Küresel dalgalarda vakum dalgalanmaları
Vacuum fluctuations in spherical waves
- Tez No: 39669
- Danışmanlar: PROF.DR. MAHMUT HORAÇSU
- Tez Türü: Doktora
- Konular: Fizik ve Fizik Mühendisliği, Physics and Physics Engineering
- Anahtar Kelimeler: Belirtilmemiş.
- Yıl: 1994
- Dil: Türkçe
- Üniversite: İstanbul Teknik Üniversitesi
- Enstitü: Fen Bilimleri Enstitüsü
- Ana Bilim Dalı: Belirtilmemiş.
- Bilim Dalı: Belirtilmemiş.
- Sayfa Sayısı: 62
Özet
ÖZET Bu doktora çalışmasında enflasyonist modele alternatif olan ve gözlem sel olarak evrendeki mikrodalga anizotropilerini ve kuasarların çift görülmeleri gibi olayları açıklayan bir model olan kozmik sicim modelinin vakum dalgalanmalarına yani parçacık yaratılmasına neden olup olmadığını çeşitli örnek fon alanları alarak inceledik, incelediğimiz örneklerde kozmik sicimlerin vakum dalgalanmalarına neden olmadıklarını gördük. Ancak tam ve kesin bir sonuca varabilmek için böyle örneklerin hem teorik hem de deneysel olarak desteklenmesi gerekmektedir. ilk bölümde bu çalışmanın amacı belirtilmiş ve konuya genel bir giriş yapılmıştır, ikinci bölümde ise Minkowski uzayı için vakum ve parçacık tanımı yapılarak skaler alan kuantizasyonu örneği verilmiştir. Üçüncü bölümde eğri uzay tanımı verilmiş ve bu uzaylarda parçacık kav ramının tam olarak anlaşılamadığı ve ancak yerel olarak tanımlanan büyüklüklerin parçacık kavramı için bize daha doğru sonuçlar verebileceği söylenmiştir. Burada örnek olarak Robertson- Walker tipi bir uzay ele alınarak bu uzayda parçacık yaratılması incelenmiştir, ikinci ve üçüncü bölümlerde Breli ve Davies'in“ Quantum Fields in Curved Space”isimli kitabından alıntı yaptık. Dördüncü bölümde Gleiser-Pullin tarafından verilen kozmik sicimlerin kopması olayına karşı gelen örnek fon uzayda parçacık yaratılması olup olmadığı incelenmiş ve parçacık tanımını yapabileceğimiz bir sonuç elde edilememiştir. Bölüm 5'de Nutku tarafından önerilen ve kopan ve uçları dönerek uzaklaşan kozmik sicim örneğine karşı gelen uzayda parçacık yaratımı gözlenememiştir. Son bölümde ise yine Nutku tarafından verilen ve kopan sicim örneğine karşı gelen metrikle işe başladık ve bu uzayda metriğin özelliğinden dolayı elde edilen ilginç bir kısıtlama bulduk.
Özet (Çeviri)
VACUUM FLUCTUATIONS IN SPHERICAL WAVES SUMMARY The most serious difficulty in discussing quantum field theory in curved space-time is that there does not exist a quantum field theory formalism an arbitrary curved space-time. This problem is deep and arises from the fact that standard formalism of fi eld theory requires preferential slicing in space-time. In other words, quantum field theory as we know today is Lorentz invariant; but it is not generally covariant. This conceptual problem introduces a new level of observer depen dence in quantum theory, the implications of which axe not yet well understood. It should be noted that such a difficulty exists even in flat space when curvili near coordinates are used. Our aim to couple the particles to the field through the d'Alembertian operator, written in the background of a non-trivial metric. Here we do not discuss these problems, but study the gravitational effects in the semi-classical approximation. The possibility that the cosmic strings may be at the root of the mecha nism of galaxy formation is not still ruled out. We need further experimental data on the anisotropies in the cosmic microwave radiation, lensing of quasar images, or gravitational radiation stemming from decay of strings to accept or reject this alternative to inflationary quantum fluctuations. If cosmic strings exist they may give rise to vacuum fluctuations which in turn, may result in particle production. The exterior space-times of the cosmic strings are exact solutions of Ein stein's equations. We use the topological properties of fiat but conical exterior space-time to derive the physical effect of the vacuum fluctuation of a confor- mal scalar field outside such a string. Vacuum polarization effects due to scalar field arise in a fiat space-time when the topology is unusual at the boundaries present. There can also be vacuum polarization in a flat space-time even if the manifold is complete without boundaries, in this thesis we study the first case. The exterior space-time is locally flat, a piece of Minkowski space-time, but globally it has a conical structure. viTo search for vacuum, fluctuations, the usual method is the calculation of the energy momentum tensor of a scalar field in a background of a metric proposed for the cosmic string. Alternatively, in the presence of a time-like Killing vector, one can define in and out states and calculate the Bogolubov coefficients to see whether particle production actually occurs. A third possib le method would be an approximate field theory calculation which has been applied to gravitational particle production during string formation. In this study we apply the first and the second methods to investigate whether particle production occurs when the cosmic strings snap and their ends move at the speed of light in different metrics given by Nutku and Glei- ser-Pullin. In Chapter 1, we want to explain our aim in studying these problems. In Chapter 2, we define.introduce the particle concept for flat, i.e; Minkowski space-time. In this space all inertial observers agree on the particle concept. The Minkowski time coordinate assures the existence of a global rime-like Kil ling vector in fiat space-time. Whenever a global time-like Killing vector field is available it may be used to provide a natural definition for the particle. But not all space-times permit this luxury. If our space-time asymptotically approaches static limits, once again a particle definition can be based on these asymptotic domains. An arbitrary curved space-time which does not possess any of the above mentioned features defies a field theorv formalism. While the above fact may be somewhat discouraging, it should not be surprising. Even in nongravitational field theories the particle content of the field can be defined only for free fields. Since quantum theory in a curved space-time is equivalent to a quantum theory interacting with a gravitational field, it is only natural that the concept of particle is not well defined. In the third chapter we introduce some features of curved space-time and discuss the quantization of a scalar field in such a background. In these spa ces there is a serious problem in defining particle and detecting the particles. We have to define local terms instead of global ones like particle and partic le detectors. Since the curved space-times do not have Poincare' invariance, we can not define global vacuum and particle concepts. So we define local terms like < 0|TM“|0 >. This is achieved by calculating the two point function, Gf{x,x ) for the model and differentiating according to coordinates after the coincidence limit is taken. Then we give the second, Bogolubov coefficients method to calculate particle production. Lastly as an example of a curved space-time, we calculate particle production for a Robertson- Walker type met- Vllric. We see that particle production occurs due to cosmic expansion which is the character of the R-W universe. These three chapters are meant to be of introductory nature, and are taken from standard textbooks. In the latter parts of the thesis we apply these methods to new problems. In Chapter 4, we apply the first and the second methods to investigate particle production during the snapping of the cosmic string. The background metric is choosen as the Gleiser-Pullin solution, which describes the snapping of a cosmic string whose ends expand with the velocity of light. We start with the metric given by Gleiser and Pullin ds2 = Adudv - A2d$2 - B2dz2 where A = u - [/32e{-v) + e(v)]v B = u + [p2e(-v) + 6{v)]v and here 0 is the Heavyside step function. The Ricci tensor equals zero, but one component of the Weyl tensor is proportional to the Dirac delta functi on. We calculate Bogolubov coefficients with four different coordinate frames. First we change to variables with u = ^^ v - j£ x = r cos @ y = r sin Ş. For the massless scalar field the equation of motion n-ud2 id \d2 d2 a* M [dt* tdt (Pt2dz2 dx2 dy2İ When we calculate in and out states we find Hankel functions and for the Bogo lubov coefficient 0k we find zero. In the second case t = u+/32v r = u -fi2v here and z are the same. Now the field equation is U,dt tds dr2 (32r2d2 f32t2 dz2* For this coordinate system we find Hankel functions as solution. In the third case we use the 2u - p{\ + cos 9) 2ft2v - p(l - eos#) and z are not changed. In this coordinate system the field equation is d2f 2 cot 9 df \df_ 1 df cot 0 5/ dp2 p d82 p dp p2 sin 9 cos 9 39 p2 39 i d2f i d2f f32 cos2 9 3>j>2 /32 dz' = 0 VHlWe get in this coordinate system the Legendre functions. In the fourth case coordinates are u = p cosh 6 {32v = psinhd and the field equation is { - - sinh2Ö- ? + - cosh 20- - - sini 20- 1 2 dp2 p odop 2p2 o62 7,2 + - (cosh 20 - sinh 26) } 9 = 0. P Now the solutions are Hypergeometric functions and we get again a null result for the particle production coefficient. In Paragraph 4.5 we calculate the Green's function for a scalar mas- less particle in this background. We change to variables u - ^p v = |rjj? x - r cos d, y = r sin^©. The equation of motion for the Green's function in this background is ? [dt2 + tdt t2p2dz2 dx2 dy2^ F -8JLz±}6{x-x')6{y-y')6(z-2'}. Then we find Green's function as 1 Cflnt'/t) Gf = 167r2/3(t + 1' )(t -*')_( l2il/i)2 _ (z ^2 But we see that there is a serious problem in formulating a well-defined va cuum, and another in taking the coincidence limit. There is an ambiguity in the renormalization procedure. We see that we have to be careful in the regularization of the stress-energy tensor arising from C^°' metrics. In Chapter 5, we use the metric given by Nutku ds1 = 2Pdudv + 2uPçdC>dv + 2uP^dÇdv - 2u2d(dÇ Although the Ricci scalar is zero in this metric, we have a discontinuity in the Weyl tensor, which is proportional to the Heavyside step function. Spa ce-time is locally flat, but globally it is a cone. In this chapter we calculate the Green's function corresponds to the metric given above. We know that if the expectation value of the energy- momentum tensor is equal to zero then we can conclude that there is no vacuum fluctuation in this background. We see that the Green's function is n 4tt2Wv/HX1/2) (uu{2x - ly) 3yNir(2),0 f ~ ~ Gf = -. tt(,v 2, 2A-, rr-T~+- )Ho (2mJ(u ~u)(v-v)) \u - u \(u - u ){x2 + y2) [u - u )ul u * IXfor this background: get zero for the energy-momentum tensor. So no vacu um fluctuations occur. The disappearance of the trace of TM”suggests the existence of conformal symmetry. Although our metrics are not conformal symmetric the null result may be due to symmetry after the string snapped. Chapter 6, we use another form of Nutku 's metric and calculate the Green's function but here the metric is a bit different from Chapter 5. The P function is now equal to P = i + !N and k = ±1. Here we calculated the Green's function of a flat solution. The parameters of our metric are chosen to give us the trivial solution. The trivi ality of the metric should be reflected in the solutions of the wave equation, and finally in the Green's function. Insisting on the flat space solutions means terminating the perturbation series. This requirement, however is only satisfi ed by a special relation among our free parameters, which reduces the degree of the freedom of the problem. This amounts to reducing the dimension of the space-time of the model by one. Although our example is a simple one it may be of importance in showing how one can reduce the dimension of space-time by constrains, here arising from physical conditions. Another immediate result is that we do not get any nonzero result if k equals +1. One can check easily that all of our terms in Green's function restric the initial metric to one value of k. The peculiar behaviour of our model is probably due to the fact our metric is C^°\ given in two patches. x
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