tunnels from a sonic

 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 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 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 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 .

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 (). 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 Now 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 ω and note that ω ranges from 0 to ω in the tunneling process, we have

rfωdH′ r f ω d ω ′ 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 . Utilizing the complex singular integral formula, we have ω π πω ImS = d ω′ = . (12) ∫0 α α Thus we get the phonon tunnel ling probability , i.e. , the rate of emission Γ∼exp( − 2Im S) = e −2πω α . (13)

The equation above shows that the radiation spectrum is of black body spectrum with temperature  TH = α . (14) 2π kB

CONCLUSIO N

For a sonic black hole, Hawking’s formula was given by [ 7,10]

  d( v− c ) TH =   , 2π KB  dx

d( v− c ) Where the surface gravity κ in Hawking temperature κ is replaced by . 2π KB dx

d( v− c ) From Eq. ( 3), and notice the inverse of the sign between c and c , we can get that s dx just equals the constant α . So we confirm the result we obtain in Eq. (1 4) agrees with that o b- tained by calculating the su rface gravity [7, 1 0], the method we take is correct . More, the black body spectrum in Eq. (1 3) means the acoustical information lost completely. But if we allow for the higher orders in Eq. (3), the emission spectrum will deviate from exact black body spectrum. The correction of thermal spectrum implies the information at least partly conserv ed during the radiation of acoustic black hole. ACKNOWLEDGEMENT

This research is supported by the Natural Science Foundati on of China (10847166).

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