Hawking Radiation as Tunneling

Transcription

1 Hawking Radiation as Tunneling Chis H. Fleming May 5, 5 Abstact This is a pesentation of a pape by Paikh and Wilczek[] wheein they deive Hawking adiation fom paticles tunneling though the event hoizon of a black hole. The motivation is to povide a moe mechanistic o heuistic-fiendly deivation of Hawking adiation as a tunneling phenomena. A key featue of this teatment is that a dynamical geomety is used the black hole mass is allowed to vay) with consevation of enegy stictly enfoced. This gives ise to highe ode tems in the Hawking adiance! It is thought that a black hole cannot be pecisely themal, because the mass of the black hole changes as it adiates. These highe ode tems impose enegy coections fo ω not so small, whee ω is the enegy of M the adiated paticle and M is the mass of the black hole. Intoduction The common heuistic explanation of Hawking adiation being caused by pai poduction nea the hoizon whee a negative enegy paticle falls in and the positive enegy paticle is adiated out is used as the key mechanism in this deivation of black hole adiation. Beneath the event hoizon thee is a space-like killing vecto. This allows negative enegy states. These states ae classically esticted to the inteio of the event hoizon. But they can tunnel out Quantum Mechanically. This causes pai ceation with a positive enegy paticle outgoing and a negative enegy antipaticle ingoing[]. Paikh and Wilczek conside two possible scenaios. Pai poduction can occu just inside the hoizon with a positive enegy paticle tunneling out and the pai poduction can occu just outside the event hoizon with a negative enegy paticle tunneling in. Classically speaking a paticle inside the event hoizon of a blackhole is tapped within and cannot escape. We conside a thin shell of enegy ω tunneling though the event hoizon and escaping to an outgoing geodesic. The I do not stictly follow the method of Paikh and Wilczek. Instead of taking the WKB esult to be tue, namely ψ exp ī h dxp, I deive the fomula beginning with a minimally coupled scala field. As a esult I immediately get the ingoing and outgoing, paticle and antipaticle wavefunctions and am able to use fewe ticks initially.

2 tunneling pocess will be teated semiclassically with the tansmission coefficient detemined, via WKB method, fom the classical action of the paticle. Method. Nonsingula Coodinates The fist step is to choose coodinates which ae not singula acoss the hoizon. A tansfomation is made fom Schwazschild to Painlevé coodinates. This was ou fist homewok assignment. Beginning with Schwazschild coodinates, we shift Schwazchild time t S by a function of, f). ds = ) dt S + ) d + dω ) t S = t + f) ) ds = ) dt f ) ) dtd 3) + ) f ) ) ) d + dω 4) Next the metic is made to be spheical fo constant time slices. This fixes f) and ids us of the singulaity. f ) = ) = ds =.. Radial, Null Geodesics ) f ) ) 5) 6) ) dt + dtd + d + dω 7) With the new metic we can now solve fo the only cuves that ae both adial and null. ) ds = 8) dt = ) + ṙ + ṙ 9) ṙ = ± )

3 Whee the ± accounts fo the outgoing ṙ > ) and ingoing ṙ < ) geodesics when outside the event hoizon. When inside the event hoizon, both geodesics ae ingoing. These cuves ae obviously null ays in the asymptotically flat space fa away fom the black hole.. WKB Appoximation Conside the s-wave of a minimally coupled scala field fo an abitay metic. We begin the WKB appoximation by casting the field as an exponential and explicitly sepaating the eal and imaginay pats which give the amplitude and the phase. h φ = m φ ) φx) = e T x)+ısx) ) T + ıs) + T + ı S) = m h 3) T + T ) S) = m h 4) S + S T = 5) As this is a semiclassical appoximation, we expand both T and S as a powe seies in h. They must stat at least with an h tem o the equations cannot be satisfied. T x) = h T x) + ht x) + h T x) + ) Sx) = h S x) + hs x) + h S x) + ) 6) 7) The zeo th ode tems ae as follows. T ) S ) = m 8) T S = 9) In the WKB appoximation we set the amplitude to be slowly vaying as compaed to the phase, T =. S ) = m ) Now let us solve fo S fo a massless field given ou choice of metic. g µν = ) ) 3

4 g µν = = ) S S t = S t + ± ) ) S S ) 3) 4) = S t + ṙ S ) d S = ±ω t ṙ 5) 6) Whee the tem in equation [4] we ecognize as ṙ fo the adial, null geodesic. Via saddle point appoximation the semiclassical kenel K, that popagates the paticle between x and x in configuation space, can now be evaluated. K = N e ī h S x x 7) S hee acts as the classically evaluated action note that it does indeed have units of action). Paikh and Wilczek will efe to it as such. We can use standad esults of the WKB method fo the calculation of the tansmission coefficient fo tunneling though a potential baie that would be fobidden by classical law. The WKB method is used to find solutions befoe, in, and afte the classically fobidden egion. The coefficients ae matched fo continiuty and the esulting tansmission coefficient is Γ = e h Im S x x 8) Whee x and x denote the beginning and the end of the classically fobidden egion..3 Tunneling.3. Paticle Channel Conside pai poduction occuing just beneath the event hoizon with the positive enegy paticle tunneling out. Hee is a diagam of the heuistic pocess. 4

5 + ω IN OUT 7 R With espect to the vacuum, this is an occupied negative enegy state that is tunneling out causing the pai ceation pocess.[] We ae looking fo the imaginay pat of the action ove the classically fobidden egion. out in d p 9) out d p in dp 3) Fist the classical momentum is expanded into an integal. Next Hamilton s equation is used to tansfom vaiables fom momentum to enegy. dh dp = ṙ 3) out in d M ω M dh ṙ 3) In this last equation we have used the fact that if the paticles have tunneled out, then the black hole will have lost some enegy ω. Next we switch integation vaiables fom H to the paticle enegy ω. H = M ω 33) dh = dω 34) out in d ω dω ) ṙ 35) Fom my deivation of the WKB solution, we could have come to equation [35] in two steps. Nomally it would not occu to expand the enegy out into an integal. What this will do is continuously tunnel the paticle though the event hoizon as opposed to all at once. The authos do not mention this and because of thei diffeent oute in deivation I am inclined to believe that they 5

6 may not be awae of it. The affect of this will be coect highe ode tems in the Hawking adiance. Im S = ±Im Im S = ± ω out in d ω ṙ out dω d ṙ in 36) 37) Now because the paticle is tunneling out, we use the outgoing null geodesic and what is the closest thing to an outgoing geodesic within the event hoizon. It is almost as if thee is a classical tuning point just beneath the hoizon, asymptotically in the infinite past. Though tuely these ae not classical tuning points, but the WKB appoximation does not cae. The action becomes imaginay and we can match coefficients in the thee egions. That is all that mattes. We also take the initial point to be just inside the event hoizon. We take the final point to be just outside the event hoizon as well, but it must be taken into account that with the black hole losing enegy, the event hoizon is in a diffeent location afte the paticle has tunneled out. Finally, self gavitation of the shell of enegy is taken into account eplacing M with M ω in the metic in which the paticle tavels. This esult is deived by Kaus and Wilczek[3]. M ω ) ṙ = + 38) in = ɛ 39) out = M ω) + ɛ 4) Im S = Im ω M ω) d ω dω M ω ) 4) Im S = π dω 4M ω ) 4) Im S = 4πω M ω ) 43) In equation [4] the contou integal is a ight-handed semicicle aound the pole by defoming the contou down on the E plane. This gives a pefacto of πı. This gives a positive tempeatue. A second calculation will now be pefomed whee the integals will be evaluated in evese ode. This will easset the choice of out as the coect choice. out in d M ω M de E 44) 6

7 out Im S = π d ) 45) ɛ out = 4M ImS) π 46) out = M ω)) 47) In equation [44] the contou integal is a ight-handed semicicle aound the pole by defoming the contou down on the E plane. This gives a pefacto of πı. This calculation shows that the value of out is consistent with the pevious calculation of the imaginay pat of the action. Note that the fist calculation did not depend on pecise location of out, but only that it encompassed a pole along with in. Equation [8] of Wilczek and Paikh[] contains an typesetting eo. The last tem has both the ı stiped out and Im applied. Doing both sets the tem to zeo and nullifies the equality..3. Antipaticle Channel Now conside pai poduction occuing just outside the event hoizon with the negative enegy paticle tunneling in, in evese time. Mathematically this is the exact same pocess as the fist, the integals will come out to be exactly the same. Physically thee is no eason that this would be esticted to antipaticles and the othe channel esticted to paticles. I think the pape is confusing how negative enegy paticles ae intepeted in QFT with the eally negative enegy states that can occu when you have a space-like killing vecto fo t. This is the evese-time diagam fo the actual calculation that is made. + ω IN OUT 7 7 R Hee is what I think would actually have to happen, in fowad time, fo any of this to make sense. 7

8 + ω IN 7 OUT 7 R Positive enegy must be coming out in fowad time fo the black hole to be shinking. The pai ceation is also stated to occu outside the event hoizon. This leaves the negative enegy paticle to necessaily fall, not tunnel, but fall into the black hole. And now on to the calculation. Fist let us switch to the evesed time metic. ds = ) dt + dtd + d + dω 48) t R = t 49) ds = ) dt R dt Rd + d + dω 5) ṙ R = ± + 5) Now we must choose the incoming geodesic. And we must note to add to the mass of the black hole to take into account the self gavitation of the paticle. ṙ R = + Im S = Im M + ω ) in out d in out d M+ω M ω M+ω) ω d dh ṙ dω ) ṙ ω dω + M+ω ) 5) 53) 54) 55) Im S = π dω M + ω ) 56) Im S = 4πω M ω ) 57) 8

9 .4 Hawking Radiation To combine the two contibutions Paikh and Wilczek simply add the amplitudes of the Feynman diagams. Howeve the channels ae stated to be fo paticle and antipaticle, so that wouldn t be allowable. But eally both channels could be fo eithe and both. But none of this mattes any way because we will get the same exponential tem. Γ e Im S T otal 58) e 8πωM ω ) 59) Fo small enegy ω, this educes to, e 8πMω, a Boltzmann facto fo enegy ω at the Hawking tempeatue 8πM.4. The Highe Ode Tem The highe ode tem enfoces enegy consevation. Fistly conside the effective tempeatue. The negative sign ensues that the tempeatue 8πM ω ) inceases with adiation. Secondly conside if the emitted paticle takes all of the mass of the black hole with it. This would have a tansmission ate of Γω = M) e 4πM 6) Thee can only be one of these outgoing states. But thee ae e S BH, whee S BH is the Bekenstein-Hawking entopy, states in total, so the pobability of finding that states is one in that numbe o P ω = M) e S BH 6) P ω = M) e A 4 6) P ω = M) e 4πM 63) So the highe ode coection is in ageement with the Bekenstein-Hawking entopy..5 Chaged Black Holes It is a tivial extension to conside unchaged adiation fom chaged black holes. We begin with the coesponding Panlevé coodinates and then calculate thei adial, null geodesics. ds = ) + Q dt + Q dtd + d + dω 64) 9

10 is H = M ± M Q 65) ṙ = ± Q 66) The imaginay pat of the action fo the outgoing, positive enegy paticle ω out dω ) in d 67) M ω ) Q A change of vaiables befoe the esidue is taken can simplify this integal geatly. u = M ω ) Q 68) du = u dω 69) M ω) Q out u du d 7) Q in u M ω)+ M ω) Q Im S = π d 7) M+ M Q Im S = π M+ M Q 7) M ω)+ M ω) Q Im S = π ω M ω ) 73) M ω) M ω) Q M )) M Q 74) Γ e Im S 75) Γ e π ωm ω ) M ω) )) M ω) Q M M Q 76) This equation [76] is incoectly multiplied by in Paikh and Wilczek. In the next step we will taylo expand the exponent and take the linea tems to find the Hawking tempeatue. We have Γ = e βω+α ω +α 3 ω 3 + M + ) M Q 77) β H = π M Q 78).5. Highe Ode Tems A simila analysis of the highe ode tems was not given in the pape. I attempting to calculate the shell enegy ω associated with a pobability of in

11 the numbe of states, one will find that it must be the case that Q = o the conditions cannot be satisfied. Such a pocess of unchaged adiation fom a chaged black hole cannot happen. 3 Discussion 3. Results Using the humble tools of WKB tunneling, the Hawking tempeatue is deived fo chaged and unchaged black holes. The dynamical geomety, along with continuous tunneling eveals highe ode tems which enfoce enegy consevation and ae in ageement with the Bekenstein-Hawking entopy. 3. Futhe Reseach It may o may not be impotant to note that the limit of ω compaable to M has been investigated, and yet the paticle is teated semiclassically while the metic is only teated classically. Pehaps it would be wise to conside using the Wheele-DeWitt equation, combining the Hamiltonians fo the metic and paticle, given that the metic can only vay by M. I have just begone the most udimentay investigation and have discoveed this so fa. The Painlevé coodinates aleady natually foliate space and time in the manne needed to use the WDW equation. Howeve the 3-metic q ij is simply the flat spheical metic. The Hamiltonian constaint is identically zeo even though you may vay M in time. What is lost ae the suface integals fom the action. This contibutes to the WDW equation in a most cuious manne. This has all been investigated befoe by Teitelboim and I am just beginning to ead his papes. Refeences [] Maulik K. Paikh and Fank Wilczek Hawking Radiation As Tunneling axiv:hep-th/997 [] Thibaut Damou and Remo Ruffini Black-hole evapoation in the Klein-Saute-Heisenbeg-Eule fomalism Physical Review D, Volume 4, Numbe, page 33 [3] Pe Kaus and Fank Wilczek Self-Inteaction Coection to Black Hole Radiance axiv:g-qc/9483