Hawking radiation in NS5 and little string theory

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In the using or citation of parts of the thesis it’s obliged to indicate the name of the author UNIVERSITAT POLIT` ECNICA DE CATALUNYA Departament de F ´ ısica i Enginyeria Nuclear Hawking radiation in NS5 and Little String Theory Mem`oria presentada per Oscar Lorente Esp´ın per optar al grau de Doctor en Ci`encies Barcelona, Juny de 2012 Programa de F ´ ısica Computacional i Aplicada Mem`oria presentada per Oscar Lorente Esp´ın per optar al grau de Doctor en Ci`encies Director de la tesi Oscar Lorente Esp´ın Dr. Pere Talavera S´anchez Membres del tribunal de tesi President: Dr. Francisco Fayos Valles. Secretari: Dr. Jaume Garriga Torres. Vocal: Dr. Josep Maria Pons R`afols. Vocals suplents: Dr. Jaume Haro Cases i Dr. Bartomeu Fiol N´u˜nez. Experts Externs: Dr. Xavier Calmet i Dr. Elias C. Vagenas. A mis padres, Jos´e y Anselma Contents 1 Introduction 1 2 Semi-classical emission of Black Holes 7 2.1 Hawkingradiation ............................ 12 2.2 Euclidean path integral and Hawking temperature . . . . . . . . . . . 17 2.3 Hawking radiation as tunneling . . . . . . . . . . . . . . . . . . . . . 19 2.4 The complex path method . . . . . . . . . . . . . . . . . . . . . . . . 24 3 Hawking radiation in Little String Theory 29 3.1 LST, thermodynamics overview . . . . . . . . . . . . . . . . . . . . . 29 3.2 Semi-classical emission in NS5 . . . . . . . . . . . . . . . . . . . . . . 36 3.3 Hawking radiation via tunneling . . . . . . . . . . . . . . . . . . . . . 42 3.3.1 Tunneling approach in LST . . . . . . . . . . . . . . . . . . . 44 3.3.2 Locking information at the Hagedorn temperature . . . . . . . 47 3.3.3 Hawking emission via tunneling: Wrapped fivebranes . . . . . 50 3.4 Complex path and anomalies in LST . . . . . . . . . . . . . . . . . . 51 3.4.1 Complex path method . . . . . . . . . . . . . . . . . . . . . . 52 3.4.2 Anomalies............................. 57 3.5 Validity of the Semi-classical approaches . . . . . . . . . . . . . . . . 60 3.6 Further thermodynamic relations . . . . . . . . . . . . . . . . . . . . 61 3.7 Discussion and remarks . . . . . . . . . . . . . . . . . . . . . . . . . . 63 3.8 Spectrum ................................. 66 3.8.1 Blackbody spectrum . . . . . . . . . . . . . . . . . . . . . . . 67 3.8.2 Hawking radiation flux . . . . . . . . . . . . . . . . . . . . . . 68 3.8.3 Back-reaction spectrum . . . . . . . . . . . . . . . . . . . . . 69 3.9 Greybodyfactor.............................. 70 v vi CONTENTS 3.10Quasinormalmodes............................ 73 4 Emission of fermions in LST 75 4.1 Emissionprobability ........................... 76 4.2 Fermion modes and greybody factor . . . . . . . . . . . . . . . . . . . 78 5 Back-reaction and quantum corrections 83 5.1 Quantum correction on the metric . . . . . . . . . . . . . . . . . . . . 84 5.2 Back-reaction viewed as a quantum correction . . . . . . . . . . . . . 86 5.3 An example in string theory . . . . . . . . . . . . . . . . . . . . . . . 89 5.3.1 Quantum corrections at action level . . . . . . . . . . . . . . . 89 5.3.2 Quantum corrections on the metric . . . . . . . . . . . . . . . 92 5.3.3 Discussion............................. 93 6 Einstein and conformal frame 95 7 Summary, conclusions and outlook 99 A Calculus tools and notation conventions 103 B N-sphere area 105 C Komar integral and ADM energy 107 D Gamma matrices 109 E Average number of emitted bosons 111 Bibliography 113 Chapter 1 Introduction Since ancient times some people have been interested in the world where they live and its environment. However, the unlimited human curiosity does not stop here, and goes beyond to the asymptotic limits of the universe. Questions about kinematics and dynamics of bodies, i.e. questions about motion, lead us to crucial responses dressed in consistent scientific theories. In this sense, gravitation has always been an special topic of study: from the physical philosophy of Aristotle to the free falling experiments of Galileo; from the Newton’s law of universal gravitation to the Einstein’s theory of general relativity. At the end of XVIII century, the mathematician physicist Pierre Simon Laplace and the cleric John Michell were influenced by the scientific ideas of Newton concerning the gravitation and light built by corpuscles. They were considering how gravitation would affect light, and if it would be possible that existed a star so massive and dense that light could not escape from its surface. Effectively, for a spherical star of fixed mass exists a minimum radius that acts as a frontier. For a values of radius lower than the minimum radius nothing can escape from the gravitational force at the star surface even the light. This star is named dark star. One century later, Einstein announced his theory of relativity changing our perception of the nature of space and time. A few years later, Schwarzschild found an intriguing solution to the Einstein’s equations of general relativity. For a spherically symmetric body of fixed mass, neither with angular momentum nor electric charge, in vacuum, there exists a minimum radius known since then as Schwarzschild radius under which the body would collapse gravitationally to a space-time singularity. This object was called by John Wheeler, somewhat joking, black hole, nevertheless the astronomers have shown that such objects could exist in 1 2 Chapter 1. Introduction our universe. When an extremely massive compact object gravitationally collapses it could form a neutron star, however if it reaches the Chandrasehkar’s limit nothing can stop the collapse and it will form a black hole. Another interesting scenario is the string theory framework, more concretely the AdS/CFT correspondence, where black holes are viewed as thermal states of a conformal field theory. Nevertheless this thesis is basically founded in the semi-classical theory of black holes. It sheds some light over problems like the information loss or thermodynamical aspects of NS5 and LST black holes that although being constructed in string theory they will be studied using semi-classical methods. What we call semi-classical approach is: the background space-time is described by the Einstein’s theory of general relativity, whereas the content of matter fields will be described by quantum field theory. Looking at the Einstein’s equation of gravitation without the cosmological term Rµν −1 2Rgµν = 8πGTµν ,(1.1) on the left hand side it is seen the background geometry described by the general metric gµν, the Ricci tensor Rµν and the scalar curvature R; while on the right side one sees the energy matter content included in the energy-momentum tensor Tµν. A black hole is a classical solution of the equations of motion (1.1) in which there is a region of space-time that is causally disconnected from asymptotic infinity [1]. If we consider a spherically symmetric, non-rotating and uncharged distribution of matter collapsing under self-gravitation, when its radius is lower than the critical Schwarzschild radius the collapse cannot be stopped. The final result will be the matter ending up in an infinite density singular point, while the background metric will be the Schwarzschild metric, ds2=−(1−2GM rc2)dt2+1 1−2GM rc2 dr2+r2dΩ2 2,(1.2) and Mis the black hole mass. The event horizon radius is r0= 2GM. Hereafter we adopt the Planck units convention: ~=c=G=kB= 1, except in some cases where we will restore some units for convenience. The Schwarzschild solution is the unique spherically symmetric solution of the vacuum Einstein’s equations (Rµν = 0) [2]. The singularity theorems of Hawking and Penrose [3, 4] guarantee the existence of singularities once the collapse of a body, not necessarily spherical, reaches a certain point. Geodesic incompleteness, i.e. a geodesic that cannot be extended within the manifold but ends at a finite value of the affine parameter, lead us to the singularity hidden behind a trapped surface, something like a no return barrier. The 3 cosmic censorship conjecture preserves us to observe naked singularities formed in a gravitational collapse from generic initially non-singular state in an asymptotically flat space-time obeying the dominant energy condition. Thus the singularity of a black hole will be hidden behind a null-like hypersurface causally disconnected from the out space of the black hole called the event horizon. All the relevant physics of black holes take place on the event horizon, consequently all the work developed in this thesis is concerned with the event horizon of the studied black holes. For a deep technical study about the above topics of classical black holes we refer the readers to [1] and [5]. Another important characteristic of black holes is the no hair theorem that states: four-dimensional stationary, asymptotically flat, black hole solutions coupled to electromagnetic fields are fully characterized by three parameters, i.e. mass, angular momentum and electric charge. In the seventies, Bekenstein stated that black holes have entropy and this is proportional to the area of the event horizon [6], SBH =A 4G.(1.3) The second law of thermodynamics states that the entropy of the Universe never decreases. However one could imagine a quantity of gas around a black hole, which has a certain entropy, falling towards the black hole. An observer only would see the gas outside the black hole, then only accounts for the entropy of this gas that it is vanishing from the view of the external observer. In order to save the second law, Bekenstein associated an entropy to the black hole proportional to its surface area. Furthermore, one observes that the surface area of a black hole never decreases, that is the area theorem. Therefore if two black holes merge, the area of the final black hole will be equal or greater than the sum of the area of the two initial black holes. This behavior is reminiscent of the second law of thermodynamics applied to the black hole area. Eventually it is fulfilled the generalized second law, which states that the sum of the entropy of the black hole plus the matter surrounding never decreases, d dt(SBH +Smatter)≥0.(1.4) One can establish a direct relation between the laws of thermodynamics and the mechanics laws of black holes through the relations [7], E↔M , S ↔A 4G, T ↔κ 2π,(1.5) where Ais the event horizon area, κis the surface gravity of the black hole and E,S,Tare the usual thermodynamical variables respectively energy, entropy and 4 Chapter 1. Introduction temperature. The first law of black hole mechanics can be identified with the first thermodynamics law, dM =κ 8πdA + Work terms dE =TdS −PdV . (1.6) However, after the work of Bekenstein, Hawking found that actually one can speak about the thermodynamics of black holes. In [8] Hawking found that black holes radiate a thermal spectrum of particles, since then called Hawking radiation, at a temperature TH=~κ 2π. However, this result shows that the temperature of a black hole is inversely proportional to its mass, having thus a negative specific heat. Therefore when a black hole radiates it loses its mass, it evaporates and eventually disappears, and this fact will lead us to the information loss problem. Following an heuristic picture, Hawking radiation is produced by vacuum quantum fluctuations around the black hole where the gravitational field is strong. For black holes of large masses the curvature invariants are sufficiently small, hence one can work in a semi-classical regime where a theory of quantum gravity is not needed. Moreover, due to the no-hair theorem one can only know three charge parameters as mentioned above, thus all physics will be independent of the details of the initial configuration of matter that will forms the black hole. If the black hole completely evaporates away with a thermal spectrum, the final state of the radiation cannot have any information of the initial matter state. This violates the principle of information conservation, which it is fulfilled both in classical and quantum mechanics. In classical mechanics this principle is embodied in Liouville’s theorem on the conservation of phase space volume. In quantum mechanics the principle of information conservation is expressed as the unitarity of the S-matrix. We have seen that it is not the case for the black hole evaporation, where the final state will not be related in a one-to-one to the initial state, thus violating the unitarity of the time evolution operator. Furthermore, the final state cannot be entangled if the black hole has completely evaporated. Initially the outgoing particle created by the quantum fluctuations was in a mixed state with the ingoing particle, and the outgoing radiation was entangled with the ingoing particles. Therefore, the final system, after the evaporation, will be described not by a pure quantum state but by a mixed state. There are a few alternatives in order to avoid this situation. One of them is the existence of a remnant of Planck size to which the outgoing radiation would be entangled. However the entanglement entropy is larger than the black hole 5 entropy, which is really a huge number 1, thus the number of possible microstates of the remnant goes to infinity as the mass of the initial black hole increases. Another alternative, proposed by Hawking, is that black holes completely evaporate and the initial pure state evolves to a final mixed state in a theory of quantum gravity. In this case the description of the states is in terms of density matrices. Nevertheless this approach did not convince the quantum physicists community [9], due to the violation of quantum unitarity. It has been argued that the Hawking radiation carry out somehow the information of the collapsing matter, so that the black hole could completely evaporate and the process would not violate the unitarity. Since then a lot of work has been done in order to solve the information paradox. A good candidate is string theory, more concretely the holographic conjecture and its AdS/CFT realization [10]. For example, counting microstates of the black hole in [11] the authors obtained a microscopic derivation of black hole entropy. For a good reviews on black holes in string theory see for example [12, 13, 14, 15]. Nevertheless, as we have mentioned above, in this thesis we will focus on semi-classical methods that enable us to obtain non-thermal spectra for the vast majority of black holes. This fact is due to taking into account the back-reaction of the metric and imposing energy conservation when the black hole radiates particles. Specifically, we have studied NS5 and Little String Theory (LST) black holes. We have calculated the Hawking radiation for both models, obtaining a non-thermal spectrum for NS5, whereas a purely thermal spectrum for LST. 1Taking into account the expression of the entropy using statistical mechanics S=klogΩ, where Ω is the number of microstates accessible to the macroscopic system, in this case a black hole of area A. For a black hole of solar mass one finds 101078 states. 6 Chapter 1. Introduction Chapter 2 Semi-classical emission of Black Holes Despite the absence of a complete theory of quantum gravity, one may hope to be able to say something concerning the influence of the gravitational field on quantum phenomena, for example the radiation emission carried out by black holes. One can study the quantum aspects of gravity in which the gravitational field is retained as a classical background, adopting the Einstein’s general theory of relativity as a description of gravity, whereas matter fields are quantized in the usual way. The Planck scale: Planck length lP=√G~ c3≈10−35mand Planck time tP= √G~ c5≈10−44s; establishes the frontier at which a full theory of quantum gravity is necessary. Unlike the QED coupling constant e2 ~cthe Planck length has dimensions, hence the effects become significant when the length and time scales of quantum processes of interest fall below the Planck scale. Thus the higher orders of perturbation theory become comparable with the lowest order. Nevertheless, when the distances and times involved are much larger than the Planck scale, the quantum effects of the gravitational field will be negligible, and a semi-classical theory appears to be valid. However, according to the equivalence principle all matter and energy, included the gravitational energy, couple equally strongly to gravity, thus the graviton is also subjected to an external gravitational field as could be a photon. Therefore quantum gravity will enter in a non-trivial way at all scales whenever interesting quantum field effects occur. 7 8 Chapter 2. Semi-classical emission of Black Holes It may still be possible to work with a semi-classical approach. In the same way that in classical relativity one studies the propagation of gravitational waves in curved space-time, one can consider the graviton field as a linearized perturbation on the background space-time: gµν =g(0) µν + ˆgµν. The contribution of the dilaton to the left-hand side of Einstein’s equations, can be casted in a form that might be included along with all the other quantum fields in the right-hand side of Einstein’s equations, being part of the matter rather than the geometry. On the other hand, the fact that the gravitational constant Ghas units of length square gives rise to a non-renormalizable theory of gravitation. hence the quantization of gravitational field has not been already accomplished. Nevertheless, one can truncate the expansion of the semi-classical theory (classical gravity plus quantum matter fields) at one-loop level for example. In this way, the finite number of divergences can be removed by renormalization of a finite number of physical quantities, thus the truncated theory could be considered renormalizable. Since important gravitational effects occur in quantum field modes for which the wavelength is comparable with some characteristic length scale of the background space-time, only in the vicinity of the microscopic black holes or in the early epochs of the Big Bang we can expect such gravitational effects. Otherwise, in the rest of the phenomenology one can study quantum field theory in curved space-time, i.e. in a semi-classical way. The fundamental Hawking’s discovery of thermal emission by black holes [8] is a clear example of how gravity, quantum field theory and thermodynamics are closely interwoven. Henceforth all the work developed in this thesis gravitates in some way around such discovery. We will see how curved space-time can create particles, henceforth we will not ever consider particle as a fundamental fixed concept, otherwise it might be considered as an observer-dependent object. For this study we have followed the notes in [16]. We consider space-time to be a C∞n-dimensional, globally hyperbolic, pseudo-Riemannian manifold [5]. We write the background metric gµν associated with the line element as ds2=gµνdxµdxν;µ, ν = 0,1, ..., (n−1) ,(2.1) where xµ,xνare the coordinates. We define the determinant as g≡| detgµν |.(2.2) Now we want to consider the quantization of a field in the classical curved space-time 2.1. Hawking radiation 15 Figure 2.3: Parallel transport of the unitary vectors nand lthrough the ingoing null geodesic. an ingoing null ray starting at I−with v > 0 never reaches I+since it crosses the event horizon H+. Thus the outgoing mode on I−is φω(v)∼{0v > 0 exp (iω κlog(−v))v < 0.(2.29) By Fourier transforming ¯ φω=∫∞ −∞ eiω0vφω(v)dv , (2.30) one obtains the following relation demonstrated in [18] ¯ φω(−ω0) = −e−πω κ¯ φω(ω0), ω0>0.(2.31) Eventually, a positive definite frequency mode on I+becomes a mixed positive and negative frequency mode on I−. Thus identifying the Bogoliubov coefficients as αωω0=¯ φω(ω0) βωω0=¯ φω(−ω0),(2.32) it is accomplished the following relation between the Bogoliubov coefficients βij =−e−πω καij .(2.33) 16 Chapter 2. Semi-classical emission of Black Holes Now taking into account the relation (2.19) ∑ k (αikα∗ jk −βikβ∗ jk) = (e π(ωi+ωj) κ−1)∑ k βikβ∗ jk =δij ,(2.34) and taking i=j ∑ k|βik|2=1 e2πωi κ−1.(2.35) Actually the inverse process is needed, namely start with a positive frequency mode on the past null infinity I−that propagates until it becomes a mixed positive and negative frequency mode on the future null infinity I+. The final result for the expectation value of the number of particles created and emitted to I+is hNiI+=1 e2πω κ−1.(2.36) This result corresponds to a Planck distribution for black body radiation at the Hawking temperature TH=~κ 2π.(2.37) So far we have considered that all the thermal radiation emitted by the black hole arrives to the future null infinity I+without any change in the amplitude of the wave function. However, some emitted radiation will be partially scattered back to the event horizon. This fact is due to the gravitational potential barrier around the black hole, where some fraction of radiation will be reflected back to the hole, acting thus as a filter for the emitted radiation. Taking into account this effect we have to modify the orthonormal condition (2.19) by ∑ k (αikα∗ jk −βikβ∗ jk) = Γ ,(2.38) where Γiis known as the greybody factor and it accounts for the deviation from pure black body spectrum, then the number of emitted particles will be hNiI+=Γ e2πω κ−1.(2.39) Greybody factors have a relevant importance because successful microscopic account of black hole thermodynamics should be able to predict them. For example, it is shown in [19] that D-branes provide an account of black hole microstates which is successful to predict the greybody factors. There exists a vast literature on how to compute greybody factors in the context of the quantum field theory in curved space-time, e.g. [20, 21, 22, 23, 24, 25, 26]. 2.2. Euclidean path integral and Hawking temperature 17 2.2 Euclidean path integral and Hawking temperature One can better understand the result that black holes radiate thermally appealing the Euclidean path integral formalism [27]. If one works with imaginary time coordinate setting t=iτ , (2.40) for a four-dimensional spherically symmetric black hole we obtain a positive definite metric known as Euclidean metric, ds2 E=f(r)dτ2+1 f(r)dr2+r2dΩ2 2,(2.41) where f(r) is a metric function defined as f(r)≡(1−r0 r)being r0the radius of the event horizon. This metric still presents a coordinate singularity at r=r0, so that one performs a change of coordinates going to the Rindler sector. For a general four-dimensional spherically symmetric background we define the proper length as ρ=∫√grr dr . (2.42) Expanding around r0we write the metric function f(r) near horizon as f(r) = f(r0)0(r−r0).(2.43) Then the new Rindler radial coordinate will be ρ= lim r→r0[2√(r−r0) f(r)0].(2.44) Thus for the Schwarzschild black hole, i.e.: grr =1 f(r)with f(r) = (1−2M r)and r0= 2Min Planck units; the Euclidean Schwarzschild metric is ds2 E=ρ2(κdτ)2+dρ2+r2dΩ2 2,(2.45) where κis the surface gravity 2and equals to κ=1 4M.(2.46) 2The surface gravity of a black hole is defined as the acceleration of a static particle near the event horizon measured by an asymptotic observer. It can be calculated with the formula κ2=−1 2(∇µζν) (∇µζν) evaluated at the event horizon, where ζνis a Killing vector. See [28] for a rigorous study. 18 Chapter 2. Semi-classical emission of Black Holes The metric in the ρ−τplane is just the plane polar coordinates if one identifies τ with period 8πM. In general, the coordinate singularities on the horizon (conical singularities) of Euclidean black hole metrics can be removed by identifying τ→τ+2π κ,(2.47) so that the imaginary time coordinate τis periodic with period 2π κ. Therefore the Euclidean functional integral must be taken over fields that are periodic in τwith period 2π κ. The Euclidean path integral is Z=∫D[φ]e−SE[φ],(2.48) where SEis the Euclidean action. Taking the integral over fields that are periodic in imaginary time with period ~β, one can write (2.48) as Z=tr e−βH ,(2.49) which is the thermodynamic partition function corresponding to a quantum system with Hamiltonian Hat the temperature given by β=1 kBT, being kBthe Boltzmann constant. In order to see this last result we consider the probability amplitude to go from an initial field configuration φ1on the space-like hypersurface at t1to a field configuration φ2on the hypersurface at t2. This amplitude is determined by the matrix element eiH(t2−t1)[29]. Also we can calculate the amplitude as a path integral over all fields φbetween t1and t2with φ1and φ2as fields on the initial and final hypersurface respectively. Thus hφ2, t2|φ1, t1i=hφ2|eiH(t2−t1)|φ1i=∫D[φ]eiS[φ].(2.50) Then if one considers that the interval time is imaginary and equal to β, t2−t1=iβ . (2.51) Choosing as boundary conditions, φ1=φ2(2.52) on the two hypersurfaces, and summing over all field configurations φn, one obtains on the left of (2.50) the partition function Zof a quantum system, i.e. the 2.3. Hawking radiation as tunneling 19 expectation value of e−βH summed over all states, at a temperature kBT=β−1. Furthermore, using the Euclidean action on the right of (2.50), we finally obtain Z=∑ nhφn|e−βH|φni=∫D[φ]e−SE[φ].(2.53) Therefore the partition function for the field φat temperature Tis given by a path integral over all fields in Euclidean space-time, which is periodic in the imaginary time direction with period β= (kBT)−1. So that fields in Schwarzschild space-time in particular, and in curved background in general, will behave as if they were in a thermal state with temperature TH=~κ 2πkB, or using Planck units, TH=κ 2π(2.54) where THis the Hawking temperature of a black hole at which the quantum field theory is in equilibrium. We point out that the equilibrium of a Schwarzschild black hole at Hawking temperature is unstable. From (2.46) and (2.54) we see that a black hole that emits radiation loses its mass hence its temperature increases, therefore the specific heat capacity of Schwarzschild black hole is negative. 2.3 Hawking radiation as tunneling One way to solve semi-classically the information loss paradox is proposed in [30], where the authors obtains a non-thermal emission spectrum corresponding to a Schwarzschild black hole. The problem is addressed considering the emission of radiation by a black hole as a tunneling process. The key idea is that the energy of a particle changes its sign as it crosses the event horizon. The heuristic picture [31] shows a virtual pair of particle and antiparticle created just inside the horizon. Then the positive energy virtual particle can tunnels out, it materializes as a real particle and propagates to the infinity. These particles will be seen by an asymptotic observer as Hawking flux radiation. Conversely, the virtual pair could be created just outside the horizon, in that case the negative energy particle can tunnels inwards the black hole. In both cases the negative energy particle is absorbed by the black hole, thus the mass of the black hole decreases in the same amount of the positive energy released out through the emitted particle. The fact that black hols decreases its mass supports the quantum gravity idea that black holes can be regarded as highly excited states. Anyway the total energy of the system is conserved. The idea that black holes lose mass by absorbing negative energy is studied in [16]. 20 Chapter 2. Semi-classical emission of Black Holes In the WKB approximation the tunneling rate probability is related to the imaginary part of the action for the classically forbidden path, Γ∼e−2ImS.(2.55) The tunneling is between two separated classical turning points which are joined by a complex path. Nevertheless in this case it does not preexist a barrier, but it is just created by the outgoing particle itself. As the total energy must be conserved during the emission of radiation by the black hole, when particles are emitted the hole loses mass. Therefore, if the black hole loses mass it shrinks its event horizon to a new small radius, and the contraction will depends on the energy of the outgoing emitted particle [32]. We introduce the method of tunneling emission considering at first a line element of a four-dimensional spherically symmetric black hole. ds2=−f(r)dt2+f(r)−1dr2+r2dΩ2 2,(2.56) where the metric function is f(r)≡1−r0 r,(2.57) being r0the event horizon radius. In order to avoid coordinate singularities at the event horizon we will write the metric in regular Painlev´e coordinates [33], thus we obtain a smooth behavior through the horizon. Just to say that the Painlev´e time coordinate is nothing more than the proper time of a radially free-falling observer [34]. Then if we shift the time coordinate to proper time coordinate t→t−g(r),(2.58) where g(r) is a function that depends only on the radial coordinate, we can write the new metric as ds2=−f(r)dt2+ 2f(r)g(r)0dtdr +(f(r)−1−f(r)g(r)02)dr2+r2dΩ2 2,(2.59) Also demanding that the constant time slices be flat, f(r)−1−f(r)g(r)02= 1 ⇒g(r)0=√1−f(r) f(r).(2.60) Eventually the metric, written in Painlev´e coordinates, is ds2=−f(r)dt2+ 2√1−f(r)dtdr +dr2+r2dΩ2 2.(2.61) 2.3. Hawking radiation as tunneling 21 Considering the Schwarzschild solution in Planck units with f(r)≡1−2M r,(2.62) being Mthe mass of the black hole, one obtains for the metric in Painelev´e coordinates, ds2=−(1−2M r)dt2+ 2√2M rdtdr +dr2+r2dΩ2 2.(2.63) We can see that these coordinates are stationary and not singular through the horizon. Then we can define a vacuum state demanding that it annihilates the modes with negative frequency. Now consider a radial null geodesic ˙r=±1−√2M r,(2.64) with the plus (minus) sign corresponding to outgoing (ingoing) geodesics respectively. However, we have to modify the geodesic equation when self-gravitation is included, then we will not consider the emission of point particles but the propagation of shell particles. Self-gravitating shells in Hamiltonian gravity were studied in [35]. Keeping fixed the ADM mass [36] and allowing the black hole mass to vary, we see how the metric backreacts due to the emission of a shell particle, hence M→M−ωin order to keep energy conservation. The wavelength of the radiation is of the order of the size of the black hole. Nevertheless, when we trace back the outgoing wave, we point out that the wavelength is blue-shifted, justifying thus the use of the WKB approximation (2.55). In order to simplify, one could consider the propagation of an s-wave, neglecting then the angular part of the background metric (2.63). Thus using the Birkhoff’s theorem one can decouple gravity from matter. Therefore, the imaginary part of the action for an s-wave outgoing positive energy particle will be ImS= Im ∫rout rin prdr = Im ∫rout rin ∫pr 0 dprdr . (2.65) The particle crosses the horizon from rin to rout with rin > rout due to the shrinking of the horizon when the particle is emitted and the metric backreacts. Making use of the Hamilton’s equation ˙r=dH dprand writing the Hamiltonian as H=M−ω, we obtain ImS= Im ∫M−ω M∫rout rin dr ˙rdH = Im ∫ω 0 (−dω)∫rout rin dr 1−√2(M−ω) r .(2.66) 22 Chapter 2. Semi-classical emission of Black Holes Figure 2.4: Diagram picture of the tunneling approach. A particle of energy +ωis emitted by a black hole of initial mass Mand initial radius rin. After the emission the event horizon shrinks, , and the black hole loses mass. In the last integral there is a pole in the upper half plane of integration. In order to perform the integral we use the Feynman prescription to displace the pole from ω to ω−i deforming the contour around the pole. We just can see how the particle tunnels along forbidden classical path between rin = 2M−just inside the horizon and rout = 2(M−ω) + just outside. Hence the imaginary part of the action will be ImS= 4π(Mω −ω2 2).(2.67) Finally from (2.55) the emission rate of the tunneling process is Γ∼e−8π(Mω−ω2 2).(2.68) We point out that we can write the above result in a statistical mechanics fashion in terms of the change of the entropy as Γ∼e∆SBH ,(2.69) where ∆SBH is the change of the Bekenstein-Hawking entropy according to the area law, being the entropy before the emission Si= 4πM2and after the emission Sf= 4π(M−ω)2. It is very interesting to notice from the expression (2.68) that the 2.3. Hawking radiation as tunneling 23 emission is not purely thermal. Taking into account the backreaction of the metric and imposing energy conservation we obtain a non-thermal emission reflected in the presence of the ω2-term. This fact leads us to think that some sort of correlations exist between the emitted particles, carrying out some degrees of freedom that enables us to recover the information lost in the black hole. Of course, if we neglect the quadratic energy term we obtain the Planck spectrum ρ(ω) = Γω (eω/T −1) dω 2π,(2.70) at a Hawking temperature TH=1 8πM , where Γωis the greybody factor. One can perform the same analysis in the Reissner-Nordstrom black hole obtaining similar conclusions. However, in order to simplify, we only consider the emission of uncharged particles, otherwise we must consider the electromagnetic interactions between the particles and the electromagnetic field of the black hole. The line element for the Reissner-Nordstrom charged black hole is ds2=−(1−2M r+Q2 r2)dt2+1 (1−2M r+Q2 r2)dr2+r2dΩ2 2,(2.71) being Qthe charge of the black hole. In Painlev´e coordinates this metric is written as ds2=−(1−2M r+Q2 r2)dt2+ 2√2M r−Q2 r2dtdr +dr2+r2dΩ2 2.(2.72) A radial null geodesic for a outgoing uncharged particle is ˙r= 1 −√2M r−Q2 r2.(2.73) As in the Schwarzschild case we compute the imaginary part of the action for the emission of a shell of energy ω ImS=∫M−ω M dH ∫rout rin dr ˙r=∫ω 0 d(−ω)∫rout rin dr 1−√2M r−Q2 r2 .(2.74) In order to evaluate the radial integral we perform the following change of coordinates u=√2Mr −Q2⇒du =M udr , (2.75) thus the radial integral in terms of the ucoordinate is ∫u(u2+Q2) M(u(u−2M) + Q2)du . (2.76) 24 Chapter 2. Semi-classical emission of Black Holes The integral has a pole at u=M±√M2−Q2, where plus/minus sign corresponds to the outer/inner horizon position, effectively if we apply the Cauchy’s theorem we obtain a residue value of (M+√M2−Q2)2 √M2−Q2. Now if we take into account the self-gravitation [37], then replacing Mby M−ωand integrating the energy, the imaginary part of the action becomes ImS=−2π∫ω 0((M−ω) + √(M−ω)2−Q2)2 √(M−ω)2−Q2d(−ω) (2.77) = 2π[M(M+√M2−Q2)−(M−ω)(M−ω+√(M−ω)2−Q2)]. Eventually we can also see quadratic energy terms, thus the emission rate (2.55) is non-thermal, Γ∼e−4π[M(M+√M2−Q2)−(M−ω)2−(M−ω)√(M−ω)2−Q2].(2.78) 2.4 The complex path method Another semi-classical method in order to calculate the particle production near the event horizon of black holes was proposed in [38]. The complex path method has the advantage that avoids the Kruskal extension of the space-time thus one can work with the usual spherical coordinates, and hence it is not needed to compute the Bogoliubov coefficients. We will show the method in the simple case of fourdimensional spherically symmetric background (2.56) and (2.57). We only consider the r−tsector relevant for the emission process, so that the effective two-dimensional metric is ds2 eff =−f(r)dt2+1 f(r)dr2.(2.79) Now we consider the propagation of a massless scalar field in this two-dimensional background, then the Klein-Gordon equation of motion gµν∇µ∇νφ= 0 ,(2.80) can be written as [1 f(r)∂2 t−∂r(f(r)∂r)]φ= 0 .(2.81) Then if we take the WKB ansatz solution φ∼ei ~S(t,r),(2.82) 3.1. LST, thermodynamics overview 31 equation (6.11) for a exact relation in string frame and Einstein frame respectively. We define the parameter χwhich takes the values 1 for NS5 model and 0 for LST, these are the unique values for which exist a supergravity solution. In addition to the previous fields there is an NS −NS H(3) form along the S3,H(3) = 2N3. According to the holographic principle the high spectrum of this dual string theory should be approximated by certain black hole in the background (3.5). The geometry transverse to the 5-branes is a long tube which opens up into the asymptotic flat space with the horizon at the other end. In the limit r→r0appears the semi-infinite throat parametrized by (t, r) coordinates, in this region the dilaton grows linearly pointing out that gravity becomes strongly coupled far down the throat. The string propagation in this geometry should correspond to an exact conformal field theory [50]. The boundary of the near horizon geometry is R5,1×R×S3. The geometry (3.5) is regular as long as r06= 0. We are going to reduce the metric (3.5) to the r−tsector, relevant for the forthcoming sections. At first step we take the scalar field action S=1 2k2 10 ∫M d10x√−g(R−1 2∂µφ∂µφ−1 12e−ΦH2 (3)).(3.9) Performing a change to tortoise coordinate, see (3.5): dr∗=√A(r) f(r)dr, we expand the ten-dimensional action as S=1 2k2 10 ∫dtdr∗dθdϕdψ 5 ∏ j=1 dxjr3A(r)2sin2θsinϕ(gse−Φ)5/2[f(r) √A(r)× ×(R−e−Φ 12 H2 (3))+(1 2√A(r)(∂2 t−∂2 r∗)−f(r) 2r2A3/2(∂2 θ+1 sin2θ∂2 ϕ+ +1 sin2θsin2ϕ∂2 ψ)−f(r) 2√A(r) 6 ∑ j=2 ∂2 j)φ(t, r)S(Ω3)ei∑kjxj], (3.10) where we have decomposed the scalar field into r−t, 3-angular and 5-brane parts. Our following approximations are based on three main steps: 1. We only consider the propagation mode of an s-wave. 2. We only take into account a subset of states of the Hilbert space such that the eigenstates of momentum parallel to the NS5-brane vanish. 3. We take the near horizon limit, r→r0. 32 Chapter 3. Hawking radiation in Little String Theory Eventually we come back to the original rradial coordinate, obtaining for the action S=Vol(S3)Vol(R5) 2k2 10 ∫dtdrA(r)2e−2Φ (−1 f(r)∂2 t+f(r) A(r)∂2 r)φ(t, r),(3.11) where Vol(R5) stands for the volume of the NS5-branes and Vol(S3) is the volume of the 3-sphere. From (3.11) we find out that the scalar field can be seen as (1 + 1)- dimensional scalar field φ(t, r) propagating in the background ds2 eff =−f(r)dt2+A(r) f(r)dr2,(3.12) together with an effective dilaton field e2Φ =g2 sA(r).(3.13) Henceforth we are going to work with this two-dimensional effective metric. Concerning the black hole thermodynamics we will construct the thermal states of the black hole following the same analysis of Chapter 2, Section 2.2. Working in imaginary time coordinate t=iτ, we will write the positive Euclidean metric in Rindler coordinates. The radial Rindler coordinate is ρ= lim r→r0[2√A(r)(r−r0) f(r)0].(3.14) Thus the Euclidean metric in Rindler coordinates is ds2 E=ρ2(κdτ)2+dρ2+A(r)r2dΩ2 3+ 5 ∑ j=1 dx2 j,(3.15) where we have defined κas κ=f(r0)0 2√A(r0),(3.16) which it is precisely the surface gravity of the NS5 and LST black holes. In the footnote of Section 2.2 we had given a simple explicit formula in order to calculate the surface gravity [28], κ2=−1 2(∇µζν) (∇µζν) (3.17) 3.1. LST, thermodynamics overview 33 evaluated at the event horizon, where ζνis a Killing vector. For the NS5 and LST stationary black holes we choose the Killing vector ζν= (−∂t, ζi) with ζi= 0 , i = 1, ..., 9 ; (3.18) and its covariant form ζν=gνλζλ=gtt (−∂t).(3.19) Calculating ∇µζν=gµλ∇λζν=grr∇rζt,(3.20) and ∇µζν=∇rζt,(3.21) it is obtained the expression κ=1 2√−gtt ·grr (∂rgtt).(3.22) Evaluating this expression at the event horizon r0, it is obtained the surface gravity. Concretely for NS5 and LST we obtain κ=f(r0)0 2√A(r0)=1 √χr2 0+N m2 s .(3.23) Then identifying the period of the Euclidean time with τ→τ+2π κwe avoid the conical singularity in (3.15), thus the imaginary time is periodic with period β= 2π κ. As it was demonstrated in Section 2.2 we can identify β−1with the Hawking temperature TH, thereby calculating the value of the surface gravity (3.16) we obtain the Hawking temperature for the NS5 and LST black holes, TH=~ 2π√χr2 0+N m2 s .(3.24) Notice that this value for LST (χ= 0) is independent of the black hole radius, that is fixed even if many particles impinge on the black hole. This results holds at all orders in α0(inverse string tension) corrections, but receives modifications from higher genus [51, 52]. We would like to address the question whether an observer in a moving frame observes a temperature above the Hagedorn temperature. We know that in the near horizon limit of NS5, i.e. LST, the system reaches the maximum temperature, namely the Hagedorn temperature. One could think that a boosted observer may 34 Chapter 3. Hawking radiation in Little String Theory observe a temperature higher than the Hagedorn one, for this reason we want to verify the validity of this statement. We have evaluated the simplest case, namely a scalar particle-like observer who moves on an NS5-brane with constant velocity at a fixed distance rfrom the horizon of the LST black hole. We consider the orbit for which x1=vt. Relating the time coordinate twith the proper time τ(this is not the imaginary time) through dτ2=−gµνdxµdxν, one obtains dτ dt =√f(r)−v2.(3.25) The velocity is bounded by the local velocity of light thus we have to impose the constraint v2≤f(r). This relation brings us to a new coordinate of the horizon position seen by the moving particle, r=r0 √1−v2. Furthermore the Killing vector relevant for the process is ζ=−∂t+v∂x1. Therefore evaluating the surface gravity using this new coordinate r, we obtain the local temperature for the moving scalar particle ¯ T=~(1 −v2) 2π√χr2 0 (1−v2)+N m2 s ,(3.26) where we have used natural units, c= 1 and v < 1. We notice two important features. First of all, we see that in the v→0 limit we recover the result (3.24). Secondly, comparing the temperature for the particle-like observer (3.26) with the temperature defined by (3.24) for an asymptotic static observer, we see that the former is lower than the later. We conclude that the Hawking temperature of LST is a maximum bound and corresponds to the Hagedorn temperature. Unfortunately, we are not able to perform the same analysis for an accelerating particle-like observer. The main problem is that the path which the particle follows is not generated by a Killing vector field, this fact prevent us from using the surface gravity method in order to calculate the temperature. Next, we calculate the entropy using the area law through the BekensteinHawking entropy relation SBH =AH 4G(10)~,(3.27) where AHis the area of the event horizon and G(10) is the ten-dimensional gravitational constant. Working in string frame the area of the event horizon is AH=∫√−g(8) dθ dϕ dψ djx= Vol(R5) 2π2(χr2 0+N m2 s)3/2 ,(3.28) 3.1. LST, thermodynamics overview 35 the factor 2π2accounts for the volume of the 3-sphere, see Appendix B, and −g(8) is the determinant of the induced metric on the event horizon d˜s2=A(r)r2dΩ2 3+ 5 ∑ j=1 dx2 j.(3.29) Then the Bekenstein-Hawking entropy is SBH =AH 4G(10)~= Vol(R5)π2(χr2 0+N m2 s)3/2 2G(10)~.(3.30) We have seen that the LST temperature is independent of the black hole radius and therefore of the black hole mass. We could identify this temperature with the Hagedorn temperature. It has been argued, [53], that the energy, entropy and temperature of a CFT at high temperatures can be identify with the mass, entropy and Hawking temperature of the dual black hole. The Euclidean action for a LST black hole solution gives a vanishing contribution to the Helmholtz free energy: log Z=−I = 0, with Zbeing the string partition function. In that precise case the entropy and energy density are directly proportional to each other and the Bekenstein-Hawking entropy relation is fulfilled. Otherwise, one can compute the Komar energy Efor the LST background, see Appendix C, either in Einstein frame [51, 54] or in string frame [46, 49] which satisfies the usual thermodynamic relation S=βE. This relation implies that the free energy of the system F=E−TS vanishes. This behavior suggests that at leading order the Hagedorn density of states at very high energies grows as ρ(E) = eS(E)∼eβE [45, 55]. At first sight one could think that a phase transition is present when the system evolves from NS5 to the near horizon limit of NS5, i.e. LST, but we have checked that it is not the case. Plotting the entropy (3.30) versus the temperature (3.24) we do not detect any critical point (Davies point) [56] that would signal a phase transition. Even working in thermodynamic geometry [57], writing the LST metric as a Ruppeiner metric ds2=−3√πG ~2MdS2, we do not detect any divergence in the scalar curvature that would signal a possible phase transition. However calculating the specific heat as C=T∂S ∂T , we have found that it has a negative value: −3S, showing that the theory is unstable. In the work [45], the authors show that loop/string corrections to the Hagedorn density of states of LST were of the form ρ(E)∼EαeβE(1+O(1 E)), where αis a correction factor. The temperature-energy relation thus becomes β= ∂log ρ ∂E =β0+α E+O(1 E2), where β0=T−1 H. The authors found that since αis negative the high energy thermodynamics corresponding to near-extremal 5-branes 36 Chapter 3. Hawking radiation in Little String Theory is unstable, the temperature is above the Hagedorn temperature and the specific heat is negative. This instability would be associated to the presence of a negative mode (tachyon) in string theory. The high temperature phase of the theory yields the condensation of this mode. The authors are lead again to the conclusion that the Hagedorn temperature is reached at a finite energy, being associated with a phase transition. 3.2 Semi-classical emission in NS5 We are going to calculate the Bogoliubov coefficients, partially following [58], corresponding to the NS5 black hole. We want to stress that it is not possible to make this computation for the LST black hole because it is not asymptotically flat. The creation of particles in the vacuum is due to a mixing of positive and negative frequency modes near the event horizon where the gravitational field is strong. Somehow we are observing the evolution of an initial positive frequency state in one definite vacuum to a final negative frequency state in another vacuum. Working in Heisenberg picture and evaluating the number operator between initial vacuum state, one observer in the final vacuum state will detect a number of particles created in the process. We are interested in the propagation of a massless scalar field φin a geometry which is asymptotically flat. We decompose the scalar field into positive frequency ingoing modes {fi}in the past null infinity hypersurface I− φ=∑ i (fiai+f∗ ia† i).(3.31) The set {fi}form a complete orthonormal basis with well inner defined product (2.6). The operators aiand a† ican be interpreted as annihilation and creation operators respectively that fulfill the usual commutation relation [ai, a† j] = δij and [ai, aj] = [a† i, a† j] = 0. Thus one obtains a well defined vacuum state on I− ai|0−i= 0 .(3.32) At some point a black hole is formed and an event horizon appears. On the future horizon H+there is not Cauchy data coming from the future null infinity hypersurface I+, in the same way on I+we do not have Cauchy data coming from H+. Therefore the scalar field can also be decomposed as φ=∑ i (pibi+p∗ ib† i+qici+q∗ ic† i),(3.33) 3.2. Semi-classical emission in NS5 37 where {pi}are positive frequency outgoing modes defined on I+with its corresponding creation and annihilation operators biand b† i. The modes {qi}are absorbed by the future event horizon H+thus cannot escape to I+. These modes are not-well positive frequency defined modes because on H+we cannot define positive (or negative) frequencies, having thus a mixing of positive and negative frequency modes. Nevertheless the choice of {qi}does not affect the calculation at the asymptotic limit since they are zero at I+. Also we can define a vacuum state on I+ bi|0+i= 0 .(3.34) The modes {pi}on I+can be decomposed in terms of the incoming modes {fi} pi=∑ j (αijfj+βijf∗ j).(3.35) In the same way we can relate the corresponding operators bi=∑ j (α∗ ijaj−β∗ ija† j), b† i=∑ i (αija† j−βijaj).(3.36) These relations are known as Bogoliubov transformations and relate different modes expressed in different basis. We can see that operating with annihilation operator biin the vacuum state defined on I−the result will be different from zero, if the coefficients β∗ ij are non-zero. Thus one has a mixing between positive and negative frequency modes. It can be calculated the number of particles created, i.e. the number of particles measured by an observer in the future null infinity in the vacuum defined on I−, h0−|Ni|0−i=h0−|b† ibi|0−i=∑ j|βij|2.(3.37) In order to calculate the scalar field modes we must solve the Klein-Gordon equation for a massless particle φ= 0. In the background (3.5) this equation is written as [−A(r)∂2 ∂t2+f(r)A(r)∂2 ∂y2+f(r) r3 ∂ ∂r (r3f(r)∂ ∂r)+f(r) r2L2]φ= 0 ,(3.38) where we have defined the angular momentum operator as L2≡1 sin2θ ∂ ∂θsin2θ∂ ∂θ +1 sin2θsinϕ ∂ ∂ϕsinϕ∂ ∂ϕ +1 sin2θsin2ϕ ∂2 ∂ψ2.(3.39) 38 Chapter 3. Hawking radiation in Little String Theory We are looking for a solution of type φ=(Ae−iωt +A∗eiωt)R(r)(Beikx +B∗e−ikx)Yl,m,m0(θ, ϕ, ψ),(3.40) where we have decomposed the solution into stationary part, a pure radial term, the propagation through the flat-space brane directions xand the angular part in which the 3-dimensional scalar spherical harmonics satisfy L2Yl,m,m0(θ, ϕ, ψ) = −l(l+ 2)Yl,m,m0(θ, ϕ, ψ).(3.41) Thus the equation of motion can be written as [A(r)ω2−f(r)A(r)k2+f(r) r3 ∂ ∂r (r3f(r)∂ ∂r)−f(r) r2l(l+ 2)]R(r) = 0 .(3.42) Performing a change to tortoise coordinate dr∗=√A(r) f(r)dr , (3.43) and the standard functional change R(r) = R(r∗) r, [39], we obtain a Schrodinger-type equation [∂2 ∂r2 ∗ +(ω2−f(r)k2−f(r) A(r)r2L2)]R(r∗) = 0 .(3.44) Considering the propagation of an s-mode in the asymptotic limit we find the solution R(r) = 1 r(C1e−i√ω2−k2r∗+C2ei√ω2−k2r∗)(3.45) where C1and C2are constants. Eventually the scalar field takes the form φ=(Ae−iωt +A∗eiωt)×(Beikx +B∗e−ikx)× ×1 r(C1e−i√ω2−k2r∗+C2ei√ω2−k2r∗)Yl,m,m0(θ, ϕ, ψ).(3.46) If we only consider a subset of states of the Hilbert space such that the eigenstates of momentum parallel to the NS5-brane vanish, i.e. k= 0, the scalar field solution will be φ=1 r(AC1e−iω(t+r∗)+AC2e−iω(t−r∗)+A∗C1eiω(t−r∗)+A∗C2eiω(t+r∗))× ×Yl,m,m0(θ, ϕ, ψ).(3.47) We introduce the advanced and retarded null coordinates v=t+r∗, u =t−r∗,(3.48) 3.2. Semi-classical emission in NS5 39 and we use them as a canonical affine parameters in order to define the positive frequency modes. Substituting (3.48) in (3.47), we obtain for the incoming modes defined at I−, fω0lmm0∼1 √2πω0 Fω0(r) reiω0vYl,m,m0(θ, ϕ, ψ),(3.49) and for the outgoing modes defined at I+, pωlmm0∼1 √2πω Pω(r) reiωu Yl,m,m0(θ, ϕ, ψ).(3.50) Fω0(r) and Pω(r) are integration variables that contain tiny effects depending on r since this modes are calculated at the asymptotic. The normalization constant 1 √2πω is frequently used in the Klein-Gordon equation. Next, we transform the discrete expressions (3.35), (3.36) and (3.37) to the continuous limit integrating the energy ωand considering the same value for the l,m, m0numbers, thereby we obtain pω=∫∞ 0 (αωω0fω0+βωω0f∗ ω0)dω0,(3.51) bω=∫∞ 0 (αωω0aω0−β∗ ωω0a† ω0)dω0,(3.52) and Nω=∫∞ 0|βωω0|2dω0.(3.53) We calculate the Bogoliubov coefficients αωω0and βωω0by Fourier transforming (3.51). Then substituting (3.49) into (3.51) and multiplying both sides by ∫∞ −∞ e−iω00vdv, we obtain ∫∞ −∞ pωe−iω00vdv =1 √2πω0 Fω0(r) r∫∞ 0 (αωω02πδ(ω0−ω00) + β∗ ωω02πδ(−ω0−ω00)) dω0, (3.54) where the second term of the integral must be zero due to the properties of the delta distribution. The Bogoliubov coefficients can be written as αωω0=r√ω0 Fω0(r)√2π∫∞ −∞ pωe−iω0vdv , (3.55) βωω0=r√ω0 Fω0(r)√2π∫∞ −∞ pωeiω0vdv . (3.56) 40 Chapter 3. Hawking radiation in Little String Theory Figure 3.1: On the left: a Penrose diagram which shows the future and past null infinity and the future and past event horizon. On the right: a null ray is traced back in time from the future null infinity using the parallel transporting of two unitary vectors nand l. It is obtained a relation between the null coordinates vand uin order to express the outgoing modes pωin terms of the advanced null coordinate v. In order to evaluate the integral (3.55) we must express the outgoing modes pωin terms of the advanced null coordinate v. It is considered a light ray traced backward from I+as proposed in [8] and it is taken the Kruskal coordinate as the affine parameter on the past event horizon H−. We consider a mode pωpropagating backward from the future null infinity hypersurface I+and with zero Cauchy data on H+. Some part of this mode solution will be scattered over the black hole potential and will eventually emerge at I−with the same frequency ω. On the other hand, some part of pwwill be partially scattered and reflected, eventually emerging at I−. This second part will produce the creation of some new particles measured by the asymptotic observer at I+. The modes will be extremely blue-shifted at H+ because the outgoing null coordinate tends to infinity, therefore we are able to use the optical theorem, which states that only the reflected part of the wave will be significant. Now consider a point xon the horizon, a null tangent vector lµon the horizon at x, and a future directed null vector nµat xwhich is normal to the horizon and directed radially inwards. This two vectors are normalized: lµ·nµ=−1, see figure 3.1. Then we consider a null geodesic γthat starts at a point u0, goes along H+, is reflected at the center, and emerges towards I−along a path defined by v0. 3.3. Hawking radiation via tunneling 47 As the radiation comes always as a pure state, the Hilbert space can be factorized into two disjoint parts, H=Hin ⊕Hout, which correspond to states located at the inner and outer sides of the event horizon, respectively. It will follow from the superposition principle that the state inside the horizon must be a unique state carrying no information at all. Summing up, this can be expressed in a somewhat muted fashion as: the black hole at the Hagedorn temperature does not interact with its environment and hence we can represent a state of the entire space as |ψ(t)i=|ψin(t)i⊗|ψout(t)i. 3.3.2 Locking information at the Hagedorn temperature That the result (3.75) gives the correct behaviour for the LST system is intuitively clear in the semi-classical approach from the very beginning since in this type of black holes the temperature is not related with the black hole mass. It is precisely this fact which encodes the ultimate reason for the non-thermal behaviour in the model of [30]. To make this point more clear, instead of using the field content of LST we retain the full asymptotic, ten-dimensional CHS background [65] ds2=−f(r)dt2+ 5 ∑ j=1 dx2 j+A(r) f(r)dr2+A(r)r2dΩ2 3,(3.76) and dilaton e2φ=χ+N m2 sr2with χ≡1 in (3.8). One then sees that the temperature depends on the black hole mass [66]. In this case the Hawking temperature can be determined by the surface gravity method at the event horizon and is given by βCHS =β0√1 + χr2 0/N , (3.77) notice that it provides an infra-red cutoff for the radial coordinate. We have used χas an eventual continuous variable that parameterizes the geometry (3.76). By no means, one should not understand that all the intermediate values correspond to supergravity solutions. Its utility is twofold: first the near horizon limit is recovered setting χ= 0, and second it will also control the temperature; for instance χ→0 increases the temperature to the Hagedorn one. The basic tenant is that (3.77) relates the temperature with the size of the black hole, thus as the black hole emits, not only the radius shrinks but also the temperature increases. This fact relates the emission with the thermodynamic properties of the black hole and contrary to the previous situation we expect that the radiation provides information on the black hole state. 48 Chapter 3. Hawking radiation in Little String Theory As previously, the geometry at the horizon can be brought to a smooth form with a Painlev´e-like change of coordinates t→ˆ t−r√A(r)−χf(r) arctanh (r r0√1−χf(r) A(r))+ +r0√A(r) log [2r(√χ+√A(r))] .(3.78) After using (3.78) the metric field (3.76) is reduced to ds2=−f(r)dˆ t2+ 5 ∑ j=1 dx2 j−2√A(r)r0 rdˆ tdr +A(r)(dr2+r2dΩ2 3).(3.79) A calculation similar to (3.73) leads to the probability for a CHS black hole of mass Mto emit a shell of energy ω Γ∼exp (−2π√N+Mχ ω +χ ω2 4√N+Mχ +. . .),(3.80) where the ellipsis stand for terms proportional to higher powers of ω. Now for χ→1 (3.80) is clearly non-thermal while for χ→0 we recover once more the thermal emission (3.75). In view of this fact it seems wholly tenable that as the temperature is increased, βCHS →β0, the system evolves from non-thermal to thermal, and as a consequence an asymptotic observer could conjecture that the black hole internal degrees of freedom are reduced during the evaporation process and eventually one remains with a single state. The very same conclusions can be traced back from a stringy point of view if one consider the strings as the fundamental degrees of freedom of the black hole. In a flimsy language: as one approaches the Hagedorn temperature strings condense leaving a residual single one, a unique state that contains no information at all [67]. To substantiate this point we have computed, in the spirit of [68], some properties of a classical string located at the stretched horizon, i.e. a time-like curve slightly outside the global event horizon, that is of relevance in describing the evaporation process. We expect that for sufficiently large black hole masses both the proper distance between the stretched and the event horizon, ∼∫s.h. e.h.dr√grr, and the local Unruh temperature, Tloc(r) = 1 β√f(r),(3.81) are ballpark of the Planck order (up to a numerical factor of order 1). This implies that the stretched horizon must be almost coincident with the event horizon, rP≈ 3.3. Hawking radiation via tunneling 49 r0+δfor some positive and infinitesimal constant δ. Using (3.81) at the Planck radius and the Planck temperature, TP∼G−1/2, we obtain δ≈G√GM β2 0+ 4GMχ ,(3.82) where we have momentally reinstated the Newton constant Gin the proper spacetime dimension. For the CHS model δ∼√G/M, thus for large black hole masses one can consider that the stretched horizon is almost on top of the event horizon. As we increase the temperature the distance δalso increases up to reaching δ∼ G√GM/β2 0at the Hagedorn temperature. At this point the stretched horizon is displaced towards the distant observer and swallows up all of space, provided we ensure the validity of the supergravity approximation M∼r2 0N1.(3.83) In the CHS model all the thermodynamic quantities on the stretched horizon can be identify as those of the event horizon, with additional subleading terms suppressed by the black hole mass. This is in contrast with the outcome at the Hagedorn temperature where subleading contributions are no longer suppressed. Let us continue examining the classical behavior of the stretched horizon and visualize the “number of states”. For that purpose we calculate, in the two-dimensional flat Minkowski space at the Planck temperature TP, the mass of a ring shaped string located between the boundary and the event horizon. It reads m=∫√GM+δ √GM 2πrρPdr ≈       1 GM ,if χ= 1; M β2 0+O(GM β4 0),if χ= 0 (3.84) where we have used the behavior ρP∼G−2. Notice that (3.84) matches the result below (3.80): For the background (3.76) the string mass can be considered residual and in accordance the black hole mass remains to be almost ∼GM. Furthermore, the whole mass is localized inside the event horizon. As we increase the temperature the mass of the string forming a ring of radius rPis of the order of the black hole mass and hence there must be only a residual mass in the interior of the event horizon. With the expectation of a small distortion with respect to the flat Minkowski space the approach of (3.84) is fully justified in this latter case. One can regard this phenomenon as a progressive melting of the strings as they encounter Hagedorn temperature conditions [69]. The energy of the strings states is so large when the 50 Chapter 3. Hawking radiation in Little String Theory Hagedorn temperature is approached, that the strings on the horizon will tend to join forming a single one [70]. Thus the system evolves to a single state and consequently the entropy is reduced. This picture matches the view where black hole states at the Hagedorn temperature are in one to one correspondence with single string states. 3.3.3 Hawking emission via tunneling: Wrapped fivebranes The metric (3.5) is the ultraviolet completion of a large family group of regular nonabelian monopole solutions in N= 4 gauged supergravity, interpreted as 5-branes wrapped on a shrinking S2[62]. In the following we shall deal with a thermal deformation of one of such metrics dual to N= 1 SQCD with a superpotential coupled to adjoint matter [71]. Analyzing the emission problem with the method outlined in Section 3.3.1 leads to the same result obtained in (3.24), i.e. a constant outward flux of particles independent of the black hole characteristics. The metric field in Einstein frame is given by ds2=eφ0 2r[−K(r)dx2 1+ 4 ∑ j=2 dx2 j+Nα0(4 r2K(r)dr2+1 ξdΩ2 2+1 4−ξd˜ Ω2 2)+ +Nα0 4(dψ + cos θdϕ + cos ˜ θd ˜ϕ)2], K(r) = 1 −(r0 r)4.(3.85) In addition we have a dilaton field which is linear φ=φ0+rand a RR 3-form field. First of all we truncate the theory to two dimensions: the radial and temporal one. To cast (3.85) in Painlev´e coordinates we chose the function f(r) in (3.71) as f(r) = √Nlog K(r).Then the truncated theory equivalent to (3.85) is rewritten as ds2=eφ0 2r(−K(r)dx2 1+ 4Nα0dr2 r2K(r)−4√Nα0r2 0 r3dx1dt).(3.86) To calculate the semi-classical emission one needs the radial null geodesics of the back-reacted metric. The mass scales as M∼r4 0, then the emission of a shell with energy ωtranslates in a shift in the radius, so M−ω∼r4 1. This leads, after the emission, to the geodesic ˙r=1 2√Nα0r(r2 1 r2±1).(3.87) 3.4. Complex path and anomalies in LST 51 Its solutions are r2=r2 1(e±x1/√Nα0∓1), and one finds for timings the very same pattern as in the LST case. Inserting the outgoing solution of (3.87) in (3.117) one obtains ImS = π√Nω, from where follows once more the behavior (3.75). Thus, most probably, all metrics which asymptotic completion is LST will emit thermally. As in the LST case, one can check that using the mass density m=r4 0e2φ0N5/2 and entropy density s=r4 0e2φ0N2[54] the emission entropy in (3.75) turns to be directly related with Hawking-Bekenstein entropy, e−β0ω=e∆SBH . 3.4 Complex path and anomalies in LST In this section we reproduce the work [72] where we have studied the Hawking radiation of NS5 and LST using two semi-classical methods: the complex path and the gravitational anomaly. As in the previous section NS5 exhibits non-thermal behavior that contrasts with the thermal behavior of LST. We remark that energy conservation is the key factor leading to a non-thermal profile for NS5. In contrast, LST keeps a thermal profile even considering energy conservation because the temperature in this model does not depend on energy. Since the pioneering proposal of Hawking that black holes can radiate [8], much work has been done in order to obtain a complete theory of quantum gravity. When Hawking announced his amazing results, a new powerful paradox emerged. The information loss paradox with the apparent violation of unitarity principle has consequences on well-established quantum mechanics. A recent effort in order to solve this paradox has been done studying different semi-classical approaches such as the tunneling method, studied in the preceding section, proposed by Parikh and Wilczek [30, 32], the complex path analysis [38, 73, 74] or the cancellation of gravitational anomalies [75, 76, 77]. In order to develop our study we have reduced the ten-dimensional metric of LST to two-dimensional one, see (3.10−3.12). Momentally we make a comment on the validity of this truncation of the metric (3.5) to two dimensions. The interesting points concern: i) the fate of dimensional and field content reduction on the S3 modes is consistent [78]. ii) Furthermore, both the R5and S3wrap factors are 52 Chapter 3. Hawking radiation in Little String Theory independent of the (t, r) coordinates. As a consequence the equation of motions of these modes can be taken static and rindependent, i.e. the emission in the t−r plane does not alter the dynamics in the transverse coordinates to it. Hence all the physics will be analyzed within the propagation of massless particles in the r−t sector of the metric. We have verified that the NS5 model shows a non-thermal emission whereas LST shows a thermal emission. This last conclusion matches with the Hagedorn properties of LST, namely the temperature of LST corresponds to the Hagedorn temperature. Complex path method and anomalies yields the same results as the tunneling method, analyzed in [59], for the temperature and the emission rate. It is worth to mentioning that in the classical computation of the Bogoliubov coefficients all the results for emission rates shows thermal profiles due to the lack of energy conservation. This fact had driven Hawking to state that all the information that falls into the black hole is lost for ever, establishing in this way the information loss paradox. Nevertheless, one hopes to overcome this weird conclusion using semi-classical methods. 3.4.1 Complex path method The complex path method has been developed in [38], in order to calculate particle production in Schwarzschild-like space-time and it was extended for different coordinate representations of the Schwarzschild space-time [73, 74]. Nevertheless complex path analysis had already been discussed by Landau and Lifshitz [79], where it was used to describe tunneling processes in non-relativistic semi-classical quantum mechanics. We will follow the reference [38] in which the authors avoid to work in the Kruskal representation. They use the standard coordinates in the r−tsector. However the method presents a disadvantage because one finds a coordinate singularity at the horizon. Nevertheless using the techniques of complex integration one bypasses the singularity. We also want to mention that the method of complex path leads to the same results with those in [27]. In both methods, for the Schwarzschild space-time and also as we will see in this section for NS5 and LST space-time, it has been found that the relation between emission and absorption probabilities is of the form 3.4. Complex path and anomalies in LST 53 Pe=e−βωPa,(3.88) where ωis the energy of the emitted particles. We are tempting to compare this relation with the standard thermal Boltzmann distribution for blackbody radiation where β−1is identified with the Hawking temperature. We have verified that this is the case, if we compare our results with the temperature calculated using the definition of surface gravity for example. It is noteworthy to say that this method allows one to derive temperatures for black holes comparing probabilities of emission and absorption but it is not able to calculate the spectrum of thermal radiation. In that sense the tunneling method is so far incomplete. To remove this shortcoming the authors in [80] presented a new mechanism. In order to apply the complex path method to NS5 and LST we have constructed the semi-classical action obtained from Hamilton-Jacobi equations. Then we have computed the semi-classical propagator K(r2, t2;r1, t1). Eventually we have calculated the emission and absorption probabilities. We consider the equation of motion of a massless scalar particle φ= 0 in the background (3.12), −A(r)∂2 ∂t2φ(t, r) + f(r) r3 ∂ ∂r [r3f(r)∂ ∂rφ(t, r)]= 0 .(3.89) Using the standard ansatz solution φ(t, r)∼ei ~S(t,r),(3.90) and substituting in (3.89) we get an expression in terms of the action S(t, r), −A(r)(∂S ∂t )2 +f(r)2(∂S ∂r )2 + +~ i[−A(r)∂2S ∂t2+f(r)2∂2S ∂r2+f(r) r3 d(r3f(r)) dr ∂S ∂r ]= 0 ,(3.91) where we have collected the terms with ~dependence. The following step is to write the action as an expansion in a power series of (~ i), S(t, r) = S0(t, r) + (~ i)S1(t, r) + (~ i)2 S2(t, r) + ... . (3.92) 54 Chapter 3. Hawking radiation in Little String Theory Substituting the above expansion in (3.91) and neglecting terms of order (~ i) and higher, we obtain a non-linear first order partial differential equation which corresponds to the Hamilton-Jacobi equation of motion to the leading order in the action S, −A(r)(∂S0(t, r) ∂t )2 +f(r)2(∂S0(t, r) ∂r )2 = 0 .(3.93) We are interested in the evaluation of the semi-classical propagator which inform us about the amplitude for a particle going from r1at time t1to r2at time t2. In the saddle point approximation we get K(r2, t2;r1, t1) = Nexp [i ~S0(r2, t2;r1, t1)],(3.94) where Nis a normalization constant. Applying separation of variables in (3.93) we get S0(r2, t2;r1, t1) = −ω(t2−t1)±ω∫r2 r1√A(r) f(r)dr , (3.95) the plus/minus sign corresponds to ingoing/outgoing particles respectively and ωis the energy of the emitted or absorbed particle. The integral (3.95) is not well behaved if the horizon r0is within the region of integration. This turns to be the case since we are interested in the emission of particles through the event horizon, so the region of integration runs from inside the horizon to outside. First we consider the propagation of an outgoing particle in the inner region r1< r0. Applying the usual complex analysis tools, we deform the contour of integration around the pole r0in the upper complex half-plane. Obtaining for the radial part of (3.95) Se 0=iπω 2r0√A(r0).(3.96) We will call it emission action because we simply consider the emission of an outgoing particle propagating from inside the horizon to the outside. In the same way one proceeds with analogous analysis to evaluate the action at lowest order for absorbed particles. In that case we are considering the propagation of an ingoing particle in the outer region, r0< r2. Deforming the contour of integration in the upper complex half-plane, eventually we obtain the same result as the emission process up to a change of sign. Now we are obtaining the absorption 3.4. Complex path and anomalies in LST 55 action for a particle that propagates from the region outside of the horizon to the inside Sa 0=−iπω 2r0√A(r0).(3.97) We are interested in the expressions (3.96) and (3.97) in order to evaluate the probabilities of the emission and absorption processes. Thereby using the definition of the probability: P=|K(r2, t2;r1, t1)|2, and substituting the expression for the corresponding actions, we finally obtain for the emission and absorption probabilities Pe∼exp [−π ~ωr0√A(r0)], Pa∼exp [π ~ωr0√A(r0)],(3.98) where we have omitted the normalization constants. Eventually we are interested in writing the relation between emission and absorption probabilities, Pe= exp [−2π ~ωr0√A(r0)]Pa.(3.99) At first sight we observe that the absorption process dominates over the emission, it is easier for the system to absorb than to radiate particles. Also we note some misleading behavior in the expression for the absorption probability (3.98), because we could think that one might get a probability absorption greater than 1. However we only have considered the spatial contribution of the action in order to calculate the probabilities of emission and absorption processes. Instead of this we must also have considered the time contribution as proposed in the work [81]. Comparing (3.99) with the same relation in a thermal bath of particles (3.88), we can identify the temperature of our system (taking ~= 1 and ms= 1) as T=1 2πr0√A(r0)=1 2π√χr2 0+N,(3.100) that coincides with the value of temperature obtained in (3.24). So far we have studied NS5/LST systems without backreaction. The next step is to consider the backreaction of the metric due to the emission process. Our starting point in the evaluation of the backreaction is the expression of the action for the emission process (3.96). In our NS5/LST model we have the following relation between the event horizon and the mass of the black hole: r2 0∼M, where Mis the mass of the black hole and the factors omitted here are not relevant to our study. 56 Chapter 3. Hawking radiation in Little String Theory When the metric backreacts in the emission process the energy conservation implies that r2 0→r2 0−ω. The shrink of the event horizon rides the tunneling emission between turning points defined just inside and just outside of the event horizon. Once the emission has been carried out we perform the previous change in (3.96), Se 0=iπ 2ω√χ(r2 0−ω) + N . (3.101) Expanding in low energies we get Se 0=iπ 2(ω√χr2 0+N−χω2 2√χr2 0+N+O(ω)3).(3.102) Calculating the emission probability for both models we obtain Pe∼         exp [−π ~(ω√r2 0+N−ω2 2√r2 0+N+...)] if χ= 1 (NS5); exp [−π ~ω√N]if χ= 0 (LST). (3.103) We see higher-order correction terms corresponding to the NS5 emission probability, which indicate that the emission is not purely thermal. On the other hand the emission probability expression corresponding to the LST model is exact, which indicates that the emission is purely thermal. In this work, we have concluded that the results obtained from the tunneling formalism in [30] are nothing more than an extension of the Hamilton-Jacobi formalism taking into account the energy conservation, which induces the backreaction of the event horizon. In our particular case we are facing with an anomalous model, in the sense that it does not fulfill the previous expectations about non-thermal emission. The LST model emits thermal radiation irrespective whether the energy conservation holds, or not. In order to analyze the deviations from the thermal behavior of the NS5 model it would be relevant to perform the computation of the greybody factors. So that we must solve the radial part of the equation of motion (3.89). As far as we know this equation cannot be solved analytically, therefore a numerical analysis is needed in order to show up the non-thermal character of the NS5 model. Even so, we can elucidate that the non-thermal behavior of the NS5 model comes from the throat region. In this region the dilaton grows linearly pointing out that gravity becomes 3.7. Discussion and remarks 63 density are proportional, hence it follows that e−βω =e∆SBH . This matches the statistical picture in which large fluctuations are suppressed and supports the idea that in this background the Bekenstein-Hawking area-entropy relation, SBH =A/4, can be obtained by counting the degeneracy states [11]. 3.7 Discussion and remarks We have reviewed briefly some aspects about LST thermodynamics. We have exposed the thermal emission of LST due to the non-energy dependence of the Hagedorn temperature. In addition, we have evaluated the temperature experienced by a scalar particle-like observer, thereby we have verified that the Hagedorn temperature of LST is a maximum bound. Furthermore we have studied the Hawking radiation of the NS5 and LST black hole models using two semi-classical emission methods: the complex path method and the cancellation of the gravitational anomaly. It shall be stressed that using both methods we have recovered our previous results derived in [59] where we worked using the tunneling formalism. The complex path method [38, 74] shows how to evaluate the emission rate in the framework of the HamiltonJacobi formalism. We have proved that imposing the energy conservation, in order to take into account the backreaction of the metric during the emission process, we reproduce exactly the same results with those derived in the tunneling formalism [30, 32]. We would like to point out the advantage of the complex path method over the tunneling method. First of all, we avoid heuristic explanations about the tunneling mechanism in the process of the emission. Secondly, we work with the well-known Hamilton-Jacobi equations plus the imposition of the energy conservation. Finally, it is not necessary to change the standard coordinates of the metric into Painlev´e coordinates. We conclude that the tunneling method is nothing more than the complex path method plus energy conservation. We have verified that another successful method to evaluate Hawking radiation in NS5 and LST models is that of the cancellation of the gravitational anomaly [75]. However, it is argued that the method fails for non-asymptotically flat space-times like de Sitter space-time and Rindler space-time. In [85] it is proposed a new method based on the chiral nature of field theories in the near horizon region of the black hole, but does not depend on the existence of a chiral anomaly. It defines a new effective energy-momentum tensor (either in consistent or in covariant form same 64 Chapter 3. Hawking radiation in Little String Theory results are obtained) Tµ ν=Tµ ν∓Nµ νthat is conserved ∂µTµ ν= 0. The physics in the whole part of the manifold external to the horizon can be described without the need to decompose the background in two pieces, i.e. near horizon and asymptotic. Thus the new method works even for non-asymptotically flat space-times as de Sitter and Rindler space-times. Summarizing, we have shown that all the above methods lead to a non-thermal emission for the NS5 black hole and to a thermal one for the LST black hole, see (3.103). The latter can be interpreted as the thermal limit of the former. The entire process of black hole evaporation, except for the final period when the black hole is of Planckian size, can be summarized according to the following patterns: Starting from the NS5 system at a given temperature we checked, in a semi-classical approximation, that the black hole emission is non-thermal (3.80). The black hole contains many degrees of freedom coupled with its environment. At this point the system is thermodynamically irreversible, and the entropy of the surrounding increases as the black hole emits. As the emission takes place the black hole temperature increases while, both the mass and the emission rate, decrease and the latter becomes pure thermal at the Hagedorn temperature (3.75). The interference term vanishes at this point and the black hole system is thermodynamically reversible and consists of a single state. This single state radiates, while the black hole temperature remains completely independent of its mass. Thus, as the LST black hole evaporates, its energy flux is exactly constant. Once this point is reached, one could think that we deal with a stable remnant with zero entropy. That this is not the case can be inferred from the stringy corrections to the entropy as a function of the energy. This gives a thermodynamically unstable system [45] which in turn implies that the probability of emission diverges. In order to have a rough idea of the latter effect we use the area law relation but incorporating its first quantum corrections Sc=Area 4+αlog (Area 4)+γ Area +. . . . (3.125) Taking into account the relations of the mass and energy densities, the black hole emission (3.75) is replaced at leading order by Γ∼(Area1 Area0)α e∆SBH =(1−ω M)αe−β0ω.(3.126) The above expression together with the fact that the value of αis negative –the system is unstable– shows that the trend in (3.126) is that as the system evolves in 3.7. Discussion and remarks 65 time the emission increases, i.e. without further considerations at play the system would fully evaporate without leaving any relic behind it. This fact is clearly driven by the sign of α, which is negative. For a more complete discussion see [86]. The above picture relies in a truncation of (3.125) and as one approaches Planck scales one must consider that subleading contributions in (3.125) are enhanced and they wash out any solid conclusion. We also have found that for theories which their ultraviolet completion is LST, the radiation is also that of a blackbody at a fixed temperature (3.24). Finally, the cluster decomposition principle is a crucial physical requirement which states that very distant experiments produce uncorrelated results, thus establishing the local behavior of the field theory. Cluster decomposition principle states that if multi-particle processes are performed in Nvery distant laboratories, then the S-matrix element for the overall process factorizes. This factorization ensures a factorization of the corresponding transition probabilities, corresponding to uncorrelated experimental results. In the line of the works [87, 88] where the authors linked the existence of correlations among tunneled particles and the entropy conservation of the full system (black hole plus Hawking radiation), we have calculated the successive emission probabilities for two particles of energies ω1and ω2using (3.103) for each model, respectively. We have found that the NS5 model does not satisfy cluster decomposition ln |Γ(ω1+ω2)| −ln |Γ(ω1)Γ(ω2)|=ω1ω2 2√N+r2 0 .(3.127) On the other hand we have found that the LST model satisfies cluster decomposition as we expected ln |Γ(ω1+ω2)| −ln |Γ(ω1)Γ(ω2)|= 0 ,(3.128) where Γ(ω1) and Γ(ω2) are the emission probabilities corresponding to a particle of energy ω1and ω2, respectively and Γ(ω1+ω2) is the emission probability of a particle with energy ω1+ω2. We have found that Γ(ω1|ω2) = Γ(ω1+ω2) is fulfilled at low energies, namely, the emission probability of a particle ω2conditioned by the previous emission of a particle ω1is the same as the emission of a single particle of energy ω1+ω2. With these results at hand we can conclude that NS5 black hole there are correlations among emitted particles. This fact is intimately related with the 66 Chapter 3. Hawking radiation in Little String Theory non-thermal emission rate (3.103). Regarding (3.127) one hopes that the successive Hawking emissions could preserve unitarity avoiding in such a way the information loss paradox. However it is not the case for the LST black hole where the thermal emission rate (3.103) leads us to cluster decomposition. Therefore the successive emissions of particles are independent among them, and thus the information of the initial states remains hidden. 3.8 Spectrum So far we have calculated the temperature and the emission rate of NS5 and LST black holes. Nevertheless, in order to obtain a complete study we must evaluate the emission spectrum even taking into account the back-reaction of the metric. In [80], the authors calculated the Hawking radiation spectrum corresponding to a spherically symmetric static black hole. It is our aim to perform a similar analysis corresponding to NS5 and LST black holes. Our starting point will be the twodimensional action (3.11) at leading order (3.95). This action can be written as S0(r, t) = ω(t±r∗),(3.129) where r∗is the tortoise coordinate defined as dr∗=√A(r) f(r)dr . (3.130) Then if we consider the outgoing/ingoing null coordinates u=t−r∗, v =t+r∗,(3.131) we can define the right/left modes inside and outside of the black hole in the following way, φR in =e−i ~ωuin , φL in =e−i ~ωvin , φR out =e−i ~ωuout , φL out =e−i ~ωvout .(3.132) The Kruskal coordinates corresponding to the inside and outside of the NS5 and LST black holes are defined, see [1], as Tin =eκr∗ in cosh(κtin), Xin =eκr∗ in sinh(κtin), Tout =eκr∗ out sinh(κtout), Xout =eκr∗ out cosh(κtout),(3.133) 3.8. Spectrum 67 where κis the surface gravity corresponding to each model. The two sets of Kruskal coordinates are then connected: Tin →Tout and Xin →Xout, by the following transformation relation between the coordinates tand r tin →tout −iπ 2κ, r∗ in →r∗ out +iπ 2κ.(3.134) Moreover, the null coordinates are also transformed as uin →uout −iπ κ, vin →vout .(3.135) Eventually we have obtained a transformation relation between the left/right modes inside and outside the black hole, φR in →φR out e−πω ~κ, φL in →φL out .(3.136) This last relation is precisely the relation that one obtains between the Bogoliubov coefficients in the standard study of the emission of Hawking radiation (2.31) and (3.66). 3.8.1 Blackbody spectrum We are going to construct the density matrix operator corresponding to an outside observer for an nnumber of bosons and fermions. In this way we will be able to compute the average number of particles (bosons or fermions) emitted by the black hole. We will see that the emitted spectrum corresponds to the blackbody spectrum without taking into account the back-reaction of the metric. For a more detailed calculations see Appendix E. We construct the physical state associated to a system of nnumber of noninteracting virtual pair of particles created inside the black hole, |ψi=N∑ n|nL ini⊗|nR ini.(3.137) Applying the relations between modes (3.136), we get the physical state measured by the observer outside of the black hole which is necessary to evaluate the density matrix operator, |ψi=N∑ n e−πωn ~κ|nL outi⊗|nR outi.(3.138) 68 Chapter 3. Hawking radiation in Little String Theory The normalization constant Nis determined using the orthonormalization condition hψm|ψni=δmn , N=(∑ n e−2πωn ~κ)−1 2 .(3.139) For bosons (n= 0,1,2, ...), and fermions (n= 0,1), the normalization constant is respectively Nb=(1−e−2πω ~κ)1 2, Nf=(1 + e−2πω ~κ)−1 2.(3.140) Henceforth we will only perform the calculations for the boson state system, and one can proceed in analogous way for the fermion system. We write the density matrix operator for the boson system as ρb=|ψbihψb|=(1−e−2πω ~κ)∑ n,m e−πω(n+m) ~κ|nL outi⊗|nR outi hmL out|⊗hmR out|,(3.141) where |noutiand |moutiare orthonormalized outside eigenstates. Then tracing over the left modes we get the matrix density operator in terms of the right eigenmodes, ρR b=(1−e−2πω ~κ)∑ n e−2πωn ~κ|nR outihnR out|.(3.142) Now we compute the average number of particles detected at asymptotic infinity using the relation hni=Tr(nρR), where the trace is taken over |nR outieigenstates. We have obtained for bosons and fermions respectively hnbi=1 e2πω ~κ−1,hnfi=1 e2πω ~κ+ 1 .(3.143) Both distributions correspond to a blackbody spectrum with Hawking temperature defined as TH=~κ 2π. We can identify the Bose-Einstein distribution for bosons and the Fermi-Dirac distribution for fermions. 3.8.2 Hawking radiation flux Integrating over all energy range the expression for the average number of particles (3.143), we will give the flux of bosons and fermions, respectively, seen by an asymptotic observer, F∞=1 2π∫∞ 0hnbiω dω =~2κ2 48π=π 12T2 H, F∞=1 2π∫∞ 0hnfiω dω =~2κ2 96π=π 24T2 H.(3.144) 3.8. Spectrum 69 3.8.3 Back-reaction spectrum During the emission of the Hawking radiation we have not considered the backreaction of the metric. In this section our aim is to evaluate it. In [72] we had showed that the back-reaction effect was introduced imposing the energy conservation in the framework of the Hamilton-Jacobi formalism. We need to know how the modes are affected by the back-reaction of the metric. Looking at the transformation expression between the modes (3.136), we write the new transformation relation between the modes taking into account the back-reaction as ˜ φR in →˜ φR out e−πω ~˜κ,˜ φL in →˜ φL out ,(3.145) where ˜ φR,L in,out are the back-reacted modes and ˜κ=1 √χ(r2 0−ω) + N m2 s ,(3.146) is the back-reacted surface gravity for the NS5 and LST models. In order to calculate the emission probability corresponding to a mode that tunnels through the event horizon and emerges to the outside of the black hole, we use the above relation between the right (outgoing) modes (3.145), obtaining Pe=|˜ φR in|2→ |˜ φR out e−πω ~˜κ|2=e−2πω ~˜κ.(3.147) Therefore if we make an expansion at low energies we obtain for the emission probability observed by an asymptotic observer Pe= exp   −ω TH  1−χω 2(χr2 0+N m2 s)−χ2ω2 8(χr2 0+N m2 s)2+...    ,(3.148) where THis the Hawking temperature (3.24). This result coincides with [59, 72] but now we have avoided the problem of the temporal term showed in [81]. Moreover, we have evaluated the absorption probability corresponding to an incoming mode; in this case the left (incoming) mode that propagates toward the center of the black hole does not change, see (3.145), whereby the absorption probability will be Pa=|˜ φL in|2→ |˜ φL out|2= 1 .(3.149) Thus using the principle of detailed balance, Pe=e−βωPa, we see from (3.147) and (3.149) that ˜ β=2π ~˜κ. Then we can conclude that there exists an effective temperature 70 Chapter 3. Hawking radiation in Little String Theory for the back-reacted NS5 and LST black holes whose value is ˜ T=˜ β−1=~ 2π√χ(r2 0−ω) + N m2 s .(3.150) Of course the effective temperature (taking into account higher order terms in ω) that appears in the emission rate (3.148) is nothing more than the expansion at low energies of the effective temperature (3.150). Eventually we can write the emission probability (3.148) in a more compact form as Pe=e−ω/ ˜ T,(3.151) which resembles the thermal emission corresponding to a perfect blackbody at temperature ˜ T. In the spirit of the work [89], we could think that when we do not consider the back-reaction of the metric the emission is purely thermal, Pe∼e−ω/TH, with a spectrum corresponding to a blackbody with temperature (3.24). On the other hand, when we are taking into account the back-reaction, i.e. energy conservation, the emission is not strictly thermal. Nevertheless, we can define an effective temperature (3.150) and consider that the black hole is emitting as a black body with this effective temperature, (3.151). We also see that the deviation from pure thermal behavior of the spectrum is ˜ T TH = √1−χω χr2 0+N m2 s  −1 .(3.152) Furthermore the results for the number of emitted particles (3.143) and fluxes (3.144) are subjected to the back-reaction effect through the factor e2πω ~˜κ, in which the backreacted surface gravity, ˜κ, appears. We may also expand at low energies the equations (3.143) and (3.144), obtaining higher order energy terms and thus the new spectrum deviates from the pure blackbody radiation spectrum. 3.9 Greybody factor In this section we will compute the decay rate of an excited black hole into neutral scalars, Γ = σabs ρ(ω T)d4k (2π)2,(3.153) 3.9. Greybody factor 71 given in terms of the thermal factor ρ(ω T)=1 eω/T −1,(3.154) and the classical absorption cross section, which corresponds to the greybody factor. In order to calculate the greybody factor of LST black hole we have basically followed the works [21, 23, 24]. We start considering the Klein-Gordon equation φ= 0 describing the propagation of a massless s-wave scalar particle minimally coupled to the fixed background (3.5). We obtain the absorption cross section as the ratio of the flux into the black hole at the future horizon to the incoming flux from the infinity, σabs =Fabs Fin .(3.155) Since we are interested in the r−tsector of the metric we must solve the equation (3.89) in the background (3.12). In terms of the new variable z=f(r), the equation (3.89) becomes z∂ ∂z (z∂ ∂z φ(z))+α (1 −z)2φ(z) = 0 ,(3.156) with α≡ω2N 4m2 s ,(3.157) hereafter for simplicity we take ms= 1. There exist two possible approximations: i) the low-energy regime, ω√N1 and ii) the dilute gas region, r0N, for which the system resembles the D1−D5 system in the limit r0, r1, rnr5. Performing a function substitution of the form φ(z)≡zα(1 −z)βF(z), the equation (3.156) can be reduced to an hypergeometric equation, see e.g. [24]. Provided we choose α±=±iω√N 2, β±=1 2(1±√1−ω2N),(3.158) the solution of (3.156) becomes φ(z) = C1zα+(1 −z)β±F(α++β±, α++β±; 1 + 2α+;z) + C2zα−(1 −z)β±F(α−+β±, α−+β±; 1 + 2α−;z),(3.159) where C1and C2are constants. The boundary conditions are •At the event horizon (z→0): purely ingoing waves. •At spatial infinity (z→1): purely outgoing waves. 72 Chapter 3. Hawking radiation in Little String Theory Taking into account the first boundary condition we pick the first term on the right hand side of (3.159) as solution. Furthermore, both roots of βgive the same result, thus henceforth we drop the subindex. Expanding the solution for large r(or equivalently z→1) and neglecting the divergent solution, we obtain an asymptotic solution in the inner region φa(r) = Ca Γ(1 −iω√N) Γ(−√1−ω2N) Γ(1−iω√N−√1−ω2N 2)2(r0 r)1+√1−ω2N,(3.160) where Cais a constant. Evaluating the asymptotic solution in the outer region directly from equation (3.156), and using the Frobenius method it is found, in terms of the rvariable, φa(∞) = √π 2(A1r−1−√1−ω2N+A2r−1+√1−ω2N),(3.161) where A1and A2are constants. Then if we match both solutions we find a relation between the constants, Ca=√π 2r−1−√1−ω2N 0 Γ(1−iω√N−√1−ω2N 2)2 Γ(1 −iω√N) Γ(−√1−ω2N)A1, A2= 0 .(3.162) Imposing the second constraint one neglects the divergent modes at asymptotic infinity. In order to obtain the behavior near the event horizon we expand the ingoing mode solution (3.159) around r0(or equivalently z→0), φh(r) = Ch(1−r2 0 r2)−iω√N 2 ,(3.163) where Chis a constant. Then we match the solutions (3.160) and (3.163) at the matching point rm, which fulfills r0rmr5=√N, see [21], we thus obtain a relation between constants, Ca=(r0 r)−1−√1−ω2N(1−r2 0 r2)−iω√N 2Γ(1−iω√N−√1−ω2N 2)2 Γ(1 −iω√N) Γ(−√1−ω2N)Ch.(3.164) Eventually comparing (3.162) with (3.164), we obtain the desired relation between the constant in the asymptotic solution and the constant of the near horizon solution, A1=√2 π(1−r2 0 r2)−iω√N 2(1 r)−1−√1−ω2N Ch.(3.165) 4.2. Fermion modes and greybody factor 79 equivalently we could also study the spin-down case. Therefore the Dirac equation becomes (−i √f(r)∂t+m)Ψ+(t, r)−i[√f(r) A(r)∂r+f0(r) 4√A(r)f(r)]Ψ−(t, r) = 0 , (i √f(r)∂t+m)Ψ−(t, r) + i[√f(r) A(r)∂r+f0(r) 4√A(r)f(r)]Ψ+(t, r) = 0 . (4.18) Next, we consider the following ansatz for the spinor field Ψ+(t, r) = φ+(r)e−iωt ,Ψ−(t, r) = iφ−(r)e−iωt .(4.19) Substituting this expressions into (4.18) and after doing algebra we obtain the following set of equations ∂rφ−(r) + f(r)0 4f(r)φ−(r) + (m√A(r) f(r)−ω√A(r) f(r))φ+(r) = 0 , ∂rφ+(r) + f(r)0 4f(r)φ+(r) + (m√A(r) f(r)+ω√A(r) f(r))φ−(r) = 0 . (4.20) We can solve this set of coupled equations. If we define η±(r)≡m√A(r) f(r)±ω√A(r) f(r),(4.21) we will obtain η−1 +(r)φ+(r)00 +(∂rη−1 +(r) + η−1 +(r)f(r)0 2f(r))φ+(r)0+ +(∂rη−1 +(r)f(r)0 4f(r)+η−1 +(r)∂r(f(r)0 4f(r))+η−1 +(r)(f(r)0 4f(r))2 −η−(r))φ+(r) = 0 . (4.22) In order to simplify the resolution of the above equations, we consider the propagation of a massless fermion through the LST background (3.12). Substituting the values of f(r) and A(r) given in (3.8), into (4.21) and (4.22), eventually we obtain the propagation equation for a massless fermion mode, 4r2(r2−r2 0)2φ+(r)00+4r(r2−r2 0)(r2+2r2 0)φ+(r)0+(4ω2Nr4−4r2 0r2+5r4 0)φ+(r) = 0 . (4.23) This equation admits the following solution φ+(r) = √r(r2−r2 0)−1/4(C1(r2−r2 0)−i 2ω√N+C2 2iω√N(r2−r2 0)i 2ω√N),(4.24) 80 Chapter 4. Emission of fermions in LST where C1and C2are arbitrary constants. The gravitational potential barrier around the black hole acts as a filter for the emitted radiation, therefore the spectrum detected at the asymptotic infinity is not a pure Planckian spectrum. The greybody factor accounts for this deviation from the purely blackbody spectrum, see (2.39). However LST exhibits a different behavior; the non-dependence of its temperature on the black hole mass leads to the fact that the emission is purely thermal, even taking into account back-reaction effects. Therefore, one expects that the spectrum shall be purely Planckian and the greybody factor takes the value 1, as we verified in Section 3.9. for massless scalar particles. But now, we are going to verify this assumption computing explicitly the greybody factor corresponding to the emission of a massless fermion in a twodimensional effective background (3.12). We will follow the method of matching the solutions at asymptotic infinity of the black hole and near the horizon at a matching point rm, see references [21, 22, 23, 24]. Basically we must calculate the flux F=1 2i(φ∗ +(r)r3f(r)∂rφ+(r)−c.c.)(4.25) near the horizon of the black hole and at the asymptotic infinity. The ratio of the two fluxes is the absorption cross section, i.e. the greybody factor of the black hole. The mode solution at the near horizon limit is obtained imposing the propagation of ingoing modes as a boundary condition. Then if we expand the solution (4.24) near the horizon, we obtain φh(r) = Ch(r−r0)−1 4−i 2ω√N,(4.26) where we have collected all the terms that are independent of the radial coordinate in the constant Ch. The flux (4.25) calculated at the near horizon limit is Fh=|Ch|2 2ω√Nr(r+r0) √r−r0 .(4.27) Next we calculate the mode solution at the asymptotic limit. We must take into account that in this limit the metric function f(r) fulfills the relation lim r→∞f(r) = 1 .(4.28) Then, we solve equation (4.22) for the massless case using (4.28), and we obtain for the modes solution at the asymptotic limit φ∞(r) = C∞riω√N.(4.29) 4.2. Fermion modes and greybody factor 81 Now, the flux (4.25) computed in the asymptotic limit is F∞=|C∞|2 2ω√N r2.(4.30) In order to find a relation between the constants Chand C∞we match both solutions at the matching point rm, which fulfills r0<< rm. Hence imposing the matching condition: φh(rm) = φ∞(rm), we find the following relation between the constants |C∞|2=|Ch|2 √rm−r0 .(4.31) Finally, if we calculate the greybody factor as the ratio of the ingoing flux through the horizon, Fh, to the outgoing flux at the asymptotic limit, F∞, we obtain Γω≡|Fh| |F∞|= 1 .(4.32) This result indicates that for LST we will obtain a pure Planckian spectrum, ρ(ω) = 1 (eω/T −1) dω 2π,(4.33) in accordance with the result of Hawking, see (2.36). Effectively, one would expect this result since we have demonstrated how LST exhibits a purely thermal behavior, even taking into account the back-reaction of the metric. 82 Chapter 4. Emission of fermions in LST Chapter 5 Back-reaction and quantum corrections In a recent work we have shown how the back-reaction can be treated as a quantum correction, [96]. The novel semi-classical approach which will be presented here consists of the introduction of adequate quantum corrections into the r−tsector of the black hole metric. Thus, we will obtain corrected values for the temperature, entropy and emission rate, which at leading order coincide with the results derived in the tunneling approach. Comparing this approach some semi-classical methods as: the tunneling method, the complex path analysis or the cancellation of gravitational anomalies; we conclude that we obtain similar results for the emission rate and Bekenstein-Hawking entropy, however we alsonotice the appearance of new terms. We also apply this technique to the Little String Theory. Interestingly, we find similar results for the entropy with those using string one-loop calculations, e.g. we have found the classical Bekenstein-Hawking entropy plus a logarithmic correction term. We have seen in the previous chapters that during the radiation emission of black holes we enforce energy conservation, thus the metric back-reacts and the event horizon shrinks. When the black hole radiates the total ADM mass [36] is conserved, whereas the mass of the black hole decreases by the same amount of the energy that has been released by emission. According to the heuristic picture most commonly considered [31], the quantum vacuum fluctuations generate a pair of virtual particles; one member of the pair, for example the anti-particle, falls into the black hole while the other member of the pair, i.e. the particle, escapes towards 83 84 Chapter 5. Back-reaction and quantum corrections the asymptotic infinity. The net effect would be as if the black hole had emitted a particle at the expenses of slowly decreasing its mass. Accordingly, we must consider the quantum nature of the emission process; thereby, we have been led to introduce quantum perturbations into the original static metric of the black hole in order to evaluate the back-reaction. In this work we have considered a general metric with some sort of perturbations of quantum character. Eventually, we want to show that the back-reaction of the metric, imposing energy conservation, can be viewed as a quantum perturbation. Furthermore, we have analyzed the same sort of perturbations in LST. 5.1 Quantum correction on the metric Consider a general metric in conformal-string frame with spherical symmetry defined in a d-dimensional space-time, ds2=−f(r)dt2+g(r) f(r)dr2+h(r)r2dΩ2 d−2.(5.1) The event horizon is found at the radial coordinate position r0and dΩ2 d−2defines the (d−2)-sphere. Since the radiation emission depends only on the r−tsector of the metric, we are going to slightly modify those terms of the metric, furthermore we want that these changes on the metric accounts for quantum effects. In [97] the authors introduced quantum corrections considering all the terms in the expansion of a single particle action. Motivated by this work, we introduce the following perturbations on the radial and time part of the metric (5.1), δgtt =−f(r)∑ i ξi~i ξi~i+r(d−2)i 0 , δgrr =g(r) f(r)∑ i ξi ~i r(d−2)i 0 ,(5.2) thus the slightly perturbed metric, ˆgµν =g(0) µν +δgµν, can be written as ˆ ds2=−f(r)(1 + ∑ i ξi ~i r(d−2)i 0)−1 dt2+g(r) f(r)(1 + ∑ i ξi ~i r(d−2)i 0)dr2+ +h(r)r2dΩ2 d−2,(5.3) where ξiare positive dimensionless parameters. This choice of the perturbations has been motivated by dimensional analysis. The reduced Planck length (˜ lP=lP 2π) in a 5.1. Quantum correction on the metric 85 d-dimensional space-time is defined as ˜ ld−2 P=~G(d) c3, where G(d)is the d-dimensional Newton’s constant. In natural units (G=c= 1) we obtain the following dimensional relation [˜ ld−2 P] = [~]. Since for the black hole metric (5.1) we have only one parameter with length dimensions, i.e. the event horizon r0; we conclude that rd−2 0must be proportional to ~. The perturbed metric expression (5.3) deserves a few comments. Firstly, we should verify whether it is a solution of the Einstein equations. In fact, we notice that this is the case since the perturbations are independent of any of the coordinates. Secondly, we point out the modification of the particles velocity in the region near the event horizon. Causal propagation is limited to time-like and null particle trajectories with respect to the background (5.3), therefore in the case of null coordinates we find that the maximum velocity of photons has been shifted to a new value, ˆc=c(1 + ∑ i ξi ~i r(d−2)i 0)−1 .(5.4) In any case, we do not obtain superluminal propagation velocities. Eventually, we verify that the null energy condition is not affected by the inclusion of quantum perturbations, thus Tµνeµeν≥0, or equivalently Rµνeµeν≥0 for any null vector eµ, is fulfilled near the event horizon. Next, we are interested in studying how the Hawking temperature of the black hole is modified by the above perturbations. As usual, if we introduce the euclidean time, τ=it, we get the corresponding Euclidean positive definite metric. Furthermore, taking into account the definition of the proper length, dρ2=grrdr2, together with the expansion of the metric function near the event horizon, f(r) = f0(r0)(r−r0), we can define a new radial coordinate as ρ= 2√g(r)(r−r0) f0(r)r→r0(1 + ∑ i ξi ~i r(d−2)i 0)1/2 .(5.5) We write the metric in Rindler coordinates, ˆ ds2 E=ρ2  f0(r) 2√g(r)r→r0(1 + ∑ i ξi ~i r(d−2)i 0)−1 dτ  2 +dρ2+h(r)r2dΩ2 d−2,(5.6) where we point out the presence of the modified surface gravity due to the correction 86 Chapter 5. Back-reaction and quantum corrections terms, ˆκ=f0(r) 2√g(r)r→r0(1 + ∑ i ξi ~i r(d−2)i 0)−1 .(5.7) We can remove the apparent conical singularity at the event horizon in (5.6) by identifying the imaginary (Euclidean) time coordinate with the period β=2π ˆκ. We find that the effective temperature corresponding to the perturbed black hole is ˆ T=~ˆκ 2π.(5.8) In this equation it is easily seen that the new temperature is just the standard Hawking temperature TH=~ 4π f0(r) √g(r)r→r0,(5.9) corrected by quantum perturbations. 5.2 Back-reaction viewed as a quantum correction We would like to analyze how the metric is affected by the back-reaction, and consequently if we can consider such back-reaction of the metric as a quantum effect. Motivated by the idea that the emitted particles are quantum fields whose energy, ωin natural units (~= 1), is also quantized; our aim is to show if we can treat the back-reaction of the metric as a quantum perturbation. In order to interpret properly the quantum perturbation of the back-reacted metric, it is useful to show the relation between the mass and the event horizon of the black hole. For that purpose we have calculated the Komar integral, see Appendix C, associated with the time-like Killing vector Kν. For the background (5.1) we have found the following relation, M=Vol(Sd−2) 8(d−3)πG(d) f(r)0 √g(r)(r√h(r))d−2r→r0,(5.10) where Vol(Sd−2) stands for the volume of the (d−2)−sphere and all quantities are evaluated at the event horizon. Moreover, we also impose the following three conditions on the space-time metric: 5.2. Back-reaction viewed as a quantum correction 87 1. Spherical symmetry. 2. The background is asymptotically flat. 3. The metric function f(r) is expressed as f(r) = 1 −(r0 r)d−3, depending on the mass through the event horizon r0. For future convenience we write the metric functions g(r) and h(r) as (1 + r2 i,j r2), depending on the charges riand rj, respectively, which are different from the mass charge. With this choice for the metric functions we see from the relation (5.10) that M∝rd−3 0. Taking into account the above three conditions, and expanding in the energy of the emitted particle ω, we eventually write (5.1) as ˜ ds2=−˜ f(r)dt2+g(r) ˜ f(r)dr2+h(r)r2dΩ2 d−2,(5.11) where we have defined the new metric function f(r) as ˜ f(r) = f(r) + 1 rd−3∑ i ωi r(d−3)(i−1) 0 .(5.12) We motivate this expression for the expansion in the energy ωof the particle based on dimensional analysis, since we have just seen that rd−3 0has energy-mass dimension. Working as in the above section we find the effective temperature, which is ˜ T=~ 4π ˜ f0(r) √g(r)r→r0,(5.13) and taking the derivative of (5.12) at the event horizon we eventually obtain ˜ T=TH−~(d−3) 4πrd−2 0√g(r0)∑ i ωi r(d−3)(i−1) 0 .(5.14) Since the heat capacity is negative, we can verify that this expression for the temperature works properly increasing its value when the black hole emits a particle of energy ω. To see this, we can rewrite equation (5.14) using the definition of the Hawking temperature (5.9) and imposing the above third condition, hence we get for the effective temperature ˜ T=~(d−3) 4π√g(r0)(1 r0−1 rd−2 0∑ωi r(d−3)(i−1) 0).(5.15) 88 Chapter 5. Back-reaction and quantum corrections Since the event horizon shrinks proportionally to ω1/(d−3), we see from this last expression that at low energies the temperature increases with respect to the standard Hawking temperature (5.9). Finally, if we compare the two expressions for the temperatures (5.8) and (5.14), we obtain definite values for the dimensionless parameters, ξi, in terms of the released energy, ω, ξi=(rd−2 0 ~)iωi r(d−3)i 0−ωi.(5.16) Therefore, looking at the metric (5.3) and its corresponding temperature (5.8), we conclude that back-reaction can be treated as a quantum perturbation leading us to the following expressions for the perturbed metric and effective temperature respectively, ˆ ds2=−f(r)(1 + ∑ i ωi r(d−3)i 0−ωi)−1 dt2+g(r) f(r)(1 + ∑ i ωi r(d−3)i 0−ωi)dr2+ +h(r)r2dΩ2 d−2,(5.17) ˆ T=TH(1 + ∑ i ωi r(d−3)i 0−ωi)−1 .(5.18) We are going to specify all the aforesaid expressions in a simple four-dimensional, static and spherically symmetric background. Therefore we consider a Schwarzschild black hole which is asymptotically flat, the metric functions are defined as: f(r) = 1−2M r,g(r) = h(r) = 1 and the event horizon is at r0= 2Min natural units. From (5.17) we write the perturbed back-reacted metric as, ˆ ds2=−(1−r0 r)(1 + ∑ i ωi ri 0−ωi)−1 dt2+1 (1−r0 r)(1 + ∑ i ωi ri 0−ωi)dr2+ +r2dΩ2 2.(5.19) The Hawking temperature corresponding to a Schwarzschild black hole is TH= 1 8πM . When the black hole emits a single particle with energy ω, the new effective temperature at first order in energy expansion is ˆ T=1 8π(M−ω)(1 + ω 2M−ω)−1 .(5.20) Likewise at semi-classical level we calculate the Bekenstein-Hawking entropy using the area law, SBH =A 4, in the presence of back-reaction effects, and obtain ˆ SBH = 4π(M−ω)2.(5.21) Chapter 6 Einstein and conformal frame In the previous chapters we have calculated some physical magnitudes as the temperature or the entropy of NS5 and LST black holes working mainly in string (or conformal) frame. However, in some cases, we have specified that we were working in Einstein frame. Actually we understand that physics must be frame independent, thus we guessed that some sort of scale factor should relate the two frames. In this section we have obtained the precise scale factor that relates the two frames. Aconformal transformation is a local change of scale (or geometry but not a change of coordinates) that leaves light cones invariant. Such transformations are defined as ˜gµν = Ω2(x)gµν ,(6.1) where Ω(x) is a space-time non-vanishing function. We then say that the physical quantities are expressed in the conformal frame. So far we have worked in string frame (3.5). On the other hand there exists a conformal transformation that relates the string action with the standard EinsteinHilbert action [10]. The low energy action for ten-dimensional type IIB string theory can be written as I=1 16πG(10) ∫M d10x√−g[e−2Φ(R+ 4(∇φ)2)−1 12H2 3].(6.2) Whereas the action in Einstein frame takes the standard Einstein-Hilbert form, IE=1 16πG(10) ∫M d10x√−g(R−1 2∂µφ∂µφ−1 12e−ΦH2 (3)),(6.3) 95 96 Chapter 6. Einstein and conformal frame where Φ is the dilaton scalar field, H3is the NS −NS form along the S3and φis the scalar field. Thus the metric in string frame can be written in Einstein frame using ds2 E=√gse−Φds2,(6.4) where gsis the string coupling. Both metrics are related by a Weyl rescaling given by the dilaton. In the string frame the scalar field play the role of a spin-0 component of gravity, whereas in the Einstein frame the scalar field plays the role of a source matter field. The question arises: ’Which frame is the physically relevant frame?’ With the aim to answer this question we are going to calculate the entropy of the NS5 and LST black holes in Einstein frame, and compare it with the expression obtained in string frame, Chapter 3, Section 3.1. We start writing the metric (3.5) in Einstein frame by using the relation (6.4) and also using the definition of the dilaton (3.7), ds2 E=−f(r) A1/4(r)dt2+A3/4(r) f(r)dr2+A3/4(r)r2dΩ2 3+ 5 ∑ j=1 dx2 j A1/4(r).(6.5) If we then calculate the temperature we will notice that it is frame independent. Next, we are going to calculate the area of the event horizon corresponding to NS5 and LST black holes defined in the induced Einstein metric dˆs2 E=A3/4(r)r2dΩ2 3+ 5 ∑ j=1 dx2 j A1/4(r),(6.6) for which the determinant is √−ˆgE=r3√A(r) sin2(θ)sin(ϕ).(6.7) The area of the event horizon is AH=∫√−ˆgEdθ dϕ dψ 5 ∏ j=1 djx=V52π2r3 0√A(r0) = Vol(R5) 2π2r2 0√χm2 sr2 0+N ms . (6.8) Comparing with (3.28) we can see that the value of the area of the event horizon is frame dependent. Then the Bekenstein-Hawking entropy is SBH =AH 4G(10)~=Vol(R5)π2r2 0√χm2 sr2 0+N 2G(10)~ms ,(6.9) and differs from (3.30) calculated in the string frame. 97 In order to calculate the total energy of the NS5 and LST black holes we use the Komar integral (C.1), see Appendix C. We choose as normal vectors in Einstein frame et=−√f(r) A1/4(r), er=√A3/4(r) f(r).(6.10) The total energy (mass) is given by E=Vol(R5)πr2 0 4G(10) ,(6.11) which differs from (C.5), therefore the Komar integral is also frame-dependent. Thus comparing the expressions of the entropy (6.9) and the energy (6.11) in Einstein frame with the entropy (3.30) and the energy (C.5) in string frame, we see that they are related by the metric function A(r0) at the event horizon, S(E) BH =A(r0)−1SBH , E(E)=A(r0)−1E . (6.12) Therefore A(r0) acts as a scale factor for physical extensive quantities like entropy, whereas the temperature is an intensive quantity and its value does not change under conformal scalings. We can conclude that physical laws are invariant under conformal scalings. Only the values of extensive quantities change by a fixed scale factor. 98 Chapter 6. Einstein and conformal frame Chapter 7 Summary, conclusions and outlook After a brief outline in Chapter 1 about the properties of black holes, where we have introduced the information loss paradox, we have reviewed in the Chapter 2 how curved space-time, e.g. black hole backgrounds, creates particles. Hawking demonstrated that black holes with temperature THemit thermal radiation, and calculated its flux without taking into account the back-reaction of the metric. Afterwards we have presented two semi-classical methods, i.e. the tunneling approach and complex path method, that somewhat solve the information loss paradox stated by the work of Hawking. In Chapter 3, we have applied both semi-classical methods plus the covariant anomaly method in NS5 and Little String Theory (LST) black holes. We have calculated some thermodynamical quantities as the temperature and the entropy; furthermore, after reducing the ten-dimensional theory to a two-dimensional effective theory, we have calculated the emission rate and the corresponding fluxes taking into account the back-reaction of the metric. In Chapter 4, we have calculated the emission probability of fermions by NS5 and LST black holes obtaining identical results as for scalar particles. In Chapter 5, we have presented a novel method in order to introduce quantum perturbations directly in the black hole metric, that accounts for back-reaction effects. This method has been applied to a general stationary spherically symmetric metric, recovering similar results with the results of the semi-classical methods presented in the previous chapters. Moreover, when we have applied this method in LST’s black hole we have obtained similar results with those derived in string one-loop theory. Finally in Chapter 6, we have calculated and compared some thermodynamical quantities as the entropy, using both Einstein frame and conformal frame. 99 100 Chapter 7. Summary, conclusions and outlook In previous sections we have already been discussing some conclusions, now we will outline the most general salient features for the NS5 and LST black holes. In general we have seen: •The Hawking radiation as tunneling approach solves partially the information loss paradox, since the emission rates obtained for a general class of black holes, e.g. Schwarzschild, Reissner-Nordstrom, stringy black holes as NS5, etc., are non-thermal. The lack of thermal behavior in the emission spectrum lead us to establish some correlations between the emitted particles, recovering at least all the information stored in the initial configurations that originate the black hole. This approach takes into account the back-reaction of the metric when a scalar massless shell particle is emitted by the black hole, imposing energy conservation. The emission rate is in accordance with the statistical mechanics results. •The complex path semi-classical method allow us to calculate the emission probability of a black hole. Imposing energy conservation in order to implement the back-reaction of the metric we obtain again a non-thermal spectrum. This method, compared with the tunneling approach, has the advantage that avoids any heuristic interpretation of the emission mechanism. Moreover it is not needed to go to Painlev´e coordinates, since one can works directly with spherical Schwarzschild coordinates. •The factor that cancels the gravitational anomaly in a two-dimensional effective black hole metric is exactly the Hawking radiation flux of the black hole. We have calculated explicitly the spectrum in (3.144) and we have verified that this result matches the anomaly result (3.114) for massless scalar particles. •The general clue for obtaining non-thermal spectra, in the vast majority of the black holes studied in the literature, is the imposition of energy conservation when one takes into account the back-reaction of the metric. •We can define an effective temperature Teff , which consists basically of the standard Hawking temperature of the black hole corrected by a factor depending on the energy of the particle emitted by the black hole when the back-reaction is taking into account. Then the emission rate will be the standard Boltzmann factor at a temperature Teff , (3.151). 101 •Introducing some sort of quantum corrections in the r−tsector of the metric we are able to analyze the back-reaction as a quantum perturbation. This corrections are built on dimensional grounds and satisfy the Einstein’s equations, since are independent of any system of coordinates. Regarding the concrete case of NS5 and LST we would like to remark the following aspects: •The NS5 black hole shows the expected non-thermal spectrum, thus all the above conclusions are verified in this case. On the other hand, LST keeps its thermal behavior even taking into account the back-reaction of the metric, hence the greybody factor for LST is 1. NS5 does not accomplish cluster decomposition, therefore it would be possible to recover the information of the initial configurations that formed the black hole. This information could be encoded in the correlations between the emitted particles and would be released out when the black hole evaporates. Nevertheless LST satisfies cluster decomposition, and the emitted particles are not correlated. In fact, we point out the notorious property that the LST Hawking temperature is independent of its mass, thus this temperature is constant even if the black hole is emitting. Therefore the information remains hidden behind the event horizon of LST until it evaporates. •LST is the thermal limit of NS5. When we explore the region near the event horizon of an evaporating NS5 black hole the temperature increases until it reaches the maximum temperature, i.e. the Hagedorn temperature, becoming then a single pure thermal state. This single state, i.e. LST, radiates a constant flux of energy at a constant temperature. •The emission spectrum of fermions and scalar particles is the same either for NS5 or LST black holes. In general, even for other black holes, working with an effective theory in the r−tsector of the metric the emission will be independent of the spin degrees of freedom. •Introducing quantum corrections that accounts for the back-reaction in the metric of LST, and then studying the thermodynamics, we have obtained the same kind of logarithmic entropy corrections that are also found working in one loop string theory. The thermodynamics of LST presents a Hagedorn 102 Chapter 7. Summary, conclusions and outlook behavior, its specific heat is negative, hence the thermodynamics is unstable. Therefore, we can conclude that LST consists of a single unstable state. We would like to finish with a brief outlook. One decade ago a new theory proposed a higher dimensional mechanism for solving the hierarchy problem [99]. In that framework the Planck scale can be reduced considerably [100, 101] until it reaches the TeV scale. There also exists some four-dimensional models which are able to reduce the Planck mass to the TeV scale [102, 103]. Then considering the results of the present thesis, where we have presented different semi-classical methods which lead us to non-thermal spectra results, it should be interesting to focus our attention to the study of the emission of gravitons at the LHC (Large Hadron Collider). Furthermore, it should be very interesting the study of quantum production [104] of small quantum black holes in scattering processes. Appendix A Calculus tools and notation conventions We use two kinds of tensor indices: greek index (µ, ν, ρ, ...) in general curved spacetimes and latin index (a, b, c, ...) in flat space-time. We also use the Einstein notation of summing over repeated indices. We choose the metric signature (−+...+). The expression ηab represents the components of the Minkowski metric and gµν the general components of a curved space-time metric. We define the vielbeins as eµ aeν bgµν =ηab , ea µeb νηab =gµν .(A.1) The expressions ∂ ∂xµor ∂µor , µ represent a partial derivative. Whereas ∇µor ; µ represents a covariant derivative. A prime over a function means partial derivative with respect to the radial coordinate: f0≡∂f ∂r , whereas a point over a function means derivative with respect to time coordinate: ˙ f≡∂f ∂t . And finally Dis the covariant Lorentz derivative. These derivatives are defined over the tensors and spinors ψas ∇µkν=∂µkν+ Γν µρkρ, ∇µψ=∂µψ−1 4ωab µΓabψ , Dµka=∂µka+ωa µbkb,(A.2) where Γab is the antisymmetric product of two gamma matrices. The connections are related by ωb µa = Γb µa +eν a∂µeb ν,(A.3) 103 104 Chapter A. Calculus tools and notation conventions where the affine connection Γρ µν are the Christoffel symbols Γρ µν =1 2gρσ (∂µgνσ +∂νgµσ −∂σgµν).(A.4) We use for the anticommutator the relation [A, B] = AB −BA , (A.5) and for the commutator {A, B}=AB +BA . (A.6) For the majority of the cases presented in this work we use Planck units ~=c=G= 1 .(A.7) However in some cases we write the units explicitly for convenience. Appendix E Average number of emitted bosons In this appendix we have explicitly calculated some expressions of the Section 3.8.1. concerning to the blackbody spectrum of a black hole. We start with a state that describes a system of nvirtual pair of particles inside the black hole, |ψi=N∑ n|nL ini⊗|nR ini.(E.1) We want to write this physical state in terms of the out eigenstates. The reason is that outside of the black hole we can carry out observations. Thus taking into account the relation (3.136) between the modes inside and outside of the black hole, we obtain |ψi=N∑ n e−πωn ~κ|nL outi⊗|nR outi.(E.2) In order to calculate the normalization constant Nwe make use of the orthonormalization condition between the orthonormalized states hψm|ψni=δmn .(E.3) Thus considering two states |ψniand hψm|we construct hψm|ψni=(N∑ m e−πωm ~κhmL out|⊗hmR out|)·(N∑ n e−πωn ~κ|nL outi⊗|nR outi) =N2∑ m,n e−πω(m+n) ~κhmL out|nL outi⊗hmR out|nR outi.(E.4) Then taking into account (E.3) we obtain 1 = N2∑ n e−2πωn ~κ.(E.5) 111 112 Chapter E. Average number of emitted bosons The normalization constant corresponding to bosons (n= 0,1,2, ...) and fermions (n= 0,1) is respectively Nb=(1−e−2πω ~κ)1 2, Nf=(1 + e−2πω ~κ)−1 2.(E.6) Eventually a state associated to a system of bosons inside the black hole can be written as |ψbi=(1−e−2πω ~κ)1 2∑ n e−πωn ~κ|nL outi⊗|nR outi.(E.7) The density matrix for a boson system is ρb=|ψnihψm| =(Nb∑ n e−πωn ~κ|nL outi⊗|nR outi)·(Nb∑ m e−πωm ~κhmL out|⊗hmR out|) =(1−e−2πω ~κ)∑ n,m e−πω(n+m) ~κ(|nL outi⊗|nR outi)·(hmL out|⊗hmR out|) =(1−e−2πω ~κ)∑ n,m e−πω(n+m) ~κ(|nL outihmL out|)⊗(|nR outihmR out|).(E.8) Tracing over the left modes hmL out|(|nL outihmL out|)|nL outi,(E.9) and taking into account the orthonormalization condition (E.3) we obtain ρR b=(1−e−2πω ~κ)∑ n e−2πωn ~κ|nR outihnR out|.(E.10) This expression corresponds to the density matrix for bosons in terms of the right outgoing modes. This modes will be detected at asymptotic infinity as the Hawking radiation. 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