Skip to main content

Advertisement

Log in

On gravity’s role in the genesis of rest masses of classical fields

  • Research Article
  • Published:
General Relativity and Gravitation Aims and scope Submit manuscript

Abstract

It is shown that in the Einstein-conformally coupled Higgs–Maxwell system with Friedman–Robertson–Walker symmetries the energy density of the Higgs field has stable local minimum only if the mean curvature of the \(t=\mathrm{const}\) hypersurfaces is less than a finite critical value \(\chi _c\), while for greater mean curvature the energy density is not bounded from below. Therefore, there are extreme gravitational situations in which even quasi-locally defined instantaneous vacuum states of the Higgs sector cannot exist, and hence one cannot at all define the rest mass of all the classical fields. On hypersurfaces with mean curvature less than \(\chi _c\) the energy density has the ‘wine bottle’ (rather than the familiar ‘Mexican hat’) shape, and the gauge field can get rest mass via the Brout–Englert–Higgs mechanism. The spacelike hypersurface with the critical mean curvature represents the moment of ‘genesis’ of rest masses.

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

Access this article

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

Instant access to the full article PDF.

Institutional subscriptions

Fig. 1
Fig. 2
Fig. 3

Similar content being viewed by others

Notes

  1. The signature of the spacetime metric \(g_{ab}\) is chosen to be \((+,-,-,-)\).

  2. The potential U is called a local, algebraic expression of the field \(\phi \) if its value \(U (\phi )(p)\) at any spacetime point p is completely determined by the value \(\phi (p)\) of the field there, and \(U(\phi )(p)\) is an algebraic function, e.g. a polynomial, of \(\phi (p)\). Thus the derivatives of U with respect to \(\phi \) at the point p are simply the derivatives of \(U(\phi )(p)\) with respect to \(\phi (p)\). These derivatives yield genuine smooth fields on M rather than distributions.

  3. Latin indices from the beginning of the alphabet are abstract tensor indices, and the underlined indices are name indices, referring to some basis and taking numerical values, e.g. \({\underline{a}\,}=0,\ldots ,3\).

  4. Thanks are due to one of the referees for suggesting to discuss the issues of this subsection in more details.

  5. Analogous instantaneous vacuum states in the quantum theory of linear scalar fields in FRW spacetimes have been introduced recently in [14].

  6. In the particle physics literature, instead of the 4-covariant connection 1-form \(\omega _a\) the 4-potential \(\varpi _a:=\omega _a/g\) is used, where \(g>0\) is the coupling constant; and the corresponding rest mass is defined by the second derivative of the Lagrangian with respect to \(\varpi _a\) rather than to \(\omega _a\). With this convention \(m_\varpi =g\vert \Phi _v\vert \).

References

  1. Penrose, R.: Cycles of Time. The Bodley Head, London (2010). ISBN 9780224080361

    MATH  Google Scholar 

  2. Arnold, V.I.: Mathematical Methods of Classical Mechanics. Springer, New York (1989). ISBN 0-387-96890-3

    Book  Google Scholar 

  3. Hawking, S.W., Ellis, G.F.R.: The Large Scale Structure of Spacetime. Cambridge University Press, Cambridge (1973). ISBN 0-521-09906-4

    Book  MATH  Google Scholar 

  4. Higgs, R.W.: Broken symmetries and the masses of gauge bosons. Phys. Rev. Lett. 13, 508–509 (1964). https://doi.org/10.1103/PhysRevLett.13.508

    Article  ADS  MathSciNet  Google Scholar 

  5. Englert, F., Brout, R.: Broken symmetry and the mass of gauge vector mesons. Phys. Rev. Lett. 13, 321–323 (1964). https://doi.org/10.1103/PhysRevLett.13.321

    Article  ADS  MathSciNet  Google Scholar 

  6. Abers, E.S., Lee, B.W.: Gauge theories. Phys. Rep. 9, 1–141 (1973). https://doi.org/10.1016/0370-1573(73)90027-6

    Article  ADS  Google Scholar 

  7. Cheng, T.-P., Li, L.-F.: Gauge Theory of Elementary Particle Physics. Clarendon Press, Oxford (1984). ISBN 0-19-851961-3

    Google Scholar 

  8. Penrose, R., Rindler, W.: Spinors and Spacetime, vol. 1. Cambridge University Press, Cambridge (1982). ISBN 0-521-24527-3

    MATH  Google Scholar 

  9. Bjorken, J.D., Drell, S.D.: Relativistic Quantum Mechanics. McGraw-Hill, Yew York (1964). ISBN 07-005493-2

    MATH  Google Scholar 

  10. Hawking, S.W.: Stable and generic properties in general relativity. Gen. Relativ. Gravit. 1, 393–400 (1971). https://doi.org/10.1007/BF00759218

    Article  ADS  MathSciNet  MATH  Google Scholar 

  11. Adams, R.A., Fournier, J.J.F.: Sobolev Spaces. Academic Press, Amsterdam (2003). ISBN 0-12-044143-8

    MATH  Google Scholar 

  12. Penrose, R., MacCallum, M.A.H.: Twistor theory: an approach to the quantisation of fields and space-time. Phys. Rep. 6, 241–316 (1972). https://doi.org/10.1016/0370-1573(73)90008-2

    Article  ADS  MathSciNet  Google Scholar 

  13. Szabados, L.B.: Gravity, as a classical regulator for the Higgs field, and the genesis of rest masses and charge. arXiv:1603.06997v4 [gr-qc]

  14. Agullo, I., Nelson, W., Ashtekar, A.: Preferred instantaneous vacuum for linear scalar fields in cosmological space-times. Phys. Rev. D 91, 064051 (2015). https://doi.org/10.1103/PhysRevD.91.064051, arXiv:1412.3524v2 [gr-qc]

  15. Newman, E.T., Penrose, R.: New conservation laws for zero rest-mass fields in asymptotically flat spacetimes. Proc. R. Soc. A 305, 175–204 (1968). https://doi.org/10.1098/rspa.1968.0112

    Article  ADS  Google Scholar 

  16. Szabados, L.B., Wolf, G.: Singularities in Einstein-conformally coupled Higgs cosmological spacetimes. arXiv:1802.00774 [gr-qc]

  17. Isenberg, J., Nester, J.: Canonical gravity. In: Held, A. (ed.) General Relativity and Gravitation, vol. 1. Plenum Press, New York (1980). ISBN 0-306-40365-3(v.1)

    Google Scholar 

  18. Weinberg, S.: General theory of broken local symmetry. Phys. Rev. D 7, 1068–1082 (1973). https://doi.org/10.1103/PhysRevD.7.1068

    Article  ADS  Google Scholar 

Download references

Acknowledgements

The author is grateful to Árpád Lukács, Péter Vecsernyés and György Wolf for the numerous and enlightening discussions both on the structure of the Standard Model and on various aspects of the present suggestion. Special thanks to György Wolf for the careful reading of an earlier version of the paper, his suggestions to improve the text at several points and for drawing the figures; and to Helmut Friedrich and Paul Tod for their remarks on both the conformal cyclic cosmological model and the present suggestions. Thanks are due to the ‘Geometry and Relativity’ program at the Erwin Schrödinger International Institute for Mathematics and Physics, Vienna, for the support and hospitality where the final version of the present paper was prepared. Funding was provided by Erwin Schrödinger International Institute for Mathematics and Physics (Grant No. Geometry and Relativity CBS 2017).

Author information

Authors and Affiliations

Authors

Corresponding author

Correspondence to László B. Szabados.

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Szabados, L.B. On gravity’s role in the genesis of rest masses of classical fields. Gen Relativ Gravit 50, 34 (2018). https://doi.org/10.1007/s10714-018-2340-1

Download citation

  • Received:

  • Accepted:

  • Published:

  • DOI: https://doi.org/10.1007/s10714-018-2340-1

Keywords

Navigation