Algebraic classification of the Weyl tensor: selected applications. (English) Zbl 1313.53029

Pravda, Vojtěch

Brandts, J. (ed.) et al., Proceedings of the international conference ‘Applications of mathematics’, Prague, Czech Republic, May 2–5, 2012. In honor of the 60th birthday of Michal Křížek. Prague: Academy of Sciences of the Czech Republic, Institute of Mathematics (ISBN 978-80-85823-60-8/pbk). 214-223 (2012).

This is a review paper which summarizes certain recent results in the context of Lorentzian geometry in more than four dimensions, which is of interest for higher-dimensional theories of gravity.
First, it describes results about spacetimes for which all Riemann polynomial invariants vanish, in relation with the algebraic classification of tensors based on the concept of null alignment (previously developed by the author himself and collaborators).
Next, extensions to higher dimensions of the Newman-Penrose and Geroch-Held-Penrose formalism are outlined, with some applications to the case of algebraically special spacetimes (integration of the Sachs equations under certain assumptions, and properties of type-N spacetimes that follow from the Bianchi equations).
Finally, it is shown how the equations of quadratic gravity can be simplified thanks to suitable assumptions of algebraically special types for the Weyl tensor. In particular, the result that type-N Einstein spacetimes are exact solutions to this theory is reviewed.
For the entire collection see [Zbl 1277.00031].

Reviewer: Marcello Ortaggio (Praha)

MSC:

53B30	Local differential geometry of Lorentz metrics, indefinite metrics
53B50	Applications of local differential geometry to the sciences
83C20	Classes of solutions; algebraically special solutions, metrics with symmetries for problems in general relativity and gravitational theory
83C60	Spinor and twistor methods in general relativity and gravitational theory; Newman-Penrose formalism
53A55	Differential invariants (local theory), geometric objects
83E99	Unified, higher-dimensional and super field theories

Keywords:

Lorentzian geometry; Newman-Penrose; Geroch-Held-Penrose; Weyl tensor; polynomial invariants; algebraically special spacetimes; quadratic gravity