Abstract
Recently Bonilla and Senovilla studied factorizations of the symmetric and tracefree rank four Bel-Robinson tensor Tabcd into two symmetric tracefree rank two tensors. While the Bel-Robinson tensor has the dimension of energy density squared, each of these factors has the dimension of energy density. When the two factors can be chosen to be equal they are called the “square root” of Tabcd. The approach used was purely tensorial. In this paper we use spinors and show that the factors can be found in a very simple way using the principal null directions of the Weyl tensor. We obtain a factorization of the Weyl spinor into two symmetric rank two spinors, which when multiplied by their complex conjugates give the tracefree and symmetric factors of Tabcd. The factorization is immediately seen to be non-unique in most cases and the number of essentially non-equivalent factorizations becomes clear. It also becomes obvious that the square root only can exist in spacetimes of Petrov types N, D and O, in which cases one can equally well speak about the “square root” of the Weyl spinor. Explicit formulas for the factors of the Weyl spinor are given for all Petrov types.
Similar content being viewed by others
REFERENCES
Bonilla, M. A. G., and Senovilla, J. M. M. (1997). Gen. Rel. Grav. 29, 91.
Penrose, R., and Rindler, W. (1984). Spinors and Spacetime (Cambridge University Press, Cambridge), vol. 1.
Bergqvist, G. (1997). “Positivity properties of the Bel-Robinson tensor”, Preprint.
Bonilla, M. A. G., and Senovilla, J. M. M. (1997). Phys. Rev. Lett. 11, 783.
Penrose, R., and Rindler, W. (1986). Spin or sand Spacetime (Cambridge University Press, Cambridge), vol. 2.
Rights and permissions
About this article
Cite this article
Bergqvist, G. Spinor Factorizations of the Bel-Robinson Tensor. General Relativity and Gravitation 30, 227–238 (1998). https://doi.org/10.1023/A:1018844727612
Issue Date:
DOI: https://doi.org/10.1023/A:1018844727612