Abstract
We prove existence of spherically symmetric, static, self-gravitating photon shells as solutions to the massless Einstein–Vlasov system. The solutions are highly relativistic in the sense that the ratio 2m(r) / r is close to 8 / 9, where m(r) is the Hawking mass and r is the area radius. In 1955 Wheeler constructed, by numerical means, so-called idealized spherically symmetric geons, i.e., solutions of the Einstein–Maxwell equations for which the energy momentum tensor is spherically symmetric on a time average. The structure of these solutions is such that the electromagnetic field is confined to a thin shell for which the ratio 2m / r is close to 8 / 9, i.e., the solutions are highly relativistic photon shells. The solutions presented in this work provide an alternative model for photon shells or idealized spherically symmetric geons.
Article PDF
Similar content being viewed by others
References
Akbarian, A., Choptuik, M.: Critical collapse in the spherically symmetric Einstein–Vlasov model. Phys. Rev. D 90, 104023 (2014)
Andréasson, H.: Sharp bounds on 2m/r of general spherically symmetric static objects. J. Differ. Equ. 245(8), 2243–2266 (2008)
Andréasson, H.: On the Buchdahl inequality for spherically symmetric static shells. Commun. Math. Phys. 274, 399–408 (2007)
Andréasson, H.: On static shells and the buchdahl inequality for the spherically symmetric Einstein–Vlasov system. Commun. Math. Phys. 274, 409–425 (2007)
Andréasson, H., Fajman, D., Thaller, M.: Static solutions to the Einstein–Vlasov system with a nonvanishing cosmological constant. SIAM J. Math. Anal. 47(4), 2657–2688 (2015)
Andréasson, H., Kunze, M., Rein, G.: Existence of axially symmetric static solutions of the Einstein–Vlasov system. Commun. Math. Phys. 308, 23–47 (2011)
Andréasson, H., Kunze, M., Rein, G.: Rotating, stationary, axially symmetric spacetimes with collisionless matter. Commun. Math. Phys. 329, 787–808 (2014)
Andréasson, H., Rein, G.: A numerical investigation of the stability of steady states and critical phenomena for the spherically symmetric Einstein-Vlasov system. Class. Quant. Grav. 23, 3659–3678 (2006)
Andréasson, H., Rein, V.: On the steady states of the spherically symmetric Einstein–Vlasov system. Class. Quant. Grav. 24, 1809–1832 (2007)
Anderson, P.R., Brill, D.R.: Gravitational geons revisited. Phys. Rev. D 56(8), 4824–4833 (1997)
Bartnik, R., McKinnon, J.: Particlelike solutions of the Einstein–Yang–Mills equations. Phys. Rev. Lett. 61, 141–144 (1988)
Brill, D.R., Hartle, J.B.: Method of the self-consistent field in general relativity and its application to the gravitational geon. Phys. Rev. 135(1B), B271–B278 (1964)
Dafermos, M.: A note on the collapse of small data self-gravitating massless collisionless matter. J. Hyp. Differ. Equ. 3(4), 589–598 (2006)
Martín-García, J., Gundlach, C.: Self-similar spherically symmetric solutions of the massless Einstein–Vlasov system. Phys. Rev. D 65, 084026 (2002)
Makino, T.: On spherically symmetric stellar models in general relativity. J. Math. Kyoto Univ. 38, 55–69 (1998)
Ramming, T., Rein, G.: Spherically symmetric equilibria for self-gravitating kinetic or fluid models in the non-relativistic and relativistic case - a simple proof for finite extension. SIAM J. Math. Anal. 45(2), 900–914 (2013)
Rein, G.: Static solutions of the spherically symmetric Vlasov–Einstein system. Math. Proc. Camb. Philos. Soc. 115, 559 (1994)
Rein, G.: Static shells for the Vlasov–Poisson and Vlasov–Einstein system. Indiana Univ. Math. J. 48, 335–346 (1999)
Rein, G., Rendall, A.: Global existence of solutions of the spherically symmetric Vlasov–Einstein system with small initial data. Commun. Math. Phys. 150, 561–583 (1992)
Rein, G., Rendall, A.: Smooth static solutions of the spherically symmetric Vlasov–Einstein system. Annales de. l*I.H.P., section A, tome 59 4, 383–397 (1993)
Rein, G., Rendall, A.: Compact support of spherically symmetric equilibria in non-relativistic and relativistic galactic dynamics. Math. Proc. Camb. Philos. Soc. 128, 363–380 (2000)
Rendall, A.D., Velázquez, : A class of dust-like self-similar solutions of the Massless Einstein–Vlasov system. Ann. Henri Poincaré 12, 919–964 (2011)
Schaeffer, J.: A class of counterexamples to Jeans’ Theorem for the Vlasov–Einstein system. Commun. Math. Phys. 204, 313–327 (1999)
Smoller, J., Wasserman, A., Yau, S.-T., McLeod, J.: Smooth static solutions of the Einstein/Yang–Mills equations. Commun. Math. Phys. 143, 115–147 (1991)
Wheeler, J.A.: Geons. Phys. Rev. 97, 511 (1955)
Author information
Authors and Affiliations
Corresponding author
Additional information
Communicated by James A. Isenberg.
Rights and permissions
Open Access This article is distributed under the terms of the Creative Commons Attribution 4.0 International License (http://creativecommons.org/licenses/by/4.0/), which permits unrestricted use, distribution, and reproduction in any medium, provided you give appropriate credit to the original author(s) and the source, provide a link to the Creative Commons license, and indicate if changes were made.
About this article
Cite this article
Andréasson, H., Fajman, D. & Thaller, M. Models for Self-Gravitating Photon Shells and Geons. Ann. Henri Poincaré 18, 681–705 (2017). https://doi.org/10.1007/s00023-016-0531-4
Received:
Accepted:
Published:
Issue Date:
DOI: https://doi.org/10.1007/s00023-016-0531-4