The Unruh Effect 1. Paul de Lange 2 Onder begeleiding van prof. dr. E.P. Verlinde. July 2, 2009

Transcription

1 The Unruh Effect 1 Paul de Lange 2 Onder begeleiding van prof. dr. E.P. Verlinde July 2, Verslag van Bachelorproject Natuur- en Sterrenkunde, omvang 12 EC, uitgevoerd tussen 12 mei en 30 juni 2 No , Instituut Theoretische Fysica Amsterdam, Faculteit Natuurwetenschappen en Informatica

2 Chapter 1 Layman summary It is hard to imagine a box containing absolutely nothing at all. If you agree with this sentence, don t be embarrassed: the empty box, or vacuum, has been one of the most fascinating subjects throughout the entire history of physics and philosophy and still is a concept not fully understood by anyone. Not only the conception of the vacuum is a fairly hard deal: it s physical description has turned out to be a real conundrum. Where physicist used to define the vacuum indeed as a space containing no energy, particles or matter whatsoever, this prescription turned out to be impossible to maintain: this classical picture of the vacuum is impossible to prepare in the laboratory. But things got even worse. Since the introduction of quantum mechanics it has become clear that there is some thing as vacuum energy - a certain amount of energy contained in this so-called vacuum. It has ever since become clear that there is more to this concept op the vacuum then absolutely noting at all. In this thesis I investigate this concept even further, by asking the following question: Consider a volume containing no particles. Is this property no particles one that all observers will agree on? If, for example, Leucippus is standing in front of this volume and sees no particles, will Democritus - standing at the back of the volume - will agree with Leucippus observation? More generally I wonder wether the property no particles is an invariant one with respect to the properties of the observer. Due to William Unruh it turns out not to be. In fact, when Leucippus is standing still with respect to this volume and detects no particles, Democritus, having a constant acceleration with respect to Leucippus, will detect radiation of particles streaming out of the volume with a temperature proportional with his accelartion! This most counter-intuitive result is called the Unruh effect - the title of my thesis Paul de Lange, Amsterdam 1

3 Contents 1 Layman summary 1 2 Introduction 3 3 Quantum Field Theory Brief introduction to Quantum Mechanics The Klein-Gordon Field Fock space Bogolyubov Transformations The Particle Concept Curved Coordinates Rindler Space The m = 0 Free Field The Massive Free Field Rindler Thermodynamics Black Hole Mechanics The Schwarzschild solution Hawking Radiation Discussion 30 2

4 Chapter 2 Introduction To write a thesis on the vacuum is, from a naive point of view, to write a thesis on nothing at all. Classically one could define the vacuum as a box containing no matter or particles whatsoever or, more rigorously, a subspace V of the R 3 such that N(V ) = 0, where N denotes the number of particles detected by an observer in the exterior of V. Intuitively this function N : R 3 N is an invariant under coordinate transformations, since it shouldn t matter how one looks at V : when there are no particles, there s nothing to detect by anyone! Even though the geometry of V is not an invariant (due to e.g. Lorentz contraction), I couldn t, a priori, imagine how this would account for the creation of particles. One of the most exhilarating results of Quantum Field Theory is the observation due to Unruh[1] that N is not an invariant, i.e. there are coordinate transformations τ such that N(τV ) 0: the notion of the vacuum becomes ambiguous. In fact, a uniformly accelerating observer will detect a thermal spectrum radiating out of V. So where does our intuitive definition of the vacuum break down? When taking into account the study of quantum mechanics, first of all the notion of a particle is blurred out, and we are to think about a wave-particle dualism. States are represented by wave vectors ψ so our volume V previously defined should be replaced by such a vector, living in Hilbert space. Simultaneously, observables are represented by operators. The function N is turned into a number operator ˆN which acts on the state of interest, and one is to calculate the expectation value ˆN = ψ ˆN ψ of this number operator to obtain the expectation value of the number of particles contained in the state ψ. Now let us define two new operators, a(x µ ) and a (x µ ), the annihilation and creation operator respectively, in terms of a set of local coordinates x µ. They are defined so that if n = ˆN is the number of particles in state n, then a ψ n 1 and a ψ n + 1, where the constants of proportionality are obtained upon normalization. Note that N(x µ ) = a (x µ )a(x µ ). With the aid of these new operators, the vacuum state 0 is defined such that a 0 = 0, where the state 0 is expressed in terms of the set x µ. Vacuum states with respect to different coordinates need not equal each other, which is to say that 0 a need not equal 0 b where a and b are two different sets of coordinates. With this new definition of the vacuum, the question of wether or not the previously defined function N(V ) is invariant under coordinate transformations is turned into the question wether the outcome of 0 N 0 is altered upon trans- 3

5 formation of x µ, or: Does 0 N(x µ ) 0 equal 0 N(x µ ) 0 for every coordinate transformation? Ever since Unruh answered this question with a surprisingly no we have to once more change our intuition on the concepts of a particle and empty space. Making a leap from the microscopic to the macroscopic, we encounter a likewise counterintuitive result in the study of black hole mechanics. Where one would expect a black hole to indeed be entirely black, i.e. not to radiate any form of particles, Zel dovich[2][3] proposed on heuristic grounds that there might be such radiation, a statement made rigorous by Hakwing[4][5]. This remarkably discovery, now referred to as Hawking radiation, turns out to depend, to some extend, on the Unruh effect. In this thesis I will examine these canonical field theoretic results and compare the Unruh effect with Hawking radiation. After having introduced the technical tools of quantum field theory required to derive the results stated above I will follow a subtle argument due to Unruh to end up with an expression for the number of particles in a vacuum as observed by a uniformly accelerating observer. Finally I will inexhaustively study the black hole and relate the Unruh effect with the phenomenon of Hawking radiation. Throughout this thesis I employ a system of units wherein c = = k b = G = 1. 4

6 Chapter 3 Quantum Field Theory The quantum theory of fields is the extension of quantum mechanics, which is confined to the quantization of particles, to the realm of fields and their quantizations. The urgency to introduce this field theory is perhaps best motivated by the inconsistencies one encounters when trying to treat proper quantum mechanics relativistically. When, in the language of quantum operators, the classical energy E is replaced by the relativistic E = (p 2 +m 2 ) 1/2, obtaining the bosonic Klein-Gordon wave equation, we immediately encounter an infinite set with negative-energy solutions, excluding the existence of stable-state bosons. That is, in the Schrödinger picture, we write down the Schrödinger equation: i φ = Hφ. (3.1) t Identifying the hamiltonian with the total energy of the system, we would like to replace H with the relativistic expression for the energy, E. Getting rid of the square root, we take the square of (3.1) and substitute E for H, obtaining: 2 t 2 φ = (p2 + m 2 )φ. Identifying the momentum p with the canonical p = i, we readily obtain: ( 2 2 t 2 m2) φ = 0, (3.2) the Klein-Gordon equation or, written more compactly: ( m 2 ) φ = 0, with = 2 t + 2, the d alembertian. 2 This linear equation in φ can be solved with a Fourier transformation: φ = e i k, k x = k 0 t k x, k 2 0 k 2 = m 2 (3.3) Realizing that E = ω = k 0, we find a that half of the solutions in (3.3) carry negative energy k 0 = ω k, ω k = k2 + m 2. This is a severe problem, since the spectrum in not bounded from below any longer: an arbitrary large amount of energy can be extracted from the system, in contrast with the observation of existence of stable stationary bosons. 5

7 3.1 Brief introduction to Quantum Mechanics To understand the concepts of quantum field theory, one is to have a proper understanding of quantum mechanics itself. In the microscopic world measurements have, inheritably, a probabilistic character. Whereas in classical mechanics, two independent parameters in your system can always be measured with arbitrary accuracy, this is not the case for a quantum system: observables need not commute with each other. Quantum mechanics can be, grosso modo, developed in two ways: the Schrödinger picture, starting with the Schrodinger equation (3.1), and the Heisenberg picture, starting with the equation of motion: dô dt = i[ô, H], with Ô an operator representing an observable and H the hamiltonian of the system. Although the latter approach is the custom starting point for developing a quantum field theory, we shall first briefly run trough the former, especially to consider the model of the harmonic oscillator. In a quantum system states are represented by a wave vector ψ. The probability of finding this state in a fixed volume V is evaluated trough: P (V ) = ψ 2 dτ, or P = ψ ψ, in Dirac notation. A state ψ needs to be normalized, that is: P (whole space) = 1. Observables, at the same time, are represented by hermitian operators 1, acting on states in the Hilbert space. Their expectation values are obtained trough: O = ψ O ψ. When two operators A and B have a nonvanishing commutator AB BA =: [A, B], then you can t simultaneously make arbitrary accurate measurement of the observables A and B: they obey a uncertainty relation (since in classical mechanics obervables are represented by real-valued functions, every pair of observables commutes, and every commutator vanishes). Quantum states obey the Schrödinger equation, that is, for every state ψ, one has equation (3.1). In free space (i.e. in the absence of, say, an electromagnetic field), the Hamiltonian equals: H = 1 2m 2 + V, obtaining: V i t ψ = 1 2m 2 ψ + V ψ (3.4) In the case of the one-dimensional harmonic oscillator, setting m = 1, the potential is set to V = 1 2 ω2 x 2, and the Schrödinger equation reads: Ψ(x, t) i t = 1 2 Ψ(x, t) 2 x ω2 x 2 Ψ(x, t) (3.5) In the case that H itself is independent of time, we can seek for solutions of 3.5 using separation of variables, that is look for solutions of the form Ψ(x, t) = ψ(x)θ(t). One finds that θ(t) = e iet and Hψ = Eψ, 1 An operator Ô is called hermitian if Ô = Ô. Henceforth we shall omit the hat on operators 6

8 the time-independent Schrödinger equation: Our problem reduces to finding the eigenstates of the operator H. In the case of the harmonic oscillator we have: d 2 Eψ = 1 2 dx 2 ψ ω2 x 2 ψ We can solve this differential equation analytically, in terms of Hermite polynomials: ψ n = A n H n ( ωx)e ωx2 /2 Here A n is determined trough normalization and the eigenvalues are E n = (n )ω. There is, however, an algebraically and more constructive way to tackle this problem, which is of more interest bearing in mind that we are interested in quantization. Recalling the identity p = i we write the Hamiltonian as H h.o. = p ω2 x 2. To determine the eigenvalues of this operator algebraically we define two new operators a and a, the annihilation and creation operator respectively: a = 1 2ω (p iωx), a = 1 (p + iωx) 2ω (3.6) With it s inverse expressions: x = 1 2ω (a + a ), ω p = i 2 (a a ) (3.7) The canonical commutation relation [p, x] = i is now equivalent to [a, a ] = 1. With the aid of these operator the hamiltonian is written as: H h.o. = ω ( a a The vacuum state 0 is an eigenstate of H h.o. with eigenvalue E 0 = 1 2 ω and a 0 = 0. The spectrum of normalized eigenstates of H h.o. is completely described by the set: n = 1 ( a ) n 0, n! where n is an integer and the state n has eigenvalue E n = (n )ω. This process, of finding annihilation and creation operators, is identified with the quantization of particles in a field theory: particles are read as excitations of the field. ) 3.2 The Klein-Gordon Field The most elementary field is the Klein-Gordon field: the field φ which obeys the field equation ( m 2 )φ = 0. (3.8) 7

9 To quantize the Klein-Gordon Field I will compare it with an expression of the harmonic oscillator and identify it s proper creation and annihilation operators. We first recall some basic statements from classical field theory. The most fundamental statement in a field theory is the principle of least action. When, for one or more fields φ, we have a Lagrangian L = L(φ, µ φ)d 3 x, the principle of least action states that for the action S = Ldt we have: δs = 0. This condition naturally leads to the so-called Euler-Lagrange equations: ( L ) µ L ( µ φ) φ = 0. Here, summation over µ is understood. For a discrete system the Hamiltonian is defined to be: H = x p(x) φ(x) L, with φ = d L dtφ. Defining the momentum density π(x) = (x) this becomes, in φ the continuum limit: (π(x) ) H = φ(x) L = H. The Hamiltonian has the concrete physical interpretation of the total energy contained in the system described by the Lagrangian. For the Klein-Gordon field we have the Lagrangian density: L = 1 2 ( µφ) m2 φ 2, where equation (3.8) occurs as a solution of the Euler-Lagrange equation, and we have obtain the Hamiltonian: H = 1 2 π2 + 1 ( ) 2 1 φ m2 φ 2. (3.9) To quantize this still classical field, we want to interpret the dynamical variables φ and π as operators, obeying the canonical commutation relations, i.e.: [φ(x), π(y)] = iδ(x y), [φ(x), φ(y)] = [π(x), π(y)] = 0. (3.10) When we switch over to the momentum space, via Fourier transformation, the analogy between equation (3.8) and the harmonic oscillator becomes clear: φ(x, t) = (2π) 3 d 3 p e i p x φ(p, t) The field is expanded in terms of it s it s momentum modes. Note that equation (3.8) becomes: ( 2 t + p 2 + m 2) φ(p, t) = 0. Compare this with equation of motion for the harmonic oscillator: ( 2 t + ω 2) x = 0. We identify the angular frequency as: ω p = p 2 + m 2. Taking the analogy even further, we expand the field in terms of annihilation and creation operators, c.f. equation (3.7), but we treat every momentum mode φ(p) and an individual oscillator. We may write: 1 φ(x, t) = (2π) 3 d 3 p ( e ip x a (p) + e ip x a(p) ) (3.11) 2ω p 8

10 The factor of 1 2ω p in the measure is justified in (3.14) but not important. We now need to check whether the operators a (p) and a(p) are subject to the right commutation relations, that is we need to control if [a (p), a(p )] = δ(p p ), modulo a normalization constant. To do so we first find an expression for the conjugate momentum in a and a: π(x, t) = t φ = i (2π) 3 ωp 2 (e ip x a (p) e ip x a(p) ). We are now ready, using (3.9) to write down the Hamiltonian in terms of the set a and a: H = 1 ) 2 (2π) 3 d 3 p ω p (a (p)a(p) + a(p)a (p) (3.12) To find an expression a and a in terms of φ and π we exsert inverse-fourier transformation on the expressions found above, yielding: d 3 x e ip x φ(x, t) = 1 ( e iω pt a ( p) + e iωpt a(p) ) 2ωp d 3 x e ip x π(x, t) = i ωp 2 ( e iω pt a ( p) e iωpt a(p) ) Solving for a and a we obtain: a (p) = e iωpt d 3 xe ip x( ω p i ) φ(x, t) π(x, t) 2 2ωp a(p) = e iωpt d 3 xe ip x( ω p i ) φ(x, t) + π(x, t) 2 2ωp Where, recalling the commutation relations (3.10), a final straightforward calculation on the commutator, taken on equal time, gives us: [a(p), a (p)] = (2π) 3 δ(p p ), [a(p), a(p )] = [a (p), a (p )] = 0 (3.13) as desired. 3.3 Fock space Having quantisized the Klein-Gordon field, I will construct a Hilbert space on which the operators φ and π (which where previously scalar functions) act. Since, in the construction of these operators, we made a analogy with a field of uncoupled oscillators, we would expect a likewise analogy in the Hilbert space. Starting with the state which is, for our purposes, of most interest, the vacuum is defined as the state 0 such that: a(p) 0 = 0, p That is, the vacuum is the state annihilated by the annihilation operator. Using the expression (3.12) for the Hamiltonian we find: H 0 = E 0 0, E 0 = 1 2 (2π) 3 d 3 p ω p [a(p), a (p)] 9

11 Therefore, 0 is a proper eigenstate of the Hamiltonian. Using the commutation relation for a(p), the eigenvalue is written as: E 0 = d 3 p 1 2 ω pδ(0), making clear that E 0 is an infinite c-number. The infinite value of this vacuum energy looks somewhat disturbing. However, upon close inspection of (3.11), we picture the field as one described by infinitely many oscillators, distributed trough an infinite region of space. Since the vacuum energy of one such oscillator is non-zero, we would indeed expect the sum over all the energies to diverge. Fortunately, this infinite energy doesn t affect our local measurements since we only measure the difference with the ground-state energy. Therefore we can shift, or renormalize, our Hamiltonian by E 0, leading to the normal- or Wickordered 2 Hamiltonian: : H : = H E 0 = H = (2π) 3 d 3 p ω p a (p)a(p) As for all Wick-ordered operators we now indeed have H 0 = 0. 3 I must stress that this renormalization is permitted only if we do not take relativistic effects - equivalence of mass and energy - in account: we may only apply this renormalization in weak gravitational fields. On a global scale the vacuum energy does effect our measurement. In fact, the vacuum energy density Λ = E 0 /V, with V the volume of our universe, is from a quantum field theoretical point of view interpreted as the cosmological constant - the factor which, from a cosmological point of view, accounts for the accelerated expansion of our universe. In fact astronomical observations yield the numerical value of Λ From the vacuum state we can construct particle states by acting on it with the creation operator: 1 p = C p (1) a (p) 0 or, more generally: n p = C p (n) ( a (p) ) n 0 To determine the normalization constant we ll first fix C p (1), and obtain the general constant trough induction. First, note that: 1 p 1 p = Cp C p 0 a p a p 0 = Cp C p 0 a p a p a pa p 0 = Cp C p[a p, a p] 0 0 = C p 2 (2π) 3 δ 3 (p p ). The normalization constant is obtained by demanding: 1p 1 p dµ = 1. At this stage we only need to find the proper Lorentz-invariant measure. This measure is found by restricting the standard measure d 4 p to the hyperboloid p 2 0 p i p i = m 2 (summation over spatial coordinates). This results in[6]: d 3 µ = d 3 p (2π) 3 2ω p. (3.14) 2 An operator is Wick-ordered if all annihilation operators are to the right of the creation operators 3 Henceforth I will omit the ordering symbol : : 4 This numerical value stands in painful contrast with the theoretical prediction: QFT predicts a value of Λ This discrepancy, called the cosmological catastrophe, is yet to be resolved. 10

12 We conclude: C p (1) = 2ω p. A multi-particle system of N identical, linear independent particles in the same state is described by: N N 1 pi = C a (p i ) 0 (3.15) i=1 If p j p k 1 j, k N, j k, then C 2 = (2ω p1 ) (2ω pn ) Rewriting (3.15) and exploiting the commutativity of a (p j ) and a (p k ) we observe: i=1 i=1 i=1 i=1 N N N N 1 pi = C a (p i ) 0 = C a (p σ(i) ) 0 = i=1 N 1 pσ(i) for all σ S N : the particles are subject to Bose-Einstein statistics, to conclude that the Klein-Gordon equation is a bosonic field equation. More generally, a multi-particle system of N particles in different states is described by: N i=1 n (i) p i = C i=1 N ( a ) n (i) (p i ) 0 (3.16) i=1 Note the intuitively obvious isomorphism: 1 p 1 p 2 p The Hilbert space span by all states as in (3.16) is called Fock space. I emphasize that the state space of N particles is N L 2 µ, the symmetrized space of N L 2 µ, where µ is the measure as in (3.14) and L 2 µ the Hilbert space of square integrable functions over µ. These definitions are rigorous enough for our purposes, and I refer to [7] for a mathematical complete treatise on this subject. There is one operator acting on Fock space that is of specific significance. Define: N(p) a (p)a(p), the number operator. This nomenclature is justified by evaluating the expectation value of N(p) on a state in Fock space: ( N N(p k ) = n (i) p i i=1 ) ( N N(p k ) i=1 ) n p (i) i = (0,..., n (k),..., 0) (3.17) Upon evaluation on a multi-particle state, the number operator N(p k ) tells you how many quanta with momentum p k occupy this state. The total number of particles is obtained by evaluating 5 : N tot k N(p k ) 5 By abuse of notation I ve set N(p k ) = n (k). 11

13 3.4 Bogolyubov Transformations In section 2 we expanded the field φ in equation (3.11). Let us write this as: ( ) φ = d 3 p a(p)u p (x) + a (p)u p(x). (3.18) The momentum modes u(x) = c p e ip x form a complete orthonormal basis of L 2 µ. Now suppose we have a second orthonormal set {v v L 2 µ}, defining a new Fock space. We have a second decomposition of the field: ( ) φ = d 3 q b(q)v q (x) + b (q)vq (x). (3.19) The operators b and b define a new Fock space and a new vacuum state: b(q) 0 (v) = 0 q In this expression, n (v) denotes a state in the Fock space generated by the basis {v q }. Since {u p } constitutes a complete orthonormal set, we can decompose v q in terms of u p : v q (x) = d 3 p α pq u p + β pq u p, (3.20) Equating (3.18) and (3.19) and plugging in equation (3.20) for v yields the relation: [ d 3 p au p + a u p = d 3 p d 3 q α pq b + β pq b ] [ u p + d 3 q β pq b + αpqb ] u (3.21) Due to completeness we read off the relation between a and b: a = d 3 q α pq b + βpqb (3.22) The relation (3.20) is a so named Bogolybov transformation[8]. For the sake of completeness I state the inverse transformations of (3.20) and (3.22): u p = d 3 q αpqv p β pq v q (3.23) b = d 3 p αpqa βpqa (3.24) Close inspection of relation (3.22) reveals the first counterintuitive result to come across: if we can find a Bogolyubov transformation with β pq 0, the vacuum state of the two Fock spaces, expressed in the different bases as above, need not equal eachother: a(p) 0 (v) = d 3 q βpq 1 (v) p. Simultaneously we already observe that in this situation the expectation value of the number of particles in the vacuum is altered: N p (v) = d 3 q β pq 2. 12

14 However, just as in the case of vacuum-energy, expect the above expression to give a divergent outcome: it is interpreted as the total number of quanta in the whole of space. We once again renormalize. By separation of the volume of space we obtain the finite quantity of density of particles: d 3 q β pq 2 = n p δ(0), where n p denotes the number density of quanta carrying momentum p. At this stage I would like to recapitulate the obtained results briefly. In (3.11) we decomposed the field φ(x, t) in terms of the momentum modes u(x) = c p e ip x. In this section we observed how changing the basis of our Fock space, or introducing a new set of mutually orthonormal modes, can alter the expectation value of the number of particles in the vacuum state. One way to find such a new set is by making a coordinate transformation in x. The question remains whether there is a non-trivial Bogolybov transformation (with β pq 0) and if so, whether there is one with an elementary expression for the altered expectation value in the vacuum state. 3.5 The Particle Concept In the last section we ve seen how the notion of a vacuum can become ambiguous. Two independent observers can disagree on the expectation value of the number of quanta contained in a vacuum state. Before seeking a non-trivial Bogolybov transformation, it is necessary to reset our intuition on the concept of a particle and define properly what the detection of a particle really means. The root of the ambiguity of the vacuum really lies, I think, in the non-local character of the particle concept. As a particle, or an excited state in Fock space, is smeared out over the whole of space-time, it is unlikely that an observer - making a local measurement - will receive complete and nonspeculative information about this state. In fact, the result of this local measurement will have to depend on the location of the observer in space-time, and even it s history therein. Therefore, it is no longer interesting to speak about the presence of particles in some region, without explicitly stating with respect to what observer and it s world-line. However ill-defined the number of particles in a given state is, it has to stay an invariant property for all inertial observers. Physical measurements are still to be un-altered under Lorentz-transformations. That this is guaranteed is seen best by writing the Klein-Gordon equation in co-variant notation: ( µ µ m 2) φ = 0. One might wonder now what is the source of energy, from a conservation argument needed to provide for the creation of particles. This energy comes from the agent that accelerates the particle detector trough space. Note that, to be able to detect particles, temperature, or volume in the first place, the detector is to be in thermal contact with the region of interest. 13

15 Chapter 4 Curved Coordinates In the previous chapter we equipped ourselves with a basic set of quantum field theoretic tools. As a remarkable feature of the construction of Fock space, we noticed how the introduction of two competing decomposition of the field can lead to an ambiguity of the vacuum state. We are ready to seek an explicit expression for the Bogolyubov transformations as introduced in 3.4. As a result of Lorentz-invariance, we noticed how two inertial observers will never disagree on the vacuum state and the number of particles therein. We will investigate the simplest example of a non-inertial case, shining our light on the uniformly accelerating observer. In this chapter I agree with Misner et al.[9] on the positive signature of the metric: ( + ++). 4.1 Rindler Space Our aim is to evaluate the field equation in a coordinate system for the uniformly accelerating observers. To do so we shall first have to find a suitable set of coordinates and a metric describing his trajectory in space-time. To describe the motion and world-line of a uniformly accelerating observer we consider the flat Minkowskian space-time, with the metric defined by: ds 2 = dt 2 + dx 2 + dy 2 + dz 2 = η µν dx µ dx ν. (4.1) Let the trajectory of an observer be described by x µ (τ), τ denoting the observer s proper time. It s 4-velocity and 4-acceleration are defined by: u µ = dxµ dτ (4.2) a µ = duµ dτ. (4.3) Proper time is related to the coordinates x µ by: dτ 2 = η µν dx µ dx ν 1, hence we have the condition: u µ u µ = 1. (4.4) 1 This is a direct result of de identity τ = γt, with γ = (1 v 2 ) 1/2 the Lorentz factor 14

16 Let the observer have 4-velocity u µ at some time. Then clearly in the co-moving frame 2, it s spatial velocity components vanish, to end up with: u µ = (1, 0, 0, 0). Differentiation of (4.4) with respect to τ yields: a µ u µ = 0. We conclude that the time-component of the 4-acceleration vanishes and we have the Lorentz-invariant identity: a µ a µ = a i a i a 2 (4.5) Let s take without loss of generality the spatial acceleration to point in the positive x-direction, that is, set a 2 = a 3 = 0 and a 1 = a > 0. This leaves us with the differential equation: (du 0 ) 2 (du 1 ) 2 = a 2 (4.6) dτ dτ ( u 0 ) 2 ( u 1 ) 2 = 1 (4.7) Solving this equation while posing the boundary condition of u 1 (0) = 0 yields the equations: u 0 = cosh(aτ), u 1 = sinh(aτ), and after a simple integration we find the parametrization of x µ in terms of τ: x 0 (τ) = a 1 sinh(aτ) (4.8) x 1 (τ) = a 1 cosh(aτ) (4.9) Here I choose x 0 (0) = 0 and x 1 (0) = a 1. The trajectory of the observer is described by a hyperbola, with asymptotes at x 1 = ±x 0, in the first and second quadrant of Minkowski space. Notice how u µ approaches unity as x 0 runs up to infinity. The two asymptotes cut up the Minkowski space-time into four regions, which we will denote with the roman numbers: I for the right region (x 1 > x 0 ), and counting counter clockwise to the upper region IV (x 0 > x 1 ). Equations (4.8) describe the word-line of the observer from the point of view of the co-moving inertial frame. Now we want to find a description of this word-line from the point of view of the accelerating observer itself, that is in terms of it s proper time and length. This is to say that we are interested in a relation between (x 0, x 1 ) and (τ, ξ), with ξ denoting proper length. To find en expression for the proper time, we consider a stick of proper length ξ held by the uniformly accelerating observer described by the proper coordinates (τ, 0) and (τ, ξ), the end held by the observer and the far end respectively. Now consider the co-moving frame at time τ. For him, the stick is represented by (0, ξ) = σ µ. Now this co-moving frame and the ground frame σ O are connected by a Lorentz-boost: ( σ 0 σ 1 ) O = ( γ γ dx1 dt γ dx1 dt γ ) ( σ 0 2 The co-moving frame is the inertial frame with 4-velocity u µ σ 1 ) (4.10) 15

17 Recalling the identity: γdτ = dt, this becomes: ( ) σ 0 σ 1 = O ( u 0 u 1 u 1 u 0 ) ( σ 0 σ 1 ) = ξ ( u 1 u 0 ) (4.11) We are now ready to write down the expressions for ( x 0 (τ, ξ), x 1 (τ, ξ) ) : x 0 (τ, ξ) = x 0 (τ) + σ 0 O, x 1 (τ, ξ) = x 1 (τ) + σ 1 O Introducing the coordinate ρ = 1 + aξ, these expressions turn into: The metric (4.1) becomes: x 0 (τ, ρ) = a 1 ρ sinh(aτ) (4.12) x 1 (τ, ρ) = a 1 ρ cosh(aτ) (4.13) ds 2 = ρ 2 dτ 2 + a 2 dρ 2 + (dx 2 ) 2 + (dx 3 ) 2 (4.14) Space-time described by (4.12)-(4.14) is called Rindler space-time[10]. In fact, it is just region I of Minkowski space, with a different parametrization (this region is therefore sometimes referred to as the Rindler wedge. The fact that that this parametrization does cover only a quarter of the Minkowski space needs to be treated delicately. Note how the horizon of Rindler space-time lies at the line ρ = 0. Since e.g. the point (x 0, x 1 ) = (0, 0) corresponds to (τ, ρ) = (, 0): it takes an infinite amount of time for an event at (x 0, x 1 ) = (0, 0) to intersect the horizon line ρ = 0. 16

18 4.2 The m = 0 Free Field Now that we have found a suitable parametrization for the trajectory of the uniformly accelerated observer, we will investigate what from his point of view the field looks like. Before considering the massive field we start off by considering the massless scalar free (no interactions) Klein-Gordon field, that is, the field that obeys the equation: µ µ φ = 0. (4.15) To develop a field theory in a Rindler space-time it is convenient to write the metric (4.14) in a conformally flat form, that is to show that upon a proper re-parametrization, the metric is angle-preserving (or a re-scaling). The parametrization which shows this is: The metric becomes: ξ = a 1 ln ρ. ds 2 = e 2aξ ( dτ 2 + dξ 2 ). (4.16) The massless scalar field in conformally invariant in 1+1 dimensions. This means precisely that, when there is a conformal connection between two metrics: g αβ g αβ = Ω 2 (x 0, x 1 )g αβ, the action and hence the field equation are unaltered. Thus, the conformal invariance of the massless field, and the conformal connection between the metrics (4.14) and (4.1) trough Ω(τ, ρ) = e aξ, guarantees that from the point of view of the accelerated observer the field equation has the same form as for the inertial observer: 2 φ τ φ ξ 2 = 0 (4.17) There is however one subtle, but important difference between this equation and the inertial Klein-Gordon field equation: upon decomposing the field in Rindler coordinates, we will only find wave equation defined in region I of Minkowskian space time. Before being able to find the right Bogolyubov transformations connecting the competing compositions and the corresponding Fock spaces these decompositions construct, we will have to seek for continuations of the found wave solution into the the remaining three regions. First of all we find the first straightforward wave equation: ( ) φ = d 3 p b(p)r p (τ, ξ) + b (p)rp(τ, ξ) (4.18) Let s pick out one momentum mode and try to extend it to the regions II IV. A momentum mode of the field equation is described by: r p (τ, ξ) = 1 2ωp e i(ωpτ kξ) It admits positive energy and is defined only in region I. Recalling that ω = k, we first of all look at waves traveling to the right, that is waves with k > 0. It is instructive to work in light-cone coordinates: ξ ± = ξ ± τ. (4.19) 17

19 In terms of these coordinates this right-traveling wave is written as: Whereas a left-traveling wave corresponds to: r + = 1 2ω e iωξ (4.20) r = 1 2ω e iωξ+ (4.21) These (analytical) wave functions are confined to region I, and want to (analytically) continue them beyond this region. To do so, we recall the relations between (x 0, x 1 ) and (τ, ξ), introducing the Minkowskian light-cone coordinates x ± = x 1 ± x 0. A straightforward manipulation yields: ξ ± = a 1 ln(ax ± ) e iωξ± = (ax ± ) iωa 1 (4.22) Using these expressions for the wave modes in terms of the Minkowskian lightcone coordinates, we may analytically extend the modes beyond region I. Note, however, how the function e iωρ is a function only of x, and is confined to the upper-half x + -plane (x + > 0). The function may only be continued to region II. Likewise, the mode e iωρ+ is confined to the upper-half x -plane (x > 0) and may only be continued to region IV. What remains is to find an extension to tack-on region IV. To do so we reverse the signs in place and time ((x 0, x 1 ) ( x 0, x 1 )), which is equivalent with interchanging region I with IV, and II with III. This offers us the well defined analytical function in region IV 3 : r IV = 1 2ω e iωξ (4.23) In summing up, we have found two analytical wave functions, which taken together can be continued to cover the entire of the space-time. These functions are called the supports of their continuations: r I = r IV = 1 2ω e i(ωτ kξ), region I (4.24) 1 2ω e i(ωτ kξ), region IV (4.25) Once again I stress the footnote to avoid confusion in the above expressions. Now we are ready to expand the field: ( ) φ = dk b I r I + b I r I + b IV r IV + b IV r IV (4.26) Here, I absorbed the normalization constants in the momentum modes. The operators b i (i = I, IV ) define a new Fock space with a vacuum defined by b i 0 R = 0, with 0 R denoting the Rindler vacuum. It is are task to find the right Bogolyubov transformations, connecting the two expansions. The straightforward approach is quite elaborate and cumbersome, but there is a 3 Here I made a slight abuse of notation. This coordinate ξ is actually different from the ξ in 4.22 and obtained by setting x x 18

20 trick due to Unruh to avoid hard labor. Let s look more closely to the continuation of the mode r IV : r IV = 2π 2ω ( ax ) iωa 1 (4.27) Due to the factor of ( 1) iωa 1, this is a multi-valued function. To fix it, let s demand that Arg ( ) ( 1) iωa 1 = π. This choice places the branch-cut in the lower half x -plane. We obtain x = e iπ x and end up with: r IV = 2π 2ω e ωπa 1 (ax ) iωa 1, (4.28) which is unambiguously defined in the regions III and IV. Now we may construct one wave function which is spread out over the whole of space-time: R 1 = r I + e ωπa 1 r IV (4.29) By the same line of reasoning we simultaneously find a likewise spread out function: R 2 = r IV + e ωπa 1 r I (4.30) We find a new decomposition as proposed by an inertial observer: φ = dk B 1 R 1 + B 1 R 1 + B 2 R 2 + B 2 R 2 (4.31) The Bogolyuov transformation connecting b i (i = I, IV ) and B j (j = 1, 2) is straightforward: b I b I b IV b IV = The normalization of B j and B j e πω 0 1 e πω 0 0 e πω 1 0 e πω of b i and b i : [b i (p), b i (p )] = δ ii δ(p p ). A straightforward calculation yields: B 1 B 1 B 2 B 2 (4.32) is determined by the commutation relations [B j (p), B j (p )] = δ jj δ(p p ) 1 e 2πωa 1 (4.33) Having found the right transformations we may express the number operator 4, N(p) = b I b I in terms of B j : N = b I b I = B 1 B 2 + e πωa 1 (B 1 B 2 + B 2B 1 ) + e 2πωa 1 B 2 B 2 (4.34) 4 Notice how we only need b I, since the trajectory is in region I and b IV excites modes which vanish in this region 19

21 Consider the Minkowksi vacuum, 0 M. Evaluation of the number operator as above gives us the expectation value of the number of particles as detected by a Rindler observer: 0 M N 0 M = e 2πωa 1 0 M B 2 B 2 0 M δ(0) 2πωa 1 = e 1 e 2πωa 1 δ(0) = e 2πωa 1 1 (4.35) This last term is recognized as the distribution of a blackbody spectrum. The Rindler observer detects particles, distributed according to a Planck factor (e ω/t 1) 1. He detects a thermal bath, with a mean temperature of T = a 2π or, restoring the classical factors, T = a 2πck b. This temperature is referred to as the Unruh temperature. It is the main result of this thesis. 20

22 4.3 The Massive Free Field Having computed the Bogolybov transformations for the free Klein-Gordon field with m = 0, we wonder whether this result can be generalized to a massive field theory. The first problem one encounters when seeking such a generalization is that in a massive field theory the conformal invariance is lost: the field equation with respect to the Rindler observer differs from that of the Minkowskian. Due to this loss we might as well focus on a massive field theory. The approach in this case is somewhat more analytical (in the technical instead of functional sense) then the m = 0 case. In the treatment of the massive case I mainly follow the line of thought as given by t Hooft[11]. We consider a field that obeys the Klein-Gordon equation: ( µ µ m 2) φ = 0 (4.36) It is expanded as in (3.11), where the mass term is incubated in: p 0 = p 2 + m 2. As we do not use conformal invariance, we stick to our initial description of Rindler space, eqs. (4.12)-(4.14). We shall, for convenience, work in a parametrization with a = 1, making the time coordinate τ dimensionless. In the end we will plug it back in while interpreting our result. For now we will work with the description: x 0 = ρ sinh(τ) x 1 = ρ cosh(τ) (x 2, x 3 ) = ( x 2, x 3 ) ds 2 = ρ 2 dτ 2 + dρ 2 + (d x 2 ) 2 + (d x 3 ) 2 In terms of these coordinates, the Klein-Gordon equation reads: ( ) 2 τ 2 + +ρ2 ( 2 x ρ 2 m2 ) φ = 0 (4.37) To describe the solutions of this partial differential equation we introduce a special function: K(ω, α, β) = 0 ds s iω 1 e isα+iβ/s (4.38) Using a partial integration in s, it is readily verified that, in terms of K, a solution to 4.37 is given by: φ(τ, ρ, x) = K(ω, 1 2 µρ, 1 2 µρ)ei k x iωτ (4.39) Here, µ = k 2 + m 2. Before identifying and interpreting the creation and annihilation operators in this solution, I list some helpful properties of the introduced function (4.38): K (ω, α, β) = K( ω, α, β) (4.40) K(ω, σα, β/σ) = σ iω K(ω, α, β) (4.41) K( ω, α, β) = e πω K (ω, α, β), α > 0, β > 0 (4.42) K( ω, α, β) = e +πω K (ω, α, β), α < 0, β < 0 (4.43) 21

23 The first two equalities are straightforward. The last two exploit the fact that the integrand of (4.38) is bounded for Im(s) 0, allowing for the rotation of the contour trough s se iπ. We look for an expansion of the solutions similar like that of (3.11). Taking the Fourier-transform of (4.39) with respect to τ we find the field decomposition 5 : φ(τ, ρ, x) = A(τ, ρ, x) + H.c 1 A = dω 2(2π) 4 d 2 k K(ω, 1 2 µρ, 1 2 µρ)ei k x iωτ a r ( k, ω) (4.44) We would like to interpret a r ( k, ω) as an annihilation operator, but before doing so we should note a subtlety in the mode-expansion (4.44): The integration in 4.44 runs over both positive and negative values of ω. In the negative regime, a r ( k, ω) annihilates a negative amount of energy, or putting it the other way around: it creates a positive amount of energy. So when the integration runs over negative values of ω, a r ( k, ω) is interpreted as a creation, instead of an annihilation operator. To find the Bogolyubov transformations, we should first order the operators and integration in such a way that all the creation and annihilation operators are collected together. To do this, we make use of the equalities (4.42) and (4.43), finding the properly ordered expansion. For the plane ρ > 0 this becomes: 1 2(2π) 4 0 dω e iωτ d 2 k e i k x K(ω, 1 2 µρ, 1 ) (a 2 µρ) r ( k, ω) + e πω a r( k, ω) + H.c (4.45) while for ρ < 0 we get: 1 dω e iωτ 2(2π) 4 0 d 2 k e i k x K(ω, 1 2 µρ, 1 ) (a 2 µρ) r ( k, ω) + e +πω a r( k, ω) + H.c (4.46) Note that so far we did nothing but rearranging the terms of (4.44). Identifying the integration variable s in the definition (4.38) with k0 k 3 µ we identify the operator a r ( k, ω) as: dk 3 a r ( k, ω) = a(k)( k 3 + k 0 ) iω (4.47) 2πk 0 µ Recall that (k 0 ) 2 = (k 3 ) 2 + µ 2. Realizing how ( ) ( ) k 3 +k 0 iω k µ = e iω ln 3 +k 0 µ, we readily obtain the inverse Fourier-transform of (4.47): a(k) = 1 2π ( dω a r( k, ω)e iω ln k 0 ) k 3 +k 0 µ (4.48) The commutation relations as in (3.13) give us likewise commutation relations for a r : [a r ( k, ω ), a r( k, ω )] = δ(ω ω )δ( k k ) (4.49) 5 Sometimes I will write φ = A + H.c instead of φ = A + A 22

24 Recalling the normalization in (4.33), we are now invited to apply the following Bogolyubov transformations: 1 e 2πω a I ( k, ω) a I ( k, ω) a II ( k, ω) a II ( k, ω) = e πω 0 1 e πω 0 0 e πω 1 0 e πω a r ( k, ω) a r( k, ω) a r ( k, ω) a r( k, ω) (4.50) By the virtue of the square root normalization factor, the annihilation and creation operators commute canonically, with the extra: [a I, a II ] = [a I, a II ] = [a I, a II] = 0 When we calculate the Hamiltonian, the importance of the reordering in (4.45) and 4.46 comes best to light: H = dω d 2 k ωa r a r = dω d 2 k ( ω a I ( k, ω)a I ( k, ω) a II ( k, ω)a II ( k, ω) ) 0 =: H I H II (4.51) Similar tot the massless case, we have split our operators into two groups: group I, which only creates and annihilates states in the wedge ρ > 0, and group II, which does so only in the regime ρ < 0. Looking at the expression for the Hamiltonian, we see that even more generally every observable constructed out of φ in region I commutes with H II and vice versa. Especially we have: [H I, H II ] = 0. Now let 0 M be the vacuum as proposed by a Mikowskian observer, that is, let it be the state such that a(k) 0 M = 0. Let s split the basis of the Rindler Fock space in two pieces in the following way: let n I a I a I and n II a II a II. Let n I and n II be defined in the usual way, then the basis of Fock space consists of all combinations n I n II. The consequence of our Bogolyubov transformation is: ( ) a I ( k, ω) e πω a II ( k, ω) 0 M = 0 (4.52) ( ) a II ( k, ω) e πω a I ( k, ω) 0 M = 0 (4.53) Or: n I 0 M = e πω a I a II 0 M = e πω a II a I 0 M = n II 0 M (4.54) Thus, the Minkowskian vacuum state is built up out of all Rindler states with n I 0 M =n II 0 M and completeness of Fock space gives us: 0 M = n C n n n (4.55) Here n n denotes the state with n I = n II. To determine the constants C n we make use of the relation (4.52), finding a 23

25 recursion in n: C n n n 1 I n II n = e πω C n n + 1 n I n + 1 II n C n+1 = e πω C n (4.56) We find: 0 M = C 0 e nπω n n (4.57) To determine C 0 we demand that 0 0 M = 1, yielding: n 0 M = k,ω 1 e πω e πnω n, n ± k (4.58) We calculate the probability for a Rindler observer in region I to detect n I particles with energy ω and momentum + k while looking at the Minkowskian vacuum: P (n I ) = 0 ( n I n II n I n II ) 0 M = C ni 2 = (1 e 2πω ) 1 e 2πn I ω n II (4.59) Whereas the derivation of the Unruh effect for the massless cases seemed to lean completely on the analytical continuation of the wave solutions beyond the first region, this does not appear, at first sight, to be the (crux) in the massive case. The most important step here was to rearrange the operators properly. Note however that this operation hinges entirely on the properties (4.42) and (4.43) of the expansion in Rindler space. Due to this analytical rewriting an rearranging of the wave modes, this method in fact does come down to that employed in the massless case. n=0 4.4 Rindler Thermodynamics The Planckian distribution of the particle density for the Rindler observer hints towards a connection between the quantum behavior of particles and their thermodynamics. In this section I will give a derivation which even clearer points towards this intersection. Although the methods exploited below are somewhat more heuristic then the quite mathematical derivation given in the last two chapter, it does beautifully combine elements from QFT, General Relativity and Thermodynamics 6. Consider a set of wavefunctions { ψ n }, each carrying an energy of H n = ψ n H ψ n. From the theory of thermodynamics we have: P i = P ( ψ i H ψ = H i ) = 1 Z e βhi (4.60) Here, P i denotes the probability for ψ to be in the state ψ i, β = 1/T and Z is the partition function: Z = e βhi (4.61) i 6 This section is inspired on a talk I had with prof. E.P. Verlinde 24

26 Naturally, we have: P i = 1 (4.62) To define an operator, let us introduce the density matrix 7 : The equivalent of (4.62) is now: The eigenfunctions and -values are found by: i ρ = 1 Z e βh (4.63) Z = tr ( e βh) (4.64) tr(ρ) = 1 (4.65) ρ ψ i = P i ψ i (4.66) Consider at the same time the time-evolution operator, satisfying: e ih t ψ i (t) = ψ i (t + t) (4.67) Let an observer be uniformly accelerated, having his world-line described by the Rindler space-time: ds 2 = dt 2 + dx 2 ds 2 = ρ 2 dτ 2 + a 2 dρ 2 (4.68) We re-parameterize the time-coordinate, introducing so-called Euclidian time: The metric becomes: τ iτ τ E. (4.69) ds 2 = ρ 2 dτ 2 E + a 2 dρ 2 and we see how the time-coordinate is periodic (this is also seen by τ E = ρa 1 sin(a 1 τ E )): τ E = τ E + 2π a Simultaneously, the time-evolution operator is turned into e teh and we are invited to make the identification: β t E. Thus, the evolution operator sends: τ E τ E + β (4.70) We finally conclude: β = 2π a or: T = a 2π, the Unruh temperature. First of all we note how we didn t use the concept of a particle density whatsoever. In fact, this result states that the whole set { ψ n } is distributed among Rindler space-time with a Planckian spectrum of T = a/2π. A closer inspection of the more rigourously derivation of the massless and massive case reveals 7 For an n n matrix A = (a j i ), we define tra = n i=1 ai i, the trace of the matrix 25

27 however that the introduction of the number operator was not mandatory to get to the result. In fact, any operator constructed out of the annihilation and creation operators would have been sufficient. Second of all, the notion of the vacuum is not introduced above. This however is not problematic. The Unruh effect is formulated for the vacuum merely out of an anecdotic consideration: the result is the most counter-intuitive when formulated for the vacuum, and the expressions and derivations are somewhat less cumbersome. However, it does apply just as well to a many-particle state, see e.g. [12], p.55. However, we did not need the notion of an analytical continuation of wave mode expansion of any kind. We actually didn t introduce quantum field theoretic tools at all: we worked in a canonically Heisenberg picture, without introducing a field equation, let alone it s quantization. To what extend were these methods mandatory then? I think that to this extend the derivation given above should not be seen as a benchmark but merely as a hint towards a connection between QFT and spacetime geometry trough thermodynamics. By studying the geometry op space-time around a black hole, we shall see how this link is further strengthened. 26

28 Chapter 5 Black Hole Mechanics From a strictly formalistic point of view a black hole is nothing but an exact solution to Einstein s field equation, the equation Einstein formulated to describe the geometry op space-time and the gravitational force. Fortunately there is more to physics then formalism and mathematics, and by the virtue of their strong imaginative appeal, black holes are most famous objects of interest not only in contemporary theoretical physics, but also in the entire of popular science. Classically, we can picture a black hole as a gravitational object with an escape velocity exceeding that of the speed of light. With the aid of plain Newtonian mechanics we immediately find a characteristic length-scale for such an object, known as the Schwarzschild radius: r Sw = 2M (5.1) For an object with fixed mass M, this radius is the point of no return: once you found yourself inside the Schwarzschild radius, the gravitational force will prevent you from ever getting out, wether you are a space-ship or a photon. Not only are these objects physically allowed: astronomers have, indirectly, observed objects in the sky whose properties completely match the theoretical predictions of black holes and can t be identified with anything else. In the treatment of the vacuum we learned how one should be careful with definitions. Just as Unruh showed how a vacuum can not be defined as a region with no particles, Hawking has shown how a black hole can not be defined as a region in space time out of which nothing can escape. The first step in investigating the properties of a black hole more rigorously is stepping away from the Newtonian picture, entering the realm of Einstein s theory of General Relativity. 27

29 5.1 The Schwarzschild solution The most compact summary of Einstein s theory of gravity has perhaps been given best by Wheeler [9]: Spacetime tells matter how to move Matter tells spacetime how to curve This statement immediately reveals the dualistic character of gravitation, or Geometrodynamics. Together with the equivalence principle 1, the realization of curvature of space-time led Einstein to set up his General relativity. I will not go in to full detail on the field equation and will only investigate the properties of it s solutions. I refer to [13],[14] for compact yet exhaustive introductions. Solutions of the field equations are metrics g µν, describing the curvature and geometry of space-time. The exact solution found by Schwarzschild is given by: ds 2 = g µν dx µ dx ν = (1 2M/r)dt 2 + (1 2M/r) 1 dr 2 + r 2 dω 2 (5.2) Schwarzschild and his contemporaries were rather astound by the apparent singularity that occurs as r 2M. This limit is disturbing because it results in g 00 0 and g rr. Now we must not rush into conclusion and we should determine first wether this apparent singularity is actually a physical one or just a reminiscent of an unlucky choice of coordinates. In his famous original paper Schwarzschild re-parameterized the radial coordinate by r = ( r 3 (2M) 3) 1/3, placing the apparent singularity at the origin r = 0. This does not resolve the problem and this transformation is not used anymore. One coordinate substitution, due to Kruskal, however does reveal the nature of the disturbing point: xy = ( r 2M 1) e r/(2m) (5.3) x/y = e t/(2m) (5.4) In terms of these Kruskal coordinates, the line element becomes: ds 2 = 32M 3 e r/(2m) dxdy + r 2 dω 2 (5.5) r The apparent singularity has vanished to conclude that as r 2M there is no physical singularity. Note how there always is a true physical singularity as r 0. 1 The principle that an observer can not distinguish a uniform acceleration from a free fall in a gravitational field 28