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The Chevreton Superenergy Tensor in Einstein-Maxwell Spacetimes
Linköping University, Department of Mathematics, Applied Mathematics. Linköping University, The Institute of Technology.
2007 (English)Doctoral thesis, comprehensive summary (Other academic)
Abstract [en]

In this thesis we investigate the superenergy tensor that was introduced by Chevreton in 1964 as an electromagnetic counterpart to the Bel-Robinson tensor for the gravitational feld.

We show that in Einstein-Maxwell spacetimes with a source-free electromagnetic feld, the Chevreton superenergy tensor has many interesting properties. It is a completely symmetric rank-4 tensor and it gives rise to conserved currents for orthogonally transitive 1- and 2-parameter isometry groups.

The trace of this tensor is divergence-free and it is related to the Bach tensor. We investigate the implications for when the trace vanishes and we are able to determine the full set of such spacetimes. We use this to treat the problem of Einstein{-Maxwell spacetimes that are conformally related to Einstein spaces and we find new exact solutions with this property.

Place, publisher, year, edition, pages
Matematiska institutionen , 2007.
Series
Linköping Studies in Science and Technology. Dissertations, ISSN 0345-7524 ; 1136
Keyword [en]
Einstein-Maxwell, graviational energy, superenergy, Bel tensor, Chevreton tensor, conservation laws, Conformal Einstein space
National Category
Other Physics Topics
Identifiers
URN: urn:nbn:se:liu:diva-10093ISBN: 978-91-85895-76-2 (print)OAI: oai:DiVA.org:liu-10093DiVA: diva2:16839
Public defence
2007-12-03, Planck, Fysik, Campus Valla, Linköpings universitet, Linköping, 13:15 (English)
Opponent
Supervisors
Available from: 2007-10-24 Created: 2007-10-24 Last updated: 2009-04-27
List of papers
1. New electromagnetic conservation laws
Open this publication in new window or tab >>New electromagnetic conservation laws
2003 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 20, no 13, 2663-2668 p.Article in journal (Refereed) Published
Abstract [en]

The Chevreton superenergy tensor was introduced in 1964 as a counterpart, for electromagnetic fields, of the well-known Bel–Robinson tensor of the gravitational field. We here prove the unnoticed facts that, in the absence of electromagnetic currents, Chevreton's tensor (i) is completely symmetric, and (ii) has a trace-free divergence if the Einstein–Maxwell equations hold. It follows that the trace of the Chevreton tensor is a rank-2, symmetric, trace-free, conserved tensor, which is different from the energy–momentum tensor, and nonetheless can be constructed for any test Maxwell field or any Einstein–Maxwell spacetime.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-12678 (URN)10.1088/0264-9381/20/13/313 (DOI)
Available from: 2007-10-24 Created: 2007-10-24 Last updated: 2009-04-28
2. Conserved matter superenergy currents for hypersurface orthogonal Killing vectors
Open this publication in new window or tab >>Conserved matter superenergy currents for hypersurface orthogonal Killing vectors
2006 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 23, no 7, 2279-2290 p.Article in journal (Refereed) Published
Abstract [en]

We show that for hypersurface orthogonal Killing vectors the corresponding Chevreton superenergy currents will be conserved and proportional to the Killing vectors. This holds for four-dimensional Einstein–Maxwell spacetimes with an electromagnetic field that is source-free and inherits the symmetry of the spacetime. A similar result also holds for the trace of the Chevreton tensor. The corresponding Bel currents have previously been proven to be conserved and our result can be seen as giving further support to the concept of conserved mixed superenergy currents. The analogous case for a scalar field has also previously been proven to give conserved currents and we show, for completeness, that these currents also are proportional to the Killing vectors.

Keyword
Einstein-Maxwell, superenergy, conservation laws
National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-12679 (URN)10.1088/0264-9381/23/7/005 (DOI)
Available from: 2007-10-24 Created: 2007-10-24 Last updated: 2009-05-12
3. The Chevreton tensor and Einstein-Maxwell spacetimes conformal to Einstein spaces
Open this publication in new window or tab >>The Chevreton tensor and Einstein-Maxwell spacetimes conformal to Einstein spaces
2007 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 24, no 13, 3437-3455 p.Article in journal (Refereed) Published
Abstract [en]

In this paper, we characterize the source-free Einstein–Maxwell spacetimes which have a trace-free Chevreton tensor. We show that this is equivalent to the Chevreton tensor being of pure radiation type and that it restricts the spacetimes to Petrov type N or O. We prove that the trace of the Chevreton tensor is related to the Bach tensor and use this to find all Einstein–Maxwell spacetimes with a zero cosmological constant that have a vanishing Bach tensor. Among these spacetimes we then look for those which are conformal to Einstein spaces. We find that the electromagnetic field and the Weyl tensor must be aligned, and in the case that the electromagnetic field is null, the spacetime must be conformally Ricci-flat and all such solutions are known. In the non-null case, since the general solution is not known on a closed form, we settle by giving the integrability conditions in the general case, but we do give new explicit examples of Einstein–Maxwell spacetimes that are conformal to Einstein spaces, and we also find examples where the vanishing of the Bach tensor does not imply that the spacetime is conformal to a C-space. The non-aligned Einstein–Maxwell spacetimes with vanishing Bach tensor are conformally C-spaces, but none of them are conformal to Einstein spaces.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-12680 (URN)10.1088/0264-9381/24/13/018 (DOI)
Available from: 2007-10-24 Created: 2007-10-24 Last updated: 2009-04-23
4. Conserved matter superenergy currents for orthogonally transitive Abelian G2 isometry groups
Open this publication in new window or tab >>Conserved matter superenergy currents for orthogonally transitive Abelian G2 isometry groups
2007 (English)In: Classical and Quantum Gravity, ISSN 0264-9381, Vol. 24, no 20, 4955-4968 p.Article in journal (Refereed) Published
Abstract [en]

In a previous paper we showed that the electromagnetic superenergy tensor, the Chevreton tensor, gives rise to a conserved current when there is a hypersurface orthogonal Killing vector present. In addition, the current is proportional to the Killing vector. The aim of this paper is to extend this result to the case where we have a two-parameter Abelian isometry group that acts orthogonally transitive on non-null surfaces. It is shown that for four-dimensional Einstein–Maxwell theory with a source-free electromagnetic field, the corresponding superenergy currents lie in the orbits of the group and are conserved. A similar result is also shown to hold for the trace of the Chevreton tensor and for the Bach tensor, and also in Einstein–Klein–Gordon theory for the superenergy of the scalar field. This links up well with the fact that the Bel tensor has these properties and it gives further support to the possibility of constructing conserved mixed currents between the gravitational field and the matter fields.

National Category
Mathematics
Identifiers
urn:nbn:se:liu:diva-12681 (URN)10.1088/0264-9381/24/20/004 (DOI)
Available from: 2007-10-24 Created: 2007-10-24 Last updated: 2009-04-27

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