MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV TEORETICKÉ FYZIKY A ASTROFYZIKY Bakalářská práce BRNO 2019 MARTINA BÁRTOVÁ MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV TEORETICKÉ FYZIKY A A S T R O F Y Z I K Y Entropie černých děr v teorii strun Bakalářská práce Martina Bártová Vedoucí práce: Jörgen Linus Wulff M.Sc. Ph.D. Brno 2019 MASARYKOVA UNIVERZITA PŘÍRODOVĚDECKÁ FAKULTA ÚSTAV TEORETICKÉ FYZIKY A A S T R O F Y Z I K Y Black hole entropy in string theory Bakalářská práce Martina Bártová Vedoucí práce: Jörgen Linus Wulff M.Sc. Ph.D. Brno 2019 Bibliografický záznam Autor: Název práce: Studijní program: Studijní obor: Vedoucí práce: Akademický rok: Počet stran: Klíčová slova: Martina Bártová Přírodovědecká fakulta, Masarykova univerzita Ústav teoretické fyziky a astrofyziky Entropie černých děr v teorii strun Fyzika Fyzika Jörgen Linus Wulff M.Sc. Ph.D. 2018/2019 vři+ 38 Entropie černých děr; Entropie; Teorie strun; Bosonová teorie strun; Extra dimenze; Entropie strun; Termodynamika; Struny; D-brány; Schwarzschildovo řešení; BekensteinovaHawkingova entropie; Černá díra Bibliographic Entry Author: Title of Thesis: Degree Programme: Field of Study: Supervisor: Academic Year: Number of Pages: Keywords: Martina Bártová Faculty of Science, Masaryk University Department of Theoretical Physics and Astrophysics Black hole entropy in string theory Physics Physics Jórgen Linus WulffM.Sc. Ph.D. 2018/2019 vii + 38 Black hole entropy; Entropy; String theory; Bosonic string theory; Extra dimensions; String entropy; Thermodynamics; Strings; D-branes; Schwarzschild solution; BekensteinHawking entropy; Black hole Abstrakt V této bakalářské práci jsou nejdříve uvedeny některé základní koncepty, které se využívají v teorii strun. Následně je dvěma různými způsoby spočtena entropie extrémní pětidimenzionální černé díry - pomocí Bekensteinovy-Hawkingovy formule a využití popisu černých děr v IIB superstrunové teorii. Supersymetrie garantuje shodu obou výsledků. Abstract In this bachelor thesis are introduced some of the essential concepts that are used in the string theory. Then the entropy of an extremal five-dimensional black hole is computed using two different ways - the Bekenstein-Hawking formula and the IIB superstring theory description. At the end we will see that they are same if supersymmetry is included. MASARYKOVA UNIVERZITA Přírodovědecká fakulta ZADÁNÍ BAKALÁŘSKÉ PRÁCE Akademický rok: 2018/2019 Ústav: Ústav teoretické fyziky a astrofyziky Studentka: Martina Bártová Program: Fyzika Obor: Fyzika Ředitel Ústavu teoretické fyziky a astrofyziky PřF MU Vám ve smyslu Studijního a zkušebního řádu MU určuje bakalářskou práci s názvem: Název práce anglicky: Black hole entropy in string theory Oficiální zadání: Since the work of Bekenstein and Hawking in the 1970s it has been known that black holes carry entropy. The quantum microstates responsible for this entropy remained elusive until 1995 when Strominger and Vafa were able to calculate the entropy of certain black holes in string theory from the microscopic description. The goal of this project is to understand how black hole entropy can be calculated in string theory in simple cases and also to understand the relevant background, i.e. basic black hole thermodynamics and basic aspects of string theory needed for the calculation. Jazyk závěrečné práce: angličtina Vedoucí práce: Jórgen Linus Wulff, M.Sc, Ph.D. Datum zadání práce: 22. 11. 2018 V Brně dne: 3.1.2019 Souhlasím se zadáním (podpis, datum): Název práce: Entropie černých děr v teorii strun Acknowledgements I would like to express my deep gratitude to Jdrgen Linus Wulff M.Sc. Ph.D., my research supervisor, for his patient guidance, enthusiastic encouragement and useful critiques of this research work. Also, I wish to thank Mgr. Be. Patrik Novosad for his useful notes on this work and Samuel Valach for technical support. Declaration I hereby declare that I have composed the presented thesis independently on my own and without any other resources than the ones indicated. All thoughts taken directly or indirectly from external sources are properly denoted as such. Brno May 22, 2019 Martina Bártová Contents Introduction 1 Chapter 1. Basics 3 1.1 Thermodynamics 3 1.1.1 Review of classical thermodynamics and statistical mechanics 3 1.1.2 Black hole thermodynamics 6 1.1.3 The derivation of Bekenstein-Hawking formula 8 1.1.4 Black hole statistical mechanics 10 1.2 Extra dimensions 10 1.2.1 Gravity in the world with extra dimensions 13 1.3 String coupling constant and dilaton 15 1.4 String action and relativistic strings 16 1.5 Equations of motion and Dp-branes 18 1.6 Quantum violin string 20 1.6.1 The fermionic part 22 1.6.2 The bosonic part 23 1.6.3 The general system 24 1.7 Energy relation and the bosonic string entropy 25 1.7.1 Brief summary of steps 25 1.7.2 Energy relation 25 Chapter 2. Main part 27 2.1 Matching entropies 27 2.2 Classical solution 31 2.3 String theory 33 Conclusion 37 References 38 Introduction In all ages, physicists yearn for the theory that would describe everything - the true description of the world as it is. Since nineteenth century we have seen many remarkable unifications. The oldest example of unification is Maxwell's electromagnetism, which at that time unified the electricity and the magnetism. Maxwell's equations of electromagnetism are classical, so they do not provide quantum theory and microscopic description. After discovering quantization methods physicists suddenly have available the quantum electrodynamics QED as a quantum version of the classical electromagnetism. Another achievement was unifying the weak interaction with QED to make the electroweak theory. The quantization also provided the theory of the strong interaction, the quantum chromodynamics QCD, which describes the forces between quarks. Together QCD and electroweak theory forms the Standard Model which is essentially the summary of our present knowledge of all particle physics. However, this might sound like the final theory, the Standard Model does not include the gravity. Unfortunately, the Standard Model is a quantum theory while Einstein's general relativity, as a description of a gravity, is classical. However, the gravity must be included if we want a complete theory. One of the most successful candidates for the unified theory is the string theory. The string theory is a quantum theory which includes gravity, so where other theories failed the string theory provides the string interactions which are therefore necessary for the consistency of the quantum gravity. This provides an option of the description of black holes on quantum level. Black holes were always theoretical challenge to physicists. In general relativity black holes are just solutions of Einstein's equations. They represent a point in spacetime with an infinite density. From the early work of Jacob Bekenstein and Stephen Hawking [1], [2] we know that black holes have a temperature and also an entropy, which Einstein's description do not provide. One of the successes of the string theory is the understanding of black holes on a quantum level where do not appear infinities, which are unphysical. The string theory is able to describe the black hole entropy via counting the degrees of freedom which is done in this work. In the string theory a certain black hole can be constructed from various types of D-branes and strings and that's why different possible states emerge and hence the entropy can be calculated. -I- Uvod 2 This work is focused on a string-theory description of a particular five-dimensional black hole for which entropy can be calculated easily. The black hole will be firstly related to a bosonic string with a high degree of excitation which is just signification that the string theory goes the right way. The black hole entropy calculation at the end of this work is done using two possible ways - the first one uses thermodynamic description and Bekenstein-Hawking formula and the second one is done through the superstring theory description. But as we will see, both entropies turn out to have the same value. Chapter 1 Basics In this chapter we will introduce some of the essential knowledge to underestand the primary thoughts on black hole entropy which computation is treated in chapter two. 1.1 Thermodynamics 1.1.1 Review of classical thermodynamics and statistical mechanics Thermodynamics is a field of physics that describes the system through its macroscopic properties such as temperature, energy or entropy. The behavior of these quantities is determined by the laws of thermodynamics [4]. Thermodynamics describes the system in general, but it can be the box full of strings or a black hole. T H E Z E R O T H L A W If two systems are each in thermal equilibrium with a third, they are also in thermal equilibrium with each other. T H E FIRST L A W The conservation of the energy provides dE = 6Q-pdV T H E S E C O N D L A W The entropy of a closed system can never decrease in time. SQ < TdS or dS > 0 where T is the temperature of a system and S is an entropy. Since the entropy can never decease in time it requires a particular direction of a time and a possible measurement can distinguish the past from the future. In the thermal equilibrium the inequality saturates and we can rewrite the first law as dE - TdS - pdV and therefore the energy depends like E - E(S, V). -3- Chapter 1. Basics 4 T H E T H I R D L A W When the temperature approaches absolute zero the entropy of a system has some constant value. Sometimes is useful to rewrite the conservation of energy (the first law) in different variables. We can define thermodynamic potentials which gives the Legendre transformation of the first law. For us is the most useful the Helmholtz free energy defined as F = E-TS We can rewrite the first law and see dE = d(TS)-SdT-pdV which gives dF = -SdT-pdV The Helmholtz free energy has variables S, V so entropy is given by (1.1) More detailed description of a system provides the statistical mechanics. It uses the point of view of the quantum mechanics and uses quantum states for a description. Our statistical description starts with the ensemble. The canonical ensemble is a set of many virtual copies of a system in thermal equilibrium with a heat bath and a fixed volume. Each copy represents the possible (quantum) state of the system and we donate P[E) as the number of all possible quantum states which depends on the average energy of the system. The entropy is closely related to this number and is given by S{E) = kkiPffl If we know the energy spectrum Ea associated with a set of quantum states (which are labeled a) one can define the partition function Z as Z = ; p = .L (1.2) a The partition function is an interesting quantity and it is useful to use it to describe the thermodynamic system. It also contains all the thermodynamic information because it is used to define the thermodynamic potentials as we will see. Given by definition the partition function depends on temperature /3 and energy which depends on external parameters such as volume of a system V. Thus we can write Z = Z(/3, V). It is also related to the probability that the system will be in a state a e-PBt Chapter 1. Basics 5 The average energy of the system is then ^ M ^ e~PEa dlnZ a a ^ OP At this point we can calculate the expression for the Helmholtz free energy writing down the differential of lnZ(/3, V) and the differential of (£73) dlnZ ,„ dlnZ , ,„ dlnZ , d(lnZ) = ——dyS + — — d V = -EdB + ——dV 66 H dV H dV Ed/3 = d[E0) - 6dE => d(lnZ) = -d(Ej8) + j 8 d £ + - ^ - d V 1 , , 1 ,/• ^ , 1 dlnZ , - d ( l „ Z ) = - - d ( E f l + d B + - — I V d B - r d i H n z + ± * ) - H ^ v We can see now from the first thermodynamic law dE - TdS - pdV that 1 , dlnZ S = A ; l n Z + - £ ' p-kT T F dV From the expression for the entropy we can calculate the expression for the free energy S = k\nZ+-E T TS = kT\nZ + E E-TS = -kTlnZ thus the expression is F=-kT\nZ (1.4) In the section 1.7 Quantum violin string we estimate the entropy of a string using basic thermodynamics. Later we will see the statistical mechanics derivation of the entropy of extremal black holes using string theory. It is a big success for string theory because it describes black holes on a quantum level and calculates the entropy using the number of quantum states of a black hole. Chapter 1. Basics 1.1.2 Black hole thermodynamics 6 A black hole is a region of a space where a gravitational interaction is so huge (or the spacetime is so curved) that nothing can escape from. It can form when a massive star runs out of thermonuclear fuel and collapses under intense gravitational field into a point. This point according to general relativity has an infinite mass density which sounds unphysical. It is a consequence of failing mathematical description - the singularity. However, the general relativity is not the final theory of everything and we want to believe that black holes have no infinities in its description. This problem might be solved by the string theory according to which a black hole is a system of strings attached to Dp-branes which are wrapped around a finite volume. But in classical physics a black hole is just a solution of Einstein's equations - the solution is a metric which describes the curved spacetime around the point with infinite density. Typically the metric has two singularities: a mathematical and a physical one. A mathematical singularity is called an event horizon which encloses the physical (gravitational) singularity - a dense point. The mathematical singularity does not have to exists and there are also black hole solutions with no event horizon called the naked singularities but these are believed not to exist. This nonexistence of naked singularities is called cosmic censorship. One of the most remarkable property is that a black hole has an entropy. Historical note on black hole entropy Imagine a hot cup of tea falling into the black hole. Does it change its entropy? In 1970's John Wheeler and his graduate student Jacob Bekenstein were discussing how it really is with the entropy of a black hole. Wheeler was questioning if black holes violate the second law or not. At that time some physicists and John Wheeler thought that since the thermal equilibrium of a black hole is in zero temperature therefore it must have zero entropy. Throwing a hot cup of tea into it would violate the second law because its entropy would be lost. They look at a black hole as just a cold material which is in its ground state with zero entropy. Another point of view was that when a black hole absorbs heat (for example from a tea) it still will be in equilibrium with zero temperature and the entropy would be infinite because dS = 8Q/0. But it means that the second law of thermodynamics says dS > oo. Since an entropy is related to the informational loss Jacob Bekenstein noticed that through these infinite changes of entropy an infinite amount of information would be lost during every thermodynamic process. That sounded ridiculous - the entropy should be able to change finitely and also a finite amount of information should be lost. To get rid of the infinity Jacob Bekenstein included a quantum mechanics and postulated a generalized second law that total entropy never decreases [1]. Sfof = SbiacJchole + Sfea > A S f o f > 0 (1.5) This says that black hole have an entropy and its not infinite. The negative change of a tea entropy balances the positive change of a black hole entropy. It makes sense Chapter 1. Basics 7 - if an entropy of a cup of tea is lost in a black hole the generalized second law is not violated. This leads Jacob Bekenstein to his famous quantum mechanical formula that the entropy is proportional to its event horizon area _ r]Ac3 k ^blackhole — ^ nG where rj is some dimensionless numerical factor. This all suggests that a black hole has a small but nonzero temperature. Afterward Stephan Hawking showed [2] that black holes can radiate quantum mechanically. Using this Hawking found that rj - 1/4 . Laws of black holes Now will be introduced the laws of thermodynamics in a language of black holes that are believed that black holes satisfy. They are analogous to those we know from classical thermodynamics. THE ZEROTH LAW: The thermal equilibrium The temperature TH is a constant across the surface area of an event horizon of a stationary black hole. The temperature of a black hole TH is also related to the surface gravity K as K The surface gravity is essentially a gravitational acceleration experienced by an object on the surface of the event horizon. Then we can reformulate the zeroth law as: The surface gravity is constant on the event horizon of a stationary black hole. Black holes (actually the event horizon) also quantum mechanically radiate the Hawking radiation which is described by the Hawking temperature he3 T H = 0 (1.6) 8nGMk The Hawking radiation is essentially a thermal black body radiation with the temperature TH. THE FIRST LAW: The conservation of energy d £ = c 2 d M = THdS + ndJ + ®dq (1.7) where M is a mass, TH is the temperature, Q is an angular velocity, / is an angular momentum, <1> is an electrostatic potential and q is an electric charge. This holds for reversible (quasistatic) changes where 8Q - THdS. If we rewrite the first law as THdS - c2 dM - QdJ - <$dq it tells us how entropy changed if we throw a massive, charged object into a black hole. Chapter 1. Basics. T H E S E C O N D L A W In all physical processes the surface area of a black hole event horizon never decreases. dA>0 (1.8) Unfortunately, this is not true. Actually, it is true classically but not quantum mechanically. But generally we still have to use a generalized second law (1.5) and this formula (1.8) applies only if we have an isolated system of a black hole where is not, other matter such as a cup of tea. T H E T H I R D L A W The black hole temperature TH can't be zero (i.e. a black hole with vanishing surface gravity K - 0 doesn't exist). Actually the simplest computations are done with extremal black holes for which the surface gravity (and also temperature) vanishes and therefore they don't radiate the Hawking radiation because the Hawking temperature vanishes as well [5]. They also have the minimal possible mass that saturates the inequality 1.1.3 The derivation of Bekenstein-Hawking formula From the first law of thermodynamics we can somehow derive the Bekenstein-Hawking formula for a non-rotating (/ = 0) black hole with a constant electric charge {q - const.). We can rewrite the first law (1.7) using (1.6) as Th He dM=-^-dS= -dS (1.9) c2 8nGMk now we can separate the variables 1 , 8nG , -dS = — — M d M A: nc and solve the differential equation to get the entropy of a black hole 1 4nG ? - S = ^ M 2 + S(M = 0) A: nc assuming that entropy of a black hole with zero mass is zero we get 1 4 n G M 2 k he Note that the entropy is proportional to a mass squared and mass is proportional to the Schwarzschild radius (1.15) and its square is proportional to the area of the event horizon i.e. 1 47TG 2 AnGlc2 \ 2 lc3 A A T S = ^ M 2 = ^ — R\ =-— = — (1.10) k he he \2G j A hG A£2 p Chapter 1. Basics 9 with surface area of event horizon A - 4nR2 . When we set c - h - 1 in (1.10) we get relation SBH A — = — (1.11) k 4G Schwarzschild black hole In 1916 Karl Schwarzschild published [6] a solution to Einstein's equation 1 8nG R^v - -Rg^v + Ag^v = —f T^y (1.12) where R^v is the Ricci curvature tensor, R - g/ i V i?/ i V is the scalar curvature, A is a cosmological constant and T^v is the stress-energy tensor. Schwarzschild supposed a vacuum solution T^v - 0 and zero cosmological constant A = 0. It leads to the vacuum equation R^v - 0 Then he made an anzatz of a static and spherical symmetric metric of a form f -A(r) 0 0 0 0 B{r) 0 0 0 0 r2 0 0 0 0 r2 sin2 0 so the spacetime interval looks like ds2 = g^vdx^dxv = -A(r)c2 dt2 + B(r)dr2 + r2 [d62 + sin2 0d02 ) (1.13) The full derivation can be found in [5]. The Schwarzschild solution can be written in spherical coordinates as d 5 2 = _ L _ c2dt2 + L _ 2£M j " 1 + ^ ^ q 2 + ^ 2 ( i M ) It was the first exact solution of the Einstein's equation found. In the Schwarzschild metric (1.14) are two singularities. The first singularity is in r —• 0 which is a gravitational singularity and the center of mass. The second singularity is when 1 - = 0 and this is a mathematical singularity which defines the event horizon of a Schwarzschild black hole which is a sphere of radius 2GM RBH=^t (1.15) and this is called the Schwarzschild radius. For example the Schwarzschild radius of the Earth is about R-9mm but of the black hole in the centre of our galaxy R-1,2-1010 m. The Schwarzschild radius of the Planck mass is R - 3,2 • 10~3 5 m which is twice bigger than the Planck length £ P = 1,6 • 10"3 5 m. Chapter 1. Basics 10 1.1.4 Black hole statistical mechanics Since the thermodynamics of black holes is enough worked out, a natural question emerges: What about microscopical description? The entropy of classical matter can be seen as a logarithm of number of its quantum states. One would like to know if this understanding of an entropy applies also for black holes. The first successful calculation of the number of states of a black hole is from 10dimensional superstring theory using D-branes and weak string coupling. It was done in 1996 by Andrew Strominger and Cumrun Vafa [7] for extremal and near-extremal black holes. With weak string coupling constant, the number of quantum states can be computed using methods of statistical physics. The final formula for entropy successfully corresponds with Bekenstein-Hawking formula (1.11) with event horizon area. 1.2 Extra dimensions In a normal life the number of observable dimensions is three: length, width and height. Albert Einstein in his theory of relativity added the fourth coordinate which represents time. Basically, all the classical physics theories work in the four-dimensional spacetime. But let's suppose that there are more than four dimensions. Since any extra dimensions have not been observed yet they must be somehow hidden. For example, they are finite and very small, and they are curled up, so they occupy a small finite volume. The original idea of an extra compact dimension came up from Theodor Kaluza and Oscar Klein while they were working on gravity-electromagnetism unification [8], [9]. Let's for now follow their idea. Assume that there is one extra dimension so the world is (4 + 1)-dimensional IR3,1 x S1 . The extra dimension is a small circle S 1 with radius R and coordinate z (see Fig. 1.1). Since the coordinate is cyclic, we can apply a periodic identification for two points which differ by 2nR as so, if in the original 4-dimensional flat space is defined a scalar field 0(x/ i ), now it is extended to the cylinder as 0(x^, z). z~z+2nR A S 1 Figure 1.1: One extra dimension Chapter 1. Basics 11 To define such a field, we must do another identification. Roughly speaking we set a periodic condition on the extra coordinate z because there is no mathematical difference between a periodic behavior of the field and the periodicity of a coordinate, so we set Q>{jr,z) = {xll ,z) = Yan{xř)-eV (1.17) -oo The expansion coefficients are often called modes [10]. Figure 1.2: Some non-zero modes (pn{x^) Now consider a situation with two extra dimensions Z\,z2. We can imagine it as a finite plane (square with edge 2nR) with identified sides (zi,Z2) ~ {z\ + 2nR,Z2) ~ {z\,Z2 + 2nR) Such an identification is ambiguous and there are several ways. In Fig. 1.3 are two ways of plain identification a torus Fig. 1.3a) and the Klein bottle Fig. 1.3b). The easiest compact space is a multidimensional torus T" because it is a simple cartesian product of several circles T" = S 1 x S 1 x • • • x S1 . Chapter 1. Basics 12 Figure 1.3: Identification of a plane a) a torus b) the Klein bottle In the 10-dimensional superstring theory we need six compact dimensions. There are various ways how to compactify them but commonly used are Calabi-Yau manifolds (Fig. 1.4) because they have great mathematical properties, but they are is so complicated objects that we still don't know the explicit form of metric. The extra dimensions are in order R~ £p ~ 10~3 5 m so there is no way how to detect them directly. Figure 1.4: Some examples of 2D slices of Calabi-Yau manifolds Some compact spaces (for example sphere S2 ) generally can be constructed differently, than using identification but in string theory are usually used the identified ones because it is easier to work with. Chapter 1. Basics 13 1.2.1 Gravity in the world with extra dimensions Kaluza's and Klein's fifth dimension suggests that the world is classical but in every spacetime point is a little circle so the world is IR3,1 x S1 . With adding some extra space dimensions, a natural question emerges: Does the gravitational constant change since its unit is proportional to length cubed [G] - Lr'IMT2 ? Newton's gravitational constant is a dimensionful parameter which describes strength of gravitational interaction. Is it natural to ask if it changes when we add extra compact spatial dimension(s). To find the answer we look at Newtonian gravity. In four-dimensional Newtonian gravity as we know the gravity field is given by gradient of the potential as g = - v o 4 where gravitational potential is given by 0 4 = -^y-, the dimension isD = d+l=3+l = 4 and G is our well known classical Newton's constant. Such a potential also satisfies the Poisson equation A 0 4 = 47rGp (1.18) where p is a mass density. This equation is valid in four dimensions and we need to modify it a little for validity in more dimensions. Firstly, we look on the units on the left-hand side of (1.18) [Energy] ML2 L2 1 so, the units of left-hand side [LHS] -l/T2 will not depend on the number of the spatial dimensions because it doesn't depend on the length. But it is clear that mass density on the right-hand side of (1.18) depends on space. To balance the units M 1 Ld W = jj-> [G-pl =F — l G ] = WF2 where d is the number of spatial dimensions. Now we are able to write the Poisson equation (1.18) in various dimensions as A O D = AKGDPD (1.19) where we recall G - G4 . Let's examine the point mass in the world with one extra circle dimension with radius R (so the D - 4 + 1 = 5). We will suppose that the gravitational potential doesn't depend on the compact direction x 4 so classically 5 = O5 (x^,0). The point mass is concentrated in x1 = x2 - x3 - 0 and if we add one compact dimension the point mass become a massive ring with constant density Chapter 1. Basics 14 point mass: p4 = M6(x)6(x)6(x) ; [p4] = - r (1.20) o 1 o ? o ^ M massive ring: p5 = mo{x)5{x)5{x) ; [p5] = — (1.21) We suppose that it is a same object, so the mass of a massive ring is same as a point mass M so by integrating all over the five-dimensional space we get M-\ p$dx = \ \ \ \ mÔ(x1 )Ô(x2 )Ô(x3 )dxidx2dx3dx4 -2nRm J space J-ooJ-ooJ-ooJo Using the condition M = 2nmR and inserting it in (1.20) and (1.21) we get p5 = mS(xl )5(x2 )5(x3 ) = —8{xl )8{x2 )8{x3 ) = — p4y 2nR 2nRH So, from the multidimensional Poisson equation (1.19) Q &®5{x\x2 ,x3 ) =4^G5 p5 = 4n—— p4 Ann we get the condition G5 — =2nR G Generally when we have more extra dimensions the gravitational constants ratio is equal to the volume of the whole compact space GD •7? = V( D -4) (1.22) It is really a volume if we choose the shape of compact space as a cartesian product of identified circles V = S 1 x • • • x § D ~ 4 . We can also look to the general relativity. Kaluza and Klein added the fifth dimension and got an interesting result. As in general relativity everything could be described through a four dimensional metric g^v the unified gravity-electromagnetism will have a five dimensional metric gxp. The spacetime intervals will then look like ds2 = gxpdxx dxp ; A,p = 0,1,2,3,4 x4 = z They suggest that if they introduce a scalar field

T2 rCF2 -ex* l d&l d&*\ d d Off dr dodrd(J = 0 (1.38) The first term becomes the equations of motion — - ^ + —-^-=0 (1.39) or da if other terms vanish. The middle term is a time derivative and will depend on initial and final time as 8X^[j\,a) and 8X^{T2,a) and we will assume that 8X^[j\,a) - 8X^{j2,a) - 0. The last term since it is a spatial derivation will have something to do with endings of a string. We can write it more explicitly as f2 [8X° (T, <7i) 9% (T, O-i) - 8X° (T, 0-2) (T, <72) + + OX1 (T, (71) &g (T, <7i) - OX1 (T, 0-2) ^ (T, 02 ) + ... + + 8Xd (T,oi)^0 ff (T,oi) - 5ld (T,0-2)^0 F F (T,o2 )]dr We want this functional to be zero everywhere so we need 2{d +1) boundary conditions. For example, we fix both string endpoints in space (in all d space coordinates) by Dirichlet condition (dX^\ v dr J(T,CTl) \ — \ =0 (1.40) V dr j( T ; ( 7 2 ) Note that p ^ 0 because the string cannot be fixed in time (the string can't start and end in the same time) and time flows while T flows. The Dirichlet boundary conditions define some spatial hypersurface where endpoints are fixed. These objects are called D-branes where D stands for Dirichlet. Dbranes in bosonic string theory can have up to 25 dimensions and hence we define another object: A Dp-branes is a hypersurface in the spacetime with p spatial dimensions. It is a boundary surface where open strings can end on. Both endpoints of a string must lie on the Dp-brane so the endpoints satisfy the Dirichlet boundary conditions. For example, when a string end on a flat D2-brane it means that its endpoints can move only on some two dimensional plane. The DO-brane is just space point etc. If the D-brane has a maximal spatial dimension (p = d) it is called a space-filling D-brane. Whether the string is attached to space-filling D-brane its endpoints can move freely in space. It turns out that D-branes itself can move and change shapes dynamically. You can also have a multiple D-branes and not only one. However, degrees of freedom of such a system of branes is bigger than of same single D-branes. It is because strings can be stretched between different D-branes as shown on Fig. 1.7. Chapter 1. Basics 20 Dp Figure 1.7: Strings stretching between two different D-branes 1.6 Quantum violin string In the string theory essential objects are strings. Since the analysis of such a string is very complicated the slight approximation would be very useful. In this section we treat a quantized string to estimate its entropy in a high-temperature approximation. There will not be used any result of string theory, but we use this quantum mechanical result to estimate entropy in string theory point of view. A quantum string has an infinite set of vibration frequencies which are an integer multiple of a basic frequency a>. This could be realized as an infinite set of simple harmonic oscillators where each oscillator has different frequency a>, 2a>, 3a>... The Hamiltonian of such a system will then look like p2 mco2 x2 pi m{2o))2 x2 pi m(3co)2 x2 H= — + - +—+ - + — + -+• y Pi | m{nu))2 x2 n ( i 4 i ) 2m 2 2m 2 2m 2 _n 2m 2 Now we use the standard notation with creation and annihilation operators an, a\ ipn'mneo an = \l—zr- \xn + a 2h v mncoj V 2h Using these operators, the Hamiltonian simplifies to m n w | A ipn mnco H-heoY \ nana\ + - | = tuo \N + tA n 2} { 2 (1.42) where we recognize the number operator h n - ana}n and N is therefore the sum of number operators. We can get the spectrum of energy of a quantum violin string from the stationary Schrodinger equation i.e. eigenvalues H \if/) = E \if/). The energy spectrum takes form Chapter 1. Basics 21 where N are eigenvalues of the operator N and will be called a total occupation number because it is a sum of occupation numbers rii OO N=Y,lni (1.43) Now we take the high-temperature (high-energy) limit. The contribution of tuo/2 to energy will be small compared with tkoN so we can neglect it. The energy then takes a form oo E = ha)N = hu)J^lrii (1.44) l=i where n\ are the occupation numbers that emerges from the number operator and thus they say if the quantum state is filled. For example, if rii - 0 means that /-state in empty and no oscillator is in that state. Since we know the expression for the energy, we can compute the thermodynamic partition function which contains all the thermodynamic information, such as an entropy. In general partition function can be written as (1.2) a where we sum over all possible states a. The quantum violin string partition function will be then z=Y,LL---e ~^LT=llHl = E Y\e-*T{hwlni] ni n2 n3 mn2...l=l oo h Z=nL^l n ' (1.45) 1=1 ni Note that upper limit in the sum is not specified. The occupation number n\ goes through all possible states which differs for bosonic and fermionic strings. The Pauli exclusion principle applies for fermions so the ferminonic occupation number will go over 0 to 1 which roughly speaking means that in one state there can be only one fermion. For bosons there's not such a principle, so their occupation number will go from 0 to oo. This all holds for one-dimensional case, so a slight generalization is needed. Consider a bosonic string which can oscillate in b transverse directions. The previous case would have b = 1. If each oscillator can move in b dimensions there are more possible quantum states of a string. For describing this situation, we add a new label (q) to occupation numbers. The total bosonic occupation number is then oo b l=lq=l Chapter 1. Basics 22 Now suppose we have a fermionic string where each oscillator can move in / different directions, described with label (g). The total fermionic occupation number is oo / Thus, a multidimensional partition functions can be rewritten from (1.45) as -rLq=1lni ^ I n j 1 ' nf nf n CO OO n tn\ n E e - ^ i v ' = i „ « = o n g kT^-l=lln l . \ g kT^-l=lln l . \ g fcTM=l'" n z ^ Z=lrc,-=0 / oo 1 t r g=i /=i„«>= 0 f CO 1 t E /=1m,-=0 Ini (1.46) (1.47) The multidimensional partition functions are powers of one-dimensional case. As mentioned before all the thermodynamic information is hidden in the partition function. We can recall the entropy (1.1) can be calculated using Helmholtz free energy (1.4) as S = dF_ df (1.48) The fermionic part and the bosonic free energy will be computed separately. 1.6.1 The fermionic part We start from partition function for a fermionic string (1.47). ( oo 1 ha) Zf= ULe-^ \/ = ln,=0 The Helmholtz free energy is then Ini i=i Ff - -kTlnZf hcox oo . h \ /*oo , h \ -kTf\nZ=-kTfJ^\n[l + e-^l ]~-kTf J ln\l + e~kfx jdx kT .2 T-"2 f pookz Tz f hco jTln (l+ *id 1. In our case is 5 = 2 so we can evaluate the TJ{2) as 1 1 n2 n2 Hence the fermionic free energy in the high-temperature limit is / k2 T2 n2 Ff = -J - 6hco (1.49) 1.6.2 The bosonic part The bosonic part follows almost the same steps as the fermionic part and therefore some of the steps will be skipped. Again we will start with the partition function (1.46) where appears a geometrical series ' oo oo t ^ n E e~^ln ' yi=im=o , n hco i (1.50) 1=1 1 - e kT1 oo . h \ /*oo , h \ Fb = -kT\nZb = -kTMnZ = jfc77?£ln(l - e~^l j « kTbJ ln[l - e~^x jdx •• k2 T2 b r°°i Johco k2 T2 b hco e - f + I e - 2 f + I e - 3 f + I e - 4 f + I e - 5 f + i -£ 1 -21 1 _ e 3<+ l e - 4 < + l -H + 32 42 52 k2 T2 b hco -C(2) Chapter 1. Basics 24 The bosonic free energy is therefore Fb = -b k2 T2 7T2 6hco (1.51) 1.6.3 The general system Since the Helmholtz free energy is an additive quantity we can now treat the general system using partial partition functions (1.51) and (1.49) f) k2 T2 n2 FT = Ff + Fb = -kT(\nZf + \nZb) = - \b + • - ^—: = -kT\nZT (1.52) The entropy of such a system (1.48) is ö F r (, H 2k2 n2 S - - —^ - \b+ — \ —T dT { 2 6fuo (1.53) To get rid of the temperature we can use relation for total energy ET - tuoN. If we define ß-jf the total energy (1.3) can be computed from (1.52) as dlnZr d\nZbZf lf\ JT ET —— = —— = I b + 2 ' f\ k2 T2 n2 b+ 1 d/3 d/3 r ' 2) 6heop2 l ~ 2) 6hw Hence from (1.54) we can get the expression for the temperature (1.54) SfuoEi \ Tl2 k2 [b+{] The final formula for the entropy is , n 2A;2 ^2 6ha)ET b + — \ 2 J 6fuo W 2 f c 2 [ b + Z j 2nk ET[b+{) 6hco 2nk N hi) k\nP{N,f, b) (1.55) The logarithm of number of quantum microstates P(N,f,b) in high-temperature limit is \nP{N,b,f)~2n N [b+i) (1.56) Note that up to now we didn't use anything from string theory. Chapter 1. Basics 1.7 Energy relation and the bosonic string entropy 25 Now we want to get very important result from the string theory i.e. energy relation which are energy levels of an open relativistic string. It is important because from the expression for the energy we can then get an entropy of such a string. Since its derivation is quite complicated, we place here just some summary of the important steps. Precise derivation of the energy relation can be seen for example in [3]. 1.7.1 Brief summary of steps Now we use the bosonic string theory to estimate the entropy of relativistic open string. All the following argumentation is done only roughly. We start with Nambu-Goto action (1.34) and choose the proper time parametrization T = t so the zeroth coordinate become X ° ( T , C T ) = ct{T,a) - CT The a £ [0, it] parameter will be chosen to satisfy perpendicular relation OX OX _ da dt ~ With this parametrization the equations of motion (1.39) (with c - 1) become the wave equation X»-X»" = 0 At this point is useful to make the mode expansion (which is essentially the expansion to the Fourier series). It turns out that if we involve the quantum theory the mode expansion can be written in terms of creation and annihilation operators. The quantized string becomes the set of harmonic oscillators. Now it starts to be more complicated because by switching to light-cone coordinates Virasoro operators emerge which are quite complicated objects and not be discussed here. More on that can be seen in [3], [14]. However somewhere in this part of derivation it turns out that only possible number of space dimensions is d - 25 (D = 26) otherwise the bosonic string theory will be inconsistent. 1.7.2 Energy relation Working with string mass M2 - -p2 and in the light-cone coordinates the string theory give us an interesting result M2 = -p2 = 2p+ p~ - pl pl = — (N- 1) - pl pl a' where i = 2,..., 25 and N emerges as an eigenvalue of the sum of the number operators of harmonic oscillators and therefore it is exactly same N which emerges in quantization of a violin string (1.43). If the string has zero spatial momentum {pl =0) the energy of each state is just its rest mass and Chapter 1. Basics 26 M = — (iV-1) a' Using high-energy (high-excitation) approximation we can neglect the constant ? 1 M a — N a' and with mass-energy equivalence E-Mwe get the energy relation VN=Va>E (1.57) The condition of zero spatial momentum can be satisfied for example if the string is attached to a DO-brane (essentially the point in space). DO ^0 DO Figure 1.8: Open string attached to the DO-brane Now we can calculate the entropy of a highly excited bosonic string. Since the spacetime is 26-dimensional, the string can move in 24 transverse directions (we not allow the longitudinal oscillations) so the bosonic label become b - 24. Hence the entropy using relation (1.56) is / JV •24 i— S{E) = MnP(iV,24,0) =2nkJ—— =4nkVN Using energy relation (1.57) the entropy of a relativistic highly excited bosonic string with zero spatial momentum is S(E) = 47ikVa'E (1.58) Chapter 2 Main part 2.1 Matching entropies In this section we would like to relate a highly excited bosonic string entropy to the Schwarzschild black hole entropy. This part is important because it's not clear that we can use the model of excited string to describe a black hole. This part essentially gives the reason why we are able to claim that a black hole has something in common with excited strings. Following computations was originally done by Leonard Susskind [11]. Concider a string of a lenght L divided to n pieces each of lenght £s . Both endings are fixed to DO-branes (basically they are fixed to a space point). The distance between end points is called a size of a string Rs. The size of a string is usually much smaller than its actual length. Now assume that each piece of a string can point to 4-orthogonal directions randomly. Thus, the string can be seen as a random walk with steps of length £s and the number of steps is just the number of string pieces n. This creates a broken line chain (see Fig. 2.1). L = n£s p Rs Figure 2.1: Creating a chain using random walk -27- Chapter 2. Main part 28 To estimate a size of a string we can use a random walk formula [13] Rs = ~y/n (2.1) 71 The mass of a string can be rewritten as a length of the string times string tension L n£ s n M ~ LTQ = - = (2.2) 2na> 2n£2 2n£s and using (2.1) and (2.2) we can get the string size in terms of mass Rs~esVn~esy/I^M = e]M* (2.3) Since this string carries no spatial momentum, we can use the energy-relation (1.57) to see that Rs ~ 4 i V ^ ~ N*£s (2.4) This relation applies for a single string and zero string coupling because we didn't include string interactions. The problem is that for a black hole are interactions quite necessary and with zero string coupling the Newton's constant would vanish G ~ g2 £2 (1.23). Now will be shown that there exist some minimal mass M beyond which is a string size R S smaller than its Schwarzschild radius RBH (1-15) Rs ~ RBH 3 1=> £2 sMz ~GM~ g^rsM - ( 2 ' 5 ) The result (2.5) applies for small string coupling so this suggests that a string with a large mass (highly excited string) can form a black hole. Using the energy relation (1.57) we get - v^/V 1 M — ~ —— (2.6) N—^ (2.7) which is a large number for small string coupling. These results encourage us to think that our picture of a black hole as very excited bosonic string is roughly correct. But if the picture is correct entropies of these objects will be roughly same. Chapter 2. Main part 29 From the Bekenstein-Hawking formula (1.11) we get an entropy of a black hole as A Riff (MG)2 n 9 , 9 9 while the entropy of a string (1.58) is Ss ~ M V o 7 = - ^ M (2.9) The entropies don't agree. The black hole entropy goes quadratically with mass while string entropy goes linearly with mass. But we should not expect agreement because a black hole needs strong string coupling and the computations in a string entropy assumes zero or small string coupling. So, we try to set the string coupling to some specific finite value where both entropies are roughly same. Imagine a large black hole with some finite string coupling constant g and entropy (2.8) SBH ~ g2 £2 sM2 Now let's assume that we let a string coupling go slowly down. This could be done by changing the expectation value of a dilaton field (1.24). If the weakening of the string coupling is reversible the entropy will not change during this process. It means that therefore the mass must increase with 1/g to balance the change. The final values of quantities after dialing down are labeled *. From constancy of entropy we get sBH~g2 e2 sM2 ~g 2 j2 sMl RBH~G*M*~g2 J2 M* A minimal Schwarzschild radius should not be smaller than £s so let us set R* - £s. For this minimal radius we get relation es = R, ~ g2 j2 M* and if we insert it back to the black hole entropy, we get 1 g* Note that such an object will have a large entropy because this applies for dialed down string coupling. SBH~g2 J2 M2 ~^ (2.10) Chapter 2. Main part 30 Now consider a string with a same mass M*. The entropy becomes ss~esM* ~es——\ (2.ii) g*"s g* So we got that Ss ~ SBHThis result shows that the Schwarzschild black hole might be a highly excited bosonic string with a strong string coupling constant. What is roughly done in 26-dimensional bosonic string theory can be done more precisely in 10-dimensional IIB superstring theory which is in a simple version treated in the last section. However, we can still get some more rough results in here. Consider a large black hole at some finite string coupling g weakened nearly to zero. Using above expressions, we can get SBH ~ GMZ ~ G f \ 2 Rs 3 (RBH)2 \ G 1)2 R BH So since entropies are proportional S5 ~ SBH we can get RBH Rs g From this relation we can compute the number N that characterizes the string at zero string coupling esN* ~ Rs RBH g2 £2 sM g£\M g g and then from the first and the last term we get i M A/* ~gesM~ — mp where mp = \ / ^ ~ G ~ 2 ~ - ^ - i s the Planck mass. The number N is proportional N • MY mp) Now we can estimate N for the black hole in the center of galaxy M87. The mass of a central black hole is approximately M = 6,5 • 1O9 M0 so N • 6,5-109 -2-10™kg 2-10"8 A;g 10200 Chapter 2. Main part 31 2.2 Classical solution In this part we will seek for an area of a black hole event horizon using BekensteinHawking formula (1.11) to obtain information of its entropy. We want to investigate 5-dimensional extremal spherically symmetrical Reissner-Nordstrom black hole with three independent charges Qi, Q5 and N [14]. These charges have a clear meaning in the string theory as we will see in the next section. The solution of Einstein's equations is a 10-dimensional metric because the superstring theory is consistent only in 10 dimensions. Roughly speaking we get this solution from an anzatz of a spherically symmetric metric essentially same way Schwarzchild (1.13) found his solution (but this time is more complicated). This solution can be reduced to a 5-dimensional metric with the fifth dimension curled up into a circle of radius R. The result is written with h = c = 1 as [12] ds2 = - {fifsfp)"3 dt2 + [fifsfp)5 (dx2 + dx\ + dx2 + dx2 ) (2.12) where fl -- i ^ 4G5 i?Qi na'gr2 - l + C l Q l r2 fs ~- r2 C5Q5 = 1 + r2 2 2 2 2 2 r — x-y "tx2 "tXc^ ~\~ x^ fp -- 4G5N TiRr cpN rz Now we look for an area of a black hole event horizon. The metric is written in coordinates where you go r —• 0 you reach the event horizon. The solution is spherically symmetric, so we switch to hyperspherical coordinates x\ - rsini//sin0cos0 X2 - r sin 1//sin 0 sin 0 X3 = rsini//cos0 x4 = rcosi^ t= t the area element in (2.12) becomes dx\ + dx^ + dx2 + dx^ = dr2 + r2 [dy/2 + sin2 y/[d(f>2 + sin2