Mathematical Applications of Super-Energy Tensors (1999-2000)

Visiting Researcher: Prof. J.M.M. Senovilla

Principal Investigator: Prof. M.A.H. MacCallum

The designated Visiting Fellow, Prof. Senovilla, visited the School of Mathematical Sciences, Queen Mary and Westfield College for a period of ten weeks, from November 22nd, 1999 to January 30th, 2000

The main ideas were to explore relevance and usefulness of super-energy tensors in the studies of causal propagation and discontinuities of fields, the definitions of gravitational entropy, the global stability of non-flat spacetimes, and the construction of conserved super-energy (s-e) currents and quantities.

- Prof. Senovilla and Dr. R. Vera showed that the gravitational
s-e currents constructed in spacetimes with a
hypersurface-orthogonal Killing vector or orthogonally transitive
Abelian two-dimensional group are automatically conserved on their
own, independently of the matter content of the spacetime.
- Prof. Senovilla with Drs. R. Lazkoz and R. Vera, using the
algebraic manipulation available at QMW, found a one-parameter family of
solutions for a massless scalar field under the assumptions of an
Abelian G
_{2}acting NON-orthogonally transitively on spacelike surfaces. This constitutes the first general family of solutions with these properties. - Again with Drs. Lazkoz and Vera, Prof. Senovilla computed the
mixed s-e currents which involve both the Bel and the scalar field
s-e tensors in the previous solutions and checked explicitly that
they lead to conserved currents, and that these currents need both
contributions necessarily.
- In connection with QMW work on gravitational entropy, Senovilla suggested the use of an observer-dependent scalar constructed as a quotient of the Bel-Robinson s-e density with the pure-matter s-e density. Several ideas arose for future work.

As other new ideas arising from the visit we note:

- We believe the s-e
tensors can be of useful application in the generalizations to
non-flat metrics of the stability results by Christodoulou and
Klainerman for flat spacetime, where extensive use of the properties
of the Bel-Robinson tensor was made.
- Not related directly to the s-e tensors, but certainly of
interest regarding the previous point, Dr. Vera and Senovilla
analyzed a particular Szekeres spacetime with cylindrical symmetry
which seems to prevent the generalization of the stability theorems
for flat spacetime to non-vacuum non-asymptotically-flat global
perturbations.
- On the other hand, Senovilla realized that one can characterize
the spacetimes with vanishing second derivative of the Riemann
tensor, that is with
Ñ
_{m}Ñ_{n}R_{ablr}= 0, as those with a zero ``(super)^{2}-energy''. The question of which spacetimes had the previous property then arises, for future work. - Finally, several ideas were put forward by colleagues in Senovilla's short visits to several UK universities. In particular, he was led to the possibility of considering the relevance of the s-e tensors in the classical problem of the so-called ghost neutrinos (by Prof. Griffiths at Loughborough). His attention was also drawn to the classical mass positivity theorems, where the use of spinors combined with the s-e techniques might lead to new simplifications (by Dr. Vickers at Southampton). These new matters are also of interest to him and will be considered, if possible, in the future.

Two contributions were accepted for presentation in the theoretical cosmology workshop of the Marcel Grossman meeting, held in Rome in July 2000 (and so will appear in the proceedings). They are entitled:

*General properties of Bel currents* (this will be presented by Dr. Vera)

and

*G2 cosmologies with associated mixed super-energy currents*
(presented by Dr. Lazkoz).

A short paper has been already accepted for publication:

J.M.M.Senovilla, R. Vera, (2000) *Cylindrically symmetric dust
spacetime*, to appear in Class. Quantum Gravity.

Furthermore, the draft of a paper, which right now is in the form of a preprint, combining the above results, has been written:

R. Lazkoz, J.M.M.Senovilla, R. Vera, (2000) *Conserved super-energy
currents*, preprint.

This will be submitted as soon as possible. On the other hand, during the visit and later on, several of Senovilla's papers on the subject have been accepted or published, such as

J.M.M.Senovilla, (2000) *(Super) ^{n}-Energy for arbitrary fields
and its interchange: conserved quantities*, Mod. Phys. Lett. A

And some of them have partly benefited from his stay in QMW, namely:

J.M.M.Senovilla, (2000) *Super-Energy tensors*, to appear in
Class. Quantum Grav.

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