Hawking Radiation Seminar Talk

Transcription

1 Hawking Radiation Semina Talk Julius Eckhad, Max Lautsch June 9, 205 In this talk on Hawking Radiation we will fist motivate why we have to intoduce the counteintuitive concept of a black hole tempeatue leading to adiation. We will then sketch the oiginal calculations done by Stephen Hawking. Ultimately, seveal modifications to this simplified esult will be consideed and physical consequences will be discussed. Intoduction and motivation. Schwazschild Black Hole The unchaged non-otating black hole of mass M is descibed by the Schwazschild metic given by (in natual units): ds 2 = dt 2 + d d () 2M 2M One immediately sees the singulaity at the Schwazschild adius h =2M. It tuns out to be only a coodinate singulaity imposing no physical poblems. One can see this in Kuskal coodinates, in the lightcone vaiant, which tun out to be impotant late: u = t 2M ln 2M = t (2) v = t + +2M ln 2M = t + (3) in which the metic looks like ds 2 2M = dudv + 2 d (4) Although we got id of the singulaity, the Schwazschild adius is still a special value, so we define it as the adius of the black hole. This means, we can now define the aea A and the suface gavity apple as: A =4 h 2 = 6 M 2 (5) apple = M h 2 = 4M (6)

2 whee apple can also be defined igoously using Killing vectos..2 Black Hole Themodynamics We will now pesent a quick explanation why we need black hole themodynamics fo consistency. Imagine a mass distibution collapsing into a black hole, i.e. a stable state. Fom the 2nd law of TD we know that the entopy of the whole system can not decease. This means that the esulting black hole must cay an entopy S BH S. The change of entopy is given by the st law of themodynamics: du = T ds (7) so we have to find a compaable equation fo the black hole. Consequently, thee has to be some coesponding tempeatue. The enegy change of the black hole can be easily computed as dm = apple da (8) 8 This means, T BH apple and S BH A whee the pefactos ae yet to be detemined. But now we can use the Unuh e ect, stating that an acceleated obseve feels a tempeatue of T U = a (9) 2 Heuistically, the acceleation at the hoizon should be just the suface gavity, i.e. T BH = apple 2 = (0) 8 M S BH = 4 A =4 M2 () which actually tuns out to be the ight esult. To justify this, we will now sketch the computations done by Hawking in the next chapte. 2 Paticle ceation by a collapsing spheical body 2. Wave packages The setup fo this calculation is a finitely sized matte distibution, which collapses to a black hole ove time. In the initial state, spacetime is appoximately flat, and theefoe the metic is Minkowskian. We theefoe have a unique vacuum at time minus (past) infinity (J ) fo the eal massless scala field which is consideed hee fo simplicity. The basic appoach of this calculation is to follow the mode functions of the Minkowskian vacuum though the foming black hole up to futue infinity (J + ) in ode to find a Bogoliubov tansfomation between in and out vacuum. This bings us to an expession fo the spectum of adiated paticles. 2

3 Figue : Penose Diagam of collapsing matte distibution to a black hole. In the past infinity the metic is Minkowskian and we can thus define the scala field in the usual mode epesentation known fom standad QFT. Those mode functions f ae solutions to the classical wave equation 2 = 0: in = X l,m d! a!lm f!lm + a!lm f!lm? (2) f!lm = Y lm (, ')e +i!(t ) (3) whee Y lm ae the spheical hamonics and we chose the positive sign, because the solutions ae estained to incoming fequency modes. We see that all mode functions tavel though the cente of the black hole at = 0. Theefoe it is possible to conside a + dimensional toy-model to simplify the calculations. This mainly educes to dopping the angula dependence, which may be einseted in the end (which means, we only conside f! f!00 ). Eventually we ae inteested in the fom of the waves afte they have cossed the black hole and escaped into futue infinity (whee they can be measued), i.e. we want to wite out = d! b! p! + b!p?! (4) p! = d! 0 (!! 0f! 0 +!! 0f!? 0) (5) whee (5) defines a Bogoliubov tansfomation between the in- and out-vacuum 3

4 with popeties = N(!) = d! 0!! 0 2!! 0 2 (6) d! 0!! 0 2 (7) The function N(!) is pecisely the outgoing paticle spectum we want to calculate. 2.2 Redshifts and Blueshifts The easiest way to detemine the e ect of the black hole on the waves is to expess them in Kuskal coodinates. (3) then becomes f! 0 = F! 0() p 8 2! 0 ei!0u / e i!0 (t ) (8) whee F! 0() = fo an othewise empty univese. Now, imagine a wave which appoaches the black hole. It will be blue-shifted until it eaches the oigin. Aftewads, it will be ed-shifted again. If the system was static, then these two e ects would pecisely cancel in the futue infinity and one could expess the outgoing waves as p! = p 8 2! ei!v / e i!(t+ ) (9) which means that in-and outgoing waves only di e in the fact that they move in opposite diections measued fom the oigin. This means!! 0 (!! 0 )!! 0 = 0 (20) and thee is no paticle emission. Now we come back to the oiginal assumption that the matte distibution is actually collapsing. Thus, in the time the wave package needs to coss the black hole, the setup has slightly changed. This esults in the e ect that blue-shift and ed-shift do not cancel pecisely and the outgoing wave has a di eent stuctue. The undelying calculations ae quite ticky so we just state the esult: ( p p! = 8 2! exp ( 4Mi!ln[(v0 v)/c]) v<v 0 (2) 0 v>v 0 in the vicinity of v 0. To undestand this qualitatively, we again consult a Penose diagam. The value v 0 is defined by the ay. Thus, fo evey v>v 0 the wave is absobed by the black hole and thee is no outgoing p!. The logaithmic dependence can be undestood by the pile-up of ays nea o v 0. A elatively lage aea in J + 4

5 Figue 2: +D Plot of Setup of collapsing matte distibution. is coveed by a naow stip in J. This means that those light ays dominate the esult. The pecise fom of (2) can be found by intoducing a metic inside the black hole and equiing that the hoizon is not actually physical (i.e. thee is not discontinuity). We can now invet (5) and ead o the Bogoliubov coe cients as:!! 0 = v0! 0 dv 0v!! 0 2! e±i! exp ( 4Mi!ln[(v 0 v)/c]) (22) This can be evaulated in analogy to the Unuh e ect by the usage of and esults in: -functions 2.3 Spectum!! 0 = e 4 M!!! 0 (23) It is time to put eveything togethe. We agued that the Bogoliubov coe cients obey 23. Inseting this into the nomalization (6) this yields = d! 0!! 0 2 e 8 M! (24) {z } N(!) 5

6 ) N(!) = e 8 M! (25) which is just the Planck spectum fo tempeatue T = 8 M = apple 2. Fom the calculations leading to (23) one can eally elate this to an e ect caused by the pope acceleation apple on the hoizon. Unsupisingly, the tempeatue does not depend on the exact collapse of the matte distibution but solely on the mass appeaing in the Schwazschild metic, which is the only paamete chaacteizing the black hole. 3 Modifications 3. Gey Body factos An impotant e ect has been neglected in ou calculations. It is possible that a faction is scatteed by the collapsing body and the pobability fo a wave to each futue infinity is thus <. To include this e ect, the nomalization 6 of the Bogoliubov coe cients can be modified: = d! 0!! 0 2!! 0 2 (26) This facto natually depends on the collapse and can thus have an angula dependence. Theefoe the solution of the ceated spectum is modified to: dn d! = lm e! T BH Which is just the analogue to a non-pefect black body. 3.2 Femi statistics (27) Appaently, the statistics (25) only woks fo massless bosons. The coesponding femionic fomula is intuitive: dn d! = e!0 T BH + (28) whee the mass of the femions appeas in! 0 = p! 2 + m 2. The eason fo this change is the di eent nomalization of the Bogoliubov tansfomation. Since the ceatos and annihilatos now obey anti-commutation elations, the nomalization becomes d! 0!! 0 2 +!! 0 2 = (29) The est of the computation fo the mode functions goes though just the same way, as spino components of the mode functions ae ielevant, just like the spheical hamonics mentioned above. 6

7 3.3 Chaged and otating Black Holes To modify the spectum fo chaged and otating black holes it is easiest to look at the st law of black hole themodynamics. The Ke-Newman metic esults in a dependence of the hoizon (and thus the aea) on chage Q and angula momentum J. Thus: dm = apple da + dj + dq (30) 8 whee the angula velocity and the electic potential ae defined by this equation. One can ewite this as: dm = apple dj dq da (3) 2 dm dm 4 The atio of mass (enegy), anugla momentum and chage loss can be descibed by: dj dm = m! dq dm = e! fo a paticle with angula quantum numbe m and chage e. This means that the spectum is chaacteized by: which is no longe a black body spectum. 3.4 Etenal Black Hole (32)! T = 2! m e (33) apple One might wonde whethe this calculation holds if we conside etenal black holes without appaent dynamic popeties. In ou calculation on the collapsing matte distibution we found that the esult does not depend on the actual pocess of the collapse. The easoning done in ou calculation is not tansfeable to etenal black holes, but etenal and collapsed black holes being di eent seems unphysical. In the pocess of calculating the paticle spectum of an etenal black hole, the in-vacuum is the Minkowski-one, as incoming waves have not seen any e ect of the cuved space yet, and the outgoing vacuum choice emains fee. The choice of vacuum has to be made in consideation of physical easoning. Unde the assumption that etenal and collapsed black holes ae no di eent, we get exactly the same esult. But this choice can also be motivated di eently, as the calculated vacuum in ou solution is the lowest-enegy state fo feely falling obseves. Obseves at a fixed adius fom the black hole would choose a di eent vacuum state, but this state has infinite vacuum enegy density and is theefoe uled out. Those acceleated obseves, fo example sitting at the event hoizon of the black hole, measue the Hawking tempeatue as the tempeatue of the themal adiation of the black hole, as defined in last sessions talk on the Unuh e ect. 7

8 4 Physical Consequences 4. Evapoation As fo any black body, the total adiation powe of the black hole is dm dt = T 4 A = M 2 (34) and the evapoation time (in an empty univese) is given by t ev = M M0 074 s (35) M This explains some expeimental facts: We have not yet obseved any Hawking adiation since the expected heavy black hole (fo example in the cente of the Milky Way) ae adiating vey weakly. Thee ae no small black holes, because they have aleady dissipated. The uppe bound is given by the CMB tempeatue coesponding to the mass M CMB M moon. This togethe means that we can not expect to find Hawking adiation with T>T CMB which makes detection vey di cult. 4.2 Infomation Paadox When consideing paticles enteing the black hole, inteesting questions aise, fo example concening the infomation on the type of paticle. As we have shown in this semina talk, evey black hole adiates. Is the infomation on the fomation and content of the black hole peseved, as afte some time it would vanish (in an empty univese). Questions like these ae the topic of the next talk in the semina on QFT in cuved spacetime. Souces N. D. Biell P. C. W. Davies, Quantum Fields in Cuved Space, Cambidge Univesity Pess 982 S. W. Hawking, Paticle Ceation by Black Holes, Commun. Phys. 43, (975) math. Segei Winitzki, Lectue Notes on: Elementay Intoduction to Quantum Fields in Cuved Spacetime, held in Heidelbeg, Apil 8-2,