Phonon tunnels from a sonic black hole H. Z. Fang* and K. H. Zhou Department of Ph ysics, China University of Petroleum, Changjiang Road 66 , 266555, Qingdao , China We investigate the phonon radiation from a spherically symmetrical, stationary, viscid -free so n- ic black hole by using a semi -classical method. The backreaction of the radiate d phonon is taken into account. We obtain the phonon emission temperature , and it is consistent with the Hawking’s formula . I. INTRO DUCTION Hawking’s prediction that black hole radiates particles from its horizon with a Planckian spec- trum [1] is quite attractive and far -reaching-influenced in theoretical physics field. There are n u- merous derivations of the Hawking effect which emphasis different features of the process , and sometimes make markedly diffe rent physical assumptions. The Hartle -Hawking approach u se d analytic continuation of the propagator across the event horizon of a n eternal black hole [ 2]. D a- mour and Ruffini emphasiz ed that the outgoing modes suffer a barrier when crossing the horizon to yield the same Hawking temperature [ 3]. Robinson and Wilczek claim ed that there exists rel a- tionship between Hawking radiation and Gravitational Anomalies [4] and s ome physicists b elieve that Hawking effect can be interpret ed in string theory [5]. In recently years , one of the most inte r- est ing works on Hawking radiation derivation done by Parikh and Wilczek [6] comes to public ’s atten tion. Parikh and Wilczek believe that black hole radiation do can be regarded as particle tu n- nel ing, but they gave a different description of quantum tunneling t hrough the horizon in the pro - cess of black hole evaporation. The radiation spectrum they obtained deviates from exact thermal and they proved that the quantum information conserves in the process of black hole evaporation. As a very important way in physi cs and mathematics , analogies provide new ways of looking at problems that permit cross -fertilization of ideas among different branches of science . Si nce the study directly on Hawking radiation from naturally formed black holes, as is known, there are stil l too many difficulties today , so the analogy of gravity becomes an alternative way. The most bea u- tiful example of this was pointed out by Unruh [7] in 1981. He gave an anal ogy of light waves in curved space time with sound waves viewed as a disturbance of fluid background field in flat Mi n- kowski spacetime, then he deduced an “effective ” metric to descri be the sound propagation in fl u- id flow and obtained the phonon Hawking emission from sonic horizon formed in transonic bac k- ground field. This cr eative work was afterwards developed by Unruh, Visser, Jacobson, V olovik, Garay, Schützhold , Fischer , Barceío , Balbinot et al [8 -16]. Some of their works lead a more avai l- able way to experimental test on Hawking radiation by setting up a physical analogue gravity sy s- tem, give probable models for gravity anal ogue in other objects, and open up many opportunities to study the gravity indirectly . In this paper, we investigate the phonon radiation from a sonic black hole. The self -gravitation of the radiated phonons is tak en into account. In the calculation, the method of Wentzel -Kramers - *E-mail address: qdfang @163.com E-mail address: kaihu55889 [email protected] Brillouin ( WKB ) approximation is used. The feature of the potential barrier near the horizon and the process of the phonon tunnelling are described in detail. After calculat ing, w e obtai n the pho- non emission rate and the radiation temperature which is in agreement with that deriv ed by Unruh and Visser [7,10] by ca lculating the surface gravity . II . THE PHONON TUNNELLING FROM SONIC BLACK HOLE Sound waves propagating in m oving fluid will be dragged by the fluid flow, and if the speed of fluid e xceeds the local sound speed, the sound waves can ’t go back upstream. This implies the sound waves are trapped in the supersonic flow r egion and a sonic black hole exists. Similar to the light waves in curved spacetime, the propagation of sound waves in barotropic, inviscid, and irr o- ta tional fluid flow can be dete rmined by an effective metric (acoustic metric). For a spheri cally symmetric, stationary, convergent flow, the line eleme nt relating to sound propagating can be e x- pressed as [1 0] 2 ρ 2 c dsc2=−−0 υτ 22 ds d rr 22222 −+ dsind θθϕ , (1) ()s 2 2 () cs c s −υ where υ is the velocity o f the background fluid flow only with radial component, and ρ0 de- notes the density of the bac kground flow, and cs is the local speed of sound. The boundary of the supersonic flow region, which locates at r= R where υ is equal to cs , is the h orizon of the sonic black hole. We can find that the line element in Eq. (1) is singular at r= R , and t he existence of singula r- ity at hori zon brings troubles in trans-horizon physics. We can elimi nate it by a coordinate tran s- 2 2 formation. Let t=τ + ∫ υ( cs − υ )d r , we have ρ ds2=0 c 222 −υ d2ddd t + υ rtrr −− 222 d θθϕ + sind 22 . (2) ()s ( ) cs We can find that in the new coordinate system (,,t r θ , ϕ ) , the metric is regular at the horizon. And the metric keeps st ationary , i.e. time -translation invariant, which manifests that the generator of t is a killing vector. This coordinate covers the inside and outside of the sonic black hole. A. Radial phonon movement near the horizon Expanding υ in the form of Taylor se ries with respect to r at r= R and notice that the flow moves inward, we get 2 c rR OrR , (3) υ=−+s α ()() − +( − ) dυ where α = . Neglecting the second and higher order terms in Eq. (3), the line ele ment in d r r= R Eq. (2) can be rewritten as ρ ds2=0 2α crRt − d2 2 +−+− c α rRrtrr ddd −− 22222 dsind θθϕ + .( 4 ) s()() s () () cs Iow we study the speed of the outgoing phonon. The trajectory of a free phonon tha t moves along the radial direction is a sound-like geodesic line. So we have ρ ds2=0 2α crRt − d2 2 +−+− c α rRrtr ddd0 −= 2 . (5) s()() s () cs Solving the quadratic equation, we obtain the phonon speed d r 2 r = =−+ cα() rRc −±2 + α 2 () rR − . (6) d t s s The sign ± indicates the outgoing and ingoing phonon, respectively. B. WKB approximation Similar to the case near the gravity horizon, because of the infinite blueshift near the sonic h o- ri zon observed from the infinit y, we can get some reliable results in geometrical acoustics a p- proxima tion ( WKB approximation ) without involving the second-quantized Bogoliubov method or the others. For a steady flow, the background keeps all the time, and the lo cal fluid velocity υ , the pressure and the density are co nstant at arbitrary space point. A radiating phonon in this spheri - cally symmetric flow can be regarded as a s wave. When a phonon is emit ted from the sonic black hole, because of the total en ergy conservation, the energy inside the horizon will decrease . Then t he position of the sonic horizon should shift from the initial place ri to the final place rf . The initial positio n of the phonon corresponding to phonon pair creation is very close to the inner side of horizon ri , and the final position of the phonon is nearly outside the final hori zon rf . Note how self -gravitation of the phonons is essential to the tunneling picture. Without self - gravitation, phonons created just inside the horizon would only have to tunnel just across an infin i- tesimal separation, so there would be no any barrier. But backreaction results in a shift of the hori- zon, i t is the gap between ri and rf that forms the classical forbidden region, the pote ntial bar - rier. According to the WKB approximation, the s wave of the radiated phonon can be expressed as Ψ(r ) = e iS( r ) , where S is the classical action. For an outgoing phonon across the horizon, the imag inary part of the ac tion rf r f p r ImS= Im pr d = Im d pr′ d , (7) ∫rr ∫ r ∫ 0 r i i where pr is radial momentum of the outgoing phonons. Eliminating the momentum in favor of energy by using Hamilton’s equation dH∂ H = = r , (8) dpr ∂ r where the Hamiltonian, H , is the generate of time t . When the horizon emits phonons with energy ω , t he remaining energy inside the sonic black hole becomes H′ = H − ω because of the energy conserva tion. So, rep lac ing pr with and note that ranges from 0 to in the tunneling process, we have rfdH r f d ImS= Im d r = − Im d r . (9) ∫∫ri0r ∫∫ r i 0 r Substitute Eq. (6) to Eq. (9) with signature + , we have rf ω dω′ ImS= − Im d r , (10) ∫ri ∫ 0 2 2 2 −+csα() rR −+ c s + α () rR − then exchange the integral order and rearrange the expression as 2 2 2 ω rf cs+ c s +α () rR − 1 ImS= − Im dω′ − d r . (11 ) ∫0 ∫ r i 2αcrR()− 2 c s s We can find that the integrand has a singularity at r= R in the integral interval from ri to rf .