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Examensarbete Lanczos Potentials in Perfect Fluid Cosmologies David Examensarbete Lanczos Potentials in Perfect Fluid Cosmologies David Holgersson LiTH-MAT-EX{04/12{SE Lanczos Potentials in Perfect Fluid Cosmologies Till¨ampad matematik, Link¨opings tekniska h¨ogskola David Holgersson LiTH-MAT-EX{04/12{SE Examensarbete: 20 p Level: D Supervisor & Examiner: Fredrik Andersson, Matematiska institutionen, Till¨ampad matematik, Link¨opings tekniska h¨ogskola Link¨oping: 13 October 2004 Datum Avdelning, Institution Date Division, Department Matematiska Institutionen 13 October 2004 581 83 LINKOPING¨ SWEDEN Spr˚ak Rapporttyp ISBN Language Report category ISRN Svenska/Swedish Licentiatavhandling x Engelska/English x Examensarbete LiTH-MAT-EX{04/12{SE C-uppsats Serietitel och serienummer ISSN D-uppsats Title of series, numbering Ovrig¨ rapport URL f¨or elektronisk version http://www.ep.liu.se/exjobb/mai/2004/tm/012/ Titel Lanczos Potentials in Perfect Fluid Cosmologies Title F¨orfattare David Holgersson Author Sammanfattning Abstract We derive the equation linking the Weyl tensor with its Lanczos potential, called the Weyl-Lanczos equation, in 1+3 covariant formalism for perfect fluid Bianchi type I spacetime and find an explicit expression for a Lanczos poten- tial of the Weyl tensor in these spacetimes. To achieve this, we first need to derive the covariant decomposition of the Lanczos potential in this formalism. We also study an example by Novello and Velloso and derive their Lanczos potential in shear-free, irrotational perfect fluid spacetimes from a particular ansatz in 1+3 covariant formalism. The existence of the Lanczos potential is in some ways analogous to the vector potential in electromagnetic theory. Therefore, we also derive the electromagnetic potential equation in 1+3 co- variant formalism for a general spacetime. We give a short description of the necessary tools for these calculations and the cosmological formalism we are using. Nyckelord Keyword Weyl-Lanczos equation, Lanczos potentials, 1+3 covariant formalism, perfect fluid, cosmology, Bianchi type I models, shear-free and irrotational models. vi Abstract We derive the equation linking the Weyl tensor with its Lanczos potential, called the Weyl-Lanczos equation, in 1+3 covariant formalism for perfect fluid Bianchi type I spacetime and find an explicit expression for a Lanc- zos potential of the Weyl tensor in these spacetimes. To achieve this, we first need to derive the covariant decomposition of the Lanczos potential in this formalism. We also study an example by Novello and Velloso and derive their Lanczos potential in shear-free, irrotational perfect fluid space- times from a particular ansatz in 1+3 covariant formalism. The existence of the Lanczos potential is in some ways analogous to the vector potential in electromagnetic theory. Therefore, we also derive the electromagnetic potential equation in 1+3 covariant formalism for a general spacetime. We give a short description of the necessary tools for these calculations and the cosmological formalism we are using. Keywords: Weyl-Lanczos equation, Lanczos potentials, 1+3 covariant for- malism, perfect fluid, cosmology, Bianchi type I models, shear-free and irrotational models. Holgersson, 2004. vii viii Acknowledgements I would like to thank my supervisor Fredrik Andersson for helpful suggestions and discussions. I would also like to mention my opponent Jonas Jonsson Holm, who had the strength to proofread the manuscript. Holgersson, 2004. ix x Contents 1 Introduction and Basic Relations 1 1.1 Introduction . 1 1.2 Manifolds . 2 1.3 Conventions and Notation . 3 1.4 Derivative Operators . 3 1.5 The Curvature Tensor . 4 1.6 The Weyl Tensor . 4 1.7 The Electromagnetic Field Tensor . 5 1.8 The Lanczos Potential . 6 1.9 The Einstein Field Equations . 7 2 Geometry of Cosmology Models 9 2.1 Covariant Formalism . 9 2.2 Kinematical Quantities . 12 2.3 The Source Terms . 14 2.4 Covariant Propagation and Constraint equations . 15 3 The Maxwell Case 17 3.1 Splitting the Electromagnetic Field . 17 3.1.1 Electric Part . 19 3.1.2 Magnetic Part . 20 3.2 Summing-up . 21 4 The Decompositions of the Lanczos Tensor 23 4.1 Splitting the Gravitational Field . 23 4.2 The Spacetime Decomposition . 24 5 Shearfree and Irrotational Models 27 5.1 Introduction . 27 5.2 Constructing the Weyl-Lanczos Equation . 28 5.2.1 A-terms . 28 5.2.2 C-terms . 33 5.3 Summing-up . 38 Holgersson, 2004. xi xii Contents 6 Spatially Homogeneous Universes 39 6.1 Bianchi Type I Models . 39 6.2 Constructing the Weyl-Lanczos Equation . 40 6.2.1 A-terms . 40 6.2.2 C-terms . 42 6.2.3 S-terms . 44 6.2.4 P-terms . 45 6.3 Summing-up . 46 7 Conclusions and Future Work 49 7.1 Conclusions . 49 7.2 Future Work . 52 Chapter 1 Introduction and Basic Relations 1.1 Introduction In the lifetime of human civilization we have contact with only a small spactime region of our universe. Even while our telescopes can observe ob- jects remarkably far away by human scales, the information we get is only a portion of our past light cone. Therefore, it is hard to prove theories by ap- pealing only to observation data of objects falling in from space. Theoretical cosmology, where we search for a framework within which to comprehend the information from our observations, started with the cosmology models proposed by A. Einstein and W. de Sitter in 1917, based on Einstein´s the- ory of general relativity. If you check up the word cosmology in Chambers´s Dictionary, it says: 'Cosmology is the science of the universe as a whole'. The realization that space and time considered as a single whole - a four- dimensional manifold called spacetime - is one of the greatest intellectual achievements of the twentieth century. In modern study of cosmology, several kinds of physics is required. Since the dominant force on the cosmic scale is gravitation, this is the basic ingre- dient. We assume this is given by the Einstein´s theory of general relativity. The matter distrubution of e.g. fluids, gases and fields in spacetime is given by Einstein field equations. In many situations this includes the physics of electromagnetism. On top of this, we sometimes take thermodynamics into consideration and even particle physics. A cosmological model is a model of our universe which predicts the ob- served properties of the universe, and explains the phenomena in the early universe. A model must also explain why the universe was so homoge- neous and isotropic at the epoch of last scattering of the cosmic microwave background, and how and when inhomogeneities such as galaxies and stars arose. In more restricted sense cosmological models are exact solutions of Holgersson, 2004. 1 2 Chapter 1. Introduction and Basic Relations the Einstein field equations for a perfect fluid. The mathematics which is often used to describe curved spacetime is differential geometry. The global geometry of the spacetime is determined by the Riemann curvature tensor, a tensor with rather complicated symme- tries. By decomposing this tensor into simpler parts, the Weyl curvature tensor arises. In 1962 Lanczos introduced a new tensor, now called the Lanczos tensor, which is a potential for the Weyl tensor. To study potential equations can sometimes be easier, or can give additional physical under- standing. In general relativity, the existence of the Lanczos potential is in some ways analogous to the vector potential in electromagnetic theory. Therefore, we first begin with electromagnetism which is much easier and later study gravitation using the Lanczos potential. The aim is to find a procedure to derive the equation linking the Weyl tensor with its Lanczos potential, called the Weyl-Lanczos equation, using a cosmological formal- ism in a cosmology model. In this thesis, we want to go into two examples of such cosmological models. Firstly, we study shear-free and irrotational models (these were also studied by Novello and Velloso, [11], and an explicit example of a Lanczos potential was found; this potential will later act as a check on the Weyl-Lanczos equation that we derive in this thesis), and secondly we will investigate Bianchi type I models. The following basic relations, covariant descriptions and equations in chapter 1 and 2 are briefly introduced. The reader is assumed to have some knowledge about Einstein´s theory of general relativity and some knowledge about vectors, covectors and tensors. For more thorough introduction to basic relations and cosmology, see [3], [5], [8], [10] and [14]. 1.2 Manifolds A manifold is a set made up of pieces of topological space that looks locally like the Euclidean space Rn, such that these pieces can be "sewn" together smoothly. However, this set may have quite different global properties. An example is the surface of a sphere S2 such as the Earth, which is not a plane, but small patches of S2 are homeomorphic to patches of the Euclidean plane. Each patch is called a coordinate systems or a chart. The local charts can be smoothly "sewn" together and we can define directions, tangent spaces, and differentiable functions on that manifold. In special relativity and gen- eral relativity, time and three-dimensional space are treated together as a four-dimensional manifold called spacetime. The spacetime geometry is de- termined by this manifold on which is defined a metric of Lorentz signature. The metric tensor, gab, is a tensor of rank 2 that is used to measure dis- tance and angles in a space or spacetime. A point of spacetime represents an event which has four coordinates. A sequence of events that a particle occupies during its lifetime is represented by a curve called its worldline. A 1.3. Conventions and Notation 3 congruence of worldlines is represented by the 4-velocity field u, which is everywhere tangent to the congruence.
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