PHYSICAL REVIEW D, VOLUME 60, 083002 Dynamics of black hole motion P. S. Custódio and J. E. Horvath Instituto Astronômico e Geofı́sico, Universidade de São Paulo, Avenida Miguel Stéfano, 4200—Agua Funda, 04301-904 São Paulo, São Paulo, Brazil ~Received 14 December 1998; published 29 September 1999! We evaluate in the proper and cosmological frames the effects of nonzero velocities on the mass gain or loss of primordial black holes. An upper limit of twice the initial Lorentz factor is derived from the mass gain of the black holes, a value that may be enough to preclude their evaporation. Next, we analyze accelerated black holes and find that the Unruh effect can delay the onset of the evaporation regime. Finally, we reassess the equilibrium between black holes and the relic thermal radiation. @S0556-2821~99!00718-3# PACS number~s!: 97.60.Lf I. INTRODUCTION As is well known, a black hole at rest relative to background radiation is subject to spontaneous emission ~the Hawking effect @1#!. In a recent paper @2# we have reexamined the question of the fate of the holes when particle absorption ~proportional to the gravitational cross section and the environment radiation density!, which contributes to the mass accretion, is included. As could be expected from simple considerations, the motion of black holes is important to determine whether the object is in the evaporation or accretion regimes. In fact, a nonzero velocity of the black hole with respect to the background radiation field may put it away from the evaporation regime. A detailed discussion of these effects and of the acceleration onto the mass gain or loss of the black holes is the subject of the present work. As in Ref. @2# we will assume a homogeneous and isotropic radiation background as viewed by observers with negligible peculiar velocity, i.e., v pec!c. However, this actual range is very large and we may find black holes in either the absorption or evaporation regimes according to their instantaneous rest mass values for a given temperature of the background. An analytical evaluation shows that any black hole in the absorption regime ~within the radiation-dominated era! can grow very little from their formation time. For that purpose we integrate Eq. ~1!, keeping only the second term and obtain @for t f !t,t c (M i )# M ~ t ! ;M i @ 111.25g ~ T !~ M i /M pl! 2 ~ t f M i ! 21 # . * ~2! Here M i defines the initial rest mass at t f and t c (M i ) is the interval for this object to cross the critical mass curve as defined in Ref. @2#. As explained in our previous work, if we put M (t)50 in Eq. ~1!, we obtain a thermodynamical parameter defining an instantaneous equilibrium between black hole and radiation given by II. EQUATIONS OF MOTION AND THE KINEMATICAL CRITICAL MASS * /27pr eff~ T !# 1/4, M c ~ T ! 5M pl@ A ee The black holes are supposed to behave as semiclassical objects, i.e., M 0 @M pl . This restriction avoids nontrivial corrections due to quantum effects from the space-time structure in very small scales. Then, the mass evolution equation is given by ~see @2#! * 5max(Aee);1076J(M ) GeV4 with 1022 ,J(M ) where A ee ,1 ~see Ref. @2#! and we consider the standard model of elementary particles emitted by hot black holes. The M c (T) parameter thus separates two regimes of mass evolution: dM /dt(M .M c ).0 ~where there is a mass gain! and dM /dt(M ,M c ),0 ~where the black hole evaporates!. The mass gain above is small when compared to the initial mass only when t f @1.25g M i /(M pl) 2 , since the me* dium is scarce in energy density from this time on. For these conditions a Newtonian approach is enough for the calculations. As described in our previous work @2# black holes with relatively large mass (M .1017 g) will be accretting energy for a long time until they eventually cross the critical mass curve. It is easily shown that the rate of growth of the critical mass is much larger than the actual mass variation for black holes above the critical mass value. The above considerations do not take into account an inhomogeneous primeval universe, which allows the existence of a more complex dynamics for the evolution of black A ee 27p dM 0 52 2 1 4 r eff~ T 0 ! M 20 , dt M 0 M pl ~1! where the zero subscript stands by the rest mass and r eff 5rrad13 P rad is the effective energy density describing the particles around the object, the rest mass M 0 is measured in GeV, t is the cosmological time, A ee measures the degrees of freedom in relativistic particles, and M pl is the Planck mass. This equation holds in the semiclassical approximation, where any black hole is assumed to satisfy M pl!M (t f ) !M hor(t f ) @where t f stands by the black hole formation time and M hor(t f ) is the cosmological horizon mass; the mass contained in the cosmological causal region since t50#. 0556-2821/99/60~8!/083002~9!/$15.00 60 083002-1 ~3! ©1999 The American Physical Society P. S. CUSTÓDIO AND J. E. HORVATH PHYSICAL REVIEW D 60 083002 holes. The kinematical effects must be taken into account when one has to evaluate the Doppler effects in an appropriate frame. There are two obvious choices of reference frame for such an evaluation; we shall proceed to analyze the rest mass evolution ~of a given black hole! in the proper frame ~an observer comoving with the black hole! and the cosmological frame @where we put an observer in rest relative to the cosmic microwave background radiation ~CMBR!#. A. Proper frame This case describes the situation where there is an observer near the event horizon of a black hole with rest mass M 0 . For this observer, the background radiation shows a dipole anisotropy in the temperature, displaying the Doppler effect. Such anisotropy is proportional to the peculiar velocity and the temperature pattern shows an angular dependence given by T~ u !5 T0 . g ~ 12 b pec cos u ! ~4! 2 21/2 ) is the Lorentz factor that connects Here, g pec5(12 b pec the measurements made from these observers and stands for the peculiar Lorentz factor associated with the proper motion of the black hole. Then, we can write, for the dynamics, dM 0 A ee 27p 52 2 1 4 ^ r eff~ T ! & M 20 , dt M 0 M pl F i5 d ~ g M ui!. d t pec 0 ~5! ~7! The equations above are difficult to solve for general results given initial and boundary conditions. However, we are interested in the quasifree case, where we may assume F i ;0, and we allow only an initial kick as the initial condition. The instantaneous equilibrium ~in the proper frame! is given by dM 0 /d t 50 and we obtain for the ~proper! critical mass * /9pr eff~ T 0 ! F D ~ g pec!# 1/4, ~ M c ! p ~ t ! 5M pl@ A ee B. Cosmic frame We can shift the description to the cosmic frame by using the g d t 5dt relation, as seen by an observer comoving with the cosmological expansion. Thus, Eq. ~5! changes to ~6! where ‘‘^ &’’ denotes the angular average taking into account the angular temperature distribution as viewed from the black hole by its associated observer, t is the proper time, the four-velocity is u i 5(1,v ), F i is the external four-vector force denoting external action on the black hole and the index i runs from 0 to 3. An explicit calculation of the angular average ^ r eff(T)& yields 2 21 ! . 3 ^ r eff~ T ! & 5 r eff~ T 0 !~ 4 g pec T 0 ( t ), g pec , and M p . Since this observer knows the value of the background temperature ~assumed to be measurable by him using the appropriate devices! and the dipole anisotropy ~due to his peculiar motion!, he is able to decide between the absorption-evaporation regime if he knows the value of the rest mass at any given instant t. From Eq. ~8! it is clear that if the motion is strongly relativistic, the Doppler effect dominates and the black hole will be accreting energy ~such a feature is, in fact, independent of the observer!. Another observer, comoving with the cosmological expansion, will conclude the same results for the same object if he knows its rest mass. However, the parameters are measured to be different because of the different reference frames. In both frames, the observers must conclude that the shape of the black hole is modified ~since that the mass gain depends on the angle u between the direction of motion and the sight of view!. Moreover, there is the Lorentz contraction as viewed by the distant observer. The only case where the Doppler effect does not contribute is ~obviously! when b pec50. In any case, the signal of mass variation does not change when one changes the reference frame or reparametrizes the time. ~8! where, F D ( g pec)5„2 g pec(t)…2 21 is the Doppler factor as measured in the black hole rest frame. We shall refer to this quantity as the kinetic critical mass hereafter. Now this hypothetical observer has to decide whether this black hole is evaporating @ M 0 ,(M c ) p # or accreting @ M 0 .(M c ) p # . For this purpose, the observer must measure three parameters ~at a given instant t as measured by its clock!: A ee dM 0 9 pr eff~ T 0 ! F D ~ g ! 2 52 M 0. 21 dt g~ t ! g~ t !M 0 M 4pl ~9! This equation describes the rest mass variation of a black hole in the cosmic frame. When g @1, the Doppler effect term dominates ~in analogy to the case g pec@1 discussed before!. However, an observer sitting in this frame would be using a different device in order to evaluate the peculiar velocity of this object ~in the previous example, an observer comoving with the black hole must measure the radiation dipole!. If we impose dM 0 /dt50, we obtain the kinetic critical mass in that frame as * /9pr eff~ T 0 ! F D ~ g !# 1/4. M c ~ T 0 , g ! 5M pl@ A ee ~10! This expression is formally identical to the previous one @Eq. ~8!#, as it should be. This relation holds when we change from the observer at rest ~comoving with the black hole! to the observer comoving with the cosmological expansion and g pec denotes the Lorentz factor, due to the black hole peculiar velocity b pec . When the observer’s peculiar velocity is not zero, we have ~generally speaking! (M c )(T, g , g pec) Þ(M c ) 0 (T, g , g pec). This stresses the importance of the 083002-2 DYNAMICS OF BLACK HOLE MOTION PHYSICAL REVIEW D 60 083002 choice of reference frame for an accurate evaluation of the ~evaporating or growing! state of the black hole. The next step consists in analyzing the black hole mass evolution taking into account kinematical effects and evaluate its behavior either in the rest frame or the cosmological one, where all these quantities are consistently defined. where K i 5(112 g i )/(122 g i ) and J( t )536p M i g i / M 4pl* tt d t r eff„T 0 ( t )…. i This solution shows an interesting feature: if g i @1 initially, the term K i exp J(t) dominates and the maximum mass gain becomes M 0 ~ t , g i @1 ! ;2M i g i . ~18! III. BLACK HOLES IN RELATIVISTIC MOTION A. Supercritical case We shall consider the black hole mass evolution for the supercritical case, i.e., M 0 .M c . The more simple and relevant case is the free-motion F m 50 with the initial conditions M 0 ~ t i ! .M c ~ t i ! ~11! v pec~ t i ! Þ0. ~12! and In this case, we can show that the absorption term dominates over the evaporation term if M 0 .M c because the ratio G 5Ṁ abs /Ṁ evap5(M 0 /M c ) 4 @1. Therefore, we may ignore the quantum evaporation term in the Eq. ~1! and obtain dM 0 9 pr eff~ T 0 ! 2 5 M 0F D~ g ! ; dt M 4pl ~13! using the equation of motion, Eq. ~6!, we obtain F 0 5 ġ M 0 1 g Ṁ 0 50, The result is the same when the medium is very energetic, i.e. as J( t )@1. Therefore we have shown that large initial kinetic energies imposed on black holes may impart a substantial mass gain relative to their initial rest mass. The solution obtained has also a consistent nonrelativistic limit g i ;1 in a rarified medium having J( t )!1, and becomes M 0 ( t );M i for this case, as derived in the previous section @see Eq. ~2!#. Let us consider an ideal observer in the cosmic frame, having a small peculiar velocity relative to the cosmic radiation v pec!c and/or v pec!H 0 . Such an observer would write down the following equation for the same black hole ~assuming that this observer knows the initial conditions and the rest mass of this object! dM 0 9 pr eff~ T 0 ! F D ~ g ! 2 M 0, 5 dt g M 4pl where M 0 (t)5M i g i / g (t). Considering the same initial conditions as reparameterized by this new observer we obtain the following solution ~14! M 0~ t ! 5 with the solution M 0~ t ! 5 M 0~ t i ! g ~ t i ! . g~ t ! ~15! The approximation used in Eq. ~14! holds only when the kinetic energy of the black hole is larger than the energy exchanged by the black hole and the environment due to evaporation or absorption. Whenever these energies are comparable, the object is not free and the radiation exerts work on the black hole motion. Strictly speaking, one can consider F 0 ;0 only when r eff(T0)!M 4ptṀ 0 /27p M 20 . However, F 0 cannot be zero if the black hole changes energy with the surrounding medium and Eq. ~14! is only an approximation for small exchanges. This case will be analyzed in the next sections with the first law of thermodynamics applied to the radiation–black-hole system. Substituting Eq. ~15! into Eq. ~13!, we obtain the following differential equation dM 0 9 pr eff~ T 0 ! 5 @ 2M 2i g ~ t i ! 2 2M 20 # dt M 4pl ~16! with M i 5M 0 ( t i ) g ( t i ). Integrating this equation with the initial condition above, we arrive at the solution M 0~ t ! 5 2M i g i @ K i exp J ~ t ! 11 # , @ K i exp J ~ t ! 21 # ~17! ~19! 2M i g i A11 ~ 4 g 2i 21 ! exp„22J ~ t ! … , ~20! where J(t) has the same functional form as in Eq. ~17! but with the proper time t replaced by the cosmic time t. We check that this solution approaches 2M i g (t i ) in the limit g i @1, or, if the medium is very dense, J(t)@1; the solution tends to M 0 (t);M i for g i →1 ~as it should be!. Therefore, it is concluded that a strong initial kick would inject energy into the black hole due to the absorption term enhanced by the Doppler effect, with a maximum gain which does not depend on the frame. Since Ṁ c @Ṁ , the critical mass curve will cross the rest mass of this black hole in a time interval specified by the initial conditions. For black holes at rest ~relative to the cosmic radiation! this time is very short as compared to black holes with large kinetic energies. The interesting topic of how elementary processes may yield nonzero peculiar velocities to primordial black holes and to which extent they could influence the final fate of the latter comes to mind. As a particular case of the many possibilities, it has been shown that fluctuations of matter on scales l may induce a peculiar velocity v pec /c ;lH 0 ( d r / r ) ~for a V;1 universe!, to any massive object imersed within these regions ~see Ref. @3# and references therein!. However, it is easy to show that the initial Lorentz factor induced by this effect must be very small. Assuming «5 d r / r ;1025 we obtain 083002-3 P. S. CUSTÓDIO AND J. E. HORVATH PHYSICAL REVIEW D 60 083002 g i ;111/2~ lH 0 ! 2 «l 2 ;115310211, ~21! where in the last evaluation we assumed l5H 21 0 . Therefore, we conclude that the main contribution for the mass variation for black holes is due to the energy density from the background and the kinematical effects ~for the uniform motion! are very small when we consider that these peculiar velocities are induced by nonuniform matter distribution. Other mechansims may be capable of changing that situation. In principle, nonperturbative effects ~collisions with bubble walls, etc.! may induce large peculiar velocities and the above calculations are useful in addressing the physical effects on the black hole masses. In fact, it can be shown that those nonrelativistic black holes @with M 0 (t i ).M c (t i ) and g (t i );1# will stay above the critical mass curve by a finite time, specified by the initial mass and the equation of state for the medium. If we put M 0 (t i );M c (t c ) the solution for t c „M 0 (t f )… that we seek is t c ~ M 0 ! ;400~ M 0 /1015g ! 2 s, ~22! for the radiation-dominated era and 10 t c ~ M 0 ! ;2310 t H ~ M 0 /M ( ! 3/2 s, ~23! for the matter-dominated era @4#. The evolution of black holes in an expanding universe has been also analyzed by Barrow et al. @5#, taking into account several physical processes, but without considering the dynamics associated to peculiar motion. The time scales will have to be modified when these black holes have large initial kinetic energies ~see Fig. 1 and discussion below!. B. Subcritical case Now, we shall demonstrate that subcritical black holes, i.e., M ,M c , need not cross the critical mass curve in order to gain mass, even taking into account the strongly kinematical effects. If we put dM 0 /d t 50 in Eq. ~5!, we obtain the minimal Lorentz factor g th necessary to hold the accretion regime for a given black hole whose rest mass is known g th~ M 0 ,T ! 5 21 A113 @ M c ~ T ! /M 0 # 4 . and using Eq. ~6!, we can write Ṁ 0 52 ~25! ġ M . g 0 ~26! Then, the temporal derivative of g th(M !M c ) is ġ th5) ~ M c /M 0 ! 2 ~24! The interpretation of Eq. ~24! is evident: any black hole with peculiar velocity such that g . g th ~relative to a comoving observer!, is able to accrete more energy than the evaporation leaks. Black holes at rest ~relative to CMBR! will go away far from the critical mass curve, losing energy and raising their temperatures. However, it is not evident if black holes with large peculiar motions could approach and cross the critical mass curve and begin to accrete energy. Note that g th(M 0 ,T) grows as M 0 diminishes. Since the term 3(M c /M 0 ) 4 quickly exceeds unity for M 0 below M c /3, we may write g th approximately as ) g th; ~ M c /M 0 ! 2 , 2 FIG. 1. Paths of primordial black holes in the mass-time plane. The curve labeled as ‘‘1’’ displays the trajectory of a quasifree black hole ~with negligible back reaction from the background! with peculiar velocity different from zero ( g pec.1). The Doppler effect enhances the mass gain rate until this object crosses the critical mass curve ~bold! calculated for the radiation-dominated era. The curve ‘‘2’’ displays a black hole with initial mass above the critical mass which begins to evaporate. The curve ‘‘3a’’ displays the behavior of a subcritical black hole in which we ignore the background radiation, showing only the Hawking evaporation. Similarly, the curve ‘‘3b’’ displays how the Doppler effect plus the background radiation back reaction help to delay the evaporation of the black hole. For the sake of example, g i (11E thermal /E rel.)55 was chosen in all cases. S D Ṁ c ġ 1 , Mc g ~27! and assuming that the critical mass evolves slowly Ṁ c ġ ! , Mc g ~28! ġ th@ ġ . ~29! then we obtain In the extreme case, when g → g th and M 0 ,M c we obtain ġ th;2 ġ . ~30! This calculation does not hold for g . g th since these black holes are in the absorption regime. Then, by Eq. ~30!, we cannot have g bh. g th as a possible solution and the black 083002-4 DYNAMICS OF BLACK HOLE MOTION PHYSICAL REVIEW D 60 083002 hole is therefore below the kinetic critical mass and must loss energy mass due to the quantum effects. Therefore, the threshold Lorentz factor g th diverges from the kinetic gain associate to the black hole velocity. If we consider the details of the equation of state of the primordial universe, we find that the rate of critical mass variation is always smaller than the rate of mass loss for black holes in evaporation, i.e., Ṁ 0 !Ṁ c . Then, kinematical effects may retard the crossing of black holes into the evaporation regime from the initial condition M (t i ).M c (t i ), but these effects cannot take them into the absorption regime if they begin with this initial condition ~a frame-independent conclusion!. These results hold for the uniform free-motion and we shall address what happens to accelerated black holes in the next sections. IV. THE EFFECTS FROM THE BACKGROUND All the results in Sec. II were obtained in the low-energy density approximation, r eff(T)!M 4plṀ /27p M 2 . The qualitative analysis of the reference frames was carried out in order to enlighten the subtle problem of frame changes. However, we know that these black holes are not free and we may expect a varying external force from the initial nonzero peculiar velocity. The external force is strictly zero only when v pec(t)50, since then the black hole sees an isotropic medium around. Now, we will proceed to deduce the effects from the absorption of energy on the dynamics ~for single black hole! and we use the energy conservation law plus the reasonable approximation of black hole dilute gas, i.e., r bh! r rad . The causality imposes a maximum volume which the black hole is able to accrete at a given cosmological time, the horizon volume. As time goes on, this physical volume is enlarged by the cosmic expansion. We may write this volume as V hor(t)5V hor(t i ) @ a(t)/a(t i ) # 3 , where a(t) is the dimensionless scale factor for a Friedmann-Robertson-Walker model of universe and we use for convenience a(t i )51. We thus consider that the thermal radiation is diluted by the cosmic expansion by the usual law r rad(t)5 r rad(t i )a 24 (t). With these considerations above, we now consider that the energy absorption by the black hole will be extracted from the variation of the kinetic energy due to the work performed by the radiation exchanged between black hole and the thermal reservoir. Thus, from the conservation of energy to the radiation and the black hole embedded in this reservoir we write r rad~ t ! V hor~ t ! 2 r rad~ t i ! V hor~ t i ! 5 12 M ~ t i !v 2 ~ t i ! 2 21 M ~ t !v 2 ~ t ! 2 d Q ~ t ! , ~31! where d Q(t)5 * tt dtṀ @ t,a(t) # is the heat absorbed from the i medium by the black hole, which is transformed in mass ~or equivalently the mass which is converted in heat by the Hawking radiation when its rest mass is below the critical mass! and M ~ t ! 5M ~ t i ! 1 E t dtṀ @ t,a ~ t !# . ~32! ti Note that Ṁ @ t,a(t) # depends on the scale factor describing the cosmological expansion throught r eff@T(a)#. The solution for the velocity of the black hole taking into account this balance is v~ t ! 5& @ 11E kin~ t i ! / d Q # $ 11 j i ~ t !@ 12a ~ t ! 21 # % 1/2 @ 11M ~ t i ! / d Q # ~33! where j i ~ t ! 5 r rad~ t i ! V hor~ t i ! / @ E kin~ t i ! 1 d Q ~ t !# ~34! and E kin(t i ) is the initial kinetic energy of the black hole. As we would have expected, when the black hole kinetic energy is much larger than the energy accreted from the reservoir, we get the free solution as described in the previous section. From the velocity and mass we can evaluate the external force imposed by the initial conditions using the Newton’s law of motion. It is easy to show that this external force will depend on the mass rate and on the initial kinetic energy. All these considerations are useful when we are taking into account the Doppler effect, because this effect connects the heat exchange between the black hole and the reservoir and the equation describing the black hole mass variation in a nontrivial form. We will study this relation in the next sections taking into account others effects, such as the Unruh effect. The relativistic generalization is immediate and its yields the relations E total~ t i ! 5 g ~ t i ! M 0 ~ t i ! 1V hor~ t i ! r rad~ t i ! , ~35! E total~ t ! 5 g pec~ t ! M 0 ~ t ! 1V hor~ t i ! r rad~ t i ! /a ~ t ! 2 d Q ~ t ! , ~36! with a(t i )51 and g pec(t) the Lorentz factor due to the peculiar velocity measured from the background radiation. These algebraic equations have a simple solution as a constraint that it would be used in a consistent analysis for the mass evolution. Since E(t i )5E(t), we obtain g pec~ t ! 5 11K ~ t i ,t ! , 11M ~ t i ! / d Q ~ t ! ~37! where K(t i ,t)5 g pec(t i )M (t i )/ d Q(t)1 @ V hor(t i ) r rad(t i )/ d Q(t) #@ 12a 21 (t) # . The cutoff introduced by the horizon volume is not at all dangerous, since the black hole will take a finite proper time until it can swallow all the initial radiative energy content within the horizon. We can show that this time is proportional to V hor(t i ) 1/3 and is also proportional to a function dependent on the adiabatic index of the medium and the initial mass of the black hole. 083002-5 P. S. CUSTÓDIO AND J. E. HORVATH PHYSICAL REVIEW D 60 083002 Moreover, in addition to the delay to enter the evaporation regime due to transient supercritical black holes, we obtain an additional delay effect to evaporation due to the kinematics effects since that subcritical black holes with large kinetic energies will be subject to the Doppler and relativistic correction effects. In order to describe the latter, we consider the mass equation evolution for subcritical black holes, i.e., M ,M c . For the sake of the argument, we drop the second term in Eq. ~31! and integrate it from an arbitrary initial time until t@t i . Then we obtain F M 0 ~ t ! 5M 0 ~ t i ! 12 3A M 0~ t i ! 3 E t ti dt/ g pec~ t ! G 1/3 , ~38! where g pec(t) is given by Eq. ~37!. Since Ṁ c !Ṁ when M !M c , we can neglect the effects * )50 the from the cosmic expansion and obtain for M 0 (t evap approximate solution * @ M ~ t i ! , g pec~ t i !# t evap F ; g pec~ t i ! 11 G E thermal~ t i ! t @ M ~ t !# , g pec~ t i ! M ~ t i ! evap 0 i ~39! where t evap(M 0 )5M 30 (t i )/(3A) is the usual time scale for evaporation described by the Hawking effect and E thermal(t i )5 r rad(t i )V hor(t i ). As we would have expected, the radiation content from the medium works against the evaporation and black holes with large kinetic energies will be delayed to enter the evaporation era. Conversely, the asymptotic solution for large black holes, say M @3M c , is obtained from Eq. ~31! when we ignore the first term associate to quantum evaporation. Then, Eq. ~38! yields M 0~ t ! ; M 0~ t i ! 4 t 129 pr rad~ t i ! M 0 ~ t i ! /M pl* t dtF D @ g pec~ t !# /a ~ t ! 4 g pec~ t ! i . ~40! This expression shows how the Doppler effect gives rise to the enhanced gain to relativistic mass for the black hole. It is apparent that this gain is dependent on the scale factor a(t). perturbation theory in the particular case of a black hole interacting weakly with its surroundings. This naive approach will rely on the fact that a general analysis shows that to a distant observer, a black hole continues to move in the external field in the same manner as a point test particle does. Within these considerations, we will assume that the same equations of motion as from the previous section describe a black hole in motion embedded in a thermal background. Due to the conservation of linear momentum, the variation of the rest mass requires that the acceleration must be nonzero. Therefore, our description of a black hole in motion ~whose mass is time dependent! is similar to a relativistic rocket. We know that small black holes lose mass in a vertiginous rate and therefore their accelerations grow substantially if their initial velocities are different from zero. However, in this case, the relativistic corrections forbid the acceleration growth at an arbitrary rate since the relativistic mass is larger ~for a distant observer!, and thus the Hawking evaporation must be corrected accordingly. For the observer comoving with the cosmic expansion these objects are not spherical in shape and they appear flattened in the direction of motion. For the observer comoving with that black hole, the background thermal radiation is not isotropic and shows a dipole due to their peculiar accelerated motion. If this observer performs specific measurements he will see that this dipole is in fact time dependent and grows as the time goes on. However, such state a conclusion is complicated due to an additional nontrivial quantum effect: the Unruh effect. As is well known, an accelerated particle sees a thermal bath whose temperature is proportional to its acceleration @9#. This is a nontrivial quantum effect that arises from the relation between the vacuum state of an accelerated frame and the vacuum state of an inertial frame ~Minkowski vacuum!. Both frames are related by a Bogoliubov transformation that is dependent on the particle’s acceleration. The thermal radiation composing the Unruh vacuum also can be absorbed by the black hole, since the acceleration is a truly dynamical effect ~another observer traveling in the opposite direction with same acceleration must see the same thermal effect in magnitude and must observe the absorption by the black hole with the same features!. If we consider the acceleration the equations of motion become more complicated and they are written as V. ACCELERATED BLACK HOLES AND THE UNRUH EFFECT According to Ref. @6#, a black hole subjected to an external force behaves as a compact elastic body. Appropriate methods for analyzing the motion of black holes in external force fields were first developed in the seventies @7#. Likewise, it can be shown @8# that when the external metric field E acts on a black hole of charge Q and mass M, the hole acquires an acceleration a5QE/M , as given by Newton’s laws of motion. Although the general problem of the motion of a black hole in an external field does not admit an analytical solution, a detailed description is possible in terms of a sort of d~ g M 0! 5 ġ M 0 1 g Ṁ 0 , dt ~41! d~ g M 0v i ! 5 ġ M 0 v i 1 g v i Ṁ 0 1 g M 0 v˙ i , dt ~42! 0 F ext 5 i 5 F ext with the initial condition v pecÞ0. Even in the simplest case, F m 50, the nonzero rest mass variation induces a temporal variation of the Lorentz factor g (t), and this forces to keep all terms in the equations of motion. We can go ahead and consider, within the same approximation, the more complete situation taking into account the 083002-6 DYNAMICS OF BLACK HOLE MOTION PHYSICAL REVIEW D 60 083002 Unruh, Hawking, and Doppler effects and considering frame changes and the classical absorption. The complete problem is given by Eqs. ~41! and ~42! plus the following differential equation for the mass 9p A ee dM 0 r * @ F ~ g ! / g # M 20 , 52 21 dt g M 0 M 4pl eff D ~43! * 5 r eff„T 0 ~ t ! …1 r Unruh„T ~ a ! … r eff ~44! where follows from our simplifying assumptions, as we explained above, and A ee (M );3.4310 26 g 3 s 21 for masses larger than ;10 16 g and A ee (M );10 28 g 3 s 21 for masses smaller than ;10 11 g. A comparison of these terms yields a minimum acceleration for the black holes to be above the critical mass, namely a min~ T ,M , g ! ; 1.2531015 F ~ M ,T ! ~ M /M ( ! F D ~ g ! S ti 4 20 p M pl 4 dta ~ t ! . 9g ~ a ! M ~ t i ! ~49! Then, in order to drive a black hole over the critical mass ~when this object was subcritical initially! we must satisfy the constraint a pec.a min and the restriction above. When this integral is equal to the right-hand side, then the black hole reaches the critical mass in t ~without crossing it!. When 0 ,a pec,a min, it can be checked that the mass rate does not reverse its signal. The final consequence is a delay on the time scale for evaporation. This evaluation made above is based on an impulsive external force in the form F ext(t) 5F 0 d (t2t kick), where t kick denotes the cosmological time for this instantaneous action. The cosmic time for crossing the critical time depends on the relative difference specified by M c (t)2M 0 (t i ) and the initial temperature of the medium. We shall study in what follows some particular simplified cases to understand the main features of the solutions. cms22 , ~45! 4 M M( 4 . ~46! We can easily show that F(M ,T)50 implies M 5M c * * ; @ (7.3310 25 )/(T/K) # g as it should be. As discussed in Ref. @2#, a relativistic growth of the black holes is not possible because of the general constraint dr g /dt,c, where r g denotes the gravitational radius. From this requirement, we easily derive an upper limit for the peculiar acceleration in order to avoid an explosive growth for a generic black hole: 1.02310 25 G ~ M ,T ! 1/4 cms22 a max~ M ,T ! ; ~ M /M ( ! 1/2 ~47! A. Accelerated supercritical black holes First we shall study supercritical black holes satisfying M 0 @M c considering the Doppler and Unruh effects. The equations of motion reduce to Ṁ 5 T Unruh~ a ! 5 * ~t! 9 pr eff M pl 4 S D ~48! A ~strong! initial external force will put the black hole into the absorption regime, in which the object will gain mass and therefore its proper acceleration must diminish until this effect is eventually damped to zero. This is a consequence of the momentum conservation for the free motion case. Note that a max(M )@a min(M ) for all black holes with M @M pl , and that for black holes with M ;M pl we must use a full quantum gravity formalism in order to describe all processes since our present approximations break down. Now, we proceed to evaluate the constraint on the initial acceleration in order to boost the black hole into the absorption region. Let us consider the nonrelativistic motion. When a pec@a min(T,M ) we can ignore the Hawking term in Eq. ~43! and obtain a time-dependent solution for the mass. Imposing M 0 (t).M c (t) from the initial time t i we easily get the following constraint ~50! 1 g A2a m a m 5 Ag 2 a 2 12 gġ ~ v i a i ! 2 ~ ġ / g ! 2 . 2p 2p ~51! T Unruh~ ġ , g ! 5 2310 38 Ṁ . 2 W D~ g ! gs 21 W D~ g ! M 2, While in the nonrelativistic limit the expression above reduces to T}a, as it should, in the relativistic case T(a) is a function of g (t) and ġ (t) through where G ~ M ,T ! 5 @ 12M ~ t i ! /M c ~ t !# . * 1/4 D S DS D g ~T! T F ~ M ,T ! 5121.7310 29 * J~ M ! K E t 1 ~ gġ ! . ~ 2 p ! ~ g 2 21 ! 1/2 ~52! This expression is easily derived from the standard relativistic formulas. Then, for the supercritical case, we can integrate Eq. ~50!, taking into account the Doppler effect, the Unruh effect, the radiative energy from the expanding background, and the dynamical constraint obtained from the conservation of energy as is given by the Eq. ~37!. Substituting and integrating we obtain M ~ t !; M ~ti! , 12Z ~ t ! ~53! with Z~ t !5 with 083002-7 3p3M ~ ti! 10M 4pl g ~ t i ! T rad ~ t i ! 4 * E t ti dt F D~ g ! 3G ~ t, ġ , g ! ga~ t !4 ~54! P. S. CUSTÓDIO AND J. E. HORVATH PHYSICAL REVIEW D 60 083002 4 g ~ a ! T Unruh ~ ġ , g ! * G ~ t, ġ , g ! 5 11a ~ t ! g ~ t i ! T rad~ t i ! 4 S 4 * As discussed in previous sections, the net effect is to delay the crossing time for an interval proportional to the initial kinetic energy. D and F D „g (t)…54 g 2 21 is the Doppler factor. The black hole acceleration is described by its rest mass variation, similarly to a relativistic rocket in the free space. This expression can be simplified for the large acceleration, ultrarelativistic regime, i.e., a@a min(T,M,g) and g @1, where the Unruh and Doppler effects dominate. In these limits T Unruh; 1 ~2p! ġ , ~55! B. Accelerated subcritical black holes Those subcritical black holes with large initial impulses will be delayed against the evaporation. For a simple illustration of this effect, let us consider the motion restricted to a small time interval, i.e., d t5t2t i !t evap(M 0 ) without loss of generality. Since the parameter q(t) becomes larger than the unity, the solution of Eq. ~32!, becomes S and substituting back into Eq. ~54!, Z(t) reduces to Z~ t !5 3M ~ t i ! g ~ a ! * 40 p M 4pl E t M ~ t ! ;M ~ t i ! 12 dt g ~ t ! 2 ġ ~ t ! 4 . g~ t !5 ti 11K ~ t ! , 11q ~ t ! ~57! q ~ t ! 5M ~ t i ! / d Q ~ t ! . ~58! Substituting these expressions we write the function Z(t) as Z~ t !5 3g ~ a ! M ~ t i ! * 40 p M 4pl E t ti 3A ~ E thermal/E kin! 1 @ 12a ~ t ! 21 # d t 2g~ ti!M ~ ti!3 ~56! From the relations obtained from the conservation of energy applied to this system, we obtain dt g ~ t ! 2 S K̇ ~ t ! 2q̇ ~ t ! g ~ t ! q~ t ! D 4 S K ~ t ! 5q ~ t ! g ~ t i ! 1 D V hor~ t i ! r rad~ t i ! @ 12a 21 ~ t !# . M 0~ t i ! Therefore, in this limit, the effects from cosmological expansion are mild and the dynamics might be dominated by the local acceleration. In the opposite case, when a!a min(T,M,g) we have, for g @1 4 Z ~ t ! ;1.2p 3 M 0 ~ t i ! g ~ t i ! T rad ~ t i ! /M 4pl * E t dt ti * L ~ M ,t ! 5 d 2E Ṁ 52 . dtdA 32p M 2 ~63! The condition that L(M 5M c ,t)50 expresses an instantaneous equilibrium between the black hole and radiation. Taking the first time derivative we arrive to 1 dt 4 . a ~t! ti E ~62! Now, we shall show that the thermodynamical equilibrium between black holes and the cosmic radiation is not possible due to the cosmological expansion. For this, it is enough to evaluate the rate of absorbed energy per unit area in terms of the mass equation given by Eq. ~1! ~here we use the natural system of units where G 5c51!. Denoting the rate for flux absorbed by d 2 E/dtdA and from the usual relations A54 p r 2g , r g 52M , E5M we easily find ~60! 4 Z ~ t ! ;0.9p 3 M 0 ~ t i ! g ~ t i ! T rad ~ t i ! /M 4pl . VI. THERMODYNAMICAL EQUILIBRIUM AND THE COSMIC EXPANSION g~ t ! , a 4~ t ! and for g ;1 D 1/3 The extra term indicates that the mass of the black hole is larger than the mass that found in the evaluation of the case with the Hawking evaporation alone applied to the same black hole in the nonrelativistic limit. This extra mass gain was obtained from the medium and due to the relativistic effects associated to the relation mass-energy given by the special relativity. It is interesting to note that the initial thermal energy of the reservoir E thermal appears explicitly, even for M 0 !M c . ~59! where 3A d t g~ ti!M ~ ti!3 t ~61! Here, a(t)}t n is the usual scale factor for the radiationdominated (n51/2) or the matter-dominated (n52/3) eras. In any case, the large initial kinetic energy give rise to a strong mass gain until the black hole is decelerated by the reaction of the medium. From this moment on, the black hole will be described by a quasistationary solution with slow accretion until it crosses the critical mass curve. dL ~ M ,t ! 1 5 @ F ~ M ,t ! 216pṙ rad# ; dt 32 ~64! where 083002-8 F ~ M ,t ! 5 2 4A ee ~M! M7 2 64p A ee ~ M ! r rad~ t ! ; M3 ~65! DYNAMICS OF BLACK HOLE MOTION PHYSICAL REVIEW D 60 083002 and F(M ,t) vanishes at M 5M c . Although this energy flux vanishes at the critical mass, its first time derivative is not zero, denoting an instantaneous departure from the initial equilibrium. In order to see this departure explicitly, and address whether it pushes the black hole into the evaporating or accreting regimes, we must integrate Eq. ~64! with the initial condition L(M 5M c ,t i )50, where we allow a black hole with instantaneous equilibrium with the medium. For this purpose, it is enough to integrate this equation in a short time interval d t, where we can neglect the term F(M ,t) @that is, for d t!16d r /F(M ) without loss of generality#. Let dr denote a small variation in the medium density within the time interval d t above. Since now F(M )5F(M c 2 d M ) !16pṙ rad the approximate solution the above equations for the radiation-dominated era is L ~ M ,t i 1 d t ! 5 r~ ti!dt t 3i , ~66! and, similarly, for the matter-dominated era L ~ M ,t i 1 d t ! 5 r~ ti!dt t 11/3 i . ~67! Therefore, in both cases we have obtained a nonzero energy flux driven by the ~infinitesimal! expansion of the universe. These results can be shown to hold for sufficiently long time intervals. Since the sign of both solutions @Eqs. ~66! and ~67!# is positive, we have shown that black holes drift into the evaporation region inevitably, and thus no thermodynamical equilibrium between the background radiation can be maintained in either the radiation-dominated or the matter-dominated eras. First we have checked that changes in the reference frame change, in fact, the definition of the critical mass, and thus just two ‘‘good’’ reference systems are at disposal: the proper frame and the cosmic frame. Given that the black hole mass changes, the transformation of the critical mass parameter between these two frames is not trivial, and is more general than the usual formula M 5M 0 g . This is a consequence of several effects that play a role: mainly the Doppler effect, energy absorption, and Hawking evaporation. We found that subcritical black holes are not able to cross critical mass curve and begin to absorb energy, even considering the blueshift in the absorbed energy due to the Doppler effect. Such result is related to the fact that small black holes with M !M c lose mass at a much higher rate than the critical mass increases it rate induced by the cosmic expansion cooling. The expansion prevents the thermodynamical equilibrium between black holes and radiation. The existence of a critical mass allows just an instantaneous equilibrium, since this parameter grows in time. Black holes located on the critical mass curve would drift into the evaporation region. We have considered an external agent imposing an initial acceleration on black holes, and found that the Unruh effect is very important to determine the fate of the objects, since these black holes are allowed to absorb such excess energy at a continuous rate. The Unruh effect may retard the time scale for evaporation, in a form that depends on the initial momentum of the black hole. An observer that loves black holes may be able to ‘‘save’’ these objects against the evaporation by imposing a timedependent force fine-tuned to its rest mass ~and temperature! in that given moment. VII. CONCLUSIONS ACKNOWLEDGMENTS We have discussed in this paper a number of features of the dynamics of black holes in its simplest, intuitive version. 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