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Perturbative Methods in General Relativity
Umeå University, Faculty of Science and Technology, Physics.
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

Einstein's theory of general relativity is a cornerstone in the process of gaining increased understanding about problems of gravitational nature. It can be applied to problems on the huge length scales of cosmology and as far as we know it does not break down before the Planck scale is approached. Irrespective of scale, a perturbative approach is often a very useful way to reduce the Einstein system to manageable complexity and size.

The projects included in this thesis can be divided into three subcategories. In the first category the keyword is photon-photon scattering. General relativity predicts that scattering can take place on a flat background due to the curvature of space-time caused by the photons themselves. The coupling equations and cross-section are found and a comparison with the corresponding quantum field theoretical results is done to leading order. Moreover, photon-photon scattering due to exchange of virtual electron-positron pairs is considered as an effective field theory in terms of the Heisenberg-Euler Lagrangian resulting in a possible setup for experimental detection of this phenomenon using microwave cavities. The second category of projects is related to cosmology. Here linear perturbations around a flat FRW universe with a cosmological constant are considered and the corresponding temperature variations of the cosmic microwave background radiation are found. Furthermore, cosmological models of Bianchi type V are investigated using a method based on the invariant scheme for classification of metrics by Karlhede. The final category is slowly rotating stars. Here the problem of matching a perfect fluid interior of Petrov type D to an exterior axisymmetric vacuum solution is treated perturbatively up to second order in the rotational parameter.

Place, publisher, year, edition, pages
Umeå: Fysik , 2008. , 53 p.
Keyword [en]
General relativity, perturbation theory, photon-photon scattering, cosmology, FRW models with cosmological constant, Bianchi type V, slowly rotating stars, Hartle formalism
National Category
Physical Sciences
Identifiers
URN: urn:nbn:se:umu:diva-1488ISBN: 978-91-7264-493-9 (print)OAI: oai:DiVA.org:umu-1488DiVA: diva2:141221
Public defence
2008-02-07, N420, Naturvetarhuset, Umeå, 14:00
Opponent
Supervisors
Available from: 2008-01-16 Created: 2008-01-16Bibliographically approved
List of papers
1. Possibility to measure elastic photon-photon scattering in vacuum
Open this publication in new window or tab >>Possibility to measure elastic photon-photon scattering in vacuum
2004 (English)In: Physical Review A. Atomic, Molecular, and Optical Physics, ISSN 1050-2947, Vol. 70, no 1, 013808- p.Article in journal (Refereed) Published
Abstract [en]

Photon-photon scattering in vacuum due to the interaction with virtual electron-positron pairs is a consequence of quantum electrodynamics. A way for detecting this phenomenon has been devised based on interacting modes generated in microwave wave guides or cavities [G. Brodin, M. Marklund, and L. Stenflo, Phys. Rev. Lett. 87, 171801 (2001)]. Here we materialize these ideas, suggest a concrete cavity geometry, make quantitative estimates and propose experimental details. It is found that detection of photon-photon scattering can be within the reach of present day technology.

Identifiers
urn:nbn:se:umu:diva-2889 (URN)10.1103/PhysRevA.70.013808 (DOI)
Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2009-06-22Bibliographically approved
2. C-infinity perturbations of FRW models with a cosmological constant
Open this publication in new window or tab >>C-infinity perturbations of FRW models with a cosmological constant
2005 (English)In: Astronomy and Astrophysics, ISSN 0004-6361, Vol. 431, no 2, 415-421 p.Article in journal (Refereed) Published
Abstract [en]

Spatially homogeneous and isotropic cosmological models, with a perfect fluid matter source and non-vanishing cosmological constant, are studied. The equations governing linear perturbations of the space-time and the variation of energy density are given. The complete solution of the problem is obtained for C∞ perturbations, using a comoving time. The Sachs-Wolfe fluctuations of the temperature of the cosmic background radiation are obtained for the relatively growing density perturbations. It is found that the observable celestial microwave fluctuation pattern underwent a reversal approximately two billion years ago. What is observed today is a negative image of the last scattering surface with an attenuation of the fluctuations, due to the presence of the cosmological constant.

Keyword
cosmology: cosmic microwave background, cosmology: large scale structure of Universe
Identifiers
urn:nbn:se:umu:diva-2890 (URN)10.1051/0004-6361:20041472 (DOI)
Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2009-10-02Bibliographically approved
3. Tilted cosmological models of Bianchi type V
Open this publication in new window or tab >>Tilted cosmological models of Bianchi type V
2006 (English)In: Physical Review D. Particles and fields, ISSN 0556-2821, Vol. 73, no 4, 044008-044019 p.Article in journal (Refereed) Published
Abstract [en]

Cosmological models of Bianchi types Vand I containing a perfect fluid with a linear equation of state plus cosmological constant are investigated. In general, these spacetimes are tilted and describe fluids with expansion, shear, and vorticity. We use a tetrad approach where our variables are the Riemann tensor, the Ricci rotation coefficients, and a subset of the tetrad vector components. This set, called S, describes a spacetime when its elements are constrained by certain integrability conditions and due to a theorem by Cartan S gives a complete local description of the spacetime. With the help of the Lie algebra, the full line element is constructed up to quadratures in terms of the elements in S. The system obtained by imposing the integrability conditions and Einstein’s equations on the elements in S can be reduced to an integrable system of five coupled first order ordinary differential equations. In general, exact solutions to this system are hard to find, but the linearized equations around the open Friedmann models are easily integrated. The full system is also studied numerically and the perturbative solutions agree well with the numerical ones in the appropriate domains. We also give some examples of numerical solutions in the nonperturbative regime.

Identifiers
urn:nbn:se:umu:diva-2891 (URN)10.1103/PhysRevD.73.044008 (DOI)
Available from: 2008-01-16 Created: 2008-01-16Bibliographically approved
4. Graviton mediated photon-photon scattering in general relativity
Open this publication in new window or tab >>Graviton mediated photon-photon scattering in general relativity
2006 (English)In: Physical Review D. Particles and fields, ISSN 0556-2821, Vol. 74, no 12, 124028-124032 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we consider photon-photon scattering due to self-induced gravitational perturbations on a Minkowski background. We focus on four-wave interaction between plane waves with weakly space and time dependent amplitudes, since interaction involving a fewer number of waves is excluded by energy-momentum conservation. The Einstein-Maxwell system is solved perturbatively to third order in the field amplitudes and the coupling coefficients are found for arbitrary polarizations in the center of mass system. Comparisons with calculations based on quantum field theoretical methods are made, and the small discrepancies are explained.

Keyword
photon-photon scattering, general relativity, gravitation, perturbation theory, Einstein-Maxwell equations, polarisation in elementary particle scattering, quantum field theory
Identifiers
urn:nbn:se:umu:diva-2892 (URN)10.1103/PhysRevD.74.124028 (DOI)
Available from: 2008-01-16 Created: 2008-01-16Bibliographically approved
5. Slowly rotating fluid balls of Petrov type D
Open this publication in new window or tab >>Slowly rotating fluid balls of Petrov type D
2007 (English)In: Physical Review D. Particles and fields, ISSN 0556-2821, Vol. 75, no 2, 024013-024026 p.Article in journal (Refereed) Published
Abstract [en]

The second order perturbative field equations for slowly and rigidly rotating perfect fluid balls of Petrov type D are solved numerically. It is found that all the slowly and rigidly rotating perfect fluid balls up to second order, irrespective of Petrov type, may be matched to a possibly nonasymptotically flat stationary axisymmetric vacuum exterior. The Petrov type D interior solutions are characterized by five integration constants, corresponding to density and pressure of the zeroth order configuration, the magnitude of the vorticity, one more second order constant, and an independent spherically symmetric second order small perturbation of the central pressure. A four-dimensional subspace of this five-dimensional parameter space is identified for which the solutions can be matched to an asymptotically flat exterior vacuum region. Hence these solutions are completely determined by the spherical configuration and the magnitude of the vorticity. The physical properties, like equation of state, shape, and speed of sound, are determined for a number of solutions.

Identifiers
urn:nbn:se:umu:diva-2893 (URN)10.1103/PhysRevD.75.024013 (DOI)
Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2009-06-22Bibliographically approved
6. Slowly rotating fluid balls of Petrov type D - additional results
Open this publication in new window or tab >>Slowly rotating fluid balls of Petrov type D - additional results
Manuscript (Other academic)
Identifiers
urn:nbn:se:umu:diva-2894 (URN)
Available from: 2008-01-16 Created: 2008-01-16 Last updated: 2010-01-13Bibliographically approved

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