Hawking radiation in a non-covariant frame: the Jacobi metric approach
aa r X i v : . [ g r- q c ] A ug Hawking radiation in a non-covariant frame:the Jacobi metric approach
Avijit Bera , Subir Ghosh , Bibhas Ranjan Majhi Physics and Applied Mathematics Unit, Indian Statistical Institute203 B.T. Road, Kolkata 700108, India Department of Physics, Indian Institute of Technology GuwahatiGuwahati 781039, Assam, India
E-mails: [email protected]; [email protected]; [email protected] 14, 2020
Abstract
The present paper deals with a reformulation of the derivation of Hawking temperature for static andstationary black holes. In contrast to the conventional approach, where the covariant form of the metricsare used, we use the manifestly non-covariant Jacobi metric for the black holes in question. In the latter,a restricted form of Hamilton-Jacobi variational principle is exploited where the energy of the particle(pertaining to Hawking radiation) appears explicitly in the metric as a constant parameter. Our analysisshows that, as far as computation of Hawking temperature (for stationary black holes) is concerned, theJacobi metric framework is more streamlined and yields the result with less amount of complications, (asfor example, considerations of positive and negative energy modes and signature change of the metricacross horizon do not play any direct role in the present analysis).
In the present article we provide yet another new approach to derive the Hawking temperature [1, 2] fora black hole. Since many facts of Hawking radiation still remain unexplained it is always good to explorenew avenues hoping for new insights, one such being that even a manifestly non-relativistic metric carriesenough information to correctly reproduce the Hawking temperature for all known examples of static blackholes. The metric in question is the Jacobi metric [3, 4, 5, 6] that has generated a lot of interest after thepioneering work by Gibbons [7]. Originally Jacobi metric formalism was exploited solely in non-relativisticNewtonian dynamical problems [3]. The significance of Gibbon’s work [7] is that it extended the workingspace of Jacobi metric from non-relativistic to relativistic domain and in fact, previously known results (inthe latter) were correctly recovered with less amount of computational complexity and newer insights [8, 9].However, it is important to emphasize that all these works primarily dealt with particle dynamics outside ofthe black hole horizon. In the present paper, for the first time, we have further extended the applicabilityof Jacobi metric formalism in discussing the Hawking radiation as tunneling of particles across the horizon.Hawking radiation as tunneling through horizon started from a heuristic picture, originally introducedby Hawking himself [2]. This is similar to the electron-positron pair creation in a constant electric field.Here the particle and anti-particle pair is crated near to the horizon; among them one is ingoing and otherone is outgoing. The ingoing mode travels inside the black hole while the outgoing candidate crosses the1orizon barrier through tunneling and then ultimately observed at the infinity. Since the time and radialcoordinates changes their signature inside the horizon, in this case the modes are allowed to travel alongthe classically forbidden path. Consequently, the allowed path becomes a complex one as inside the horizonoutgoing modes is moving opposite to the forward in time.This heuristic picture was first formalised by Hamilton-Jacobi (HJ) approach [10, 11] and later on by nullgeodesic approach [12, 13]. These were later developed and used to study several aspects of Hawking effectby many workers [14, 15, 16, 17, 18, 19, 20, 21, 22, 23] (see the review [24] for a complete list of literaturein this direction). However, in all these cases the relativistic form of Hamiltonian-Jacobi (HJ) variationalframework was used with e.g.
Schwarzschild , Reissner-Nordstrom, Kerr-Newman metrics. The novelty inour work is the adjustment of HJ for the (manifestly non-relativistic) Jacobi metric corresponding to eachof these metrics in the context of particles tunneling out across horizon in the form of Hawking radiation.Quite interestingly, our derivation will reveal that the all important particle energy, appearing in the thermalspectrum of Hawking radiation, is incorporated in the definition of Jacobi metric from the very beginning,whereas in the relativistic HJ one has to impose it from outside in the time space variable separation of HJfunction. This makes the overall computation somewhat easier simply because time seizes to be a variablein the Jacobi metric that entirely deals with spatial degrees of freedom. Moreover, it reinforces the fact that,as far as physics is concerned, the Jacobi metric contains equivalent information like the original spacetimemetric.The paper is organised as follows. In Section 2 we provide a brief derivation of Jacobi metric correspondingto a generic relativistic metric. In Section 3 we recover the Hawking temperature for static, sphericallysymmetric black hole in Jacobi metric approach in detail. Section 4 comprises calculation of Hawkingtemperature for stationary black holes. We conclude in Section 5 with a summary of the present work andfuture directions.
The principle of least action, enunciated by Maupertuis was reframed by Jacobi where the action of amechanical system in phase space, S = Z γ p i dx i = Z γ p i ˙ x i dt , (1)with coordinates x i and conjugate momenta p i , is varied along an unparameterized path γ with the constraintthe energy E is constant along the path γ . Here dot represents the derivative with respect to time t . Becausein general, the Jacobi metric can be curved, the solution of a Newtonian dynamical problem now reducesto geodesic motion in curved space characterized by the Jacobi metric for a given particle energy. Inrelativistic context, for the static space-times, it was shown in [7] that geodesics of a particle on an energy-dependent Riemannian metric (known as Jacobi metric) describe the free motion of a massive particle onthe corresponding spacetime. For massless particles this metric reduces to the energy independent Fermator optical metric. The structure of Jacobi metrics corresponding to stationary (as well as non-stationary)black hole metrics are discussed in [8]. Below we briefly describe the construction of the Jacobi metric for ageneral spacetime whose metric coefficients are independent of time. The details can be obtained from [8].Consider the following spacetime metric: ds = − f ( x k ) dt + g ij dx i dx j . (2)The action for a particle of mass m is given by S = − m Z Ldt = − m Z dt q f ( x k ) − g ij ˙ x i ˙ x j , (3)2here dot represents derivative with respect to the coordinate time t . With the canonical momenta p i = mg ij ˙ x j q f ( x k ) − g ij ˙ x i ˙ x j , (4)the Hamiltonian turns out to be H = q m f ( x k ) + f ( x k ) g ij p i p j . (5)Thus, the Hamilton-Jacobi equation becomes, q m f ( x k ) + f ( x k ) g ij ∂ i S∂ j S = E , (6)where p i = ∂ i S and E is the conserved energy. Rewriting (6) in the form1 E − m f ( x k ) f ij ∂ i S∂ j S = 1 (7)yields the Hamiltonian-Jacobi equation for geodesics on the Jacobi-metric j ij given by j ij dx i dx j = ( E − m f ( x k )) f − ( x k ) g ij dx i dx j . (8)Note that for massless particles ( m = 0), Jacobi metric is equivalent to the optical or Fermat metric f ij = f − ( x k ) g ij , up to factor of E . Hence, in contrast to the massless case, for massive particles thegeodesics depend upon energy E in a non-trivial way. We quickly recapitulate the computational scheme for deriving Hawking temperature in the conventionalHamilton-Jacobi (HJ) approach for the covariant (relativistic) spacetime (e.g. see [10]). This will clearlyreveal the difference between the covariant framework and spatial (manifestly no-relativistic) Jacobi metricscheme. This approach is based on semi-classical technique where the wave function for the particle is takento be ψ ∼ e ( i/ ~ ) S , with S is the classical HJ action. In this picture, the action is calculated by using theparticle’s quantum equation of motion with considering only radial motion. To find the explicit form oneuses the choice of S as S ( r, t ) = S ( r ) − Et . This choice is dictated by the static nature of the spacetime.After finding S for the specific underlying spacetime, one obtains the ingoing and outgoing probabilities, theratio of which leads to the tunneling rate. It turns out that the rate is thermal in nature and one identifiesthe correct Hawking temperature by comparing with the Boltzmann factor. Below, instead of using the fullspacetime metric, we will show that non-covariant Jacobi metric is sufficient to extract the correct Hawkingtemperature. Since the expressions for the Jacobi metric corresponding to the (1 + 3)-dimensional covariant metrics havealready been provided in [7, 8, 9] we simply use these results. A generic static, spherically symmetric blackhole in Schwarzschild coordinates take the following form: ds = − f ( r ) dt + f − ( r ) dr + r ( dθ + sin θdφ ) . (9)3he location of the horizon is determined by f ( r H ) = 0. The corresponding Jacobi metric, using (8) turnsout to be [7], ds = j ij dx i dx j = (cid:16) E − m f ( r ) (cid:17)(cid:16) dr f ( r ) + r f ( r ) ( dθ + sin θdφ ) (cid:17) . (10)Below we shall find the action for a particle, moving in this background. Since the Hawking radiation is anear horizon phenomenon and tunneling occurs along the radial direction, the computation of the action willbe done in this region keeping all angular coordinates fixed. This will be the main ingredient of calculatingthe tunneling probability of the particle through the horizon.Start with the reparametrization invariant action: S = − Z r j ij dx i ds dx j ds ds , (11)for a particle in the background (10). The integrand in the action (11) becomes r j ij dx i ds dx j ds = ± ( E − m f ( r )) ( f − ( r ))( drds ) . (12)We shall now show that the positive and negative signs denote the outgoing and ingoing paths, respectively.In the semi-classical picture, the wave function of the particle is given by ψ = e ( i/ ~ ) S which we shall uselater. So the radial momentum of the particle comes out to be p r = ∂ r S . We denote the outgoing particlewhich has positive momentum while the ingoing one has negative p r . Now using (12) in (11) one finds p r = ∂ r S = ∓ ( E − m f ( r )) ( f − ( r )) . (13)Since in our tunneling picture the particle is just inside the horizon, we must have f ( r ) <
0. Then thenegative sign of the above related p r > f ( r ) around the horizon r = r H : f ( r ) = f ( r H ) + f ′ ( r H )( r − r H ) + O ( r − r H ) = 2 κ ( r − r H ) + O ( r − r H ) , (14)where in second equality the expression for surface gravity of the black hole κ = f ′ ( r H ) / r j ij dx i ds dx j ds = ± E κ ( r − r H ) ( drds ) ∓ m E ( drds ) ± O ( r − r H )( drds ) . (15)Hence, close to the horizon the action for radial motion is S = ∓ E κ Z r − r H ) dr ± m E Z dr ∓ Z O ( r − r H ) dr . (16)For particle tunneling close to and across the horizon, the limits of integration are taken as r H − ǫ to r H + ǫ ,with ǫ ( >
0) very small: S = ∓ E κ Z r H + ǫr H − ǫ r − r H ) dr ± m E Z r H + ǫr H − ǫ dr ∓ Z r H + ǫr H − ǫ O ( r − r H ) dr . (17)4he first integral on the right hand side of (17), with the change of variable r − r H = ǫe iθ , yields [28], Z r H + ǫr H − ǫ r − r H ) dr = − iπ . (18)The second integral on the r.h.s. of (17) ∼ ǫ ≈ S = ± iπE κ + real part . (19)Let us now distinguish the action for ingoing and outgoing trajectory as S in and S out respectively, S out = + iπE κ + real part , S in = − iπE κ + real part . (20)At semi-classical level, the WKB wave function for the particle is given by, ψ = Ae i ~ S with A being a(unimportant) normalization constant. Thus, the wave functions for outgoing and ingoing particle ψ out and ψ in respectively are ψ out = Ae i ~ S out , ψ in = Ae i ~ S in . (21)Then the probability of the particle, coming out from the horizon is P out = | ψ out | = | A | | e i ~ S out | = | A | e − πE ~ κ . (22)This also shows that the real part of the action does not contribute in the probability. In a similar way, theprobability of the particle, going inside the horizon, is P in = | ψ in | = | A | | e i ~ S in | = | A | e πE ~ κ . (23)Therefore the cherished expression for the tunneling rate is,Γ = P out P in = e − πE ~ κ ≡ e − ETH (24)which is identical to the Boltzmann factor with the temperature is identified as the Hawking temperature T H = ~ κ π . (25)This shows that in a manifestly non-covariant framework, Jacobi metric carries sufficient amount of blackhole features so that one can correctly reproduce the Hawking temperature. This demonstration constitutesone of our main results.We will conclude this section by calculating the Hawking temperature from the derived formula (25), forsome popular static black hole metrics. Example I: Schwarzschild black hole
For Schwarzschild black hole of mass M , the metric coefficient is given by f ( r ) = (1 − Mr ) with horizon isat r = 2 M . Then the surface gravity is κ = f ′ ( r H )2 = 14 M . (26)5hus we recover the well known Hawking temperature for Schwarzschild metric, T BH = ~ πM . (27) Example II: Reissner-Nordstrom black hole
The metric coefficient for Reissner-Nordstrom black hole with mass M and charge Q is f ( r ) = (1 − Mr + Q r ).The the surface gravity is given by κ = r + − r − r . (28)where r ± = M ± p M − Q . Here r + is the event horizon we have calculated the surface gravity at thishorizon. Then by general expression (25) one obtains the correct Hawking temperature as T BH = ~ π r + − r − r . (29)So we saw that Jacobi metric also produces the correct expression for temperature through HJ quantumtunneling approach. There is a particular difference between the conventional HJ based on spacetime andthe present one based on Jacobi metric. In the conventional one, as we mentioned at the beginning of thissection, the quantum equation of motion (like Klein-Gordon equation for scalar particle) is used to calculate S with a particular decomposition of S in time and radial parts. In this case the semi-classical wave functionis used at the very beginning. Whereas, here we did a completely classical calculation to obtain it byconstructing an action of the particle in Jacobi metric. Later on, the quantum concept is incorporated byconstructing the semi-classical wave function for the modes of the particle with the use of computed classicalaction. Once the wave functions are obtained, rest of steps are identical to the conventional one in findingthe Hawking temperature.The above point regarding conceptual distinction between the conventional scheme and the present Jacobimetric approach leads to another non-trivial difference. In our formalism the fermion particle tunneling isidentical to boson particle tunneling. Recall that for fermion tunneling one has to consider Dirac equation(instead of Klein-Gordon equation for boson particle) and its analysis is different. However, the fermion-boson mismatch does not show up in our formalism simply because we start with the classical action thatdoes not distinguish between fermions and bosons. In some sense the Jacobi metric approach underlines theuniversality of Hawking radiation. So far we have discussed the obtention of Hawking temperature in tunnelling picture using the Jacobi metricscorresponding to static, spherically symmetric black holes, expressed in Schwarzschild coordinates. In orderto show that the expression for temperature (25) does not depend on what type of coordinates one is usingto express the black hole metric, in the below, we will analyze the Jacobi metric corresponding to same blackhole metrics, expressed in Painleve coordinates [8]. Just to mention, this coordinate system is backbone ofthe tunneling formalism in null geodesic approach [12]. Therefore here we concentrate on the black holemetrics, expressed in these coordinates.It is well known that due to the coordinate singularity, for calculations close or at the horizon, theSchwarzschild form (9) may not be always suitable. One uses a Painleve coordinate transformation dt → dt − s − f ( r ) f ( r ) dr , (30)6o remove the coordinate singularity. Under this the metric (9) takes the following form: ds = − f ( r ) (cid:16) dt − s − f ( r ) f ( r ) dr (cid:17) + 1 f ( r ) dr + r ( dθ + sin θdφ ) . (31)To find the Jacobi metric of the above one we start with a general discussion on finding the Jacobi metricfor a more general spacetime metric of the form ds = − v ( x )( dt + A i dx i ) + g ij dx i dx j . (32)Jacobi metric for the generic form (32) is given by [7] ds = J ij dx i dx j = E − m v ( x ) v ( x ) g ij ( x ) . (33)where E is the energy of the particle of mass m . Now comparing (31) and (32) we identify v = f ( r ); A r = − s − f ( r ) f ( r ) , g ij dx i dx j = 1 f ( r ) dr + r ( dθ + sin θdφ ) , (34)with all other A i s vanish. With this identifications, the Jacobi metric (33) corresponding to (31), reduces to ds = J ij dx i dx j = E − m f ( r ) f ( r ) [ 1 f ( r ) dr + r ( dθ + sin θdφ )] . (35)Note that the above one is identical to that (10) obtained in Schwarzschild coordinates. Therefore proceedingsimilar to earlier discussion we again get the same expression (25) for Hawking temperature. In this section, we shall extend our discussion for stationary black holes. Using the corresponding Jacobimetric we again will find the correct expression for temperature in tunneling formalism.
Example I : Kerr black hole
For Kerr Black Hole, the metric is ds = − (1 − M rρ ) dt − M ar sin θρ dtdφ + ρ ∆ dr + ρ dθ + sin θρ [( r + a ) − a sin θ ] dφ (36)where, ∆( r ) = r − M r + a ; ρ ( r, θ ) = r + a cos θ ; a = JM with J being the angular momentum. Thecorresponding Jacobi metric has been found out in [8] as ds = ( E − m (1 − M rρ ))(1 − M rρ ) − [ ρ ∆ dr + ρ dθ + sin θρ [( r + a ) − a sin θ ] dφ ] . (37)Since the tunnelling is along radial direction and if for simplicity we take θ = 0 (i.e. taking particle’s motionalong z axis) [27], then the required action will be (12) with f ( r ) = (1 − Mr ( r + a ) ). In that case κ at the eventhorizon r H = r + is given by κ = r + − r − r + a ) , (38)7ith r ± = M ± √ M − a . So here also we obtain the Hawking temperature of the form (25) whose explicitform is T BH = ~ π r + − r − ( r + a ) . (39) Example II : Kerr-Newman Black Hole
The spacetime metric is ds = − f ( r, θ ) dt + 1 g ( r, θ ) dr − H ( r, θ ) dtdφ + ρ dθ + K ( r, θ ) dφ (40)where, A a = erρ ( r, θ ) [( dt ) a − a sin θ ( dφ ) a ] f ( r, θ ) = ∆( r ) − a sin θρ ( r, θ ) g ( r, θ ) = ∆( r ) ρ ( r, θ ) H ( r, θ ) = a sin θ ( r + a − ∆( r )) ρ ( r, θ ) k ( r, θ ) = ( r + a ) − ∆( r ) a sin θρ ( r, θ ) sin θ with ∆( r ) = r + a + e − M r ; ρ ( r, θ ) = r + a cos θ ; a = JM . Here J is the Komar angular momentumand e is the charge of the blackhole. The corresponding Jacobi metric is [8] ds = ( E − m f ( r, θ )) f ( r, θ ) [ 1 g ( r, θ ) dr + ρ dθ + K ( r, θ ) dφ ] . (41)Again for radial tunnelling and choosing θ = 0 (i.e. taking particle’s motion along z axis) [27] one identifiesaction as (12) with f ( r ) = g ( r ) = r − M r + a + e r + a . (42)Then the expression for Hawking temperature of the horizon r + will be again given by (25) with κ willbe calculated from the above f ( r ). It comes exactly identical to the form (39); but in this case we have r ± = M ± √ M − a − e .The restriction to two specific values of θ is because of the presence of the ergosphere. The calculationbreaks down because f ( r, θ ) is actually negative elsewhere at the horizon (i.e. inside the ergosphere) andthen t is not properly timelike there . The two values θ = 0 , π correspond to where the event horizon andergosphere coincide [27]. To summarize, we have correctly reproduced the Hawking temperature for all known forms of static as wellas stationary 3 + 1-dimensional black holes by using, in each case, the manifestly non-covariant (spatial)3-dimensional Jacobi metric corresponding to the covariant 3 + 1-dimensional conventional forms of the met-rics in question. As we have demonstrated the computations using Jacobi metric is considerably simpler.8ut, more significantly there are important conceptual differences as pointed out below:(i) A simplification occurs in the action functional itself. One need not consider relativistic form of Hamilton-Jacobi equation with both time and space derivatives and subsequently introduce the particle energy to makea variable separation of the action in to time and space part. The beauty of Jacobi metric approach is thethe energy of the particle appears explicitly (as a parameter) in the Jacobi metric and thus the Hamilton-Jacobi equation consists of only spatial derivatives from the very beginning. The theory lives in a manifestlynon-covariant space.(ii) In the covariant metric there exist complications since the interpretation of space and time-like coordi-nates gets interchanged as the horizon is crossed. Inside the horizon time coordinate behaves as spacelikewhereas space coordinates play the role of timelike coordinates. As a result the nature of Killing vector alsochanges from timelike to spacelike as one crosses the horizon. This forces one to use Kruskal time that is wellbehaved inside and outside the horizon. But all these complications are simply absent in the Jacobi metricformalism since the Jacobi metric is manifestly three (space) dimensional and so the question of changingover from timelike to spacelike coordinates (and vice versa) does not arise.(iii) An interesting observation is that the Jacobi metric approach underlines the universality of Hawkingradiation more strongly than the conventional covariant Hamilton-Jacobi formalism. In the latter fermionsand bosons need to treated differently since they obey Dirac and Klein-Gordon equation respectively. Onlyafter performing the analysis one finds Hawking temperature comes out the same for fermions and bosons.But this fermion-boson distinction does not appear in Jacobi metric framework since we consider a classicalparticle in constructing the action and only later the quantum concept is incorporated by constructing thesemi-classical wave function for the modes of the particle with the use of classical action computed earlier.An open problem in this context is the derivation of Hawking temperature for non-stationary or timedependent metrics. Primarily this is because the Jacobi metric corresponding to a time dependent metriccontains additional structure (see for example [8]) and care is needed to treat this in a consistent way.
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