Hawking Radiation. Alex Fontana

Transcription

1 Hawking Radiation Alex Fontana In this epot the aim is to undestand the behaviou of a Schwazshild Black Hole in a context whee quantum field fluctuations ae taken into account. We ae going to see how to intoduce biefly the idea of Black Holes, how extaction of enegy fom them is possible and the Schwazschild solution fo the metic seen by a distant obseve. Then we ae going to change coodinates to a nea hoizon obseve to demonstate that that point is not a singulaity, and daw the Kuscal Diagam. In the second pat we conside scala field fluctuations aound the hoizon as seen by the two obseves, define two diffeent vacuums that lead a non-zeo numbe paticle density. This means that the Black Hole emits adiation, and in the empty space ultimately evapoates. In the last pat we give some numbes of lifetime of Black Holes and intoduce the Black Hole Themodynamics. 1

2 Chapte 1 Black Holes The fist intuitive idea of a Black Hole is a highly massive object with a stong gavitational field that even light light cannot escape fom. Thee ae seveal solutions fo Black Holes, depending on the chaacteistics they have momentum, chage. The easiest one is the non-chaged, non-otating solution, epesented by the Schwatschild metic: 1 ds 2 = 1 2GM/c2 dt 2 1 2GM/c2 d 2 2 dω This is the metic measued by a fa distant obseve in pola coodinates in espect to the Black Hole cente = 0, with M its mass and dω the angula coodinates. Hee we see that g 2GM/c 2 is a paticula adius called event hoizon, whee the metic diveges and it epesents a no-tun-aound point. Once an object cosses this distance, it s doomed not to come out again, and fall inside the cental point at = 0, the othe singulaity. Accoding to Geneal Relativity a Black Hole can only absob matte o equivalently adiation, thus its mass can only incease. This solution is called etenal because the object will neve disappea. Let us now conside Quantum Mechanics. It is known that vacuum isn t exactly empty, each instant vitual couples of paticle-antipaticle ae ceated by vacuum fluctuations and annichiliate between themselves. In 1974 Stephen Hawking linked these concepts. He consideed quantum fields in a Black Hole backgound, and he discoveed that a Black Hole emits themal paticles and in this way it can evapoate. Why is this concept new and supising? 1.1 Rotating Black Holes In pinciple the idea of a Black Hole emitting enegy is not stange. If we conside a Rotating Black Hole, descibed by the Ke metic, we can see that thee 2

3 ae negative enegy egions outside the event hoizon, so that the gavitational field can convet a vitual couple paticle-antipaticle into a eal pai. In this way, because of the total enegy of the paticle is to be zeo, one paticle with negative enegy falls inside the black hole while the othe with positive enegy can escape to infinity. Let us explain futhe about how it is possible fo the pai to gain enegy fom Ke s Black Hole. In Classical Mechanics a otating body with a medium fo example ai exitates the paticles aound, tansfeing to them enegy thus losing it. In this way the medium acquies enegy and the body ultimately stops its otation. If we ae in vacuum, no enegy loss is possible. In Quantum Mechanics a otating body in vacuum can amplify the fluctuations and they gaining enegy fom the body become a eal adiation that an obseve aound the body can see. In this way the gavitational tidal foces aound the Black Hole pull apat the vitual pais that would quickly annichiliate if not fo these foces and ceate a population of eal paticles. 1.2 Non-Rotating Black Holes We have seen that a adiation coming fom a otating Black Hole is easonable. Poblems aise when we conside no angula momentum, because no negativeenegy egions exist outside the event hoizon. Nevetheless we ae going to see why a poduction of paticles is still possible, and a fist pictoial explanation can be given immediately. We can see that negative enegy egions ae inside the hoizon, so a couple ceated in the poximity of it has this situation: Negative enegy paticle is ceated inside the Black Hole, it falls inside Positive enegy paticle is ceated outside the Black Hole, it escapes to infinity In this way the Black Hole can emit adiation, and its mass evapoates. We can give a ough estimate of the outgoing adiation: its enegy fom uncetainity pinciple E c/ g lead us to see that the De Boglie wavelengh is bigge than the event hoizon, so the pobability to escape is bigge than zeo. The tempeatue is T H c/ g k B = c 3 /GMk B. We point out that this is a vey naive guess of T H! We need to go moe in detail though the Schwazshild solution and the quantum fields to have a bette view of the Hawking adiation. 3

4 Chapte 2 Schwazschild solution In this chapte we ae going to study the metic of a Schwazschild Black Hole. We intoduce natual units: G = = c = 1 and we estict the 1.1 to 2-dimensional case, consideing just the time and adius-coodinates. This is because the 2 addictional angula dimensions don t add physical infomation fo the pocess we ae inteested. So the 2 d Schwazshild metic: ds 2 = 1 g dt g d with g = 2M. As we have seen befoe, this metic is singula in: = g, event hoizon. This is not a physical singulaity, since one can compute the scala cuvatue in that point and see that it is finite. The metic is well defined just fo > g. = 0, tue singulaity. We want to show that = g is just a coodinate singulaity, so we have just to change them. 2.1 Totoise coodinates We would like to have d = 1 1 g d 2.2 so we define the Totoise coodinates: = g + g ln / g These ae still defined fo > g, but now < < + with g 4

5 + + The metic now is: ds 2 = 1 g dt 2 d whee comes fom 2.3. Intoducing now the lightcone coodinates: ũ = t ṽ = t + the metic becomes: ds 2 = 1 g dũdṽ 2.5 ũ, ṽ At this point we still have a singulaity at = g, and we can cove just the exteio of the Black Hole. We need one moe change of coodinates to get id of it. 2.2 Kuscal-Szekees coodinates Using 2.3 and the new lightcone coodinates we find an useful elation: 1 g = g We then descibe the metic as: ds 2 = g e 1 g e e 1 g ṽ 2g e e ṽ ũ 2g ũ 2g 2.6 dũdṽ 2.7 We define then the Kuscal-Szekees lightcone coodinates, of a fee falling object nea the hoizon: u = 2 g e v = 2 g e and the metic: ũ 2g ṽ 2g ds 2 = g u, v e 1 u,v g dudv 2.8 We can see now that it is egula in = g, so the event hoizon was just a coodinate poblem, and a fee falling obseve doesn t feel anything paticula cossing it. We can see that these coodinates ae still defined fo > g but now can be natually extended eveywhee: 5

6 < v < 0 < v < + 0 < u < + < u < + Now using 2.3 and 2.6 we find the elation between the old fa away obseve coodinates t, and the fee falling ones u, v: uv = 4ge 2 g = 4 2 g 1 e g g v u 2 = e 2t g 2.10 Hee the 2.9 tells us that at the hoizon = g u = 0 o v = 0, so we have two hoizons. The 3.10 tells us that: v = 0 t = past hoizon u = 0 t = + futue hoizon We can intoduce timelike and spacelike coodinates: u = T R v = T + R Now it can be useful to see a gafical epesentation of the spacetime we ae descibing, a Schwazschild Black Hole can be potaited by the Kuscal Diagam. We can see that the two hoizons divide the spacetime in 4 diffeent egions, in egion I the tajectoies with constant distance fom the cental singulaity ae timelike, while in egion II they ae spacelike, and we can see the = 0 singulaities in the past and in the futue. A quick emak: the Black Holes we find in natue ae ceated duing the collapse of a sta, so no past singulaity should exist. Theefoe just the I-II egions should be physically elevant, and the III-IV can be ignoed in this case. 6

7 Figue 2.1: Shaded egions IIV ae the diffeent asymptotic domains of the spacetime. Dashed lines epesent the hoizon u = 0 and v = 0. Dotted lines ae sufaces of constant. Thick dotted lines epesent the singulaity = 0. 7

8 Chapte 3 Field Quantization and Hawking adiation We have seen how the Schwazshild Black Hole can be descibed by diffeent obseves in diffeent coodinates, and how the event hoizon is completely diffeent fo the two. Now we can stat to add to this backgound the quantum fields fluctuation, to see what these two obseve nea the Black Hole hoizon. We conside fo simplicity a fee scala field, but this desciption can be epeated fo othe paticles, like femions. The action: S[φ] = 1 g αβ α φ β φ g d 2 x This is confomally invaiant, which means that it doesn t change if we change the coodinates in which we expess the field. Solving the equation of motion we find the solution fo the field with the two sets of coodinates ũ, ṽ and u, v: φ = Ãũ + Bṽ φ = Au + Bv with Ã, B, A, B abitay smooth functions. We can expess the field to espect to the modes Fouie tansfomation as: φ e iωũ = e iωt φ e iωu iωt R = e It s now the point we should ask ouselves what a paticle is: positive fequency modes with espect to the pope time of the obseve. So: A distant obseve measues a t he sees a vitual adiation 8

9 A nea-hoizon obseve measues a T he sees a eal adiation! We ae going to see that this last obseves a themal bath with finite tempeatue that pop out of the local acceleation hoizon, tun aound and fee-fall back in. Let s distinguish now. 3.1 Nea-hoizon obseve Fo g a nea-hoizon obseve measues a metic: ds 2 dudv = dt 2 dr so as said he measues a time T. The field opeato expansion: ˆφ = 0 dω 1 [ e iωuâ ω + e +iωu â + ] ω + [v modes] 3.3 2π 2ω whee we don t wite down the incoming v modes fo simplicity. We then define a vacuum 0 K > that is annichilated by: â ω 0 K >= called Kuscal vacuum. This state contains paticles fo the distant obseve we ae going to see the numbe density. The vacuum is well-defined, because thee is no singulaity at the hoizon = g, so we can think this as a physical vacuum. 3.2 Distant obseve Fo a distant obseve measues a metic: ds 2 dũdṽ = dt 2 d so as said he measues a time t. The field opeato expansion: ˆφ = 0 dω 1 ] [e iωũˆb Ω + 2π e+iωũˆb+ Ω + [v modes] 3.6 2Ω whee again we don t wite down the incoming v modes fo simplicity. We then define a vacuum 0 B > that is annichilated by: ˆb Ω 0 B >= called Boulwae vacuum. This state contains no paticles fo this obseve the numbe density is 0. The poblem is that this vacuum is not physical. In ode to undestand why, we should emembe that this set of coodinates 9

10 is singula in the hoizon, so at this point the expectation value of the enegy density diveges. Fom: < 0 K u ˆφ 2 0 K >=< 0 B ũ ˆφ 2 0 B > 3.8 because of the confomal invaiance of the action, then: < 0 B u ˆφ 2 0 B >= ũ u 2 < 0 B ũ ˆφ 2 0 B > 3.9 We have calculated the expectation value of the enegy density fo a neahoizon obseve in this vacuum, but the Jacobian ũ u 2 diveges at u = 0, so the enegy density. 3.3 Hawking adiation We have seen the diffeence between the vacuums of the two obseves. Now that we have seen that ou tue physical vacuum is Kuscal s, we can ask ouselves the numbe of paticles in this vacuum fo a distant obseve: < ˆN Ω >=< 0 K ˆb + Ωˆb Ω 0 B >= [ e 2πΩ k 1] 1 δ with k = 1/2 g called suface gavity and the δ0 simply epesenting the volume of the space, so the numbe density is: < ˆn Ω >= [ e 2πΩ k 1] which is the Planckian distibution, so we can see that the adiation is themal with tempeatue that depends just on the Black Hole mass: T H = k 2π = πM We have seen that 0 K > contains the ight-moving, so outgoing paticles fom the Black Hole, but also left-moving, so ingoing ones, with the same themal spectum. The fact that Black Holes has a population of eal paticles popping out and going back inside the hoizon means that quantum fields in a Black Hole backgound ae consistent only if the body is in a themal esevoi with a tempeatue T H. Then since a Black Hole absobs paticles has to emit to mantain equilibium so one in empty space must evapoate emitting themal adiation. 3.4 Non-etenal Black Holes A slightly diffeent appoach is to be made if we conside Black Holes as final esult of a stella collapse. In fact, in this case the v = 0 hoizon doesn t exist, 10

11 so thee is no physical need of choosing fo the v-modes the Kuscal vacuum. To undestand why, in the past befoe the collapse the spacetime was almost flat, so the Boulwae vacuum wasn t singula because no hoizon existed. Then a consistent way of choosing the vacuums should be: ight-moving modes: â ω that annichilates 0 K > left-moving modes: ˆb Ω that annichilates 0 B > dimensions We can take a bief look on what happens if we conside again 3+1-dimensions. The desciption of the poblem is identical at the one just seen, we can wite down the action and decompose the scala field in spheical hamonics φt,, θ, ϕ = lm φ lmt, Y lm θ, ϕ and substitute in the equation of motion φ = 0. We find: [ g ] g ll φ lm θ, ϕ = whee we ecognize the contibute of the 1+1-dimension case plus a coection tem V l that is not 0 even fo l = 0. This acts as a potential that the outgoing wave has to supass in ode to aive to infinity. In geneal this has the consequence of loweing the numbe density of paticles seen by an obseve: < ˆn Ω >= Γ l Ω [e Ω 1 T H 1] 3.14 whee Γ l Ω is a geybody facto due to V l, always < 1, that depends on the field consideed. The physics of the Hawking adiation can then be seen simply by the 1+1- dimension case, because V l acts just outside the Black Hole so it s no diect consequence of the pocess. 11

12 Chapte 4 Themodynamics of Black Holes In light of the new undestanding of the themal inteactions between a Black Hole and its suoundings, we can biefly ty to undestand how this adiation excange woks. 4.1 Mass loss When we conside a Black Hole in empty space it emits adiation and loses mass. But how? The flux of adiated enegy fom a spheical Black Hole with suface aea Σ = 4π 2 g = 16πM and tempeatue T H = 1/8πM is given by the Stefan-Boltzmann law: L = ΓγσTHΣ 4 Γγ = 15360πM with Γ coects fo the geybody factos, γ is the numbe of massless degees of feedom and σ is the Stefan-Boltzmann constant in Planck units. Then the mass deceases as: dm dt Γγ = L = 15360πM 2 Mt = M 0 1 t/tl 4.2 with t L = 5120πM0 3 /Γγ. We see then that the lifetime of a Black Hole is t L M0 3 so the mass to the thid. So since its mass it s enoumous, it takes a long time to evapoate, and while this happens the tempeatue inceases. To give some numbes, let s imagine a M 0 = M, then t L yeas which is fa bigge than the age of the univese yeas. This means that at ou cuent age it s not possible to obseve the evapoation of such massive objects, then we should look fo M g Black Holes to see something 12

13 today. The poblem is that these objects ae not poduced by cuently known stella collapse. It is impotant to point out that the evapoation pocess can change dastically because when we each M 10 5 the scale of the Black Hole is the Planck Lenght, and a Quantum Gavity Theoy is necessay to descibe what happens. It may be possible an analogy with the Hydogen atom, whee at the lowest obit the electon doesn t adiate anymoe, so the Black Hole stop adiating and doesn t disappea. 4.2 Entopy Let s now intoduce biefly the concept of Black Hole intinsic entopy S BH. Let s say the Black Hole has S BH = 0, if it assobs matte with S 0, then the total entopy of the univese is diminished, but this is not possible. The solution is S BH 0, linked to the suface aea Σ = 16πM 2. Diffeentiating: dm = 1 8πM dσ which has the same fom of the I law of themodynamics: de = T ds 4.4 with dm = de, T H = 1/8πM so we find S BH = 1/4Σ = 4πM 2. We can set now the I law of Black Holes themodynamics: dm = T H ds BH 4.5 With a M = M we have a S so a huge numbe of micostates hidden behind the hoizon. We can see a deep contast between a classic and semiclassic theoy hee, since Geneal Relativity shows how a Schwazschild Black Hole is completely chaacteized by its mass, so should have no entopy, while hee it s enomous. This tells us that moe has to be undestood aound Black Hole s micostates. Taking into account the intinsic entopy we can expess the II law of Black Holes themodynamics: δs = δs matte + δs BH so that the total entopy of the black hole and the matte neve deceases. 13