Skip to main content
Log in

Singular hypersurfaces and thin shells in general relativity

  • Published:
Il Nuovo Cimento B (1965-1970)

Summary

An approach to shock waves, boundary surfaces and thin shells in general relativity is developed in which their histories are characterized in a purely geometrical way by the extrinsic curvatures of their imbeddings in space-time. There is some gain in simplicity and ease of application over previous treatments in that no mention of « admissible » or, indeed, any space-time co-ordinates is needed. The formalism is applied to a study of the dynamics of thin shells of dust.

Riassunto

Si sviluppa un’approssimazione alle onde d’urto, superfici di contorno e strati sottili in relatività generale in cui le loro storie sono caratterizzate in modo puramente geometrico dalle curvature estrinseche delle loro giaciture nello spazio-tempo. Si ha un qualche guadagno in semplicità e facilità di applicazione rispetto alle trattazioni precedenti in quanto non è necessaria alcuna menzione di « ammissibilità » o, nella fattispecie di alcuna coordinata spazio-temporale. Si applica questo formalismo allo studio della dinamica degli strati sottili di polveri.

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

Similar content being viewed by others

References

  1. J. R. Oppenheimer andH. Snyder:Phys. Rev.,56, 455 (1939);F. Hoyle andJ. V. Narlikar:Proc. Roy. Soc., A278, 465 (1964);C. W. Misner andD.H. Sharp:Phys. Rev.,136, B 571 (1964);G. C. McVittie:Astrophys. Journ.,140, 409 (1964);H. Nariai andK. Tomita:Progr. Theor. Phys.,34, 155 (1965).

    Article  ADS  Google Scholar 

  2. C. Lanczos:Phys. Zeits.,23, 539 (1922);Ann. der Phys.,74, 518 (1924). See alsoN. Sen:Ann. der Phys.,73, 365 (1924).

    Google Scholar 

  3. Boundary surfaces:A. H. Taub:Ill. Journ. Math.,1, 370 (1957);N. Coburn:Journ. Math. and Mech.,10, 361 (1961);R. Abraham:Journ. Math. and Mech.,11, 553 (1962);D. G. B. Edelen andT. Y. Thomas:Journ. Math. Anal. Appl.,7, 247 (1963);G. Dautcourt:Arch. Rat. Mech. and Anal.,13, 55 (1963);H. Nariai:Progr. Theor. Phys.,34, 173 (1965). Surface layers:G. Dautcourt:Math. Nachr.,27, 277 (1964). This reference also treatsnull surface layers (e.g. shells of photons), an important topic which lies outside the scope of the present work.

    MathSciNet  Google Scholar 

  4. A. Lichnerowicz:Théories Relativistes de la Gravitation et de l’Electromagnétisme (Paris, 1955). Also:J. L. Synge:Relativity: The General Theory (Amsterdam, 1960), p. 39;W. Israel:Proc. Roy. Soc., A248, 404 (1958).

  5. S. O’Brien andJ. L. Synge:Commun. Dublin Inst. Adv. Stud., A, No. 9 (1952). See alsoLanczos, ref. (2).

  6. Cf.R. K. Sachs: inRelativity, Groups, Topology, edited byde Witt andde Witt (New York, 1964), p. 534.

  7. H. P. Robertson:Proc. Edinburgh Math. Soc.,5, 63 (1937);L. Infeld andA. Schild:Rev. Mod. Phys.,21, 408 (1949);A. H. Taub:Journ. Math. Phys.,5, 112 (1964).

    Article  Google Scholar 

  8. H. Weyl:Raum, Zeit, Materie, 5th edition, Sect.36 (Berlin, 1923);L. Infeld andJ. Plebanski:Motion and Relativity, Sect.I-6, (New York, 1960);W. B. Bonnor andN. S. Swaminarayan:Zeits. Phys.,177, 240 (1964). For critical comments, seeM. Carmeli:Phys. Lett.,11, 24 (1964).

  9. Seee.g. J. L. Synge:Relativity: The General Theory (Amsterdam, 1960), p. 175.

  10. Throughout the paper Greek indices take the range 1–4, Latin indices the range 2, 3, 4.

  11. See ref. (9), p. 35.

  12. See ref. (9), p. 276.

  13. W. Israel:Gravitational Collapse of a Spherical Shell and the Problem of Causality, Preprint Series A, No. 45, Mathematics Department, University of Alberta. (October 1965).

Download references

Author information

Authors and Affiliations

Authors

Rights and permissions

Reprints and permissions

About this article

Check for updates. Verify currency and authenticity via CrossMark

Cite this article

Israel, W. Singular hypersurfaces and thin shells in general relativity. Nuovo Cimento B (1965-1970) 44, 1–14 (1966). https://doi.org/10.1007/BF02710419

Download citation

  • Received:

  • Published:

  • Issue Date:

  • DOI: https://doi.org/10.1007/BF02710419

Keywords

Navigation