Il Nuovo Cimento A (1965-1970)

, Volume 51, Issue 3, pp 744–760 | Cite as

Gravitational bounce

  • V. de la Cruz
  • W. Israel


A solution of Einstein’s field equations is derived which represents a thin spherical shell of charged dust falling in the spherically symmetric field of a charged massive body placed at its centre. Under suitable conditions the shell bounces reversibly at a nonzero minimal radius. A bounce is still possible even after the shell has collapsed inside the Schwarzschild sphere, so that the collapse as viewed externally is irreversible. The apparent paradox is explained in terms of the latticelike structure of the analytically extended Reissner-Nordström manifold. The possible relevance of the results to the problem of realistic gravitational collapse is discussed.


Manifold Event Horizon Gravitational Mass External Observer Singular Curve 
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ГравитаЦионная упрутостъ


Выводится решение полевых уравнений Эйнштейна, которое представляет тонккую сферическую оболочку зарчженной пыли, падающей в центрально-симметричном поле заряженного массивнпго тела, помещенного в центре. При подходящих условиях оболочка отскакивает обратимо к ненулевому минимальому радиусу. Упругость еще оказывается возможной, даже после того, как оболочка коллапсировала внутрь сферы Шварцшильда, так что коллапс, когда рассматривается извне, является необратимым, Кажущиися парадокс объясняется в терминах рещетчато-подобной структуры аналитически продолженного многооб-разия Рейснера-Нордстрема. Обсуждается возможная уместность этих результатов в проблеме релятивистского гравитационного коллапса.


Si deduce una soluzione delle equazioni del campo di Einstein che rappresenta un sottile strato sferico di polvere carica che cade nel campo a simmetria sferica di un corpo carico dotato di massa posto al centro. In opportune condizionilo strato rimbalza in modo reversibile a un raggio minimo non nullo. È possibile un rimbalzo anche dopo che lo strato è crollato entro la sfera di Schwarschild, cosicchè il collasso visto dall’esterno è irreversibile. Si spiega l’apparente paradosso per mezzo della struttura reticolare della molteplicità di Reissner-Nordström estesa analiticamente. Si discute la possibile influenza dei risultati sul problema del collasso gravitazionale realistico.


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  1. (1).
    See,e. g.,Ya. B. Zel’dovich andI. D. Novikov:Sov. Phys. Uspekhi,7, 763 (1965);8, 522 (1966).ADSMathSciNetCrossRefGoogle Scholar
  2. (2).
    That the unmodified Schwarzschild metric cannot be trusted near the singular curver=0 is clear from the fact that this curve is spacelike: a body which has collapsed irreversibly tor=0 would have to travel faster than light!Google Scholar
  3. (3).
    As an illustration, it may be remarked that (1) represents the external field of an (uncharged) spherical body in the theory ofF. Hoyle andJ. V. Narlikar:Proc. Roy. Soc., A294, 138 (1966).ADSCrossRefGoogle Scholar
  4. (4).
    J. C. Graves andD. R. Brill:Phys. Rev.,120, 1507 (1960).ADSMathSciNetCrossRefGoogle Scholar
  5. (5).
    I. D. Novikov:JETP Lett.,3, 142 (1966).ADSGoogle Scholar
  6. (6).
    W. Israel:Nuovo Cimento,44B, 1 (1966); ibid,W. Israel:Nuovo Cimento,48 B, 463 (1967).ADSCrossRefGoogle Scholar
  7. (7).
    We adopt the following conventions:G=c=1, signature of metric +++-, Greek indices refer to 4-dimensional, Latin indices to3-dimensional quantities. Limits of the field quantity Ы as the eventP on Щ is approached fromV ,V + respectively are denoted by Ы(P)|, Ы(P)|+. In Sect.2 square brackets denote jump discontinuities: [Ы]≡Ы|+-Ы|.Google Scholar
  8. (8).
    B. Hoffmann:Quart. Journ. Math.,4, 179 (1933), also inRecent Developments in General Relativity (Warsaw, 1962), p. 279;A. Das:Progr. Theor. Phys.,24, 915 (1960).ADSCrossRefGoogle Scholar
  9. (9).
    B. Carter:Phys. Lett.,21, 423 (1966).ADSCrossRefGoogle Scholar
  10. (10).
    This does not contradict a theorem on the inevitability of singularities due toR. Penrose:Phys. Rev. Lett.,14, 57 (1965), since two of the hypotheses of that theorem are not satisfied here. In the first place, the manifold withe=m contains no «trapped surface» (even though it contains an event horizon), since outgoing radial null geodesics have dr/dt(1-m/r)2≥0 and do not converge anywhere. Secondly, the manifold withe≤m does not admit a Cauchy hypersurface.ADSMathSciNetCrossRefGoogle Scholar
  11. (11).
    In the Newtonian description the pulsating shell of course always remains in the same space.Google Scholar
  12. (12).
    Cf.A. Komar:Phys. Rev.,137, B 462 (1965).ADSMathSciNetCrossRefGoogle Scholar
  13. (13).
    A. G. Doroshkevich, Ya. B. Zel’dovich andI. D. Novikov:Sov. Phys. JETP,22, 122 (1966).ADSGoogle Scholar

Copyright information

© Società Italiana di Fisica 1967

Authors and Affiliations

  • V. de la Cruz
    • 1
    • 2
  • W. Israel
    • 1
    • 2
  1. 1.University of AlbertaEdmonton
  2. 2.Dublin Institute for Advanced StudiesDublin

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