Skip to main content
SpringerLink
Log in

A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory

  • Published:
Letters in Mathematical Physics Aims and scope Submit manuscript

Abstract

Not only the Dirac operator, but also the spinor bundle of a pseudo-Riemannian manifold depends on the underlying metric. This leads to technical difficulties in the study of problems where many metrics are involved, for instance in variational theory. We construct a natural finite dimensional bundle, from which all the metric spinor bundles can be recovered including their extra structure. In the Lorentzian case, we also give some applications to Einstein–Dirac–Maxwell theory as a variational theory and show how to coherently define a maximal Cauchy development for this theory.

This is a preview of subscription content, log in via an institution to check access.

Access this article

Log in via an institution

Price excludes VAT (USA)
Tax calculation will be finalised during checkout.

Instant access to the full article PDF.

Institutional subscriptions

Similar content being viewed by others

Notes

  1. That is a 2:1 covering between the total spaces equivariant with respect to the respective actions of \(\widetilde{{\text {GL}}}^{+}_m\) and \({\text {GL}}^{+}_m\) via the universal covering map \(\vartheta :\widetilde{{\text {GL}}}^{+}_m \rightarrow {\text {GL}}^{+}_m\).

  2. As in our case of double bundles, the projections of the respective bundles are not uniquely given by the total spaces, we use the not so common but very practical way of writing a diagram of vector bundle homomorphims as a diagram between the projections, meaning that there is a corresponding commuting diagram between the respective total spaces preserving the and linear on the respective fibers.

  3. Here, for a bundle \(\pi :E \rightarrow M\) and a subset \(A \subset M\), we employ the definition \(\pi \vert ^A := \pi \vert _{\pi ^{-1} (A)}\).

  4. These equations are apparently not yet formulated in the first jet bundle of a geometric bundle on S itself but can, by a well-known procedure, easily be reformulated as equations in the spinor bundle on S and the first jet bundle of Maxwell and metric fields on S; for details see for instance [9, 13].

References

  1. Ammann, B., Weiss, H., Witt, F.: A spinorial energy functional: critical points and gradient flow. (2012). arXiv:1207.3529

  2. Ammann, B., Weiss, H., Witt, F.: The spinorial energy functional on surfaces. (2014). arXiv:1407.2590

  3. Bär, C., Gauduchon, P., Moroianu, A.: Generalized cylinders in semi-Riemannian and spin geometry. Mathematische Zeitschrift 249(3), 545–580 (2005). arXiv:0303095

  4. Baum, H.: Spin-Strukturen und Dirac-Operatoren über pseudoriemannschen Mannigfaltigkeiten, vol. 41. Teubner-Texte zur Mathematik [Teubner Texts in Mathematics]. With English, French and Russian summaries. BSB B. G. Teubner Verlagsgesellschaft, Leipzig, p. 180 (1981)

  5. Baum, H.: Eichfeldtheorie. Springer, Berlin Heidelberg (2009)

  6. Baum, H., Müller, O.: Codazzi spinors and globally hyperbolic manifolds with special holonomy. (2008). doi:10.1007/s00209-007-0169-5

  7. Bernal, A.N., Sánchez, M.: Smoothness of time functions and the metric splitting of globally hyperbolic spacetimes. Comm. Math. Phys. 257(1), 43–50 (2005). ISSN:0010-3616. doi:10.1007/s00220-005-1346-1

  8. Bourguignon, J.-P., Gauduchon, P.: Spineurs, Opérateurs de Dirac et Variations de Métriques. Commun. Math. Phys. 144(3), 581–599 (1992). doi:10.1007/BF02099184

  9. Choquet-Bruhat, Y.: General relativity and the Einstein equations. Oxford Mathematical Monographs. Oxford University Press, Oxford, pp. xxvi+785 (2009). ISBN:978-0-19-923072-3

  10. Choquet-Bruhat, Y., Geroch, R.: Global aspects of the Cauchy problem in general relativity. Comm. Math. Phys. 14, 329–335 (1969). ISSN:0010-3616

  11. Finster, F.: Local U(2, 2) symmetry in relativistic quantum mechanics. J. Math. Phys. 39(12), 6276–6290 (1998). ISSN:0022-2488. doi:10.1063/1.532638

  12. Friedrich, H., Rendall, A.D.: The Cauchy problem for the Einstein equations. Lect. Notes Phys. 540, 127–224 (2000). arXiv:gr-qc/0002074 [gr-qc]

  13. Ginoux, N., Müller, O: Massless Dirac-Maxwell systems in asymptotically flat spacetimes. (2014). arXiv:1407.1177

  14. Kim, E.C., Friedrich, T.: The Einstein-Dirac equation on Riemannian spin manifolds. J. Geom. Phys. 33(1–2), 128–172 (2000). ISSN:393-0440. doi:10.1016/S0393-0440(99)00043-1

  15. Lawson, H.B., Michelsohn, M.-L.: Spin Geometry. Princeton University Press (1989)

  16. Lindblad, H., Rodnianski, I.: The global stability of Minkowski space-time in harmonic gauge. Ann. Math. (2) 171(3), 1401–1477 (2010). ISSN:0003-486X. doi:10.4007/annals.2010.171.1401

  17. Loizelet, J.: Solutions globales des équations d’Einstein-Maxwell avec jauge harmonique et jauge de Lorentz. C. R. Math. Acad. Sci. Paris 342(7), 479–482 (2006). ISSN:1631-073X. doi:10.1016/j.crma.2006.01.018

  18. Maier, S.: Generic metrics and connections on spin- and spinc-manifolds. Commun. Math. Phys. 188, 407–437 (1997)

  19. Müller, O., Sánchez, M.: Lorentzian manifolds isometrically embeddable in \(\mathbb{L}^N\). Trans. Am. Math. Soc. 363(10), 5367–5379 (2011). ISSN:0002-9947. doi:10.1090/S0002-9947-2011-05299-2

  20. Müller, O.: Special temporal functions on globally hyperbolic manifolds. In: Lett. Math. Phys. 103(3), 285–297 (2013). ISSN:0377-9017. doi:10.1007/s11005-012-0591-4

  21. Müller, O.: A note on invariant temporal functions. (2015). arXiv:1502.02716

  22. Nowaczyk, N.: Dirac eigenvalues of higher multiplicity. PhD thesis. Jan. 14, 2015. http://epub.uni-regensburg.de/31209/

  23. Pérez, R.F., Masqué, J.M.: Natural connections on the bundle of Riemannian metrics. Monatsh. Math. 155(1), 67–78 (2008). ISSN:0026-9255. doi:10.1007/s00605-008-0565-x

  24. Roe, J.: Elliptic operators, topology and asymptotic methods. Second. vol. 395. Pitman Research Notes in Mathematics Series. Harlow: Longman, pp. ii+209 (1998). ISBN:0-582-32502-1

  25. Saunders, D.J.: The geometry of jet bundles. vol. 142. London Mathematical Society Lecture Note Series. Cambridge University Press, Cambridge, pp. viii+293 (1989). ISBN:0-521-36948-7. doi:10.1017/CBO9780511526411

  26. Sbierski, J.: On the existence of a maximal Cauchy development for the Einstein equations—a dezornification. (2013). arXiv:1309.7591

  27. Taylor, M.E.: Partial differential equations III. Nonlinear equations. Second. vol. 117. Applied Mathematical Sciences. Springer, New York, pp. xxii+715 (2011). ISBN:978-1-4419-7048-0. doi:10.1007/978-1-4419-7049-7

  28. Wong, W.W.-Y.: A comment on the construction of the maximal globally hyperbolic Cauchy development. J. Math. Phys. 54(11), 113511, 8 (2013). ISSN:0022-2488. doi:10.1063/1.4833375

Download references

Acknowledgements

The authors would like to thank Bernd Ammann and Felix Finster for interesting discussions and suggestions.

Author information

Authors and Affiliations

  1. Fakultät für Mathematik, Universität Regensburg, 93040, Regensburg, Germany

    Olaf Müller

  2. Maths Department, Imperial College, Huxley Building, 180 Queen’s Gate, Academic Visitor, Office 533, London, SW7 2AZ, United Kingdom

    Nikolai Nowaczyk

Authors
  1. Olaf Müller
    View author publications

    You can also search for this author in PubMed Google Scholar

  2. Nikolai Nowaczyk
    View author publications

    You can also search for this author in PubMed Google Scholar

Corresponding author

Correspondence to Olaf Müller.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Müller, O., Nowaczyk, N. A universal spinor bundle and the Einstein–Dirac–Maxwell equation as a variational theory. Lett Math Phys 107, 933–961 (2017). https://doi.org/10.1007/s11005-016-0929-4

Download citation

  • Received:

  • Revised:

  • Accepted:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/s11005-016-0929-4

Keywords

Mathematics Subject Classification

Search

Navigation