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Multipole Moments of Stationary Spacetimes
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2008 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we study the relativistic multipole moments for stationary asymptotically flat spacetimes as introduced by Geroch and Hansen. These multipole moments give an asymptotic description of the gravitational field in a coordinate independent way.

Due to this good description of the spacetimes, it is natural to try to construct a spacetime from only the set of multipole moments. Here we present a simple method to do this for the static axisymmetric case. We also give explicit solutions for the cases where the number of non-zero multipole moments are finite. In addition, for the general stationary axisymmetric case, we present methods to generate solutions.

It has been a long standing conjecture that the multipole moments give a complete characterization of the stationary spacetimes. Much progress toward a proof has been made over the years. However, there is one remaining difficult task: to prove that a spacetime exists with an a-priori given arbitrary set of multipole moments subject to some given condition.

Here we present such a condition for the axisymmetric case, and prove that it is both necessary and sufficient. We also extend this condition to the general case without axisymmetry, but in this case we only prove the necessity of our condition.

Place, publisher, year, edition, pages
Matematiska institutionen , 2008. , 18 p.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1181
Keyword [en]
multipole moments, stationary, static, relativistic, spacetime, convergence, specified
National Category
Other Physics Topics
Identifiers
URN: urn:nbn:se:liu:diva-11519ISBN: 978-91-7393-902-7 (print)OAI: oai:DiVA.org:liu-11519DiVA: diva2:17945
Public defence
2008-08-22, Planck, Fysikhuset, Campus Valla, Linköpings universitet, Linköpings universitet, 10:15 (English)
Opponent
Supervisors
Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2009-04-24
List of papers
1. Static axisymmetric spacetimes with prescribed multipole moments
Open this publication in new window or tab >>Static axisymmetric spacetimes with prescribed multipole moments
2005 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 22, no 9, 1607-1621 p.Article in journal (Refereed) Published
Abstract [en]

In this paper we develop a method of finding the static axisymmetric spacetime corresponding to any given set ofmultipolemoments. In addition to an implicit algebraic form for the general solution, we also give a power series expression for all finite sets of multipole moments. As conjectured by Geroch we prove in the special case of axisymmetry, that there is a static spacetime for any given set of multipole moments subject to a (specified) convergence criterion. We also use this method to confirm a conjecture of Hernández-Pastora and Martín concerning the monopole–quadrupole solution.

Keyword
static, axisymmetric, exact solution, multipole moments, spacetime, algebraic equation
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-13749 (URN)10.1088/0264-9381/22/9/009 (DOI)
Available from: 2006-01-12 Created: 2006-01-12 Last updated: 2009-05-07
2. Explicit multipole moments of stationary axisymmetric spacetimes
Open this publication in new window or tab >>Explicit multipole moments of stationary axisymmetric spacetimes
2005 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 22, no 17, 3585-3594 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we study multipole moments of axisymmetric stationary asymptotically flat spacetimes. We show how the tensorial recursion of Geroch and Hansen can be reduced to a recursion of scalar functions. We also demonstrate how a careful choice of conformal factor collects all moments into one complex-valued function on , where the moments appear as the derivatives at 0. As an application, we calculate the moments of the Kerr solution. We also discuss the freedom in choosing the potential for the moments.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-13750 (URN)10.1088/0264-9381/22/17/017 (DOI)
Available from: 2006-01-12 Created: 2006-01-12 Last updated: 2009-02-16
3. Calculation of, and bounds for, the multipole moments of stationary spacetimes
Open this publication in new window or tab >>Calculation of, and bounds for, the multipole moments of stationary spacetimes
2006 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 23, no 20, 5997-6006 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, the multipole moments of stationary asymptotically flat spacetimes are considered. We show how the tensorial recursion of Geroch and Hansen can be replaced by a scalar recursion on . We also give a bound on the multipole moments. This gives a proof of the 'necessary part' of a long-standing conjecture due to Geroch.

Keyword
multipole moments, stationary, spacetimes
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-13158 (URN)10.1088/0264-9381/23/20/019 (DOI)
Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2009-05-07
4. Axisymmetric stationary solutions with arbitrary multipole moments
Open this publication in new window or tab >>Axisymmetric stationary solutions with arbitrary multipole moments
2007 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 24, no 9, 2205-2215 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, the problem of finding an axisymmetric stationary spacetime from a specified set of multipole moments is studied. The condition on the multipole moments, for the existence of a solution, is formulated as a convergence condition on a power series formed from the multipole moments. The methods in this paper can also be used to give approximate solutions to any order, as well as estimates on each term of the resulting power series.

Keyword
multipole moments, statinary, axisymmetric, existence, spacetime
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-13159 (URN)10.1088/0264-9381/24/9/004 (DOI)
Available from: 2008-05-21 Created: 2008-05-21 Last updated: 2009-04-24

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Bäckdahl, Thomas

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