12)(cid:12)(cid:12) = ω →∞ O (cid:32) √ M Λ ω (cid:33) × exp ( − M P u ( ω )) , (31)where P u ( ω ) ∼ √ ω Λ in the present high frequency regime. Equation Eq. (31) is the main result of the present work.Its meaning is clear: at late time, corresponding to the emission close to the universal horizon and thus to very largevalues of P uλ ∼ Λ / ( r/M − φ u, in ω and the high momentum WKB mode with negative norm ( ψ u, in − λ ) ∗ , irrespective of the value (andthe sign) of λ . As a result, outside the shell, the state of the field is stationary, and the vacuum with respect to the1annihilation operators associated with ψ u, in λ for λ ∈ ( −∞ , ∞ ). It thus correspond to the in vacuum as describedin [21, 22]. E. Genericness of Eq. (31)
In this subsection, we distance ourselves from the model we considered to see how the above results may be affected.We first consider a modification of the mass shell trajectory close to the universal horizon. From the calculation ofAppendixes A 4 and A 5, the factor exp (cid:16) − M √ Λ ω (cid:17) in Eq. (31) comes from the fact that the phase of the modeinside the mass shell is θ int ≈ − ωT ≈ ω ( r − v ), while that of the mode outside the mass shell is θ out ≈ Λ M/x , where x ≡ ( r/M ) −
1. At fixed v , we find that the stationary phase condition applied to θ in ± θ out (the upper sign appliesto γ while the lower sign applies to δ ) gives back the large frequency limit of Eq. (27) with x real for γ , while x ispurely imaginary for δ , with a modulus (cid:112) Λ /ω . Let us now consider an arbitrary shell trajectory close to the universalhorizon. We define an affine parameter y along this trajectory. The possible saddle points are located where ddy (cid:18) ωT ∓ Λ Mx (cid:19) = 0 , (32)i.e., ω dTdy ± Λ Mx dxdy = 0 . (33)So, the location of the saddle is x ∗ = (cid:114) ∓ M dxdT Λ ω . (34)We get the same result as before, up to the factor − M dxdT . Therefore, the ratio | δ ω,λ /γ ω,λ | is still suppressed by anexponential factor in M √ Λ ω , with a coefficient depending on the velocity of the mass shell when it crosses r = M .We now consider a generalization of the dispersion relation Eq. (21) with higher-order terms. Specifically, weconsider the dispersion relation Ω c = N (cid:88) j =0 P j Λ j − j . (35)Close to the universal horizon, the divergent wave vectors follow P ≈ ± Λ N x − N − . (36)As before, the coefficient γ corresponds to ± = + in Eq. (36). The value of the saddle point is then real, and theexponential factor appearing in γ ω,λ has a unit modulus. Instead, for the coefficient δ , corresponding to the minussign in Eq. (36), the solutions of the saddle point equation are x ∗ = (cid:16) ω Λ (cid:17) − NN e iπ lN , l ∈ Z . (37)Taking only the saddle points with negative imaginary parts, we find that δ ω,λ is suppressed by a factor which isexponentially large in ω /N . Interestingly, when using the inside spatial wave number P u ( ω ) rather than the insidefrequency ω , the norm of the coefficient δ ω,λ always decreases as exp ( − M AP u ( ω )) with A >
0, which means that itis the diverging character of P u ( ω ) which guarantees that its sign does not flip when crossing the shell.Similarly, the exponential factor suppressing δ ω,λ is mildly affected by a change in the metric and/or the form ofthe æther field, provided the inside wave vector remains smooth, whereas the outside one diverges as a power law for r → r UH , where r UH is the radius of the universal horizon. This should remain valid as long as there is no divergence(or cancellation) preventing us from defining preferred coordinates in which the dispersion relation takes the formof Eq. (35) close to the universal horizon. Indeed, the construction of Appendix A 1 can be easily extended to ageneric space-time with a Killing vector χ , endowed with a generic timelike, normalized æther field u µ . This conclusion differs from that reported in [18]. We do not understand the procedure adopted there, which apparently implies thatthe leading term in Eq. (28) does not contribute to the ratio of Eq. (31), thereby giving rise to a steady thermal radiation governed bythe surface gravity of the universal horizon. Instead, the saddle point evaluation of δ ω,λ performed in Appendix A 5 establishes that theleading term of Eq. (28) gives the exponential damping in e − MP of Eq. (31). To be complete, one should propagate backwards in time the inside field configurations, and verify that they correspond to vacuum v -like configurations for r → ∞ , t → −∞ . To verify this, we computed the scattering coefficients encoding a change of the norm of the v modes when crossing the shell. We found that they also decrease exponentially in √ ω Λ for ω → ∞ . We also recall here that the Killingfrequency of the v modes engendering a stationary u mode diverges as λ in ≈ M/ ( r c − M ), where r c is the radius when the u modeexits the shell. IV. CONCLUSIONS
We computed the late time properties of the Hawking radiation in a Lorentz violating model of a black hole witha universal horizon. To identify the appropriate boundary conditions for the stationary modes of our dispersive field,we worked in the geometry describing a regular collapse, and assumed that the inside state of the field is vacuum at(ultra) high inside frequencies ω (cid:29) Λ. We then computed the overlap along the thin shell of the outwards propagatinginside positive norm modes, and the outside stationary modes. In the limit where the shell is close to the universalhorizon, we show that the overlap between modes of opposite norms decreases exponentially in the radial momentum P . This result comes from the peculiar behavior of the momentum when approaching the universal horizon with afixed Killing frequency; see Eq. (28). Although this behavior was found in a specific model, we then argued that itwill be found for generic (spherically symmetric) regular collapses and superluminal dispersion relations.As a result, irrespective of the model, at late time, the state of the outgoing field configurations is accuratelydescribed, for both positive and negative Killing frequencies, by the WKB modes with large positive momenta P (and a positive norm). In this we recover the standard characterization of outgoing configurations in their vacuumstate in the near horizon geometry. Indeed, the condition to contain only positive momenta P prevails for bothrelativistic and dispersive fields in the vicinity of the Killing horizon. The present work, therefore, shows that thissimple characterization still applies in the presence of a universal horizon.Once this is accepted, the calculation of the asymptotic flux is also standard, and shows that for large black holes thethermality and the stationarity of the Hawking radiation are, to a good approximation, both recovered. This suggeststhat the laws of black hole thermodynamics should also be robust against introducing high frequency dispersion.As a corollary of the divergence of the radial momentum on both sides of the universal horizon, noticing thatthe inside configurations are blueshifted (towards the future), and that they have no common past with the outsideconfigurations, it seems that the field state cannot satisfy any regularity condition across the universal horizon. Itwould be interesting to study the space of the field states, and determine whether some dispersive extension of theHadamard condition can be imposed on the universal horizon. In the negative case, it seems that the universal horizonwill be replaced by a spacelike singularity. ACKNOWLEDGMENTS
We thank Xavier Busch for enlightening discussions about the causal structure of spacetimes with a universalhorizon. We also thank Ted Jacobson, David Mattingly, and Sergei Sibiryakov for interesting comments and sugges-tions. This work received support from the French National Research Agency under the Program Investing in theFuture Grant No. ANR-11-IDEX-0003-02 associated with the project QEAGE (Quantum Effects in Analogue GravityExperiments).
Appendix A: Wave equation and Bogoliubov coefficients
In this appendix, we give the general formulas and main steps in the derivation of the results presented in Section III.
1. Preferred coordinates
The preferred coordinates ( t, X ) are defined by the followng four conditions • s µ ∂ µ = ± ∂ X at fixed t ; • ∂ v = ± ∂ t at fixed X ; • ∂ r T < • ∂ r X > t and X as t = v − r, v < M,v − r ∗ U , v > M ∧ r > M, − ( v − r ∗ U ) , v > M ∧ r < M, (A1)3and X = r, v < M,r ∗ U , v > M ∧ r > M, − r ∗ U , v > M ∧ r < M. (A2)In these expressions, r ∗ U = r + M ln | rM − | is the tortoise coordinate built around the universal horizon.
2. Wave equation and scalar product
The action Eq. (17) has a U (1) invariance under Φ → e iθ Φ, from which we derive the conserved current density J µ ≡ − i √− g (cid:18) Φ ∇ µ Φ ∗ − h µν ( ∇ ν Φ) ( ∇ ρ h ρσ ∇ σ Φ ∗ ) + 1Λ Φ h µν ∇ ν ∇ ρ h ρσ ∇ σ Φ ∗ (cid:19) + c.c., (A3)where “ c.c. ” stands for the complex conjugate, satisfying ∂ µ J µ = 0 . (A4)As the wave equation Eq. (18) is linear, one easily shows that J µ defines a conserved (indefinite) inner product inthe following way. Considering two solutions Φ and Φ of Eq. (18), we first define J µ (Φ , Φ ) by replacing Φ ∗ byΦ ∗ and Φ by Φ in Eq. (A3). The inner product of these two solutions is then defined by(Φ , Φ ) τ ≡ (cid:90) d x n µ J µ (Φ , Φ ) , (A5)where n µ is the unit vector perpendicular to the 3-surface defined by τ = cst , and τ is a time coordinate. Whenconsidering the 3-surfaces defined by t = cst. , the above overlap simplifies and gives the standard (Hamiltonian)conserved scalar product of Eq. (20).
3. Matching conditions on the mass shell
In order to compute the overlap of two modes defined on either side of the mass shell, we need the matchingconditions to propagate the modes from the internal region to the external one and vice versa. As we now show,they appear naturally when considering the behavior of J v ≡ J µ ∂ µ v across the shell. To see this, we first rewrite J v (Φ , Φ ) as J v (Φ , Φ ) = − i (cid:18) Φ √− g (cid:18) ∇ v + 1Λ h vµ ∇ µ ∇ ρ h ρσ ∇ σ (cid:19) Φ ∗ − √− g ( h vµ ∇ µ Φ ) ( ∇ ρ h ρσ ∇ σ Φ ∗ ) (cid:19) − (Φ ∗ ↔ Φ ) . (A6)Inspecting Eq. (18) and requiring that the second term has no singularity which cannot be canceled by the first one,we find that the quantities • Φ, • √− gh ν ∇ ν Φ, • ∇ ρ h ρσ ∇ σ Φ, and • √− g (cid:0) ∇ + h ν ∇ ν ∇ ρ h ρσ ∇ σ (cid:1) Φare continuous across v = 4 M . Since the complex conjugate of a solution of Eq. (18) is still a solution, this appliesto Φ = Φ ∗ as well as Φ = Φ . Therefore, in evaluating Eq. (A6) one can evaluate Φ ∗ and the operators acting on iton one side of the shell, v = 4 M − (cid:15) , (cid:15) →
0, while Φ and the operators acting on it are evaluated on the other side v = 4 M + (cid:15) .4
4. Calculation of γ ω,λ Let us consider two radial modes known on different sides of the mass shell: Φ is known for v < M and Φ for v > M . The complete expression of the scalar product in the v, r coordinates is somewhat cumbersome, but itgreatly simplifies in the relevant limit where • ψ has a large frequency | ω | (cid:29) Λ; • ψ has a large wave vector | k v, | (cid:29) λ, Λ.We have introduced the wave vector k v ≡ ∂ r S at a fixed v . For the modes we are interested in k v, = ( λ + P ) /x and ω are of the order Λ /x , where x = ( r/M ) −
1. Keeping only the leading terms in the inner product then gives(Φ , Φ ) v ≈ iπ Λ (cid:90) dr (cid:18) − ψ ( ∂ v + ∂ r ) ψ ∗ + (cid:18) − Mr (cid:19) ∂ r ψ ( ∂ v + ∂ r ) ψ ∗ + ψ ∗ (cid:18) − Mr (cid:19) ∂ r ψ − ( ∂ v + ∂ r ) ψ ∗ (cid:18) Mr − (cid:19) ∂ r ψ , (cid:33) (A7)with relative corrections of order x . When choosing for ψ the in mode of frequency ω , and for ψ the stationaryWKB mode of Eq. (26) with the large momentum given by Eq. (28), we get(Φ , Φ ) v ≈ πM (cid:90) x> dx (cid:18) P ω Λ ± P ω Λ x ± Λ x + P ω x (cid:19) ψ ∗ ψ ≈ M e iM ( ω − λ ) e − iM ( ω + P ω ) π (cid:113) Λ (cid:12)(cid:12) ω (cid:0) dωdP (cid:1) (cid:12)(cid:12) (cid:90) x> dx (cid:18) P ω Λ ± P ω Λ x ± Λ x + P ω x (cid:19) exp (cid:18) i (cid:18) ∓ Λ Mx + (2 λ ± Λ) M ln | x | − M ( ω + P ω ) x (cid:19)(cid:19) . (A8)In this equation, as well as in the remainder of this appendix, the sign ± discriminates between γ and δ ; see below. Inthe large frequency limit, we evaluate this integral through a saddle point approximation. The possible saddle pointsare the values of x where ddx (cid:18) ∓ Λ Mx − M ( ω + P ω ) x (cid:19) ≈ ddx (cid:18) ∓ Λ Mx − M ωx (cid:19) = 0 , (A9)i.e., x ≈ ± Λ ω . (A10)This is very similar to the saddle point condition applied to the Bogoliubov coefficients describing the scattering ofplane waves on a uniformly accelerated mirror [32, 33].The coefficient γ ω,λ is defined for ± ω >
0. Since the integral runs over x >
0, we must choose the saddle point x ∗ at x ∗ γ ≈ (cid:115) Λ | ω | . (A11)We get γ ω,λ ≈ ± (cid:115) ∓ iM π | ω | exp (cid:18) iM (cid:18) ω − λ − P ω ∓ (cid:112) Λ | ω | + 12 (2 λ ± Λ) ln (cid:18) Λ | ω | (cid:19) ∓ Λ (cid:19)(cid:19) . (A12)It is easily shown that, under these approximations, the following unitarity relation is satisfied: (cid:90) ∞ dωγ ∗ ω,λ γ ω,λ (cid:48) ≈ δ ( λ − λ (cid:48) ) . (A13)This implies that the δ ω,λ coefficients are suppressed in the limit ω → ∞ .5
5. Calculation of δ ω,λ The calculation of δ ω,λ follows the same steps. The saddle point equation now is x ∗ δ = − Λ | ω | . (A14)To be able to deform the integration contour to include the saddle point, we must choose the solution in the half-planewhere the exponential decreases, i.e., x ∗ δ = − sgn ( ω ) i (cid:115) Λ | ω | . (A15)The exponential factor in the integral then gives a suppression factorexp (cid:18) − M (cid:18) (cid:112) Λ | ω | + π (cid:18) − sgn ( ω ) λ + Λ2 (cid:19) + Λ (cid:19)(cid:19) . (A16)In addition, to the order to which the calculation was performed, the prefactor vanishes. As the first relativecorrections from neglected terms are of order O ( x ∗ ) = O ( (cid:112) Λ / | ω | ), we get δ ω,λ = O (cid:32) √ M Λ | ω | (cid:33) exp (cid:16) − M (cid:112) Λ | ω | (cid:17) . (A17) Appendix B: Acceleration of the æther field
The acceleration of the æther field is γ µ = u ν ∇ ν u µ . (B1)Using Eq. (13), this gives for v (cid:54) = 4 M γ µ γ µ = − M ( v ) r . (B2)For completeness, we now show that in 1 + 1 dimensions a stationary universal horizon requires that the æther fieldhas a nonvanishing acceleration, thereby generalizing what was found in de Sitter in [7]. We consider a stationaryspace-time with Killing vector K µ , endowed with a timelike æther field u µ . The universal horizon is defined as thelocus where K µ u µ = 0. (Notice that the Killing field must thus be spacelike on the universal horizon.) In particular, K µ cannot be aligned with u µ . Using the Killing equation, the variation of K µ u µ along the flow of u µ is u µ ∇ µ ( K ν u ν ) = K µ γ µ . (B3)If u µ is freely falling, γ µ = 0 and u µ is tangent to the hypersurfaces of constant K µ u µ . In particular, it is tangentto the universal horizon. In 1 + 1 dimensions, since K µ and u µ cannot be aligned, K µ is not a tangent vector tothe universal horizon, which is thus not stationary. Models with a stationary universal horizon are thus in a differentclass than those studied in [22].To see the combined effects of the dispersion and acceleration, we show in Fig. 6 the local value of the wave vectorin the v, r coordinates, k v , for the outgoing u mode, as a function of r . We compare three models with the sameparameters, and for λ = 10 − Λ. The blue, solid curve shows the result for the model of Section III. The green, dottedcurve shows the relativistic case. The red, dashed one shows the result for a dispersive model with a nonacceleratedpreferred frame chosen to coincide with the æther frame of Section III at r = 2 M . We see in Fig. 6 that the threemodels give very similar results for r < M . Close to r = 2 M , the relativistic wave vector diverges, while thenonaccelerated dispersive model still closely follows the accelerated one. When r is further decreased, the predictionsof the two models separate: the nonaccelerated one gives a finite wave vector at r = M while the accelerated onegives k ∝ ( r − M ) − . It must be noted that this model is not well defined for r → ∞ . The reason is that at r = 2 M , we have u · ∂ v = 1 /
2. A nonacceleratedvector field w which coincides with u at r = 2 M must thus satisfy the two conditions w · ∂ t = 1 / w · w = 1 at r = 2 M . From thefree-fall condition, these two properties extend in the whole domain where the preferred frame is defined. Since they are incompatible inMinkowski space, we deduce that the domain in which the preferred frame can be defined does not extend to r → ∞ . A straightforwardcalculation shows that it extends up to r = 8 M/
3. However, as this model is well defined close to and inside the Killing horizon, it canbe used to see the qualitative differences between the nonaccelerated and accelerated cases. rM - k Λ FIG. 6. Comparison of three wave vectors k ( r, λ ) / Λ as a function of x = r/M − λ = 10 − Λ, and the Killing horizon is located at x = 1. One clearly sees the unbounded growth ofthe relativistic wave number. More importantly, one also sees that the two dispersive wave vectors behave in the same manneracross the Killing horizon. Hence the acceleration of u µ has a significant effect on k only when approaching the universalhorizon. Appendix C: Hawking radiation in the presence of a universal horizon
In Section III, we showed that the late time emission from the universal horizon is governed by Bogoliubov coefficientswhich are exponentially suppressed when the inside frequency ω (cid:29) λ . This result was obtained using the WKBapproximation of the stationary modes just outside the universal horizon. This approximation is trustworthy as weverified that the deviations from the WKB treatment go to zero when approaching the universal horizon. This impliesthat at late time the inside vacuum is adiabatically transferred across the shell. Hence, the u part of the field state canbe accurately described by the WKB high (preferred) momentum mode ψ u, in λ for both signs of λ . In this we recoverthe situation described in [21, 22, 29]. Therefore, the nonadiabaticity that will be responsible for the asymptoticradiation will be found in the propagation from the universal horizon to spatial infinity. The value of the Bogoliubovcoefficients should essentially come from the stationary scattering near the Killing horizon. Hence, we expect to geta nearly thermal spectrum governed by the surface gravity of the Killing horizon, and with deviations in agreementwith those numerically computed in [34, 35].To verify this conjecture, we numerically propagate the outgoing mode φ uλ from a large value of r/ M down insidethe trapped region to r → M + . This mode can be written in the limits r → M and r → ∞ as φ uλ ( r ) ∼ r →∞ ψ uλ + A λ ψ vλ ,φ uλ ( r ) ∼ r → M α λ ψ u, in λ + β λ ( ψ u, in − λ ) ∗ , (C1)where the WKB modes are as described in Section III. The coefficient A λ governs the grey body factor. In our(1+1)-dimensional model, we have verified that it plays no significant role. (We found that | A λ | is bounded by 0 . u modes of opposite norms.To efficiently perform the numerical analysis, we regularized the metric and æther field. In practice we worked witha metric of the form ds = (1 − f ( r )) dv − dvdr, (C2)and a unit norm æther field u µ ∂ µ = ∂ v − f ( r ) ∂ r . (C3)These expressions generalize the model of Section III which is recovered for f ( r ) = M/r . The Killing horizoncorresponds to f ( r ) = 1 /
2, and the universal horizon to f ( r ) = 1. We can then define the preferred coordinate X along the lines of Appendix A 1. For the numerical integration of Eq. (19), it is appropriate to work with f expressedas a known function of X . A convenient choice is f ( r ( X )) = 12 (cid:18) − η tanh (cid:18) XX (cid:19)(cid:19) . (C4)7 xx K - - - x - X - X K - - P ∂ P ∂ X FIG. 7. (Left panel) Characteristics in the v − x, x/x K coordinates, where x ≡ r − r U and x K gives the location of the Killinghorizon. Only the region outside the mass shell and the universal horizon is represented. The parameters are Λ /κ ∼ . X = 0 .
5, and λ/ Λ = 0 .
01. (Right panel) Amplitude of the nonadiabatic corrections in the present model (blue, solid line)and the one from Section III (orange, dashed line) for λ/ Λ = 0 .
01, as a function of the preferred coordinate X − X K , where X K denotes the position of the Killing horizon. In this example, the surface gravity is κ ∼ Λ. We thus verify that the normof the coefficient β λ is of the order of the maximal value of the nonadiabatic parameter ∼
10, as expected from the analysis ofnonadiabaticity [24]. η is a positive parameter which must be equal to 1 to have an asymptotically flat space at r → ∞ and a universalhorizon at r → M , i.e., X → −∞ . In our numerical simulations, we worked with η < P in ± λ close to the universal horizon. We then checked that the scatteringcoefficients become independent on η in the limit η →
1, as should be the case since the WKB approximationsbecome exact on both sides. The advantage of this model is that the metric coefficients and the æther field convergeexponentially to their asymptotic values so that the asymptotic modes become exact solutions for r → ∞ , providedthe decay rate of the exponentially decaying mode is small enough. The characteristics outside the universal horizon r = r U are shown in Fig. 7. They exhibit the main properties of Fig. 4. In the right panel we show the parametergoverning nonadiabatic corrections, ( ∂ X P λ ) /P λ , as a function of the preferred coordinate X in the present modeland the one of Section III. In the present model, they go to zero exponentially both for X → ∞ and X → −∞ . Inthe model of Section III, they decay exponentially at X → −∞ but only polynomially at X → ∞ . The two modelsbecome equivalent, in the sense that the value of P uλ ( r ) follows the same law close to the Killing horizon when workingwith the same surface gravity.The field equation was integrated numerically using [36], and the same techniques as in [34, 37]. The results areshown in Fig. 8. We obtain two important results. First, at fixed Λ and κ , the effective temperature defined by | β λ | = 1 e λ/T λ − , (C5)becomes independent of the regulator η as η → − . Second at low frequencies λ (cid:28) Λ, we get a Planckian spectrum, i.e., T λ = constant , with deviations from the Hawking temperature compatible with the results of [8, 34]. This establishesthat the propagation between the two horizons does not alter the thermal character of the outgoing spectrum.To conclude this numerical analysis, we numerically verify that the WKB approximation becomes exact whenapproaching the universal horizon. To this end, we show in Fig. 9 the logarithm of the relative deviation betweenthe numerical solution and the weighted sum of the WKB waves α λ ψ u, in λ + β λ ( ψ u, in − λ ) ∗ . As X → −∞ , we clearly seethat the numerical values of the deviations decay following the rough estimation of the WKB corrections given by (cid:12)(cid:12)(cid:12) ( ∂ X P uλ ) / ( P uλ ) (cid:12)(cid:12)(cid:12) . (The relatively important spread and the plateau for X < − . , even a relatively small error in its value givesimportant and rapidly oscillating errors.) Notice that an exponential convergence for X → −∞ is required to have a universal horizon at a finite value of r = r U , such that df/dr ( r = r U ) (cid:54) = 0. λκ T λ T H FIG. 8. Plot of the effective temperature T λ of Eq. (C5) divided by the Hawking temperature as a function of λ/κ , where κ is the surface gravity. The values of Λ /κ are 1 / / η = 0 . .
94 (cyan line), and 0 .
98 (magenta line). For the largest value of Λ, these three curvesare undistinguishable up to numerical errors. This indicates that the limit of the regulator η → − is well defined, which waschecked using a larger range of values for η ∈ (0 . , . T λ closely agrees with the Hawking value κ/ π for a larger domain of Killing frequencies. - - - - - - - FIG. 9. Deviations from the WKB approximation. The solid line shows the natural logarithm of (cid:12)(cid:12) ( ∂ X P uλ ) / ( P uλ ) (cid:12)(cid:12) as a functionof the preferred coordinate X in units of 1 / Λ, and the points show the logarithm of the relative difference between the solutioncomputed numerically and the corresponding sum of the WKB modes, for Λ = 1, X = 0 . η = 0 .
99, and λ = 0 .
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