Abstract
We investigate complex quaternion-valued exterior differential forms over 4-dimensional Lorentzian spacetimes and explore Weyl spinor fields as minimal left ideals within the complex quaternion algebra. The variational derivation of the coupled Einstein–Weyl equations from an action is presented, and the resulting field equations for both first and second order variations are derived and simplified. Exact plane symmetric solutions of the Einstein–Weyl equations are discussed, and two families of exact solutions describing left-moving and right-moving neutrino plane waves are provided. The study highlights the significance of adjusting a quartic self-coupling of the Weyl spinor in the action to ensure the equivalence of the field equations.
Similar content being viewed by others
Data availibility
No new data were created or analysed in this study.
Notes
These correspond to undotted spinors in the NP formalism. In our approach dotted spinors of NP formalism are identified as minimal right ideals with a corresponding spin basis.
Here and all that follow, variations are given up to closed forms.
We thank our referee for pointing this out.
References
Close, F.: Neutrino. Oxford University Press, Oxford (2010)
Senjanovic, G.: Neutrino 2020: theory outlook. Int. J. Mod. Phys. A 36, 2130003 (2021)
Brill, D., Wheeler, J.A.: Interaction of neutrinos and gravitational fields. Rev. Mod. Phys. 29, 465 (1957)
Wainwright, J.: Geometric properties of neutrino fields in curved space-time. J. Math. Phys. 12, 828 (1971)
Trim, D., Wainwright, J.: Combined neutrino-gravitational fields in general relativity. J. Math. Phys. 12, 2494 (1971)
Griffiths, J.B.: Gravitational radiation and neutrinos. Commun. Math. Phys. 28, 295 (1972)
Madore, J.: On the neutrino in general relativity. Lett. Nuo. Cim. 5, 48 (1972)
Taub, A.H.: Empty space-times admitting a three parameter group of motions. Ann. Math. 53, 472 (1951)
Taub, A.H.: Isentropic hydrodynamics in plane symmetric space-times. Phys. Rev. 103, 454 (1956)
Taub, A.H.: Plane-symmetric similarity solutions for self-gravitating fluids. In: O’Raifeartartaigh, L. (ed.) General Relativity: Papers in Honour of J. J. Synge, pp. 133–150. Oxford University Press, Oxford (1972)
Carlson, G.T., Jr., Safko, J.L.: An investigation of some of the kinematical aspects of plane symmetric space-times. J. Math. Phys. 19, 1617 (1978)
Dereli, T., Tucker, R.W.: An intrinsic analysis of neutrino couplings to gravity. J. Phys. A 15, 1625 (1982)
Benn, I.M.: Complex quaternionic formulation of \(SL(2,{{\mathbb{C}}})\) gauge theories of gravitation. Unpublished Ph.D. thesis. Lancaster University (1981)
Davis, T.M., Ray, J.R.: Ghost neutrinos in plane-symmetric spacetimes. J. Math. Phys. 16, 75 (1975)
Hayward, S.A.: Energy of gravitational radiation in plane-symmetric space-times. Phys. Rev. D 78, 044027 (2008)
Collinson, C.D., Morris, P.B.: Spacetimes admitting neutrino fields with zero energy and momentum. J. Phys. A 6, 915 (1973)
Davis, T.M., Ray, J.R.: Ghost neutrinos in general relativity. Phys. Rev. D 9, 334 (1974)
Dereli, T., Tucker, R.W.: Exact neutrino solutions in the presence of torsion. Phys. Lett. A 82, 229 (1981)
Griffiths, J.B.: Neutrino fields in Einstein–Cartan theory. Gen. Relativ. Gravit. 13, 227 (1981)
Dimakis, A., Müller-Hoissen, F.: Solutions of the Einstein–Cartan–Dirac equations with vanishing energy-momentum tensor. J. Math. Phys. 26, 1040 (1985)
Chianese, M., Fu, B., King, S.F.: Interplay between neutrino and gravity portals for FIMP dark matter. JCAP 01, 034 (2021)
Furey, C.: Three generations, two unbroken gauge symmetries, and one eight-dimensional algebra. Phys. Lett. B 785, 84 (2018)
Jimenez, J.B., Koivisto, T.S.: Listening to celestial algebras. Universe 8, 407 (2022)
Todorov, I.: Octonion internal space algebra for the standard model. Universe 9, 222 (2023)
Acknowledgements
One of us (T.D.) thanks the Turkish Academy of Sciences (TUBA) for partial support.
Author information
Authors and Affiliations
Corresponding author
Additional information
Publisher's Note
Springer Nature remains neutral with regard to jurisdictional claims in published maps and institutional affiliations.
Appendix
Appendix
We start with the identity \( {}^*{\bar{e}} = \frac{1}{6} e^{*} \wedge e \wedge e^{*}, \) that implies
Then we use the variational field equations
where \(h = -\frac{1}{6} \xi \xi ^{\dagger }\). Then
Now we use the identities
and
to simplify the above expression:
The first term inside the big parantheses vanishes identically. The remaining term simplifies:
Now let us write \(\xi = \alpha U^1+\beta U^2\) for some complex functions \(\alpha \) and \(\beta \). After a long straightforward calculation one reaches, on-shell, the equality
Rights and permissions
Springer Nature or its licensor (e.g. a society or other partner) holds exclusive rights to this article under a publishing agreement with the author(s) or other rightsholder(s); author self-archiving of the accepted manuscript version of this article is solely governed by the terms of such publishing agreement and applicable law.
About this article
Cite this article
Dereli, T., Şenikoğlu, Y. Weyl neutrinos in plane symmetric spacetimes. Gen Relativ Gravit 55, 126 (2023). https://doi.org/10.1007/s10714-023-03175-8
Received:
Accepted:
Published:
DOI: https://doi.org/10.1007/s10714-023-03175-8