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Invariant Construction of Cosmological Models Erik Jörgenfelt

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Invariant Construction of Cosmological Models Erik Jörgenfelt Invariant Construction of Cosmological Models Erik Jörgenfelt Spring 2016 Thesis, 15hp Bachelor of physics, 180hp Department of physics Abstract Solutions to the Einstein field equations can be constructed from invariant objects. If a solution is found it is locally equivalent to all solutions constructed from the same set. Here the method is applied to spatially-homogeneous solutions with pre-determined algebras, under the assumption that the invariant frame field of standard vectors is orthogonal on the orbits. Solutions with orthogonal fluid flow of Bianchi type III are investigated and two classes of orthogonal dust (zero pressure) solutions are found, as well as one vacuum energy solution. Tilted dust solutions with non-vanishing vorticity of Bianchi types I–III are also investigated, and it is shown that no such solution exist given the assumptions. Sammanfattning Lösningar till Einsteins fältekvationer kan konstrueras från invarianta objekt. Om en lösning hittas är den lokalt ekvivalent till alla andra lösningar konstruerade från samma mängd. Här appliceras metoden på spatialt homogena lösningar med förbestämda algebror, under anta- gandet att den invarianta ramen med standard-vektorer är ortogonal på banorna. Lösningar med ortogonalt fluidflöde av Bianchi typ III undersöks och två klasser av dammlösningar (noll tryck) hittas, liksom en vakum-energilösning. Dammlösningar med lut och noll-skild vorticitet av Bianchi typ I–III undersöks också, och det visas att inga sådana lösningar existerar givet antagandena. Contents 1 Introduction 1 2 Preliminaries 3 2.1 Smooth Manifolds...................................3 2.2 The Tangent and Co-tangent Bundles........................4 2.3 Tensor Fields and Differential Forms.........................6 2.4 Lie Groups....................................... 13 3 Spacetime 15 3.1 Metric.......................................... 15 3.2 Connection and Curvature............................... 16 3.3 The Cartan Equations................................. 20 3.4 Stress-Energy Tensor and Fluid Dynamics...................... 22 3.5 Spatially-Homogeneous Solutions........................... 26 4 Invariant Construction of New Solutions 31 5 Orthogonal Bianchi Type III 35 5.1 Dust Solutions..................................... 36 5.2 Vacuum Energy Solutions............................... 39 6 Dust Solutions with non-Vanishing Vorticity 41 6.1 Bianchi II........................................ 42 6.2 Bianchi III....................................... 43 7 Conclusions 47 8 Appendix 49 1 Introduction Study of the equivalence problem in general relativity has lead to the development of a method to construct solutions to the Einstein field equations in terms of invariant objects[14]. Integrability conditions were found in [5], and in [6] the method was extended to determining the isometry group and full line element, along with application in a fixed frame. In this essay that method is applied to find orthogonal Bianchi type III solutions, whence we briefly show, in subsection 3.5, how the method can be applied to a pre-determined spatially- homogeneous isometry group algebra. This is done through choosing a preferred co-moving Lorentz frame, dependent on the tilt. The Bianchi types classify all possible simply invariant three-dimensional isometry groups into nine types, Bianchi I–IX, based on their algebra. The preferred frame determines all Ricci rotation coefficients in terms of the tilt and the normalization factors of invariant orbit frame fields, and the derivatives of both, along with the group algebra. With the method we find two classes of dust (zero pressure) solutions, and one vacuum energy solution that is found to be a special foliation of de Sitter and anti-de Sitter space. Observation of the redshift from distant objects indicates an accelerating expansion of the universe and the large scale isotropy of cosmic microwave background radiation is indicative of large scale homogeneity[10], although the latter result has been called into question[20]. There have also been reports of results indicative of large-scale anisotropy in the current state of the universe[21]. Such observations, among others produce observational evidence to compare the physical properties of any cosmological model against, and indicate deviations from the current standard model of cosmology[10]. The computer algebra programs CLASSI by Jan Åman[1] and EXTERIOR by Michael Bradley[3] are used in the application of the theory. Solution of the differential equations that arise is aided by the program REDUCE. This essay is structured as follows. In section2 we cover preliminary results in differential geometry, and introduce some necessary concepts from Lie group theory. In section3 we cover the mathematical structure of spacetime, some fluid dynamics, and some fundamental results concerning spatially-homogeneous models. In section4 we review the method of construction of solutions from invariant objects in a fixed frame, and in particular give the necessary and sufficient conditions for a set to produce a geometry. In section5 we apply the method to orthogonal Bianchi type III solutions, and in section6 we apply it to attempt to find dust solutions with non-vanishing vorticity. If the standard invariant frame fields can be assumed to be orthogonal on the orbits, it is shown that no such solution exists in Bianchi types I – III, and the method could be further applied to Bianchi types IV – IX, although the complexity of the equations is expected to rise for higher Bianchi types. These results also hold for constant, non-zero pressure. The mathematical framework laid out here is based on [11], and more stringent formalism can be found there, although we prove proposition 2.5, which is left without proof in [11]. Some inspiration is also taken from [15]. We will use units such that the speed of light c = 1, and such that 8πG = 1, where G is the gravitational constant. We shall follow the Landau-Lifshitz timelike convention, and let indices i, j, k . denote frame fields on the tangent and co-tangent bundle unless otherwise noted. For coordinate frames we will use µ, ν, σ, . .. 1 2 Preliminaries 2.1 Smooth Manifolds Smooth manifolds are the fundamental mathematical object behind the general relativistic de- scription of gravity. Although smooth manifolds are special topological manifolds we will make no further mention of this fact. Instead we will define smooth manifolds directly via the existence of an atlas of smooth charts. In this approach focus shall be placed on achieving a workable model of spacetime and much of the mathematics behind the formalism will be glanced over. Definition 2.1. Let M be a set. Then a chart of M is a bijection of an open subset U ⊂ M n n m onto an open subset of some R . Two charts x : U1 → V1 ⊂ R and y : U2 → V2 ⊂ R are said −1 to be compatible if either x ◦ y is a smooth bijection or if U1 ∩ U2 = ?. In this manner a chart x : U → V provides a corresponding set of coordinates for the subset U, namely xi := ri ◦ x, where ri denotes the i:th canonical coordinate function on Rn. The chart may be used to “chart” its domain mathematically, but obviously the coordinates themselves carry no deeper meaning. This is an important realization in general relativity, and one that must not be forgotten. Definition 2.2. An atlas of a set M together with a collection of charts (Uα, xα)α∈A of M such that S (i) i Ui = M. (ii) All charts are compatible. Here A can be any set. Thus an atlas of M provides each region of M with at least one set of coordinates, and in any overlap there exists a smooth coordinate transformation between the coordinates. Now note that chart compatibility is an equivalence relation. A chart is said to be compatible with an atlas if it is compatible with all charts in the atlas, or equivalently with any chart in the atlas. In this manner we can extend the compatibility relation to hold between atlases: Definition 2.3. Two atlases are said to be compatible if their union is also an atlas. An equiv- alence class of atlases of some set M is called a smooth structure on M. In other words, if any chart from the first atlas is compatible with any chart from the second atlas, they belong to the same differentiable structure. Each atlas thus gives rise to a unique differentiable structure. A smooth manifold will consist of a set M together with a smooth structure on M, but since the union of all atlases in such a structure is also an atlas, and is uniquely determined by the original atlas, we may use this maximal atlas interchangeably with the differentiable structure. Definition 2.4. A smooth manifold is a set M and a smooth structure on M such that the topology induced by the structure is Hausdorff1 and second countable2. If the charts are in Rn then we say that the dimension of the manifold, dim(M) = n, or simply that M is a smooth n-manifold. We will not explicitly refer to the differentiable structure, but be satisfied with that there is such a structure chosen. Instead we may refer to a chart as admissible if it belongs to some atlas in the differentiable structure. 1 See below for the definition. 2 See below for the definition. 3 Our definition of smooth manifolds also includes some topological considerations, seen above. Although strictly not necessary, and often omitted, the following result does provide an easy way to ensure that a desired model fulfills the necessary axioms. Indeed, with this result in hand it is trivial to show the necessary topological properties in nearly any actual model of spacetime. It also provides any reader unfamiliar with point-set topology with an accessible description of the topological requirements. The open sets of the induced topology are precisely those sets that are the domain of some admissible chart. Formally, the induced topology is Hausdorff if for any two points x and y there are open sets U and V such that x ∈ U, y ∈ V , and U ∩ V = ?.
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