Black hole radiation in the presence of a universal horizon
In Hořava and Einstein-æther theories of modified gravity, in spite of the violation of Lorentz invariance, spherically symmetric stationary black hole solutions possess an inner universal horizon which separates field configurations into two disconnected classes. We compute the late time radiation emitted by a dispersive field propagating in such backgrounds. We fix the initial conditions on stationary modes by considering a regular collapsing geometry, and imposing that the state inside the 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 temperature fixed by the surface gravity of the Killing horizon. In turn, this suggests that the universal horizon should play no role in the thermodynamical properties of these black holes.
pacs:04.50.Kd, 04.62.+v, 04.70.Dy
The laws of black hole thermodynamics are firmly established in Lorentz invariant theories, and they play a crucial role in our understanding of black hole physics Wald (1994). In particular, the entropy and the temperature are governed by the area and surface gravity of the event horizon. In Lorentz violating theories, the status of these laws is unclear because essential aspects are no longer present Jacobson (2001); Dubovsky and Sibiryakov (2006); Jacobson and Wall (2010); Betschart et al. (2009); Blas and Sibiryakov (2011); Busch and Parentani (2012). For instance, the thermality of the Hawking flux is inevitably lost in the presence of high frequency dispersion, although it is approximatively recovered for large black holes, i.e., when the surface gravity is much smaller than the UV scale setting the high frequency dispersion Macher and Parentani (2009).
The origin of the difficulties can be traced to the fact that the event horizon no longer separates the outgoing field configurations into two disconnected classes. In fact, when the dispersion is superluminal, it can be crossed by outgoing radiation. However, it was recently discovered that in some theories of modified gravity such as Hořava gravity Hořava (2009); Sotiriou (2011); Janiszewski (2015) and Einstein-æther Jacobson and Mattingly (2001); Eling et al. (2004); Eling and Jacobson (2006); Barausse et al. (2011), 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 allow for arbitrarily large group velocities. (The difficulty mentioned in Jacobson (2001) is thus evaded.) Following this discovery, it has been argued that the universal horizon should play a key role in the thermodynamics of such black holes. Even though they seem to obey a first law Berglund et al. (2012); Hořava et al. (2014), a key question concerns the temperature of the Hawking radiation they emit. Would it be essentially governed by the (higher) surface gravity of the universal horizon, or would it still be fixed by the surface gravity of the Killing horizon?
Two recent works concluded that the universal horizon emits a steady radiation with properties governed by its surface gravity. Because of the complicated nature of the field propagation near that horizon, this conclusion was indirectly obtained, in Berglund et al. (2013), by making use of a “tunneling method”, and, in Cropp et al. (2014), by analyzing the characteristics of the radiation field. In the present paper, we reexamine this question by performing a direct calculation and reach the opposite conclusion that no radiation is emitted from the universal horizon at late time.
We proceed as follows. As in the original derivation of Hawking Hawking (1975), we identify the boundary conditions on the outgoing modes in the near vicinity of the universal horizon by considering a simple collapsing shell geometry, and by assuming that the state of the field is vacuum inside. We then compute the mode mixing across the shell between inside modes propagating outwards , and outside stationary modes with a fixed Killing frequency . The late time behavior is obtained by sending the inside frequency . In this limit, we show that the scattering coefficients involving modes with opposite norms vanish. This result can be understood from the fact that the modes are accurately 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 . Hence, for large outgoing radial momenta, the state of the field outside the shell is the usual vacuum, as explained in Brout et al. (1995a).
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 Coutant et al. (2012) for a recent update. It establishes that large black holes emit a stationary flux which is (nearly) thermal, and with a temperature approximatively given by the standard relativistic value. In other words, the robustness of the Hawking process, i.e. its insensitivity to high frequency dispersion which was first established in Unruh (1995), 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, and they should involve the properties of the Killing horizon. This conclusion is reinforced by the fact that the field configurations propagating on either side of a universal horizon come from two disconnected Cauchy surfaces, and are highly blueshifted. Hence, it seems that no Hadamard condition of regularity Busch and Parentani (2012) could be satisfied on the universal horizon. This raises the question of the fate of the universal horizon; see Blas and Sibiryakov (2011). This difficult question shall not be discussed in the present work.
Appendix A gives the details of the calculation which is summarized in the main text. In Appendix B we compare our model with previously-studied dispersive ones without a universal horizon, and show the role of the acceleration of the preferred frame. Appendix C shows the results of numerical simulations confirming the approximately thermal character 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 collapsing geometry Hawking (1975). Although these concepts are well known, we present them in a way which prepares the more involved calculation of the late time flux when dealing with a dispersive field in the presence of a universal horizon. As explained in the Introduction, we shall use a direct calculation which consists of pasting the modes across the infalling shell. We closely follow the derivation of Massar and Parentani (1998).
For simplicity, we consider an infalling spherically symmetric lightlike thin shell. In this case, it is particularly appropriate to work with advanced Eddington-Finkelstein (EF) coordinates , where is the advanced null time. At fixed , one has , where is the usual Schwarzschild time. Hence, outside the shell, the stationary Killing field is simply . On both sides of the shell taken to be , the line element reads
where . These coordinates cover the entire space-time, shown in the right panel of Fig. 1. On the left panel, the infalling and outgoing null radial geodesics are represented in the plane. One clearly sees that the Killing horizon (where the norm vanishes) divides the outgoing geodesics into two separate classes. We work in Planck units: .
Let be a massless real scalar field with the action
We define and consider radial solutions independent of . Inside the shell, for , we introduce the null outgoing (affine) coordinate . Outside the shell and for , we introduce the null coordinate
where is the usual tortoise coordinate, which diverges on the Killing horizon. To cover the region inside the Killing horizon, one needs another coordinate . 111This sign guarantees that is positive. As we shall see in Section III, a similar sign must be taken when studying a dispersive field on both sides of a universal horizon. The field equation then reads
For simplicity, we neglect the potential engendering the grey body factor and work with the conformally invariant equations . The solutions can be decomposed as
and similarly for with replaced by . The infalling sector and the outgoing sector completely decouple. Moreover, the modes are regular across the horizon and play no role in the Hawking effect. We thus consider only the modes, and, to lighten the notations, we no longer write the upper index on outgoing modes.
To compute the global solutions, we need the matching conditions across the null shell. In the present case, is continuous along . Hence , where the relation between null coordinates is
for (). For (), one has .
To obtain the Hawking flux one needs to relate the modes characterizing the vacuum inside the shell, to the modes characterizing the asymptotic outgoing quanta with Killing frequency . In the internal region, a complete orthonormal basis of positive-norm modes is provided by the plane waves
where is the inside frequency . In the external region, the (positive-norm) stationary modes for , are
A similar equation defines in the trapped region, for . The modes and their complex conjugate form a complete orthonormal basis. One easily verifies that the conserved scalar product for the modes can be written as
The Bogoliubov coefficients encoding the Hawking flux are then given by the overlaps between the two sets of modes:
Using of Eq. (6), they can be computed explicitly; see Massar and Parentani (1998) for details. The late time behavior is obtained by sending the inside frequency . In this limit, one recovers the standard thermal result
To prepare for the forthcoming analysis, it is instructive to compute the Bogoliubov coefficients by the saddle point method Parentani and Brout (1992); Brout et al. (1995b). For the coefficient, when , i.e., at late time, the location of the saddle is given by
where is a constant which drops out of the late time flux. (In the present model, vanishes.) From this equation we recover the time-dependent redshift relating , the large frequency emitted from the collapsing star, to , the frequency received at infinity and measured using the proper time of an observer at rest. In particular, we recover the characteristic exponential law governed by the surface gravity . Had we considered a collapsing shell following a (regular) infalling timelike curve, Eq. (12) would still have been obtained at late time.
This is the kinematical root of the universality of Hawking radiation in relativistic theories. Indeed, when studying the coefficient, one finds that the saddle point is now located at . When taking into account the fact that the integration contour should be deformed in the lower -complex plane, one finds that has an imaginary part , whereas its real part in unchanged. This gives a relative factor with respect to the coefficient. Upon squaring their ratio, we recover Eq. (11). We also recover here that the Hawking temperature is fixed by the late time exponential decay rate entering Eq. (12). We finally notice that the stationarity of the flux is nontrivial. It follows from the fact that the ratio of Eq. (11) is independent of , and from the fact that for Brout et al. (1995b).
Iii Emission from a universal horizon
iii.1 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, such as Hořava-Lifschitz gravity Hořava (2009) or Einstein-æther theory Jacobson and Mattingly (2001); Eling et al. (2004). Since our aim is to study the radiation rather than the collapse, the latter shall be described by a simplified model. At the end of the calculations, we shall argue that our 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 geometry is still Schwarzschild. In this case, the metric is again given by Eq. (1), and the Penrose diagram of Fig. 1 still covers the whole space-time. To describe the (unit time like) æther field in the external region outside the shell, we adopt the solution of Berglund et al. (2012) (also used in Cropp et al. (2014)) with , , and . The Killing horizon is still at , whereas the universal horizon, where , is located at . Inside the shell, we assume that the æther 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.5, small deviations from this configuration should not significantly modify our conclusions.
In EF coordinates, on both sides of the shell, the æther field , and its orthogonal spacelike unit field are given by
where . We introduce the “preferred” coordinates by imposing that and . Their precise definition is given in Appendix A.1. In these coordinates, the metric takes the Painlevé-Gullstrand form:
At fixed , outside the shell, and only depend on . We notice that
The factor ensures that is a total differential. Moreover, as explained in Appendix B, is constant when the æther field is geodesic. Here we work with an accelerated æther, which is a necessary condition to have a universal horizon. Importantly, vanishes on the universal horizon. 222In an analogue gravity perspective Unruh (1981); Barcelo et al. (2011), 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 must be divergent at the point where . It would be interesting to find media which could approximatively reproduce this behavior. In fact, the novelties of the present situation with respect to the standard case studied in Macher and Parentani (2009) only arise from the vanishing of , and the associated divergence of the dispersive scale .
In Fig. 2 we show the lines of constant preferred time, and the direction of the aether field , in the plane. The coordinate is discontinuous across the shell trajectory, as was the null coordinate in the former section. As in the relativistic case, outside the shell we must use two coordinates and , now on either side of the universal horizon. The inside coordinate evaluated along the shell, at , is a monotonically increasing function of both for and of for . So, the foliation of the entire space-time by the inside coordinate is globally defined and monotonic.
We consider a real massless dispersive field with a superluminal dispersion relation. Its action is given by Eq. (2) supplemented by a term quartic in derivatives:
where is the covariant derivative and is the projector on the hyperplane orthogonal to . The dispersive momentum scale is given by . The field equation reads
Using a -dimensional approximation, Eq. (18) reduces to
when working in the preferred coordinates. Since this (self-adjoint) equation is second order in , the Hamilton structure of the theory is fully preserved. In particular, the conserved scalar product has the standard form
where is the momentum conjugated to . For more details; see Appendix A.2.
The Hamilton-Jacobi equation associated with Eq. (19) is
We introduce the Killing frequency , the preferred frequency , and the preferred momentum :
In these equations should be conceived as the action of a point particle; see Brout et al. (1995a); Balbinot et al. (2005); Coutant et al. (2012). As explained in these works, governs the WKB approximation of the solutions of Eq. (19). Notice that Eq. (22) only applies outside the shell, whereas all the other equations make sense on both sides.
iii.2 The modes and their characteristics
To compute the late time radiation one should identify the various solutions of Eq. (19), and understand their behavior. In the presence of dispersion, one loses the neat separation of null geodesics into the outgoing ones, and the infalling ones. In what follows, we call () 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 ; see Macher and Parentani (2009). Similarly, the corresponding modes will also carry the upper index or .
iii.2.1 The in and out asymptotic modes
In the internal region , the situation is particularly simple. Since the velocity field vanishes, the preferred frequency is , and the dispersion relation Eq. (21) becomes
This relation is shown in the left panel of Fig. 3. At fixed , the positive frequency modes with wave vectors and define the two in modes and . They both have a positive norm, which can easily be set to unity through a normalization factor. The mode is the dispersive version of the relativistic in-mode of Eq. (7).
Outside the shell, for , at fixed Killing frequency , the situation is more complicated as the number of real roots depends on . Outside the Killing horizon, for , one has . So, Eq. (21) possesses two real roots and , which describe outgoing and infalling particles, respectively. The WKB expression for the corresponding stationary modes [the solutions of Eq. (19)] is
where is a real solution of Eq. (21) at a fixed , and the corresponding preferred frequency. These WKB modes generalize the expressions of Coutant and Parentani (2010); Coutant et al. (2012) in that is no longer a constant. Using Eq. (20), one easily verifies that they have a unit norm. One also verifies that the group velocity along the ith characteristic is . When considered far away from the black hole, , the -WKB mode is the dispersive version 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). To be able to compute these overlaps, we need to construct the globally defined modes. To this end, we must study both the behavior of near the horizon and the third kind of stationary modes which propagate in this region.
iii.2.2 Near horizon modes
Inside the Killing horizon but outside the universal horizon, for , one has . As can be seen from the right panel of Fig. 3, one recovers the two roots and we just described. One notices that the root has been significantly blueshifted, whereas the infalling root hardly changed. Locally, in the WKB approximation, the corresponding modes are again given by Eq. (26).
In addition, below a certain critical frequency that depends on and , we have two new real roots we call and , where the arrow indicates the sign of the group velocity given by . (The minus signs in front of these roots and come from the fact that they have a negative preferred frequency for . Hence, for , the mirror image roots, and , have a positive .) Since , the WKB modes associated with these roots have a negative norm Coutant et al. (2012). We call the right-moving one and the left-moving one , so that the modes without complex conjugation have a positive norm. Both of them carry a negative Killing energy . Using Eqs. Eq. (26) and Eq. (20), one easily verifies that and have a negative unit norm within the WKB approximation. As we shall see they describe the negative energy partners trapped inside the Killing horizon before and after their turning point, respectively.
iii.2.3 The characteristics
As said above, the characteristics are solutions of the Hamilton-Jacoby equation . Since the frequency is a constant of motion on each side of the mass shell, they can then be computed straightforwardly. In Fig. 4, they are shown in the external region for a small value of (left panel) and a moderate one (right panel). The solid lines correspond to positive energy solutions while the dashed ones correspond to negative energy solutions.
The infalling -like characteristics corresponding to (in blue) approach the universal horizon from infinity and cross it at a finite value of . (When sending they asymptote to null infalling geodesics ) As their wave vectors are finite for , these characteristics will play no role in the sequel. As in the relativistic case, the modes act as spectators in the Hawking effect. 333Interestingly, -like characteristics have a turning point inside the universal horizon . (The presence of the turning point may be understood from the fact that, close to , and go to infinity but goes to . So, at fixed two roots merge at a point . The turning point approaches in the limit .) For later (preferred) times, they return towards the universal horizon, approach it asymptotically for , , and are highly blueshifted. In addition, for , there is a new mode with negative norm for . It is indicated by a dashed green line in Fig. 4. It emerges from the singularity and approaches the universal horizon while closely following the positive Killing frequency characteristic after its turning point. (In fact this new mode is directly related to the modes emerging from the singularity in Jacobson (2001): inside a universal horizon, and modes are swapped because of the vanishing of at .) Since some of the modes originate from the singularity, and since the blueshift they experience is unbounded for , the 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.
The -like characteristics with positive energy (in red), corresponding to the WKB modes , emerge from the universal horizon from its right () at early times. When increases, the momentum is redshifted while increases. At a finite time, the characteristics cross the Killing horizon, and go to infinity as (almost along null outgoing geodesics when ).
The third characteristics (orange, dashed line) describe the trajectories followed by the negative-energy partners. For , they also emerge from . However, when increasing they have a turning point inside the Killing horizon, after which they move towards the universal horizon, smoothly cross it, and hit the singularity at at finite values of and . Before the turning point, they are described by the WKB mode , and after the turning point by .
It is important to notice that the only novel aspect with respect to the standard dispersive case (treated in full detail in Coutant et al. (2012)) concerns the behavior near the universal horizon. To clarify these new aspects, we represent in Fig. 5 the global structure of the characteristics in the collapsing mass shell geometry.
iii.2.4 The characteristics in the collapsing geometry
In the internal region , the characteristics are straight lines. Coming backwards in time from the outside region, the inside trajectories are fixed by the value of the inside frequency which is determined (as in the relativistic case), by continuity of the field across the mass shell; see Appendix A.3 for details. As a result, the derivative must be continuous across . At the level of the characteristics (i.e., in the geometrical optic approximation), this implies that , the radial momentum at fixed , is continuous along the shell. In terms of the inside and outside preferred momenta and evaluated at and , respectively, the continuity condition gives
This equation has two solutions, but only one is well behaved as the other one gives a trajectory along which the preferred time is not monotonic. A straightforward calculation using the dispersion relation Eq. (21) also shows that the sign of is preserved. It should be noted that Eq. (27) is the dispersive version of the relativistic equation , which gives back Eq. (12) for , , and when using rather than .
It should be also emphasized that all outgoing -like characteristics originate from inside the shell, as in the relativistic case. This is shown in Fig. 5. Therefore, thanks to the universal horizon, the state of the field inside the shell determines the state of the modes. In this we avoid the problem discussed in Jacobson (2001), namely that in the absence of a universal horizon, the modes of a superluminal field originate from the singularity at . As discussed in footnote 3, these modes still exist, but they are now trapped inside the universal horizon.
Finally, we notice that the Killing frequency of the incoming modes which generate the outgoing modes exiting the shell at is very large. More precisely, when dealing with characteristics with positive (i.e., modes with positive norm), irrespective of the sign of their Killing frequency , the Killing frequency is positive. A straightforward calculation (based on the continuity of applied to the modes) shows that it scales as .
For completeness, we have also represented in Fig. 5 a couple of infalling characteristics which enter the shell for . One comes from (the dashed line), and one from (the solid line). They both reach the singularity after having bounced at inside the shell. These characteristics, although interesting, play no role in the Hawking process.
iii.3 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 the properties of the stationary modes in the immediate vicinity of the universal horizon at . For , the two roots and remain finite as . As can be seen in Fig. 4, the associated trajectories smoothly cross the horizon. They thus play no role in the large limit. In fact they describe out modes.
The two other roots and both diverge as . Importantly, they both satisfy
where the + sign applies to , and the - sign to . We have added a superscript to emphasize that this behavior is relevant at early time , just after having crossed the shell. The simple relation between and implies that for , the two WKB modes and are also related to each other by flipping the sign of . In the forthcoming discussion, to implement these points, we shall replace by , and add a superscript to the WKB modes .
The appropriate character of this superscript can be understood as follows. Although the divergence in in Eq. (28) resembles to what is found in the relativistic case, it has a very different nature due to the different relationship between and the preferred coordinate . This can be seen by looking at the validity of the WKB approximation for close to the universal horizon. Deviations from this approximation come from terms in , , and . Using
we find that these three terms go to zero as . Therefore, close to the universal horizon, the WKB approximation of Eq. (26) becomes exact for . In fact, these modes behave as the dispersive modes near a Killing horizon Brout et al. (1995a); Coutant et al. (2012). Namely, they have a positive norm for all values of and, moreover, contain only positive values of . We recall that this is the key property which also characterizes the so-called Unruh modes Unruh (1976); Brout et al. (1995b) for a relativistic field.
These are strong indications that no stationary emission should occur close to the universal horizon, as the pair production mechanism rests on deviations from the WKB approximation. This is confirmed in the next subsection.
iii.4 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 null shell. Inside the shell, one has the mode . Along the shell, for , it is a plane wave which behaves as . After having crossed the shell, for , it may be expanded in terms of the four WKB modes (which form a complete basis)
We are interested in the coefficients and which multiply the two modes with divergent wave vectors and opposite norms. It should be pointed out that the integral over runs from to . The other two coefficients and multiply the two modes which remain regular across the universal horizon in the coordinates. They vanish in the limit .
where 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 large values of ; see Eq. (28), the propagation across the shell induces no mode mixing between the inside in-mode and the high momentum WKB mode with negative norm , irrespective of the value (and the sign) of . As a result, outside the shell, the state of the field is stationary, and the vacuum with respect to the annihilation operators associated with for . 444This conclusion differs from that reported in Berglund et al. (2013). We do not understand the procedure adopted there, which apparently implies that the leading term in Eq. (28) does not contribute to the ratio of Eq. (31), thereby giving rise to a steady thermal radiation governed by the surface gravity of the universal horizon. Instead, the saddle point evaluation of performed in Appendix A.5 establishes that the leading term of Eq. (28) gives the exponential damping in of Eq. (31). It thus correspond to the vacuum as described in Brout et al. (1995a); Coutant et al. (2012). 555To be complete, one should propagate backwards in time the inside field configurations, and verify that they correspond to vacuum -like configurations for . To verify this, we computed the scattering coefficients encoding a change of the norm of the modes when crossing the shell. We found that they also decrease exponentially in for . We also recall here that the Killing frequency of the modes engendering a stationary mode diverges as , where is the radius when the mode exits the shell.
iii.5 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 of Appendixes A.4 and A.5, the factor in Eq. (31) comes from the fact that the phase of the mode inside the mass shell is , while that of the mode outside the mass shell is , where . At fixed , we find that the stationary phase condition applied to (the upper sign applies to while the lower sign applies to ) gives back the large frequency limit of Eq. (27) with real for , while is purely imaginary for , with a modulus . Let us now consider an arbitrary shell trajectory close to the universal horizon. We define an affine parameter along this trajectory. The possible saddle points are located where
So, the location of the saddle is
We get the same result as before, up to the factor . Therefore, the ratio is still suppressed by an exponential factor in , with a coefficient depending on the velocity of the mass shell when it crosses .
We now consider a generalization of the dispersion relation Eq. (21) with higher-order terms. Specifically, we consider the dispersion relation
Close to the universal horizon, the divergent wave vectors follow
As before, the coefficient corresponds to in Eq. (36). The value of the saddle point is then real, and the exponential factor appearing in has a unit modulus. Instead, for the coefficient , corresponding to the minus sign in Eq. (36), the solutions of the saddle point equation are
Taking only the saddle points with negative imaginary parts, we find that is suppressed by a factor which is exponentially large in . Interestingly, when using the inside spatial wave number rather than the inside frequency , the norm of the coefficient always decreases as with , which means that it is the diverging character of which guarantees that its sign does not flip when crossing the shell.
Similarly, the exponential factor suppressing is mildly affected by a change in the metric and/or the form of the æther field, provided the inside wave vector remains smooth, whereas the outside one diverges as a power law for , where 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 form of Eq. (35) close to the universal horizon. Indeed, the construction of Appendix A.1 can be easily extended to a generic space-time with a Killing vector , endowed with a generic timelike, normalized æther field .
We computed the late time properties of the Hawking radiation in a Lorentz violating model of a black hole with a 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 . We then computed the overlap along the thin shell of the outwards propagating inside positive norm modes, and the outside stationary modes. In the limit where the shell is close to the universal horizon, we show that the overlap between modes of opposite norms decreases exponentially in the radial momentum . This result comes from the peculiar behavior of the momentum when approaching the universal horizon with a fixed Killing frequency; see Eq. (28). Although this behavior was found in a specific model, we then argued that it will 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 accurately described, for both positive and negative Killing frequencies, by the WKB modes with large positive momenta (and a positive norm). In this we recover the standard characterization of outgoing configurations in their vacuum state in the near horizon geometry. Indeed, the condition to contain only positive momenta prevails for both relativistic and dispersive fields in the vicinity of the Killing horizon. The present work, therefore, shows that this simple 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 the thermality and the stationarity of the Hawking radiation are, to a good approximation, both recovered. This suggests that 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 that the inside configurations are blueshifted (towards the future), and that they have no common past with the outside configurations, it seems that the field state cannot satisfy any regularity condition across the universal horizon. It would be interesting to study the space of the field states, and determine whether some dispersive extension of the Hadamard condition can be imposed on the universal horizon. In the negative case, it seems that the universal horizon will be replaced by a spacelike singularity.
Acknowledgements.We thank Xavier Busch for enlightening discussions about the causal structure of spacetimes with a universal horizon. We also thank Ted Jacobson, David Mattingly, and Sergei Sibiryakov for interesting comments and suggestions. This work received support from the French National Research Agency under the Program Investing in the Future Grant No. ANR-11-IDEX-0003-02 associated with the project QEAGE (Quantum Effects in Analogue Gravity Experiments).
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.
a.1 Preferred coordinates
The preferred coordinates are defined by the followng four conditions
at fixed ;
at fixed ;
along the shell trajectory;
along the shell trajectory.
These four conditions uniquely define and as
In these expressions, is the tortoise coordinate built around the universal horizon.
a.2 Wave equation and scalar product
The action Eq. (17) has a invariance under , from which we derive the conserved current density
where “” stands for the complex conjugate, satisfying
As the wave equation Eq. (18) is linear, one easily shows that defines a conserved (indefinite) inner product in the following way. Considering two solutions and of Eq. (18), we first define by replacing by and by in Eq. (40). The inner product of these two solutions is then defined by
where is the unit vector perpendicular to the 3-surface defined by , and is a time coordinate. When considering the 3-surfaces defined by , the above overlap simplifies and gives the standard (Hamiltonian) conserved scalar product of Eq. (20).
a.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 matching conditions 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 across the shell. To see this, we first rewrite as
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
are continuous across . Since the complex conjugate of a solution of Eq. (18) is still a solution, this applies to as well as . Therefore, in evaluating Eq. (43) one can evaluate and the operators acting on it on one side of the shell, , , while and the operators acting on it are evaluated on the other side .
a.4 Calculation of
Let us consider two radial modes known on different sides of the mass shell: is known for and for . The complete expression of the scalar product in the coordinates is somewhat cumbersome, but it greatly simplifies in the relevant limit where
has a large frequency ;
has a large wave vector .
We have introduced the wave vector at a fixed . For the modes we are interested in and are of the order , where . Keeping only the leading terms in the inner product then gives
In this equation, as well as in the remainder of this appendix, the sign discriminates between and ; see below. In the large frequency limit, we evaluate this integral through a saddle point approximation. The possible saddle points are the values of where
The coefficient is defined for . Since the integral runs over , we must choose the saddle point at
It is easily shown that, under these approximations, the following unitarity relation is satisfied:
This implies that the coefficients are suppressed in the limit .
a.5 Calculation of
The calculation of follows the same steps. The saddle point equation now is
To be able to deform the integration contour to include the saddle point, we must choose the solution in the half-plane where the exponential decreases, i.e.,
The exponential factor in the integral then gives a suppression factor
In addition, to the order to which the calculation was performed, the prefactor vanishes. As the first relative corrections from neglected terms are of order , we get
Appendix B Acceleration of the æther field
The acceleration of the æther field is
Using Eq. (III.1), this gives for
For completeness, we now show that in dimensions a stationary universal horizon requires that the æther field has a nonvanishing acceleration, thereby generalizing what was found in de Sitter in Busch and Parentani (2012). We consider a stationary space-time with Killing vector , endowed with a timelike æther field . The universal horizon is defined as the locus where . (Notice that the Killing field must thus be spacelike on the universal horizon.) In particular, cannot be aligned with . Using the Killing equation, the variation of along the flow of is
If is freely falling, and is tangent to the hypersurfaces of constant . In particular, it is tangent to the universal horizon. In dimensions, since and cannot be aligned, is not a tangent vector to the universal horizon, which is thus not stationary. Models with a stationary universal horizon are thus in a different class than those studied in Coutant et al. (2012).
To see the combined effects of the dispersion and acceleration, we show in Fig. 6 the local value of the wave vector in the coordinates, , for the outgoing mode, as a function of . We compare three models with the same parameters, and for . The blue, solid curve shows the result for the model of Section III. The green, dotted curve shows the relativistic case. The red, dashed one shows the result for a dispersive model with a nonaccelerated preferred frame chosen to coincide with the æther frame of Section III at .666It must be noted that this model is not well defined for . The reason is that at , we have . A nonaccelerated vector field which coincides with at must thus satisfy the two conditions and at . From the free-fall condition, these two properties extend in the whole domain where the preferred frame is defined. Since they are incompatible in Minkowski space, we deduce that the domain in which the preferred frame can be defined does not extend to . A straightforward calculation shows that it extends up to . However, as this model is well defined close to and inside the Killing horizon, it can be used to see the qualitative differences between the nonaccelerated and accelerated cases. We see in Fig. 6 that the three models give very similar results for . Close to , the relativistic wave vector diverges, while the nonaccelerated dispersive model still closely follows the accelerated one. When is further decreased, the predictions of the two models separate: the nonaccelerated one gives a finite wave vector at while the accelerated one gives .
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 coefficients which are exponentially suppressed when the inside frequency . This result was obtained using the WKB approximation of the stationary modes just outside the universal horizon. This approximation is trustworthy as we verified that the deviations from the WKB treatment go to zero when approaching the universal horizon. This implies that at late time the inside vacuum is adiabatically transferred across the shell. Hence, the part of the field state can be accurately described by the WKB high (preferred) momentum mode for both signs of . In this we recover the situation described in Brout et al. (1995a); Balbinot et al. (2005); Coutant et al. (2012). Therefore, the nonadiabaticity that will be responsible for the asymptotic radiation will be found in the propagation from the universal horizon to spatial infinity. The value of the Bogoliubov coefficients should essentially come from the stationary scattering near the Killing horizon. Hence, we expect to get a nearly thermal spectrum governed by the surface gravity of the Killing horizon, and with deviations in agreement with those numerically computed in Finazzi and Parentani (2012); Robertson (2012).
To verify this conjecture, we numerically propagate the outgoing mode from a large value of down inside the trapped region to . This mode can be written in the limits and as
where the WKB modes are as described in Section III. The coefficient governs the grey body factor. In our (1+1)-dimensional model, we have verified that it plays no significant role. (We found that is bounded by .) Hence, as usual, the Hawking effect is essentially encoded in the mode mixing of modes of opposite norms.
To efficiently perform the numerical analysis, we regularized the metric and æther field. In practice we worked with a metric of the form
and a unit norm æther field