No data available.

Please log in to see this content.

You have no subscription access to this content.

No metrics data to plot.

The attempt to load metrics for this article has failed.

The attempt to plot a graph for these metrics has failed.

Spinors and the Weyl tensor classification in six dimensions

Rent:

Rent this article for

USD

10.1063/1.4804991

### Abstract

A spinorial approach to six-dimensional differential geometry is constructed and used to analyze tensor fields of low rank, with special attention to the Weyl tensor. We perform a study similar to the four-dimensional case, making full use of the SO(6) symmetry to uncover results not easily seen in the tensorial approach. Using spinors, we propose a classification of the Weyl tensor by reinterpreting it as a map from 3-vectors to 3-vectors. This classification is shown to be intimately related to the integrability of maximally isotropic subspaces, establishing a natural framework to generalize the Goldberg-Sachs theorem. We work in complexified spaces, showing that the results for any signature can be obtained by taking the desired real slice.

© 2013 AIP Publishing LLC

Received 25 March 2013
Accepted 29 April 2013
Published online 21 May 2013

Acknowledgments: Carlos Batista thanks to CNPq (Conselho Nacional de Desenvolvimento Científico e Tecnológico) for the financial support.

Article outline:

I. INTRODUCTION

II. FROM *SO*(6) TO *SU*(4)

A. The precise identifications

B. Isotropic structures

III. REALITY CONDITIONS AND THE SIGNATURES

A. Changing the signature

IV. ALGEBRAIC CLASSIFICATION OF THE WEYL TENSOR

A. Some algebraically special cases

B. A map from 3-vectors to 3-vectors

C. The boost weight classification

V. INTEGRABILITY OF MAXIMALLY ISOTROPIC SUBSPACES, THE GOLDBERG-SACHS THEOREM

A. Integrability condition and the map from 3-vectors to 3-vectors—Complex case

B. Integrability condition and the map from 3-vectors to 3-vectors—Euclidian case

C. Integrability condition and the map from 3-vectors to 3-vectors—Lorentzian case

D. Generalized Mariot-Robinson theorem

VI. EXAMPLES

A. 6D Schwarzschild

B. *pp*-wave

VII. CONCLUSIONS

/content/aip/journal/jmp/54/5/10.1063/1.4804991

http://aip.metastore.ingenta.com/content/aip/journal/jmp/54/5/10.1063/1.4804991

Article metrics loading...

/content/aip/journal/jmp/54/5/10.1063/1.4804991

2013-05-21

2014-04-16

Full text loading...

### Most read this month

Article

content/aip/journal/jmp

Journal

5

3

Commenting has been disabled for this content